Reflection Positive Stochastic Processes Indexed by Lie Groups Reflection
Positive Stochastic Processes
Indexed by Lie Groups

Palle E.T. JORGENSEN †, Karl-Hermann NEEB ‡ and Gestur ÓLAFSSON § P.E.T.
Jorgensen, K.-H. Neeb and G. Ólafsson

† Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
<EMAIL_ADDRESS>
‡ Department Mathematik, FAU Erlangen-Nürnberg,
‡ Cauerstrasse 11, 91058-Erlangen, Germany<EMAIL_ADDRESS>
§ Department of mathematics, Louisiana State University, Baton Rouge, LA
70803, USA<EMAIL_ADDRESS>
Received October 28, 2015, in final form June 09, 2016; Published online June
21, 2016

Reflection positivity originates from one of the Osterwalder–Schrader axioms
for constructive quantum field theory. It serves as a bridge between euclidean
and relativistic quantum field theory. In mathematics, more specifically, in
representation theory, it is related to the Cartan duality of symmetric Lie
groups (Lie groups with an involution) and results in a transformation of a
unitary representation of a symmetric Lie group to a unitary representation of
its Cartan dual. In this article we continue our investigation of
representation theoretic aspects of reflection positivity by discussing
reflection positive Markov processes indexed by Lie groups, measures on path
spaces, and invariant gaussian measures in spaces of distribution vectors.
This provides new constructions of reflection positive unitary
representations.

reflection positivity; stochastic process; unitary representations

22E45; 60G15; 81S40

###### Contents

  1. 1 Introduction
  2. 2 Reflection positive representations
  3. 3 Reflection positive Lie group actions on measure spaces
    1. 3.1 $(G,S,\tau)$-measure spaces
    2. 3.2 Stochastic processes indexed by Lie groups
    3. 3.3 Associated positive semigroup structures and reconstruction
    4. 3.4 Klein’s characterization of the Markov case
    5. 3.5 Total subsemigroups of Lie groups
    6. 3.6 The discrete case
  4. 4 Measures on path spaces for topological groups
    1. 4.1 One-parameter convolution semigroups of measures on polish groups
    2. 4.2 Standard path space structures for locally compact groups
  5. 5 Gaussian $(G,S,\tau)$-probability spaces
    1. 5.1 Second quantization and gaussian processes
    2. 5.2 Application to reflection positive representations
    3. 5.3 Equivalence of gaussians measures for reproducing kernel Hilbert spaces
    4. 5.4 Gaussian measures on distributions on Lie groups
    5. 5.5 Ergodicity of $\gamma_{\mathcal{H}}$ for $G$
  6. A Continuity of a stochastic process on $G$
  7. B Markov kernels
    1. Acknowledgements

##  1 Introduction

Reflection positivity is one of the cornerstones in constructive quantum field
theory. It was first formulated in the fundamental work of Osterwalder and
Schrader [57, 58], see also [20]. Mathematically, a quantum physical system
corresponds to a unitary representation of the corresponding symmetry group
$G$. In euclidean quantum field theory this group is the euclidean motion
group and in relativistic quantum field theory the symmetry group is the
Poincaré group. Reflection positivity enters the picture when it comes to
passing from euclidean quantum fields to relativistic ones. The time
reflection and the passage to imaginary time, sometimes called Wick rotation,
changes the standard euclidean inner product into the Lorentz form of
relativity. The time reflection corresponds to an involution $\tau$ of the
euclidean motion group $G$ and then the duality of symmetric Lie algebras
${\mathfrak{g}}={\mathfrak{g}}^{\tau}\oplus{\mathfrak{g}}^{-\tau}\leftrightarrow{\mathfrak{g}}^{c}={\mathfrak{g}}^{\tau}\oplus
i{\mathfrak{g}}^{-\tau}$ leads to a duality between the Lie algebra of the
euclidean motion group and that of the Poincaré group.

In the following we call a pair $(G,\tau)$, consisting of a Lie group $G$ and
an involutive automorphism $\tau\colon G\to G$ a symmetric Lie group. On the
group level, the aforementioned duality is implemented by the Cartan duality
(or $c$-duality for short) $G\leftrightarrow G^{c}$, where $G^{c}$ is the
simply connected Lie group with Lie algebra ${\mathfrak{g}}^{c}$. Passing from
real to imaginary time and from the euclidean motion group to the universal
covering of the (identity component of the) Poincaré group is an important
special case of $c$-duality which has been studied independently from the
quantum field theoretic context.

In addition to $c$-duality, the other three basic notions considered in the
basic theory of reflection positivity are that of reflection positive Hilbert
spaces, reflection positive kernels, and reflection positive representations.
A reflection positive representation together with the Osterwalder–Schrader
quantization leads to an infinitesimally unitary representation of the Lie
algebra ${\mathfrak{g}}^{c}$. The problem in representation theory is then to
determine if this representation arises as the derived representation of a
unitary representation of $G^{c}$, which establishes a passage from a unitary
representations of $G$ to one of $G^{c}$.

This integration process is often accomplished using reflection positive
kernels and geometric actions of the Lie group or its Lie algebra [46] or, as
in the case of the Lüscher–Mack theorem [45], using semigroups and invariant
cones. A second step is then to determine the resulting representation in
terms of decomposition into irreducible representation. For further
representation theoretic results related to reflection positivity, we refer to
[26, 33, 34, 35, 36, 49, 50, 51, 56, 60]. Our present paper concentrates on
various mathematical aspects of constructions of reflection positive
representations: infinite-dimensional analysis, functional integration and
gaussian measures, and stochastic processes. On the way we recall several
basic facts in these areas to make this article more self-contained.

Shortly after the groundbreaking work of Osterwalder and Schrader, A. Klein
and L. Landau built a bridge between reflection positivity, path spaces and
stochastic processes [40, 41, 42]. One of our goals in this article is to
connect the ideas of Klein and Landau to representation theory and our
previous work on reflection positive representations [35, 36, 49, 50, 51].
This is done by replacing the real line, viewed as continuous time, by an
arbitrary Lie group $G$, not necessarily finite-dimensional, and the positive
time axes ${\mathbb{R}}_{+}$ by a semigroup $S\subset G$ invariant under
$s\mapsto s^{\\#}=\tau(s)^{-1}$. This leads naturally to the concepts of
$(G,\tau)$-measure spaces, reflection positive measure spaces, and a positive
semigroup structures introduced in Section 3 of this article. Many concepts
extend naturally to general triples $(G,S,\tau)$. But in the generalization of
the Abel–Klein reconstruction theorem (Theorem 3.22) which reconstructs a
$(G,S,\tau)$-measure space from a positive semigroup structure, we need to
assume that $G=S\cup S^{-1}$; then $S$ is called total. It would be
interesting to see how far our techniques can be extended beyond total
subsemigroups.

In the following section we turn back to the classical case where the
symmetric Lie group is $({\mathbb{R}},{\mathbb{R}}_{+},-\mathop{{\rm
id}}\nolimits)$, but the measure space is the path space $P(Q)=Q^{\mathbb{R}}$
of a polish topological group $Q$, mostly assumed locally compact. If $\nu$ is
a measure on $Q$ and $(\mu_{t})_{t\geq 0}$ is a convolution semigroup of
symmetric probability measures on $Q$ satisfying $\nu*\mu_{t}=\nu$ for every
$t>0$, then the reconstruction theorem applies to the corresponding
${\mathbb{R}}_{+}$-action on $L^{2}(Q,\nu)$ and leads to an invariant
probability measure on the path space $Q^{\mathbb{R}}$ corresponding to a
reflection positive representation (Theorem 4.9). However, the corresponding
measure on $P(Q)$ is the product of the measure $\nu$, on the constant paths,
and a probability measure on the pinned paths

$\displaystyle P_{*}(Q):=\\{\omega\in P(Q)\colon\omega(0)=\mathbf{1}\\},$

which is also invariant under a suitable one-parameter group of
transformations. Since we presently do not know how to obtain a similar
factorization if $({\mathbb{R}},{\mathbb{R}}_{+},-\mathop{{\rm
id}}\nolimits_{\mathbb{R}})$ is replaced by some $(G,S,\tau)$, we discuss the
one-dimensional case in some detail.

One of the advantages of the reflection positivity condition is that it allows
to construct representations of a symmetric Lie group (classically the
euclidean motion group) in $L^{2}$-spaces of measures on spaces of
distributions, which after the Osterwalder–Schrader quantization lead to
unitary representations of the $c$-dual group. In this connection the physics
literature considered reflection positive distributions
$D\in\mathcal{S}^{\prime}$, invariant under the euclidean group. The Hilbert
space $\mathcal{H}_{D}\subseteq\mathcal{S}^{\prime}$ specified by the
corresponding positive definite kernel [61] leads to a reflection positive
representation of the euclidean motion group [49, 50]. This motivates our
discussion of $G$-invariant gaussian measures corresponding to a unitary
representation $(\pi,\mathcal{H})$ of $G$. The replacement for the Gelfand
triple $(\mathcal{S},L^{2}({\mathbb{R}}^{n}),\mathcal{S}^{\prime})$ is the
Gelfand triple $(\mathcal{H}^{\infty},\mathcal{H},\mathcal{H}^{-\infty})$,
where $\mathcal{H}^{\infty}$ denotes the Fréchet space of smooth vectors in
$\mathcal{H}$ and $\mathcal{H}^{-\infty}$ is it conjugate linear dual (the
space of distribution vectors). Here the natural question is for which
representations the gaussian measure of $\mathcal{H}$ can be realized on
$\mathcal{H}^{-\infty}$? A partial answer is given in Theorem 5.17, Corollary
5.19, and Proposition 5.20. All of this links naturally to the theory of
generalized Wiener spaces and white noise processes for ${\mathbb{R}}^{n}$,
indexed by $L^{2}({\mathbb{R}}^{n})$; see, e.g., [2, 3, 5, 29].

Finally we would like to mention that there is also an important branch of
applications of reflection positivity in statistical physics which does not
refer to semigroups at all; see [19] and the more recent [51], where the
corresponding group $G$ may be a torus, hence does not contain any proper open
subsemigroup. We hope to develop this point in future work.

Our article is organized as follows. In the first section we start by
recalling the basic facts about reflection positive Hilbert spaces and
reflection positive representations [49, 50]. We always have a symmetric Lie
group $(G,\tau)$ and a subsemigroup $S$ invariant under
$s^{\\#}=\tau(s)^{-1}$. Based on ideas from [42], we introduce the special
class of reflection positive Hilbert spaces of Markov type. The main result is
Proposition 2.9.

In Section 3 we discuss stochastic processes indexed by a symmetric Lie group
$(G,\tau)$, where the forward time ${\mathbb{R}}_{+}$ is replaced by a
subsemigroup $S\subset G$ invariant under $s\mapsto s^{\sharp}=\tau(s)^{-1}$.
Here we prove our generalization of the reconstruction theorem (Theorem 3.22)
which reconstructs for a so-called positive semigroup structure a
$(G,S,\tau)$-measure space from which it can be derived. From the
representation theoretic perspective, it corresponds to finding euclidean
realizations of unitary representations of the $c$-dual group $G^{c}$.
Unfortunately, the reconstruction process requires that $G=S\cup S^{-1}$, a
property which is briefly discussed in Section 3.5.

In Section 4 we build a bridge between Markov processes and the reconstruction
process. To this end, we return to the classical setting, where the symmetric
Lie group is $({\mathbb{R}},{\mathbb{R}}_{+},-\mathop{{\rm id}}\nolimits)$,
but the measure space is the path space $P(Q)=Q^{\mathbb{R}}$ of a polish
group $Q$ and the corresponding $({\mathbb{R}},{\mathbb{R}}_{+},-\mathop{{\rm
id}}\nolimits_{\mathbb{R}})$-measure space is given by a one-parameter group
$P_{t}f=f*\mu_{t}$, $t\geq 0$, of left invariant Markov kernels on some space
$L^{2}(Q,\nu)$. Here our main result concerns a factorization of the measure
space $P(Q)$ as $Q\times P_{*}(Q)$ and a corresponding factorization of the
measure preserving action of ${\mathbb{R}}$. For the special case where
$\mu_{t}$ is the gaussian semigroup on ${\mathbb{R}}^{d}$, the corresponding
measure on $P_{*}({\mathbb{R}}^{d})$ is the Wiener measure, but the measure on
$P({\mathbb{R}}^{d})$ is a product of Lebesgue and Wiener measure. The main
results in Section 4 is Theorem 4.9 which relates all this to the
reconstruction process and hence to reflection positive representations.

In Section 5 we discuss the second quantization functor and how it can be used
to derive from an orthogonal reflection positive representation a gaussian
$(G,S,\tau)$-probability spaces (Proposition 5.6). To understand the ambiguity
in this construction we also discuss equivalence of gaussian measures for
reproducing kernel Hilbert spaces, and in Subsection 5.4 we connect this issue
to our previous work [49] on distributions on $G$ and distribution vectors of
a unitary representation of $G$. In Theorem 5.17 we determine when a gaussian
measure $\gamma_{\mathcal{H}}$ of a Hilbert space $\mathcal{H}$ can be
realized in the dual of the space $\bigcap\mathcal{D}(A^{n})$ of smooth vector
of a self adjoint operator $A$. This is then applied to a unitary
representation of $G$ in Corollary 5.19.

The article ends with two appendixes where needed material on stochastic
processes and Markov semigroups is collected.

##  2 Reflection positive representations

In this preliminary section we collect some results and definitions from [49,
50] concerning reflection positive representations.

######  Definition 2.1.

A reflection positive Hilbert space is a triple
$(\mathcal{E},\mathcal{E}_{+},\theta)$, where $\mathcal{E}$ is a Hilbert
space, $\theta$ a unitary involution and $\mathcal{E}_{+}$ a closed subspace
which is $\theta$-positive in the sense that the hermitian form $\langle
v,w\rangle_{\theta}:=\langle\theta v,w\rangle$ is positive semidefinite on
$\mathcal{E}_{+}$.

For a reflection positive Hilbert space
$(\mathcal{E},\mathcal{E}_{+},\theta)$, let
$\mathcal{N}:=\\{u\in\mathcal{E}_{+}\colon\langle\theta u,u\rangle=0\\}$ and
let $\widehat{\mathcal{E}}$ be the completion of $\mathcal{E}_{+}/\mathcal{N}$
with respect to the inner product
$\langle\,\cdot\,,\,\cdot\,\rangle_{\theta}$. Let
$q\colon\mathcal{E}_{+}\to\widehat{\mathcal{E}},v\mapsto q(v)=\widehat{v}$ be
the canonical map. Then
$\mathcal{E}_{+}^{\theta}:=\\{v\in\mathcal{E}_{+}\colon\theta v=v\\}$ is the
maximal subspace of $\mathcal{E}_{+}$ on which $q$ is isometric.

######  Definition 2.2.

Let $(\mathcal{E},\mathcal{E}_{+},\theta)$ be a reflection positive Hilbert
space. If $\mathcal{E}_{0}\subseteq\mathcal{E}_{+}^{\theta}$ is a closed
subspace, $\mathcal{E}_{-}:=\theta(\mathcal{E}_{+})$, and $E_{0},E_{\pm}$ the
orthogonal projections onto $\mathcal{E}_{0}$ and $\mathcal{E}_{\pm}$, then we
say that $(\mathcal{E},\mathcal{E}_{0},\mathcal{E}_{+},\theta)$ is a
reflection positive Hilbert space of Markov type if

$\displaystyle E_{+}E_{0}E_{-}=E_{+}E_{-}.$ (2.1)

######  Lemma 2.3.

Suppose that $\mathcal{E}$ is a Hilbert space,
$\theta\colon\mathcal{E}\to\mathcal{E}$ is a unitary involution,
$\mathcal{E}_{+}$ is a closed subspace and
$\mathcal{E}_{0}\subseteq\mathcal{E}_{+}^{\theta}$. Let
$\mathcal{E}_{-}=\theta(\mathcal{E}_{+})$.

  * (i)

If $(\mathcal{E},\mathcal{E}_{0},\mathcal{E}_{+},\theta)$ satisfies the Markov
condition (2.1), then $\mathcal{E}_{+}$ is $\theta$-positive,
$\mathcal{E}_{0}=\mathcal{E}_{+}^{\theta}$,
$\mathcal{N}=\mathcal{E}_{+}\ominus\mathcal{E}_{0}$ and
$q\colon\mathcal{E}_{0}\to\widehat{\mathcal{E}}$ is a unitary isomorphism.

  * (ii)

If $q_{0}\colon\mathcal{E}_{0}\to\widehat{\mathcal{E}}$ is a unitary
isomorphism, then (2.1) holds.

###### Proof.

(i) By definition, we have $\theta E_{+}=E_{-}\theta$ so
$\mathcal{E}^{\theta}_{+}\subseteq\mathcal{E}_{+}\cap\mathcal{E}_{-}$. Let
$u\in\mathcal{E}^{\theta}_{+}\ominus\mathcal{E}_{0}$. Then the operator on the
right hand side of (2.1) reproduces $u$ while the left hand side yields $0$.
Hence $u=0$ and $\mathcal{E}_{0}=\mathcal{E}_{+}^{\theta}$. As
$\mathcal{E}_{0}\subset\mathcal{E}_{+}^{\theta}$ it follows that
$E_{0}\theta=\theta E_{0}=E_{0}$ which implies that

$\displaystyle E_{+}\theta
E_{+}=E_{+}E_{-}\theta=E_{+}E_{0}E_{-}\theta=E_{0}\theta
E_{+}=E_{0}E_{+}=E_{0}.$

It follows that, for $u\in\mathcal{E}_{+}$, we have

$\displaystyle\langle\theta u,u\rangle=\langle E_{+}\theta
E_{+}u,u\rangle=\langle E_{0}u,u\rangle=\|E_{0}u\|^{2}\geq 0.$

We obtain in particular that $\|q(v)\|=\|E_{0}v\|$ for $v\in\mathcal{E}_{+}$,
so that $q|_{\mathcal{E}_{0}}\colon\mathcal{E}_{0}\to\widehat{\mathcal{E}}$ is
a unitary isomorphism.

(ii) Since $q|_{\mathcal{E}_{+}^{\theta}}$ is isometric, we obtain
$\mathcal{E}_{0}=\mathcal{E}_{+}^{\theta}$ and thus
$\mathcal{N}=\mathcal{E}_{+}\cap\theta(\mathcal{E}_{0})^{\bot}=\mathcal{E}_{+}\ominus\mathcal{E}_{0}$.
This leads to the orthogonal decomposition
$\mathcal{E}_{+}+\mathcal{E}_{-}=\theta(\mathcal{N})\oplus\mathcal{E}_{0}\oplus\mathcal{N}$
and to $\mathcal{E}_{0}=\mathcal{E}_{+}\cap\mathcal{E}_{-}$. Now (2.1)
follows. ∎

In the following, if $(\mathcal{E},\mathcal{E}_{0},\mathcal{E}_{+},\theta)$
satisfies the Markov condition (2.1), then we will always assume that
$\mathcal{E}_{0}=\mathcal{E}_{+}^{\theta}$ and hence only write
$(\mathcal{E},\mathcal{E}_{+},\theta)$.

######  Definition 2.4.

If $\tau$ is an involutive automorphism of the Lie group $G$, then we call
$(G,\tau)$ a symmetric Lie group. A symmetric semigroup is a triple
$(G,S,\tau)$, where $(G,\tau)$ is a symmetric Lie group and $S\subseteq G$ is
a subsemigroup satisfying

  * (S1)

$S$ is invariant under $s\mapsto s^{\sharp}=\tau(s)^{-1}$,

  * (S2)

$HS=S$ for $H:=(G^{\tau})_{0}$,

  * (S3)

$\mathbf{1}\in\overline{S}$.

We define a left invariant partial order $\prec_{S}$ on $G$ by $g\prec_{S}h$
if $g^{-1}h\in S$, i.e., $h\in gS$.

######  Example 2.5.

(1) $({\mathbb{R}},{\mathbb{R}}_{+},-\mathop{{\rm id}}\nolimits_{\mathbb{R}})$
and $({\mathbb{Z}},{\mathbb{N}}_{0},-\mathop{{\rm id}}\nolimits_{\mathbb{Z}})$
are the most elementary examples of symmetric semigroups.

(2) Semigroups with polar decomposition: Let $(G,\tau)$ be a symmetric Lie
group and $H$ be an open subgroup of $G^{\tau}:=\\{g\in G\colon\tau(g)=g\\}$.
We denote the derived involution $\mathfrak{g}\to\mathfrak{g}$ by the same
letter and define
$\mathfrak{h}=\\{x\in\mathfrak{g}\colon\tau(x)=x\\}=\mathfrak{g}^{\tau}$ and
$\mathfrak{q}=\\{x\in\mathfrak{g}\colon\tau(x)=-x\\}=\mathfrak{g}^{-\tau}$.
Then $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{q}$. We say that the open
subsemigroup $S\subseteq G$ has a polar decomposition if there exists an
$H$-invariant open convex cone $C\subset\mathfrak{q}$ containing no affine
lines such that $S=H\exp C$ and the map $H\times C\to S$, $h,X\mapsto h\exp X$
is a diffeomorphism (cf. [26, 44, 48]). Typical examples are the complex
Olshanski semigroups in complex simple Lie groups such as $\mathop{{\rm
SU}}\nolimits_{p,q}({\mathbb{C}})_{\mathbb{C}}\cong\mathop{{\rm
SL}}\nolimits_{p+q}({\mathbb{C}})$. They exist if and only if $G/K$ is a
bounded symmetric domain. This is equivalent to the existence of a
$G$-invariant convex cone $C\subset i\mathfrak{g}$ such that $G\exp C$ is a
semigroup. More generally we have the causal symmetric spaces of non-compact
type like $\mathop{{\rm SO}}\nolimits_{o}(1,n+1)/\mathop{{\rm
SO}}\nolimits_{o}(1,n)$. In this case ${\mathfrak{q}}\simeq{\mathbb{R}}^{n+1}$
and $C$ can be taken as the open light-cone.

(3) The simply connected covering group $G:=\widetilde{\mathop{{\rm
SL}}}\nolimits_{2}({\mathbb{R}})$ of $\mathop{{\rm
SL}}\nolimits_{2}({\mathbb{R}})$ carries an involution $\tau$ acting on
$\mathop{{\mathfrak{sl}}}\nolimits_{2}({\mathbb{R}})$ by

$\displaystyle\tau\begin{pmatrix}x&y\\\
z&-x\end{pmatrix}=\begin{pmatrix}x&-y\\\ -z&-x\end{pmatrix},$

and there exists a closed subsemigroup $S\subseteq G$ whose boundary is

$\displaystyle\partial S=H(S):=S\cap
S^{-1}=\exp({\mathfrak{b}})\qquad\mbox{with}\quad{\mathfrak{b}}:=\left\\{\begin{pmatrix}x&y\\\
0&-x\end{pmatrix}\colon x,y\in{\mathbb{R}}\right\\}.$

This semigroup satisfies $S^{\sharp}=S$, the subgroup $H(S)$ is
$\tau$-invariant, but strictly larger than $G^{\tau}_{0}$ (see also Section
3.5 for more on this semigroup).

######  Definition 2.6.

For a symmetric semigroup $(G,S,\tau)$, a unitary representation $U$ of $G$ on
$(\mathcal{E},\mathcal{E}_{+},\theta)$ is called reflection positive if
$\theta U_{g}\theta=U_{\tau(g)}$ for $g\in G$ and
$U_{s}\mathcal{E}_{+}\subseteq\mathcal{E}_{+}$ for every $s\in S$.

######  Remark 2.7.

Let $G_{\tau}=G\rtimes\\{1,\tau\\}$. Then $\theta U_{g}\theta=U_{\tau(g)}$
holds for every $g\in G$ if and only if $U$ extends to a unitary
representation of $G_{\tau}$ by defining $U_{\tau}=\theta$.

######  Remark 2.8.

(a) If $(U_{g})_{g\in G}$ is a reflection positive representation of
$(G,S,\tau)$ on $(\mathcal{E},\mathcal{E}_{+},\theta)$, then we obtain
contractions $(\widehat{U}_{s})_{s\in S}$, on $\widehat{\mathcal{E}}$,
determined by

$\displaystyle\widehat{U}_{s}\circ q=q\circ U_{s}|_{\mathcal{E}_{+}},$

and this leads to an involutive representation
$(\widehat{U},\widehat{\mathcal{E}})$ of $S$ by contractions (cf. [35,
Corollary 3.2] and [49]). We then call
$(U,\mathcal{E},\mathcal{E}_{+},\theta)$ a euclidean realization of
$(\widehat{U},\widehat{\mathcal{E}})$. We refer to [46] and [45] for methods
for obtaining a unitary representation of $G^{c}$ from $\widehat{U}$.

(b) For $(G,S,\tau)=({\mathbb{R}},{\mathbb{R}}_{+},-\mathop{{\rm
id}}\nolimits_{\mathbb{R}})$, continuous reflection positive unitary one-
parameter groups $(U_{t})_{t\in{\mathbb{R}}}$ lead to a strongly continuous
semigroup $(\widehat{U},\widehat{\mathcal{E}})$ of hermitian contractions and
every such semigroup $(C,\mathcal{H})$ has a natural euclidean realization
obtained as the GNS representation associated to the positive definite
operator-valued function $\varphi(t):=C_{|t|}$, $t\in{\mathbb{R}}$ [50,
Proposition 6.1].

The following proposition is a generalization of [50, Proposition 5.17] which
applies to the special case $({\mathbb{R}},{\mathbb{R}}_{+},-\mathop{{\rm
id}}\nolimits_{\mathbb{R}})$.

######  Proposition 2.9.

Let $(U_{g})_{g\in G}$ be a reflection positive unitary representation of
$(G,S,\tau)$ on $(\mathcal{E},\mathcal{E}_{+},\theta)$, let
$\mathcal{E}_{0}\subseteq(\mathcal{E}_{+})^{\theta}$ be a subspace and
$\Gamma=q|_{\mathcal{E}_{0}}\colon\mathcal{E}_{0}\to\widehat{\mathcal{E}}$. If
$(\mathcal{E},\mathcal{E}_{0},\mathcal{E}_{+},\theta)$ is of Markov type, then
the following assertions hold:

  * (i)

The reflection positive function $\varphi\colon G\to B(\mathcal{E}_{0})$,
$\varphi(g):=E_{0}U_{g}E_{0}$, is multiplicative on $S$.

  * (ii)

$\varphi(s)=\Gamma^{*}\widehat{U}_{s}\Gamma$ for $s\in S$, i.e., $\Gamma$
intertwines $\varphi|_{S}$ with $\widehat{U}$.

###### Proof.

(i) Let $\mathcal{K}\subseteq\mathcal{H}$ be the $U$-invariant closed subspace
generated by $\mathcal{E}_{0}$. Let $(\mathcal{E}_{0})^{G}$ denote the linear
space of all maps $G\to\mathcal{E}_{0}$. Then the map

$\displaystyle\Phi\colon\
\mathcal{K}\to(\mathcal{E}_{0})^{G},\qquad\Phi(v)(g):=E_{0}U_{g}v$

is an equivalence of the representation $U$ of $G$ on $\mathcal{K}$ with the
GNS representation defined by $\varphi$, and the representation $\widehat{U}$
of $S$ on $\widehat{\mathcal{E}}$ is equivalent to the GNS representation
defined by $\varphi|_{S}$, where the map
$q\colon\mathcal{E}_{+}\to\widehat{\mathcal{E}}$ corresponds to the
restriction $f\mapsto f|_{S}$ [49, Proposition 1.11]. The inclusion
$\iota\colon\mathcal{E}_{0}\hookrightarrow\mathcal{H}_{\varphi}$ is given by
$\iota(v)(g)=E_{0}U_{g}v=\varphi(g)v$ for $g\in G$, and likewise the inclusion
$\widehat{\iota}\colon\mathcal{E}_{0}\hookrightarrow\mathcal{H}_{\varphi|_{S}}$
is given by $\iota(v)(s)=\varphi(s)v$ for $s\in S$. Lemma 2.3 implies the
surjectivity of the inclusion $\widehat{\iota}$. In view of [50, Lemma 10.3],
this is equivalent to the multiplicativity of $\varphi|_{S}$.

(ii) If $\Gamma$ is unitary, then [50, Lemma 5.16(ii)] implies that
$q=\Gamma\circ E_{0}|_{\mathcal{E}_{+}}$. For $s\in S$, the relation
$\widehat{U}_{s}\circ q=q\circ U_{s}|_{\mathcal{E}_{+}}$ leads to
$\widehat{U}_{s}\Gamma E_{0}|_{\mathcal{E}_{+}}=\Gamma
E_{0}U_{s}|_{\mathcal{E}_{+}}$, so that
$\Gamma^{*}\widehat{U}_{s}\Gamma=E_{0}U_{s}E_{0}=\varphi(s)$, i.e., $\Gamma$
intertwines $\varphi(s)$ with $\widehat{U}_{s}$. ∎

######  Lemma 2.10.

Assume that $U$ is a reflection positive representation of $(G,S,\tau)$ on
$(\mathcal{E},\mathcal{E}_{+},\theta)$. Let
$\mathcal{E}_{0}\subseteq\mathcal{E}_{+}^{\theta}$. If
$\varphi(g)=E_{0}U_{g}E_{0}$ is multiplicative and $\mathcal{E}_{0}$ is
$S$-cyclic in $\mathcal{E}_{+}$ then $(\mathcal{E},\mathcal{E}_{+},\theta)$ is
of Markov type.

###### Proof.

As in the proof of Lemma 2.3, this follows from [50, Lemma 10.3]. ∎

######  Remark 2.11.

If $G=S\cup S^{-1}$ and $\mathcal{E}_{0}$ is $G$-cyclic in $\mathcal{E}$. Then
$\mathcal{E}_{0}$ is $S$-cyclic in $\mathcal{E}_{+}$ and
$\mathcal{E}_{+}+\mathcal{E}_{-}$ is dense in $\mathcal{E}$, so that the
Markov condition leads to
$\mathcal{E}=\mathcal{N}\oplus\mathcal{E}_{0}\oplus\theta(\mathcal{N})$.

##  3 Reflection positive Lie group actions on measure spaces

In this section we generalize several results from [41] and [42] to situations
where $({\mathbb{R}},{\mathbb{R}}_{+},-\mathop{{\rm
id}}\nolimits_{\mathbb{R}})$ is replaced by a symmetric semigroup
$(G,S,\tau)$. This leads us to the concept of a $(G,S,\tau)$-measure space
generalizing Klein’s Osterwalder–Schrader path spaces for
$({\mathbb{R}},{\mathbb{R}}_{+},-\mathop{{\rm id}}\nolimits_{\mathbb{R}})$.
The Markov $(G,S,\tau)$-measure spaces generalize the path spaces studied by
Klein and Landau in [42]. The main result of this section is the
correspondence between $(G,S,\tau)$-measure spaces and the corresponding
positive semigroup structures. For
$(G,S,\tau)=({\mathbb{R}},{\mathbb{R}}_{+},-\mathop{{\rm
id}}\nolimits_{\mathbb{R}})$ this has been developed in [41, 42], motivated by
Nelson’s work on the Feynman–Kac formula [54].

###  3.1 $\boldsymbol{(G,S,\tau)}$-measure spaces

In this section we introduce basic concepts related to $(G,S,\tau)$-measure
spaces and draw some consequences from the definitions.

######  Definition 3.1.

Let $(G,\tau)$ be a group with an involutive automorphism $\tau$. A
$(G,\tau)$-measure space is a quadruple $((Q,\Sigma,\mu),\Sigma_{0},U,\theta)$
consisting of the following ingredients:

  * (GP1)

a measure space $(Q,\Sigma,\mu)$,

  * (GP2)

a sub-$\sigma$-algebra $\Sigma_{0}$ of $\Sigma$,

  * (GP3)

a measure preserving action $U\colon G\to\mathop{{\rm
Aut}}\nolimits(\mathcal{A})$ on the $W^{*}$-algebra
$\mathcal{A}:=L^{\infty}(Q,\Sigma,\mu)$ which is strongly continuous in
measure, i.e.111Since the $G$-action on $\mathcal{A}$ is measure preserving,
it defines natural representations on $L^{1}(Q,\Sigma,\mu)$ and on
$L^{2}(Q,\Sigma,\mu)$. However, in general we do not have an action of $G$ on
the set $Q$ itself, but $G$ acts naturally on the set $\Sigma/{\sim}$, where
$A\sim B$ if $\mu(A\Delta B)=0$. This set corresponds to the idempotents in
the algebra $\mathcal{A}$. The continuity condition (GP3) is equivalent to the
continuity of the corresponding unitary representation of $G$ on
$L^{2}(Q,\Sigma,\mu)$ which in turn is equivalent to $\mu(gE\Delta E)\to 0$
for $g\to\mathbf{1}$ and $E\in\Sigma$ with $\mu(E)<\infty$. See the discussion
in Appendix A or [64, p. 107] for more details.,

$\displaystyle\lim_{g\to\mathbf{1}}\mu(|U_{g}f-f|\geq\varepsilon)=0\qquad\mbox{for}\quad\varepsilon>0\qquad\mbox{and}\qquad
f\in\mathcal{A}\cap L^{1}(Q,\Sigma,\mu),$

and

  * (GP4)

a measure preserving involutive automorphism $\theta$ of
$L^{\infty}(Q,\Sigma,\mu)$ for which $\theta U_{g}\theta=U_{\tau(g)}$ for
$g\in G$ and $\theta E_{0}\theta=E_{0}$, where $E_{0}\colon
L^{\infty}(Q,\Sigma,\mu)\to L^{\infty}(Q,\Sigma_{0},\mu)$ is the conditional
expectation.

  * (GP5)

$\Sigma$ is generated by the sub-$\sigma$-algebras
$\Sigma_{g}:=U_{g}\Sigma_{0}$, $g\in G$.

If $\mu$ is a probability measure, we speak of a $(G,\tau)$-probability space.
If $(G,S,\tau)$ is a symmetric semigroup, then we write $\Sigma_{\pm}$ for the
sub-$\sigma$-algebra generated by $(\Sigma_{s})_{s\in S^{\pm 1}}$, and
$E_{\pm}$ for the corresponding conditional expectations.

######  Definition 3.2.

(a) A $(G,\tau)$-measure space is called reflection positive with respect to
$S$ if

$\displaystyle\langle\theta f,f\rangle\geq 0\qquad\mbox{for}\quad
f\in\mathcal{E}_{+}:=L^{2}(Q,\Sigma_{+},\mu).$

This is equivalent to $E_{+}\theta E_{+}\geq 0$ as an operator on
$L^{2}(Q,\Sigma,\mu)$ and obviously implies $\theta E_{0}=E_{0}$. If this
condition is satisfied and, in addition, $\Sigma_{0}$ is invariant under the
group $H(S):=S\cap S^{-1}$, then we call it a $(G,S,\tau)$-measure
space222Note that $E_{+}\theta E_{+}\geq 0$ is equivalent to the condition
that the kernel $K(A,B):=\mu(A\cap\theta(B))$ on $\Sigma_{+}$ is positive
definite..

(b) A Markov $(G,S,\tau)$-measure space is a $(G,S,\tau)$-measure space with
the Markov property $E_{+}E_{-}=E_{+}E_{0}E_{-}$ (cf. Definition 2.1(b)).

Proposition 2.9 now implies:

######  Proposition 3.3.

For any $(G,S,\tau)$-measure space $((Q,\Sigma,\mu),\Sigma_{0},U,\theta)$, we
put $\mathcal{E}:=L^{2}(Q,\Sigma,\mu)$,
$\mathcal{E}_{0}:=L^{2}(Q,\Sigma_{0},\mu)$ and
$\mathcal{E}_{\pm}:=L^{2}(Q,\Sigma_{\pm},\mu)$. Then the natural action of $G$
on $\mathcal{E}$ defines a reflection positive representation of $(G,S,\tau)$.
The Markov property is equivalent to the natural map
$\mathcal{E}_{0}\to\widehat{\mathcal{E}}$ being unitary and this implies that
the function $\varphi\colon S\to
B(\mathcal{E}_{0}),\varphi(s)=E_{0}U_{s}E_{0}$ is multiplicative.

######  Definition 3.4.

(a) A positive semigroup structure $(\mathcal{H},P,\mathcal{A},\Omega)$ for a
symmetric semigroup $(G,S,\tau)$ consists of

  * (PS1)

a Hilbert space $\mathcal{H}$,

  * (PS2)

a strongly continuous $*$-representation $(P_{s})_{s\in S}$ of $(S,\sharp)$ by
contractions on $\mathcal{H}$,

  * (PS3)

a commutative von Neumann algebra $\mathcal{A}$ on $\mathcal{H}$ normalized by
the operators $(P_{h})_{h\in H(S)}$, and

  * (PS4)

a unit vector $\Omega\in\mathcal{H}$, such that

    * (a)

$P_{s}\Omega=\Omega$ for every $s\in S$.

    * (b)

$\Omega$ is cyclic for the (not necessarily selfadjoint) subalgebra
$\mathcal{B}\subseteq B(\mathcal{H})$ generated by $\mathcal{A}$ and
$\\{P_{s}\colon s\in S\\}$.

    * (c)

For positive elements $A_{1},\ldots,A_{n}\in\mathcal{A}$ and
$s_{1},\ldots,s_{n-1}\in S$, we have

$\displaystyle\langle A_{1}P_{s_{1}}A_{2}\cdots
P_{s_{n-1}}A_{n}\Omega,\Omega\rangle\geq 0.$

(b) A standard positive semigroup structure for a symmetric semigroup
$(G,S,\tau)$ consists of a $\sigma$-finite measure space
$(M,{\mathfrak{S}},\nu)$ and

  * (SPS1)

a representation $(P_{s})_{s\in S}$ of $S$ on $L^{\infty}(M,\nu)$ by
positivity preserving operators, i.e., $P_{s}f\geq 0$ for $f\geq 0$.

  * (SPS2)

$P_{s}1=1$ for $s\in S$ (the Markov condition).

  * (SPS3)

$P$ is involutive with respect to $\nu$, i.e.,

$\displaystyle\int_{M}P_{s}(f)hd\nu=\int_{M}fP_{s^{\sharp}}(h)\,d\nu\qquad\mbox{for}\quad
s\in S,\quad f,h\geq 0.$

  * (SPS4)

$P$ is strongly continuous in measure, i.e., for each $f\in L^{1}(M,\nu)\cap
L^{\infty}(M,\nu)$ and every $\delta>0$, $s_{0}\in S$, we have
$\lim\limits_{s\to s_{0}}\nu(\\{|P_{s}f-P_{s_{0}}f|\geq\delta\\})=0$.

The preceding definition generalizes the corresponding classical concepts for
the case $(G,S,\tau)$ $=({\mathbb{R}},{\mathbb{R}}_{+},-\mathop{{\rm
id}}\nolimits_{\mathbb{R}})$ ([41] for (a) and [42] for (b)).

######  Remark 3.5.

(a) For an operator $P$ on $L^{2}(X,\mu)$, the condition to be positivity
preserving does not imply that $P$ is symmetric. In fact, if $(Pf)(x)=f(Tx)$
for some measure preserving map $T\colon X\to X$, then $P$ is positivity
preserving and isometric. Hence we cannot expect $P$ to be symmetric if $T$ is
not an involution.

(b) Applying (SPS3) with $h=1$, we obtain
$\int_{M}P_{s}(f)\,d\mu=\int_{M}f\,d\mu$, i.e., that the measure $\mu$ is
invariant under the operators $(P_{s})_{s\in S}$.

(c) If $P\colon M\times{\mathfrak{S}}\to[0,\infty]$ is a Markov kernel (cf.
Appendix B), then

$\displaystyle(Pf)(x):=\int_{M}P(x,dy)f(y)$

defines a positivity preserving operator on $L^{\infty}(M,{\mathfrak{S}},\nu)$
satisfying $P1=1$.

In view of [7, Lemma 36.2], kernels on $M\times{\mathfrak{S}}$ are in one-to-
one correspondence with additive, positively homogeneous maps $T$ from the
convex cone of non-negative measurable functions $M\to[0,\infty]$ into itself
which are Daniell continuous in the sense that
$\lim\limits_{n\to\infty}T(f_{n})=T\big{(}\lim\limits_{n\to\infty}f_{n}\big{)}$
for monotone sequences $f_{n}\leq f_{n+1}$.333The notion of a positivity
preserving operator on $L^{\infty}(M,{\mathfrak{S}},\nu)$ is slightly weaker
than this concept. In particular, it only operates on equivalence classes of
functions in $L^{\infty}(X,{\mathfrak{S}},\nu)$ and not on functions itself.
In concrete situations, the positivity preserving operators actually come from
kernels, which makes them easier to deal with.

######  Remark 3.6.

We consider a standard positive semigroup structure for $(G,S,\tau)$.

(a) Conditions (SPS1/2) imply that $\|P_{s}\|\leq 1$ on $L^{\infty}(M,\nu)$.
Further (SPS2/3) imply that the restriction of $P_{s}$ to $L^{1}\cap
L^{\infty}$ is measure preserving, and from [42, Proposition 1.2(i)] it
follows that $P_{s^{\sharp}s}$ defines a contraction on $L^{2}(M,\nu)$. This
implies that we obtain a $*$-representation of $(S,\sharp)$ on $L^{2}(M,\nu)$.
The proof of [42, Proposition 1.2(ii)] further shows that (SPS4) implies that
this representation is strongly continuous because we have for $f\in L^{1}\cap
L^{2}$:

$\displaystyle\|P_{s}f-P_{s_{0}}f\|_{2}^{2}$
$\displaystyle\leq\int_{\\{|P_{s}f-P_{s_{0}}f|\geq\varepsilon\\}}|P_{s}f-P_{s_{0}}f|^{2}\,d\nu+\int_{\\{|P_{s}f-P_{s_{0}}f|\leq\varepsilon\\}}|P_{s}f-P_{s_{0}}f|^{2}\,d\nu$
$\displaystyle\leq(2\|f\|_{\infty})^{2}\nu(\\{|P_{s}f-P_{s_{0}}f|\geq\varepsilon\\})+2\varepsilon\|f\|_{1}.$

For $h\in H(S)$, both operators $P_{h}$ and $P_{h^{-1}}=P_{h}^{-1}$ are
positivity preserving and fix $1$, and since $L^{\infty}(M,\nu)$ is a
commutative von Neumann algebra, they are algebra automorphisms by [12,
Theorem 3.2.3]. The relation $P_{h}(fg)=P_{h}(f)P_{h}(g)$ then leads to

$\displaystyle P_{h}M_{f}P_{h}^{-1}=M_{P_{h}(f)}$

for the multiplication operators $M_{f}$. In particular, the action of $H(S)$
on $L^{2}(M,\nu)$ normalizes $L^{\infty}(M,\nu)$.

(b) If $\nu$ is a probability measure, $\Omega:=1$ and
$\mathcal{A}:=L^{\infty}(M,\nu)$, then the preceding discussion shows that we
also have a positive semigroup structure in the sense of Definition 3.4(a) for
which $1$ is a cyclic vector for $\mathcal{A}$. Note that Definition 3.4(b) is
not a special case of (a) because it does not require the measure $\nu$ to be
finite.

The following proposition shows that the requirement that $\Omega$ is cyclic
for $\mathcal{A}$ describes those positive semigroup structures which are
standard.

######  Proposition 3.7 ([41, Proposition 3.5]).

Let $(M,{\mathfrak{S}},\nu)$ be a probability space and $(P_{s})_{s\in S}$ be
a positivity preserving continuous $*$-representation of $S$ by contractions
on $L^{2}(M,\nu)$, i.e.,

$\displaystyle P_{s}1=1\qquad\mbox{and}\qquad P_{s}f\geq
0\qquad\mbox{for}\quad f\geq 0,\quad s\in S.$

Then $(L^{2}(M),Q,L^{\infty}(M),1)$ is a positive semigroup structure for
which $1$ is a cyclic vector for $L^{\infty}(M)$.

Conversely, let $(\mathcal{H},P,\mathcal{A},\Omega)$ be a positive semigroup
structure for which $\Omega$ is a cyclic vector for $\mathcal{A}$. Then there
exists a probability space $M$ and a positivity preserving semigroup
$\widetilde{Q}$ on $L^{2}(M)$ such that
$(\mathcal{H},P,\mathcal{A},\Omega)\cong(L^{2}(M),\widetilde{Q},L^{\infty}(M),1)$
as positive semigroup structures.

###### Proof.

The first part is an immediate consequence of the definitions (see also Remark
3.6(a)), so we only have to prove the second statement. So let
$(\mathcal{H},P,\mathcal{A},\Omega)$ be a positive semigroup structure for
which $\Omega$ is a cyclic vector for $\mathcal{A}$. Let $M$ be a compact
space with $\mathcal{A}\cong C(M)$. Then we obtain on $M$ a probability
measure $\nu$, defined by

$\displaystyle\int_{M}f(x)\,d\nu(x)=\langle
f\Omega,\Omega\rangle\qquad\mbox{for}\quad f\in\mathcal{A}.$

Since $\Omega$ is $\mathcal{A}$-cyclic, the map $C(M)\to\mathcal{H},f\mapsto
f\Omega$ extends to a unitary map $L^{2}(M,\nu)\to\mathcal{H}$ intertwining
$\mathcal{A}$ with the von Neumann algebra $L^{\infty}(M,\nu)$. It is clear
that (SPS1-3) are satisfied and Lemma A.1 implies the continuity in measure. ∎

######  Example 3.8 (the real oscillator semigroup).

We consider the Hilbert space $\mathcal{H}=L^{2}({\mathbb{R}}^{d})$, with
respect to Lebesgue measure.

(a) On $\mathcal{H}$ we have a unitary representation by the group
$\mathop{{\rm GL}}\nolimits_{d}({\mathbb{R}})$ by

$\displaystyle(T_{h}f)(x):=|\det(h)|^{-d/2}f\big{(}h^{-1}x\big{)}\qquad\mbox{for}\quad
h\in\mathop{{\rm GL}}\nolimits_{d}({\mathbb{R}}),\quad x\in{\mathbb{R}}^{d},$

and we also have two representations of the abelian semigroup $\mathop{{\rm
Sym}}\nolimits_{d}({\mathbb{R}})_{+}$ (the convex cone of positive
semidefinite matrices):

  * (a)

Each $A\in\mathop{{\rm Sym}}\nolimits_{d}({\mathbb{R}})_{+}$ defines a
multiplication operator $(M_{A}f)(x):=e^{-\langle Ax,x\rangle}f(x)$ which is
positivity preserving on $L^{\infty}({\mathbb{R}}^{n})$ but does not preserve
$1$; it preserves the Dirac measure $\delta_{0}$ in the origin.

  * (b)

Each $A\in\mathop{{\rm Sym}}\nolimits_{d}({\mathbb{R}})_{+}$ specifies a
uniquely determined (possibly degenerate) gaussian measure $\mu_{A}$ on
${\mathbb{R}}^{d}$ whose Fourier transform is given by
$\widehat{\mu}_{A}(x)=e^{-\langle Ax,x\rangle/2}$. Then the convolution
operator $C_{A}f:=f*\mu_{A}$ is positivity preserving and leaves Lebesgue
measure on ${\mathbb{R}}^{d}$ invariant. For $A=\mathbf{1}$, we thus obtain
the heat semigroup as $(\mu_{t\mathbf{1}})_{t\geq 0}$.

Any composition of these $3$ types of operators $T_{h},M_{A}$ and $C_{A}$ is
positivity preserving on $L^{\infty}({\mathbb{R}}^{d})$, and they generate a
$*$-representation of the Olshanski semigroup $S:=H\exp(C)$ in the symmetric
Lie group $G:=\mathop{{\rm Sp}}\nolimits_{2d}({\mathbb{R}})$, where
$H=\mathop{{\rm GL}}\nolimits_{d}({\mathbb{R}})$, $C=\mathop{{\rm
Sym}}\nolimits_{d}({\mathbb{R}})_{+}\times\mathop{{\rm
Sym}}\nolimits_{d}({\mathbb{R}})_{+}\subseteq\mathop{{\rm
Sym}}\nolimits_{d}({\mathbb{R}})^{2}={\mathfrak{q}}$, and

$\displaystyle\tau\begin{pmatrix}A&B\\\
C&-A^{\top}\end{pmatrix}=\begin{pmatrix}A&-B\\\
-C&-A^{\top}\end{pmatrix}\qquad\mbox{for}\quad\begin{pmatrix}A&B\\\
C&-A^{\top}\end{pmatrix}\in\mathfrak{sp}_{2d}({\mathbb{R}})$

with $B^{\top}=B$, $C^{\top}=C$ (cf. Examples 2.5). The real Olshanski
semigroup $S$ is the fixed point set an antiholomorphic involutive
automorphism of the so-called oscillator semigroup
$S_{\mathbb{C}}=G^{c}\exp(W)$ which is a complex Olshanskii semigroup [24,
30]. The elements in the interior of $S$ act on $L^{2}({\mathbb{R}}^{n})$ by
kernel operators with positive gaussian kernels and the elements of
$S_{\mathbb{C}}$ correspond to complex-valued gaussian kernels. The semigroup
$S$ contains many interesting symmetric one-parameter semigroups such as the
Mehler semigroup $e^{-tH_{\rm osc}}$ generated by the oscillator Hamiltonian

$\displaystyle H_{\rm
osc}=-\sum_{j=1}^{n}\partial_{j}^{2}+\frac{1}{4}\sum_{j=1}^{n}x_{j}^{2}-\frac{d}{2}\mathbf{1},$
(3.1)

which fixes the Gaussian $e^{-\|x\|^{2}/4}$.

(b) The parabolic subgroup $P:=\mathop{{\rm
Sym}}\nolimits_{d}({\mathbb{R}})\rtimes\mathop{{\rm
GL}}\nolimits_{d}({\mathbb{R}})_{+}\subseteq\mathop{{\rm
Sp}}\nolimits_{2d}({\mathbb{R}})$ is also a symmetric Lie subgroup with
${\mathfrak{h}}=\mathop{{\mathfrak{gl}}}\nolimits_{d}({\mathbb{R}})$ and
${\mathfrak{q}}\cong\mathop{{\rm Sym}}\nolimits_{d}({\mathbb{R}})$. Here the
commutative von Neumann algebra $\mathcal{A}=L^{\infty}({\mathbb{R}}^{d})$ is
invariant under conjugation with the operators $T_{h}$, so that $(A,h)\mapsto
C_{A}T_{h}$ defines a $*$-representation of the semigroup $S:=\mathop{{\rm
Sym}}\nolimits_{d}({\mathbb{R}})_{+}\rtimes H$ that leads to a standard
positive semigroup structure on $L^{2}({\mathbb{R}}^{d},dx)$.

The preceding example naturally extends to infinite-dimensional spaces as
follows.

######  Example 3.9.

Let $\mathcal{H}$ be a real Hilbert space.

(a) In Lemma 5.5 below we show that every continuous $*$-representation of a
topological involutive semigroup $(S,\sharp)$ by contractions on a real
Hilbert space $\mathcal{H}$ defines a standard positive semigroup structure on
the gaussian probability space $(\mathcal{H}^{a},\gamma_{\mathcal{H}})$. Here
$\mathcal{H}^{a}$ is the algebraic dual space of $\mathcal{H}$ and
$\gamma_{\mathcal{H}}$ is the unique probability measure with Fourier
transform $\widehat{\gamma}_{\mathcal{H}}(v)=e^{-\|v\|^{2}/2}$ (Example 5.3).

(b) Some of the structure from Example 3.8 extends to infinite-dimensional
Hilbert spaces $\mathcal{H}$. For the multiplication operator $M_{A}$ to be
non-zero, one has to require that $A\geq 0$ is trace class (cf. [68, p. 153]).
Likewise, the measures $\mu_{A}$ exist if $A$ is trace class. In an
orthonormal basis $(e_{n})_{n\in{\mathbb{N}}}$ in which $A$ is diagonal with
$Ae_{n}=a_{n}e_{n}$, the measure $\mu_{A}$ is
$\otimes_{n=1}^{\infty}\gamma_{a_{n}t}$, where $d\gamma_{t}(x)=(2\pi
t)^{-1/2}e^{-x^{2}/2t}\,dx$ are the centered gaussian measures on
${\mathbb{R}}$. For a linear operator $h\in\mathop{{\rm
GL}}\nolimits(\mathcal{H})$ to act on
$L^{2}(\mathcal{H}^{a},\gamma_{\mathcal{H}})$, it is necessary and sufficient
that $g^{\top}g-\mathbf{1}$ is Hilbert–Schmidt, i.e., that the polar
decomposition has the form $g=ue^{X}$ with $\|X\|_{2}<\infty$ (see also
Theorem 5.7 and [52] for more details).

(c) For $\mathcal{H}={\mathbb{R}}^{d}$, we have
$d\gamma(x):=d\gamma_{\mathcal{H}}(x)=(2\pi)^{-d/2}e^{-\frac{\|x\|^{2}}{2}}\,dx$
(Example 5.4), we obtain in particular a standard positive semigroup structure
for the contraction semigroup

$\displaystyle\mathcal{C}\big{(}{\mathbb{R}}^{d}\big{)}:=\\{g\in\mathop{{\rm
GL}}\nolimits_{d}({\mathbb{R}})\colon\|g\|\leq 1\\}=\mathop{\rm
O{}}\nolimits_{d}({\mathbb{R}})\exp(-\mathop{{\rm
Sym}}\nolimits_{d}({\mathbb{R}})_{+}),$

where $\mathop{{\rm Sym}}\nolimits_{d}({\mathbb{R}})_{+}$ denotes the closed
convex cone of positive semidefinite symmetric $(d\times d)$-matrices. The
corresponding Markov operators are given by $(\Gamma(h)f)(x)=f(h^{-1}x)$ for
$h\in\mathop{\rm O{}}\nolimits_{d}({\mathbb{R}})$. Since every element
$s\in\mathcal{C}({\mathbb{R}}^{d})$ has a polar decomposition $s=h\exp(-X)$,
$X^{\top}=X\geq 0$, diagonalization of $X$ reduces the description of the
corresponding operator to the case $d=1$. For $0<c<1$ we have

$\displaystyle(\Gamma(c)f)(x)=\int_{{\mathbb{R}}^{d}}\widetilde{K}_{c}(x,y)f(y)\,d{\gamma}(y)\qquad\mbox{with}\quad\widetilde{K}_{c}(x,y)=\big{(}1-c^{2}\big{)}^{-d/2}e^{\frac{\|y\|^{2}}{2}}e^{-\frac{\|cx-y\|^{2}}{2(1-c^{2})}}$

(cf. [55, p. 218]). For $c=e^{-t}$, $t\geq 0$, we thus obtain the Mehler
semigroup, also called the Ornstein–Uhlenbeck semigroup. It can also be
described by the Mehler formula [11, Section 1.4]

$\displaystyle(\Gamma(c)f)(x)=\int_{{\mathbb{R}}^{d}}f(cx+\sqrt{1-c^{2}}y)\,d\gamma(y).$
(3.2)

These operators form a hermitian strongly continuous contraction semigroup for
which $\gamma$ is the unique invariant probability measure. In view of [11,
Section 2.9], (3.2) still holds for infinite-dimensional spaces.

(d) For $\mathcal{H}={\mathbb{R}}^{d}$, it is instructive to connect the
gaussian picture to the Lebesgue picture. To this end, we note that

$\displaystyle\Phi\colon\ L^{2}\big{(}{\mathbb{R}}^{d}\big{)}\to
L^{2}\big{(}{\mathbb{R}}^{d},\gamma\big{)},\qquad\Phi(f)(x)=(2\pi)^{d/4}e^{\|x\|^{2}/4}f(x)$

is a unitary isomorphism. Conjugating with $\Phi$, we therefore obtain a
$*$-representation $\pi$ of the real Olshanski semigroup $S$ (Example 3.8) on
$L^{2}({\mathbb{R}}^{d},\gamma)$. This transformation has no effect on the
multiplication operators $M_{A}$ because they commute with $\Phi$ but it
transforms the convolution operators $C_{A}$ into more complicated Markov
operators. For instance the Laplace operator $\Delta=\sum_{j}\partial_{j}^{2}$
transforms into

$\displaystyle\Delta^{\prime}:=\Phi\Delta\Phi^{-1}=\Delta+\frac{\|x\|^{2}}{4}-\left(E+\frac{d}{2}\mathbf{1}\right),\qquad\mbox{where}\quad
E=\sum_{j}x_{j}\partial_{j}$

is the Euler operator. The unitary representation of $H=\mathop{{\rm
GL}}\nolimits_{d}({\mathbb{R}})$ transforms into

$\displaystyle
T_{h}^{\prime}f=e^{\frac{1}{4}(\|x||^{2}-\|h^{-1}x\|^{2})}(T_{h}f)(x)=|\det
h|^{-d/2}e^{\frac{1}{4}(\|x||^{2}-\|h^{-1}x\|^{2})}f\big{(}h^{-1}x\big{)}.$

As the operators $\Phi$ and $\Phi^{-1}$ are positivity preserving, the
semigroup $S$ also acts by positivity preserving operators on
$L^{2}({\mathbb{R}}^{n},\gamma)$. If $s\in S$ preserves $1$, then the
transformed operator $s^{\prime}$ preserves the Gaussian $e^{\|x\|^{2}/4}$.

(e) The oscillator Hamiltonian $H_{\rm osc}$ from (3.1) transforms into

$\displaystyle-\Delta^{\prime}:=\Phi H_{\rm
osc}\Phi^{-1}=\Phi\circ\left(-\Delta+\frac{\|x\|^{2}}{4}-\frac{d}{2}\mathbf{1}\right)\circ\Phi^{-1}=E-\Delta=\sum_{j}x_{j}\partial_{j}-\partial_{j}^{2},$

which also makes sense in the infinite-dimensional case, where
$\Delta^{\prime}$ is called Umemura’s Laplace–Beltrami operator [23, p. 221].
It is the infinitesimal generator of the Mehler semigroup
$e^{t\Delta^{\prime}}$.

######  Example 3.10 (a finite-dimensional example).

(a) We consider the finite set $M=\\{1,\ldots,n\\}$, endowed with the counting
measure, so that $L^{2}(M,\nu)\cong{\mathbb{R}}^{d}$. Let
$S\subseteq\mathop{{\rm GL}}\nolimits_{n}({\mathbb{R}})$ be the subsemigroup
of invertible doubly stochastic matrices $S=(S_{ij})$, i.e.,
$\sum_{i}S_{ij}=\sum_{j}S_{ij}=1$ for all $i$, $j$. According to the
Birkhoff–von Neumann theorem, $S\subseteq\mathop{{\rm conv}}\nolimits(H)$,
where $H=S\cap S^{-1}\cong S_{n}$ is the group of permutation matrices. Then
$S\subseteq\mathop{{\rm GL}}\nolimits_{n}({\mathbb{R}})$ a transposition
stable subsemigroup for which the action on ${\mathbb{R}}^{n}\cong
L^{\infty}(M)$ by matrix multiplication defines a standard positive semigroup
structure.

(b) A natural generalization of the preceding example is obtained by starting
with a probability space $(Q,\Sigma,\nu)$ and the group $H:=\mathop{{\rm
Aut}}\nolimits(Q,\Sigma,\nu)$ of measure preserving automorphisms of
$(Q,\Sigma)$. Then the probability measures $\mu$ on $Q$ defining by
$P_{\mu}(f)(x):=\int_{H}f(h^{-1}x)\,d\mu(h)$ invertible operators on
$L^{\infty}(Q,\Sigma,\nu)$ naturally generalizes the semigroup
$S\subseteq\mathop{{\rm GL}}\nolimits_{d}({\mathbb{R}})$ from (a).

###  3.2 Stochastic processes indexed by Lie groups

We now introduce the concept of stochastic processes where the more common
index set ${\mathbb{R}}$, thought of as a time, is replaced by a Lie group.
The forward direction is then given by a semigroup contained in $G$.

######  Definition 3.11.

Let $(Q,\Sigma,\mu)$ be a probability space. A stochastic process indexed by a
group $G$ is a family $(X_{g})_{g\in G}$ of measurable functions $X_{g}\colon
Q\to(B,{\mathfrak{B}})$, where $(B,{\mathfrak{B}})$ is a measurable space,
called the state space of the process.

For any such process, we obtain a measurable map $\Phi\colon Q\to B^{G}$,
$\Phi(q)=(X_{g}(q))_{g\in G}$ with respect to the product $\sigma$-algebra
${\mathfrak{B}}^{G}$. Then $\nu:=\Phi_{*}\mu$ is a measure on $B^{G}$, called
the distribution of the process $(X_{g})_{g\in G}$. This measure is uniquely
determined by the measures $\nu_{\bf{g}}$ on $G^{n}$, obtained for any tuple
${\bf{g}}:=(g_{1},\ldots,g_{n})\in G^{n}$ as the image of $\mu$ under the map

$\displaystyle X_{\bf{g}}=(X_{g_{1}},\ldots,X_{g_{n}})\colon\ Q\to B^{n}$

(cf. [23, Section 1.3]). If ${\bf{g}}=(g)$ for some $g\in G$ then we write
$\nu_{g}$ for $\nu_{{\bf{g}}}$.

The process $(X_{g})_{g\in G}$ is called stationary if the corresponding
distribution on $B^{G}$ is invariant under the translations

$\displaystyle(U_{g}\nu)_{h}:=\nu_{g^{-1}h}\qquad\mbox{for}\quad g,h\in G.$

If $\tau\in\mathop{{\rm Aut}}\nolimits(G)$ is an automorphism, then we call
the process $\tau$-invariant if the distribution is invariant under

$\displaystyle(\tau\nu)_{h}:=\nu_{\tau^{-1}(h)}\qquad\mbox{for}\quad h\in G.$

The following definition generalizes some concepts from $G={\mathbb{R}}$ to
general topological groups (cf. [40, Section 2]). We only formulate the
definition for real valued functions, but will use without further comments
the complex analogue where in the $L^{2}$-inner products the second factor has
to be conjugated.

######  Definition 3.12.

(a) A linear stochastic process (indexed by the group $G$) is a stochastic
process $(\varphi(v,g))_{v\in V,g\in G}$, where $V$ is a real linear space,
$\varphi$ is linear in $v$ and each $\varphi(v,g)$ is a real valued measurable
function on a fixed probability space $(Q,\Sigma,\mu)$.

(b) A linear stochastic process is said to be of second order if each
$\varphi(v,g)$ is square integrable and the stochastic process is full in the
sense that, up to sets of measure zero, $\Sigma$ is the smallest
$\sigma$-algebra for which all functions $\varphi(v,g)$ are measurable.

(c) That $\varphi$ is continuous in the quadratic mean means that $V$ is a
topological vector space, $\varphi$ is of second order and $\varphi\colon
V\times G\to L^{2}(Q,\Sigma,\mu)$ is continuous.

(d) Assume that $\varphi$ is of second order. Then $\varphi$ is (wide sense)
stationary means that the kernel

$\displaystyle
K((v,g),(v^{\prime},g^{\prime})):={{\mathbb{E}}}\big{(}\overline{\varphi(v,g)}\varphi(v^{\prime},g^{\prime})\big{)}=\int_{Q}\overline{\varphi(v,g)}\varphi(v^{\prime},g^{\prime})\,d\mu$

on $V\times G$ satisfies the invariance condition

$\displaystyle
K((v,g),(v^{\prime},g^{\prime}))=K((v,g_{0}g),(v^{\prime},g_{0}g^{\prime}))\qquad\mbox{for}\quad
g_{0},g,g^{\prime}\in G,\quad v,v^{\prime}\in V.$

Let $\varphi$ be a stochastic process. To adapt better to the discussion in
the previous section, we assume that $\varphi(v,g)$ is complex-valued and that
$V$ is a complex vector space. The real case can be treated in the same way.
If $\varphi$ is a linear stochastic process of second order, then the square
integrability of the functions $\varphi(v,g)$ implies that the covariance
kernel

$\displaystyle
K((v,g),(v^{\prime},g^{\prime})):={{\mathbb{E}}}\big{(}\overline{\varphi(v,g)}\varphi(v^{\prime},g^{\prime})\big{)}=\int_{Q}\overline{\varphi(v,g)}\varphi(v^{\prime},g^{\prime})\,d\mu$

is defined. It is a positive definite kernel on $V\times G$, and the map

$\displaystyle\Phi\colon\
L^{2}(Q,\Sigma,\mu)\to\mathcal{H}_{K}\subseteq{\mathbb{C}}^{V\times
G},\qquad\Phi(F)(v,g):=\langle F,\varphi(v,g)\rangle$

is a partial isometry of $L^{2}(Q,\Sigma,\mu)$ onto the reproducing kernel
Hilbert space $\mathcal{H}_{K}\subseteq{\mathbb{C}}^{V\times G}$. Its adjoint
$\Phi^{*}$ is an isometric embedding of $\mathcal{H}_{K}$ whose range is the
closed subspace on $L^{2}(Q,\Sigma,\mu)$ generated by the
$(\varphi(v,g))_{v\in V,g\in G}$. Furthermore, $\varphi$ is continuous in the
quadratic mean if and only if $K$ is continuous.

We will from now on assume that $\varphi$ is continuous in the quadratic mean.
Then

$\displaystyle(v,v^{\prime}):={{\mathbb{E}}}\big{(}\varphi(v,\mathbf{1})\overline{\varphi(v^{\prime},\mathbf{1})}\big{)}=\int_{Q}\varphi(v,\mathbf{1})\overline{\varphi(v^{\prime},\mathbf{1})}\,d\mu$

defines a continuous positive semidefinite form on $V$, and in the following
we may therefore replace $V$ by the complex Hilbert space $\mathcal{F}$
obtained by completing the quotient

$\displaystyle V/\\{v\in V\colon(v,v)=0\\}$

with respect to the induced norm. We thus obtain a stochastic process
$\varphi(w,g)$ on $\mathcal{F}\times G$. Now we may consider $\mathcal{F}$ as
a closed subspace of $L^{2}(Q,\Sigma,\mu)$ and write $E_{0}\colon
L^{2}(Q,\Sigma,\mu)\to\mathcal{F}$ for the orthogonal projection.

If $K$ is continuous and $\mathcal{F}$ is a complex Hilbert space, then $K$
also defines a positive definite $B(\mathcal{F})$-valued kernel

$\displaystyle K^{\mathcal{F}}\colon\ G\times G\to
B(\mathcal{F}),\qquad\big{\langle}K^{\mathcal{F}}(g,h)v,w\big{\rangle}=K((v,h),(w,g)),$

so that we can realize the closed subspace of $L^{2}(Q,\mu)$ generated by the
$\varphi(v,g)$ as the reproducing kernel Hilbert space
$\mathcal{H}_{K^{\mathcal{F}}}\subseteq{\mathbb{C}}^{\mathcal{F}}$ of
$\mathcal{F}$-valued functions on $G$. Here $\mathcal{F}$ is isomorphic to the
closed subspace of $L^{2}(Q,\Sigma,\mu)$ generated by the functions
$(\varphi(v,\mathbf{1}))_{v\in V}$.

Consider now (d) and assume that $\varphi$ is continuous in the quadratic
mean. This condition ensures that we obtain a unitary representation
$(U_{g})_{g\in G}$ on the subspace $\mathcal{H}_{K}\subseteq L^{2}(Q,\mu)$
satisfying

$\displaystyle(U_{g}\varphi)(v,h)=\varphi(v,gh),\qquad v\in V,\quad g,h\in G.$

Under assumption (c), we thus obtain a strongly continuous
$B(\mathcal{F})$-valued positive definite function $r(g):=E_{0}U_{g}E_{0}$.
The invariance condition leads to a $\mathop{{\rm Sesq}}\nolimits(V)$-valued
positive definite function on $G$ as discussed in [51], but we will not follow
up on that in this article.

######  Example 3.13.

Let $(G,\tau)$ be a symmetric Lie group and $(X_{g})_{g\in G}$ be a
stationary, $\tau$-invariant, full stochastic process on
$(Q,\Sigma_{Q},\mu_{Q})$. Then its distribution
$(B^{G},{\mathfrak{B}}^{G},\nu)$ satisfies the conditions (GP1,2,4,5) of a
$(G,\tau)$-probability space with respect to the natural actions of $G$ and
$\tau$ on $B^{G}$, where $\Sigma_{0}$ is the $\sigma$-algebra generated by
$(X_{h})_{h\in H}$, i.e., the smallest subalgebra for which these functions
are measurable. In this context (GP3) is equivalent to the continuity of the
unitary representation of $G$ on $L^{2}(B^{G},{\mathfrak{B}}^{G},\nu)$. We
refer to Appendix A for a more detailed discussion of the continuity
condition.

######  Example 3.14 (cf. Proposition 3.3).

Suppose that $((Q,\Sigma,\mu),\Sigma_{0},U,\theta)$ is a
$(G,\tau)$-probability space, so that we obtain on
$\mathcal{E}:=L^{2}(Q,\Sigma,\mu)$ a unitary representation of $G$. The
subspace $\mathcal{E}_{0}:=L^{2}(Q,\Sigma_{0},\mu)$ is cyclic under the
algebra $\mathcal{B}$ generated by $(U_{g})_{g\in G}$ and the multiplication
operators $M_{f}$, $f\in L^{\infty}(Q,\Sigma_{0},\mu)$. The prescription

$\displaystyle\varphi(v,g):=\pi(g)v,\qquad g\in G,\quad v\in\mathcal{E}_{0}$

defines a wide sense stationary process indexed by $\mathcal{E}_{0}\times G$.

The scalar product on $L^{2}(Q,\Sigma,\mu)$ is completely determined by the
$n$-point functionals:

$\displaystyle\langle U_{g_{1}}M_{f_{1}}\cdots
M_{f_{n-1}}U_{g_{n}}M_{f_{n}}v,w\rangle_{\mathcal{E}_{0}},\qquad
v,w\in\mathcal{E}_{0},\quad g_{j}\in G,\quad f_{j}\in
L^{\infty}(Q,\Sigma_{0},\mu),$

resp.,

$\displaystyle\langle U_{g_{1}}M_{f_{1}}\cdots
U_{g_{n}}f_{n},f_{0}\rangle_{\mathcal{E}_{0}}={{\mathbb{E}}}\big{(}f_{0}\varphi(f_{1},g_{1})\varphi(f_{2},g_{1}g_{2})\cdots\varphi(f_{n},g_{1}\cdots
g_{n})|\Sigma_{0}\big{)},$

or the operators

$\displaystyle E_{0}U_{g_{1}}M_{f_{1}}\cdots M_{f_{n-1}}U_{g_{n}}E_{0},\qquad
g_{j}\in G,\quad f_{j}\in L^{\infty}(Q,\Sigma_{0},\mu).$

In general $\mathcal{E}_{0}$ is not cyclic for $G$, so the process
$\varphi(v,g)$ provides a means to construct the whole space
$L^{2}(Q,\Sigma,\mu)$ in the spirit of a GNS construction for $\mathcal{B}$.

######  Example 3.15.

In [64] free euclidean fields (scalar of positive mass $m>0$) are dealt with
as gaussian processes indexed by the Hilbert space $N_{m}$ defined by the norm

$\displaystyle\|f\|_{N_{m}}^{2}=2\int_{{\mathbb{R}}^{d+1}}\frac{|\widehat{f}(k)|^{2}}{k^{2}+m^{2}}\,d^{2}k\qquad\mbox{for}\quad
f\in\mathcal{S}\big{(}{\mathbb{R}}^{d+1}\big{)},$

so that the corresponding euclidean Hilbert space is the Fock space
$\Gamma(N_{m})$ (see Section 5.1). Here $N_{m}$ carries a unitary
representation of the euclidean group $E(d)={\rm Mot}({\mathbb{R}}^{d})$, and
since the time translation group has continuous spectrum, the corresponding
action on $\Gamma(N_{m})$ is ergodic by Theorem 5.26 below (for time
translations, space translations and the full euclidean group).

[64, Theorem III.6] provides a Feynman–Kac–Nelson formula for the gaussian
free field. In this context, Nelson’s main achievement in [54] was that he
obtained a manifestly euclidean invariant path integral.

###  3.3 Associated positive semigroup structures and reconstruction

Our goal in this section is to prove the reconstruction theorem (Theorem
3.18). This is a central result which asserts that positive semigroup
structures all come from $(G,S,\tau)$-measure spaces.

We start with the following lemma which is an adaption of [41, Lemma 2.2] and
[42, Corollary 1.5] to $(G,S,\tau)$.

######  Lemma 3.16.

Let $((Q,\Sigma,\mu),\Sigma_{0},U,\theta)$ be a $(G,S,\tau)$-measure space,
$\mathcal{A}:=L^{\infty}(Q,\Sigma_{0},\mu)$,
$\mathcal{E}:=L^{2}(Q,\Sigma,\mu)$,
$\mathcal{E}_{0}:=L^{2}(Q,\Sigma_{0},\mu)$, and
$q\colon\mathcal{E}_{+}:=L^{2}(Q,\Sigma_{+},\mu)\to\widehat{\mathcal{E}}$ be
the canonical map. Then the following assertions hold:

  * (a)

For $f\in\mathcal{A}$, let $M_{f}$ denote the corresponding multiplication
operator on $\mathcal{E}$. Then there exists a bounded operator
$\widehat{M}_{f}\in B(\widehat{\mathcal{E}})$ with $q\circ
M_{f}|_{\mathcal{E}_{+}}=\widehat{M}_{f}\circ q$ and
$\|\widehat{M}_{f}\|=\|f\|_{\infty}$.

  * (b)

$\pi(f):=\widehat{M}_{f}$ is a faithful weakly continuous representation of
the commutative von Neumann algebra $\mathcal{A}$ on $\widehat{\mathcal{E}}$.

  * (c)

In the Markov case we identify $\widehat{\mathcal{E}}$ with $\mathcal{E}_{0}$
and $q$ with $E_{0}$ $($Proposition 3.3). For
$g_{1}\prec_{S}g_{2}\prec_{S}\cdots\prec_{S}g_{n}$ in $G$, non-negative
functions $f_{1},\ldots,f_{n}\in\mathcal{A}$ and $f_{g_{j}}:=U_{g_{j}}f_{j}$,
we have

$\displaystyle\int_{Q}f_{g_{1}}\cdots
f_{g_{n}}\,d\mu=\int_{Q}\widehat{M}_{f_{1}}\widehat{U}_{g_{1}^{-1}g_{2}}\cdots\widehat{M}_{f_{n-1}}\widehat{U}_{g_{n-1}^{-1}g_{n}}\widehat{M}_{f_{n}}1\,d\mu.$

If, in addition, $\mu$ is finite and positive, then
$\Omega:=\mu(Q)^{-1/2}q(1)$ satisfies:

  * (d)

For $g_{1}\prec_{S}g_{2}\prec_{S}\cdots\prec_{S}g_{n}$ in $G$,
$f_{1},\ldots,f_{n}\in\mathcal{A}$ and $f_{g_{j}}:=U_{g_{j}}f_{j}$, we have

$\displaystyle\int_{Q}f_{g_{1}}\cdots
f_{g_{n}}\,d\mu=\big{\langle}\widehat{M}_{f_{1}}\widehat{U}_{g_{1}^{-1}g_{2}}\cdots\widehat{M}_{f_{n-1}}\widehat{U}_{g_{n-1}^{-1}g_{n}}\widehat{M}_{f_{n}}\Omega,\Omega\big{\rangle}.$

  * (e)

$\Omega$ is a separating vector for $\mathcal{A}$ and
$\widehat{U}_{s}\Omega=\Omega$ for every $s\in S$.

  * (f)

$\Omega$ is cyclic for the algebra $\mathcal{B}$ generated by $\mathcal{A}$
and $(\widehat{U}_{s})_{s\in S}$.

###### Proof.

(a, b) Since $\theta$ commutes with $E_{0}$, it preserves the subspace
$\mathcal{E}_{0}:=L^{2}(Q,\Sigma_{0},\mu)$. Reflection positivity and
$\mathcal{E}_{0}\subseteq\mathcal{E}_{+}$ now imply that $\theta(f)=f$ for
$f\in\mathcal{E}_{0}$. The preceding argument implies that $M_{f}$ commutes
with $\theta$. As $M_{f}$ and its adjoint $M_{f}^{*}=M_{\overline{f}}$
preserve $\mathcal{E}_{\pm}$, these operators also preserve
$\mathcal{N}:=\mathcal{E}_{+}\cap\theta(\mathcal{E}_{-})^{\bot}$, so that we
obtain a well-defined operator $\widehat{M}_{f}\colon
q(\mathcal{E}_{+})\to\widehat{\mathcal{E}}.$ For $v\in\mathcal{E}_{+}$, we
then have

$\displaystyle\big{\|}\widehat{M}_{f}\widehat{v}\big{\|}^{2}=\langle\theta
M_{f}v,M_{f}v\rangle=\langle\theta
M_{|f|^{2}}v,v\rangle\leq\|f\|_{\infty}^{2}\|v\|^{2}.$

Applying this estimate to the functions $(f^{n})_{n\in{\mathbb{N}}_{0}}$, we
get $\|\widehat{M}_{f}\|\leq\|f\|_{\infty}$ from [48, Lemma II.3.8]. Since $q$
is isometric on $\mathcal{E}_{0}$ and
$\|M_{f}|_{\mathcal{E}_{0}}\|=\|f\|_{\infty}$, we actually obtain the equality
$\|\widehat{M}_{f}\|=\|f\|_{\infty}$. This proves (a) and (b).

(c) ([42, Corollary 1.5]) First we observe that the operator

$\displaystyle M_{f_{g_{1}}}\cdots
M_{f_{g_{n}}}U_{g_{n}}=M_{U_{g_{1}}f_{1}}\cdots
M_{U_{g_{n}}f_{n}}U_{g_{n}}=U_{g_{1}}M_{f_{1}}U_{g_{1}^{-1}g_{2}}M_{f_{2}}\cdots
M_{f_{n-1}}U_{g_{n-1}^{-1}g_{n}}M_{f_{n}}$

on $\mathcal{E}_{+}$ is intertwined by $q$ with
$\widehat{U}_{g_{1}}\widehat{M}_{f_{1}}\widehat{U}_{g_{1}^{-1}g_{2}}\cdots\widehat{U}_{g_{n-1}^{-1}g_{n}}\widehat{M}_{f_{n}}.$
Applying this operator to the constant function $1$ and integrating yields the
assertion.

From now on we assume that $0<\mu(Q)<\infty$.

(d) With the same argument as under (c), we derive

$\displaystyle\int_{Q}f_{g_{1}}\cdots f_{g_{n}}\,d\mu$ $\displaystyle=\langle
M_{f_{g_{1}}}\cdots M_{f_{g_{n}}}1,1\rangle$ $\displaystyle=\langle\theta
M_{f_{g_{1}}}\cdots
M_{f_{g_{n}}}U_{g_{n}}1,1\rangle=\big{\langle}\widehat{M}_{f_{1}}\widehat{U}_{g_{1}^{-1}g_{2}}\cdots\widehat{M}_{f_{n-1}}\widehat{U}_{g_{n-1}^{-1}g_{n}}\widehat{M}_{f_{n}}\Omega,\Omega\big{\rangle}.$

(e) The relation $\widehat{U}_{s}\Omega=\Omega$ is a trivial consequence of
$U_{s}1=1$ for $s\in S$. Since $q|_{\mathcal{E}_{0}}$ is isometric and
intertwines the representation of $\mathcal{A}$ on $\mathcal{E}_{0}$ with the
representation of $\mathcal{A}$ on $\widehat{\mathcal{E}}$, the
$\mathcal{A}$-separating vector $1\in\mathcal{E}_{0}$ is mapped by $q$ to an
$\mathcal{A}$-separating vector.

(f) Since $\mathcal{E}_{0}$ is $U_{S}$-cyclic in $\mathcal{E}_{+}$, this
follows from the fact that $1$ is $\mathcal{A}$-cyclic in
$L^{2}(Q,\Sigma_{0},\mu)$. ∎

######  Definition 3.17.

The preceding lemma shows that, if $((Q,\Sigma,\mu),\Sigma_{0},U,\theta)$ is a
finite $(G,S,\tau)$-measure space, then
$(\widehat{\mathcal{E}},\widehat{U},\mathcal{A},q(1))$ is a positive semigroup
structure for

$\displaystyle\mathcal{A}=\big{\\{}\widehat{M}_{f}\colon f\in
L^{\infty}(Q,\Sigma_{0},\mu)\big{\\}}.$

We call it the associated positive semigroup structure.

We now come to our version of Klein’s reconstruction theorem. Note that every
discrete group is in particular a $0$-dimensional Lie group, so that the
following theorem applies in particular to discrete groups.

######  Theorem 3.18 (Reconstruction Theorem).

Let $(G,\tau)$ be a symmetric Lie group and $S\subseteq G$ be a
$\sharp$-invariant subsemigroup satisfying $G=S\cup S^{-1}$. Then every
positive semigroup structure for $(G,S,\tau)$ is associated to some
$(G,S,\tau)$-probability space $((Q,\Sigma,\mu),\Sigma_{0},U,\theta)$.

###### Proof.

We follow the proof for
$(G,S,\tau)=({\mathbb{R}},{\mathbb{R}}_{+},-\mathop{{\rm id}}\nolimits)$ in
[41, Theorem 2.4]. In view of the Gelfand representation theorem, there exists
a compact space $Q_{0}$ with $\mathcal{A}\cong C(Q_{0})$ as $C^{*}$-algebras.
Then $Q:=Q_{0}^{G}$, the space of all functions $\omega\colon G\to Q_{0}$, is
compact with respect to the product topology. The group $G$ acts on $Q$ by

$\displaystyle(U_{g}\omega)(x):=\omega\big{(}g^{-1}x\big{)}$

and, accordingly, on functions on $Q$ by
$(U_{g}F)(\omega)=F(U_{g^{-1}}\omega)$. We further obtain an involution
$(\theta\omega)(g):=\omega(\tau(g))$ satisfying

$\displaystyle\theta U_{g}\theta=U_{\tau(g)}\qquad\mbox{for}\quad g\in G.$

Let $\Sigma_{0}$ be the smallest $\sigma$-algebra on $Q$ for which
$X_{\mathbf{1}}(\omega):=\omega(\mathbf{1})$ is measurable,
$\Sigma_{g}:=U_{g}\Sigma_{\mathbf{1}}$, and let $\Sigma$ be the
$\sigma$-algebra generated by the $\Sigma_{g}$, $g\in G$. To construct a
measure $\mu$ on $\Sigma$, it suffices to construct a Radon measure on $Q$,
i.e., a positive functional on the $C^{*}$-algebra $C(Q)$. By continuity, it
actually suffices to do this on a dense unital subalgebra. Such a subalgebra
is generated by the $G$-translates of functions in $\Xi:=C(Q_{0},\\{0,1\\})$,
which is a generating subset of $\mathcal{A}$. Now let $\mathcal{R}\subseteq
C(Q)$ be the subspace spanned by functions of the form

$\displaystyle
F(\omega):=\chi_{1}(\omega(g_{1}))\cdots\chi_{n}(\omega(g_{n})),\qquad
n\in{\mathbb{N}}_{0},\quad\chi_{j}\in\Xi,\quad g_{j}\in
G,\quad\mbox{i.e.},\quad F=\prod_{j=1}^{n}U_{g_{j}}\chi_{j},$

when we identify $C(Q_{0})$ with the subalgebra $X_{\mathbf{1}}^{*}C(Q_{0})$
of $C(Q)$. Clearly, $\mathcal{R}$ is a subalgebra of $C(Q)$, and the
Stone–Weierstraß theorem implies that it is dense.

Since $G=S\cup S^{-1}$, every finite subset of $G$ is of the form
$\\{g_{1},\ldots,g_{n}\\}$ with $g_{1}\prec_{S}\cdots\prec_{S}g_{n}$, i.e.,
$s_{j}:=g_{j}^{-1}g_{j+1}\in S$ for $j=1,\ldots,n-1$. Next we show that that,
for $(A_{j},g_{j})\in\mathcal{A}\times G$ with
$g_{1}\prec_{S}\cdots\prec_{S}g_{n}$, the operator

$\displaystyle A_{1}P_{s_{1}}A_{2}\cdots P_{s_{n-1}}A_{n}$ (3.3)

does not depend on the way we enumerate the $g_{j}$. Suppose that we also have
$g_{j+1}\prec_{S}g_{j}$ for some $j$. Then $h:=s_{j}=g_{j}^{-1}g_{j+1}\in
H(S)=S\cap S^{-1}$. We put $\widetilde{g}_{i}:=g_{i}$ for $i\not=j,j+1$ and
set $\widetilde{g}_{j}:=g_{j+1}$, $\widetilde{g}_{j+1}=g_{j}$. Accordingly, we
put $\widetilde{A}_{j}:=A_{j+1}$ and $\widetilde{A}_{j+1}:=A_{j}$. This leads
to

$\displaystyle\widetilde{s}_{i}=\begin{cases}s_{i}&\text{for
}i\not=j-1,j,j+1,\\\ s_{j-1}s_{j}=s_{j-1}h&\text{for }i=j-1,\\\
h^{-1}&\text{for }i=j,\\\ s_{j}s_{j+1}=hs_{j+1}&\text{for }i=j+1.\end{cases}$

We thus obtain

$\displaystyle
P_{\widetilde{s}_{j-1}}\widetilde{A}_{j}P_{\widetilde{s}_{j}}\widetilde{A}_{j+1}P_{\widetilde{s}_{j+1}}$
$\displaystyle=P_{s_{j-1}h}A_{j+1}P_{h}^{-1}A_{j}P_{hs_{j+1}}=P_{s_{j-1}}\big{(}P_{h}A_{j+1}P_{h}^{-1}\big{)}A_{j}P_{h}P_{s_{j+1}}$
$\displaystyle=P_{s_{j-1}}A_{j}\big{(}P_{h}A_{j+1}P_{h}^{-1}\big{)}P_{h}P_{s_{j+1}}=P_{s_{j-1}}A_{j}P_{s_{j}}A_{j+1}P_{s_{j+1}},$

where we use that $P_{h}A_{j+1}P_{h}^{-1}\in\mathcal{A}$, $H(S)$ normalizes
$\mathcal{A}$, and the commutativity of $\mathcal{A}$. The above argument
shows the asserted independence of (3.3) because any increasing rearrangement
can be obtained by successive transpositions.

For $n\in{\mathbb{N}}$ and $s_{1},\ldots,s_{n-1}\in S$, the map

$\displaystyle\mathcal{A}^{n}\cong
C(Q_{0})^{n}\to{\mathbb{R}},\qquad(A_{1},\ldots,A_{n})\mapsto\langle
A_{1}P_{s_{1}}A_{2}\cdots P_{s_{n-1}}A_{n}\Omega,\Omega\rangle$

is $n$-linear, so that there exists a well-defined linear functional
$\rho\colon\mathcal{R}\to{\mathbb{R}}$ satisfying

$\displaystyle\rho\left(\prod_{j=1}^{n}U_{g_{j}}f_{j}\right)=\big{\langle}f_{1}P_{g_{1}^{-1}g_{2}}f_{2}\cdots
P_{g_{n-1}^{-1}g_{n}}f_{n}\Omega,\Omega\big{\rangle}$ (3.4)

for $f_{1},\ldots,f_{n}\in\mathop{{\rm span}}\nolimits\Xi$,
$g_{1}\prec_{S}\cdots\prec_{S}g_{n}$, $n\in{\mathbb{N}}_{0}$.

To see that $\rho$ is positive, we observe that every non-negative function
$F\in\mathcal{R}$ can be written in the form

$\displaystyle F=\sum_{k}c_{k}F_{k}\qquad\mbox{with}\quad
F_{k}:=\prod_{j=1}^{n}U_{g_{j}}\chi_{j,k},\qquad\mbox{where}\quad
F_{k}F_{k^{\prime}}=0\quad\mbox{for}\quad k\not=k^{\prime}.$

Then $c_{k}\geq 0$ for all $k$, so that

$\displaystyle\rho(F)=\sum_{k}c_{k}\rho\left(\prod_{j=1}^{n}U_{g_{j,k}}\chi_{j,k}\right)\geq
0$

by (3.4). This proves that $\rho\geq 0$, and thus, by continuity, it extends
to a positive functional on $C(Q)$, which defines a Radon measure $\mu$ on
$Q$.

Next we show that $((Q,\Sigma,\mu),\Sigma_{0},P,\theta)$ is a
$(G,S,\tau)$-measure space. The $G$-invariance of the right hand side of (3.4)
implies that $G$ acts on $Q$ by measure preserving transformations. To see
that $\mu$ is also $\theta$-invariant, we first note that $g\prec_{S}h$
implies $\tau(h)\prec_{S}\tau(g)$ because $\tau(S)=S^{-1}$. We thus obtain

$\displaystyle\theta\left(\prod_{j=1}^{n}U_{g_{j}}f_{j}\right)=\prod_{j=0}^{n-1}U_{\tau(g_{n-j})}f_{n-j}.$

Hence the $\theta$-invariance of $\mu$ follows from

$\displaystyle\begin{split}&\big{\langle}f_{1}P_{g_{1}^{-1}g_{2}}f_{2}\cdots
P_{g_{n-1}^{-1}g_{n}}f_{n}\Omega,\Omega\big{\rangle}=\big{\langle}\Omega,f_{n}P_{\tau(g_{n})^{-1}\tau(g_{n-1})}\cdots
f_{2}P_{\tau(g_{2})^{-1}\tau(g_{1})}f_{1}\Omega\big{\rangle}\\\
&\hphantom{\big{\langle}f_{1}P_{g_{1}^{-1}g_{2}}f_{2}\cdots
P_{g_{n-1}^{-1}g_{n}}f_{n}\Omega,\Omega\big{\rangle}}{}=\big{\langle}f_{n}P_{\tau(g_{n})^{-1}\tau(g_{n-1})}\cdots
f_{2}P_{\tau(g_{2})^{-1}\tau(g_{1})}f_{1}\Omega,\Omega\big{\rangle}.\end{split}$

It remains to verify $\langle\theta F,F\rangle\geq 0$ for $F\in
L^{2}(Q,\Sigma_{+},\mu)$. The functions of the form

$\displaystyle f_{g_{1}}\cdots
f_{g_{n}}=U_{g_{1}}M_{f_{1}}U_{g_{1}^{-1}g_{2}}\cdots
U_{g_{n-1}^{-1}g_{n}}M_{f_{n}}U_{g_{n}}^{-1}1,\qquad
f_{j}\in\mathcal{A},\mathbf{1}\prec_{S}g_{1}\prec_{S}g_{2}\prec_{S}\cdots\prec_{S}g_{n},$

form a total subset of $\mathcal{E}_{+}$. On pairs of such functions we have

$\displaystyle\big{\langle}\theta f_{g_{1}}\cdots
f_{g_{n}},f_{h_{1}}^{\prime}\cdots f_{h_{m}}^{\prime}\big{\rangle}$
$\displaystyle\quad{}=\big{\langle}\theta
U_{g_{1}}M_{f_{1}}U_{g_{1}^{-1}g_{2}}\cdots
U_{g_{n-1}^{-1}g_{n}}f_{n},U_{h_{1}}M_{f_{1}^{\prime}}U_{h_{1}^{-1}h_{2}}\cdots
U_{h_{m-1}^{-1}h_{m}}f_{m}^{\prime}\big{\rangle}$
$\displaystyle\quad{}=\big{\langle}U_{\tau(g_{1})}M_{f_{1}}U_{\tau(g_{1})^{-1}\tau(g_{2})}\cdots
U_{\tau(g_{n-1})^{-1}\tau(g_{n})}f_{n},U_{h_{1}}M_{f_{1}^{\prime}}U_{h_{1}^{-1}h_{2}}\cdots
U_{h_{m-1}^{-1}h_{m}}f_{m}^{\prime}\big{\rangle}$
$\displaystyle\quad{}=\big{\langle}1,M_{f_{n}}U_{\tau(g_{n})^{-1}\tau(g_{n-1})}\cdots
U_{\tau(g_{2})^{-1}\tau(g_{1})}M_{f_{1}}U_{\tau(g_{1})^{-1}}U_{h_{1}}M_{f_{1}^{\prime}}U_{h_{1}^{-1}h_{2}}\cdots
U_{h_{m-1}^{-1}h_{m}}f_{m}^{\prime}\big{\rangle}$
$\displaystyle\quad{}=\big{\langle}\Omega,\widehat{M}_{f_{n}}P_{\tau(g_{n})^{-1}\tau(g_{n-1})}\cdots
P_{\tau(g_{2})^{-1}\tau(g_{1})}\widehat{M}_{f_{1}}P_{\tau(g_{1})^{-1}}P_{h_{1}}\widehat{M}_{f_{1}^{\prime}}P_{h_{1}^{-1}h_{2}}\cdots
P_{h_{m-1}^{-1}h_{m}}\widehat{M}_{f_{m}^{\prime}}\Omega\big{\rangle}$
$\displaystyle\quad{}=\big{\langle}P_{g_{1}}\widehat{M}_{f_{1}}P_{g_{1}^{-1}g_{2}}\cdots
P_{g_{n-1}^{-1}g_{n}}\widehat{M}_{f_{n}}\Omega,P_{h_{1}}\widehat{M}_{f_{1}^{\prime}}P_{h_{1}^{-1}h_{2}}\cdots
P_{h_{m-1}^{-1}h_{m}}\widehat{M}_{f_{m}^{\prime}}\Omega\big{\rangle}.$

This implies that

$\displaystyle\Big{\langle}\theta\sum_{j}c_{j}f_{g_{1,j}}\cdots
f_{g_{n,j}},\sum_{k}c_{k}f_{g_{1,k}}\cdots f_{g_{n,k}}\Big{\rangle}$
$\displaystyle\qquad{}=\Big{\|}\sum_{j}c_{j}P_{g_{1,j}}\widehat{M}_{f_{1,j}}P_{g_{1,j}^{-1}g_{2,j}}\cdots
P_{g_{n-1,j}^{-1}g_{n,j}}\widehat{M}_{f_{n,j}}\Omega\Big{\|}^{2}\geq 0.$

We conclude that $\mathcal{E}_{+}$ is $\theta$-positive and that the canonical
map $q\colon\mathcal{E}_{+}\to\widehat{\mathcal{E}}$ is equivalent to the map

$\displaystyle q\colon\ \mathcal{E}_{+}\to\mathcal{H},\qquad f_{g_{1}}\cdots
f_{g_{n}}\mapsto P_{g_{1}}\widehat{M}_{f_{1}}P_{g_{1}^{-1}g_{2}}\cdots
P_{g_{n-1}^{-1}g_{n}}\widehat{M}_{f_{n}}\Omega.$

This map satisfies $q\circ M_{f}=\widehat{M}_{f}\circ q$ and $q\circ
U_{s}=P_{s}\circ q$ for $s\in S$. Therefore
$(\mathcal{H},P,\mathcal{A},\Omega)$ is equivalent to the associated positive
semigroup structure. ∎

######  Remark 3.19.

Theorem 3.18 implies in particular that, for every positive semigroup
structure, $\Omega$ is separating for $\mathcal{A}$ [41, Remark 2.5].

######  Remark 3.20.

We have seen in the proof of Theorem 3.18 that it is sometimes more convenient
to deal with the unital $W^{*}$-algebra
$\mathcal{A}=L^{\infty}(Q,\Sigma_{0},\nu)$ as a $C^{*}$-algebra and write it
as $C(Q_{0})$ for a compact (Hausdorff) space $Q_{0}$. Then the Riesz
representation theorem asserts that positive functionals on $C(Q_{0})$
correspond to regular Borel measures $\mu$ on $Q_{0}$. On the other hand, the
restriction of any such measure to the $\sigma$-algebra of Baire sets, i.e.,
the smallest $\sigma$-algebra for which all continuous functions are
measurable, already determines the corresponding linear functional on
$C(Q_{0})$ by integration. Therefore the positive functionals on $C(Q_{0})$
are in one-to-one correspondence with finite positive Baire measures on
$Q_{0}$.

Since the set $\Sigma_{0}/J_{\nu}$ of equivalence classes modulo $\nu$-null
sets corresponds to the idempotents in $\mathcal{A}$, resp., $C(Q_{0})$, Baire
measures on $Q_{0}$ correspond to measures on $(Q,\Sigma_{0})$ that are
absolutely continuous with respect to $\nu$.

A slight variation of the argument in the proof of Theorem 3.18 leads to the
following generalization of the Klein–Landau reconstruction theorem [42,
Theorem 1.8] which deals with standard positive semigroup structures. For the
proof of this theorem, we need a suitable version of [42, Lemma 1.9] to deal
with the case where the measure $\nu$ is infinite.

######  Lemma 3.21.

Let $(G,S,\tau)$ be a symmetric semigroup with $G=S\cup S^{-1}$. Let $Q_{0}$
be a compact Hausdorff space and $\nu_{0}$ a $\sigma$-finite measure on the
Baire sets in $Q_{0}$. Suppose that $(P_{s})_{s\in S}$ is a positivity
preserving semigroup acting on $C(Q_{0})$ leaving the function $1$ invariant.
Then there exists a unique $\sigma$-finite Baire measure $\mu$ on the compact
Hausdorff space $Q=Q_{0}^{G}$, such that, for

$\displaystyle\tau(s_{m})\prec_{S}\cdots\prec_{S}\tau(s_{1})\prec_{S}\mathbf{1}\prec_{S}t_{1}\prec_{S}\cdots\prec_{S}t_{n}$

and non-negative $g_{1},\ldots,g_{m},f_{1},\ldots,f_{n}\in C(Q_{0})$, we have

$\displaystyle\int_{Q}g_{m}(\varphi(\tau(s_{m})))\cdots
g_{1}(\varphi(\tau(s_{1}))f_{1}(\varphi(t_{1}))\cdots
f_{n}(\varphi(t_{n}))\,d\mu(\varphi)$
$\displaystyle=\int_{Q_{0}}\big{(}P_{s_{1}}M_{g_{1}}P_{s_{1}^{-1}s_{2}}M_{g_{2}}\cdots
P_{s_{m-1}^{-1}s_{m}}M_{g_{m}}1\big{)}\big{(}P_{t_{1}}M_{f_{1}}P_{t_{1}^{-1}t_{2}}M_{f_{2}}\cdots
P_{t_{n-1}^{-1}t_{n}}M_{f_{n}}1\big{)}\,d\nu(q_{0}),$

where, as before, $M_{f}$ denotes multiplication by $f$.

###### Proof.

First we assume that the measure $\nu$ is finite. Let $\mathcal{R}\subseteq
C(Q)$ be as in the proof of Theorem 3.18. We then obtain a positive functional
$\rho$ on $\mathcal{R}$, which is uniquely determined by the requirement that,
for

$\displaystyle F(q)=g_{m}(\varphi(\tau(s_{m})))\cdots
g_{1}(\varphi(\tau(s_{1})))f_{1}(\varphi(t_{1}))\cdots f_{n}(\varphi(t_{n})),$

we have

$\displaystyle\rho(F)=\int_{Q_{0}}\big{(}P_{s_{1}}M_{g_{1}}P_{s_{1}^{-1}s_{2}}M_{g_{2}}{\cdots}P_{s_{m-1}^{-1}s_{m}}M_{g_{m}}1\big{)}\big{(}P_{t_{1}}M_{f_{1}}P_{t_{1}^{-1}t_{2}}M_{f_{2}}{\cdots}P_{t_{n-1}^{-1}t_{n}}M_{f_{n}}1\big{)}\,d\nu(q_{0})$

(see the proof of [42, Lemma 1.9] for details). The Riesz–Markov theorem now
leads to a unique finite Baire measure $\mu$ on $Q$ with
$\int_{Q}F\,d\mu=\rho(F)$ for $F\in\mathcal{R}$.

If $\nu$ is not finite, we write it as $\nu=\sum_{j}\nu_{j}$, where the
measures $\nu_{j}$ are finite. If $\mu_{j}$ denotes the measure on $Q$
corresponding to $\nu_{j}$, the measure $\mu:=\sum_{j}\mu_{j}$ satisfies all
our requirements. ∎

######  Theorem 3.22 (Reconstruction Theorem – Standard Case).

If $(G,S,\tau)$ is a symmetric semigroup with $G=S\cup S^{-1}$, then every
standard positive semigroup structure for $(G,S,\tau)$ is associated to some
$(G,S,\tau)$-measure space $((Q,\Sigma,\mu),\Sigma_{0},U,\theta)$ which is
unique up to $G$-equivariant isomorphism of measure spaces.

###### Proof.

First we show uniqueness. Let
$((Q^{\prime},\Sigma^{\prime},\mu^{\prime}),\Sigma_{0}^{\prime},U^{\prime},\theta^{\prime})$
be a Markov $(G,S,\tau)$-measure space for which
$(L^{2}(M,\nu),P,L^{\infty}(M,\nu))$ is the associated standard positive
semigroup structure.

Lemma 3.16(c) implies that the integrals of products of the form
$f_{g_{1}}\cdots f_{g_{n}}$, $g_{1}\prec_{S}\cdots\prec_{S}g_{n}$ are uniquely
determined by the relation

$\displaystyle\int_{Q}f_{g_{1}}\cdots
f_{g_{n}}\,d\mu^{\prime}=\int_{Q}M_{f_{1}}P_{g_{1}^{-1}g_{2}}\cdots
M_{f_{n-1}}P_{g_{n-1}^{-1}g_{n}}M_{f_{n}}1\,d\nu.$

Since such products for $f_{j}\in L^{2}(M,\nu)\cap L^{\infty}(M,\nu)$ form a
total subset of $L^{2}(Q^{\prime},\Sigma^{\prime},\mu^{\prime})$, any
isomorphism $L^{2}(Q^{\prime},\Sigma_{0}^{\prime},\mu^{\prime})\to
L^{2}(M,\nu)\cong L^{2}(Q,\Sigma_{0},\mu)$ extends to a $G$-equivariant
isomorphism $L^{2}(Q^{\prime},\Sigma^{\prime},\mu^{\prime})\to
L^{2}(Q,\Sigma,\mu)$.

For the existence of the measure $\mu$, we now use Lemma 3.21. Here we write
$Q_{0}$ for the spectrum of the $C^{*}$-algebra
$\mathcal{A}:=L^{\infty}(M,\nu)$, so that $\mathcal{A}\cong C(Q_{0})$ and
$\nu$ defines a Baire measure on $Q_{0}$. We may then identify
$(M,{\mathfrak{S}},\nu)$ with $(Q_{0},\Sigma_{0},\nu)$, where $\Sigma_{0}$ is
the $\sigma$-algebra of Baire subsets of $Q_{0}$.

Now let $\mu$ be the Baire measure on $Q:=Q_{0}^{G}$ from Lemma 3.21. Then the
relation $P_{s}^{*}=P_{s^{\sharp}}$, as operators on $L^{2}(M,\nu)$, leads to

$\displaystyle\int_{Q}f_{g_{1}}\cdots
f_{g_{n}}\,d\mu=\int_{Q}M_{f_{1}}P_{g_{1}^{-1}g_{2}}\cdots
M_{f_{n-1}}P_{g_{n-1}^{-1}g_{n}}M_{f_{n}}1\,d\nu$

for $0\leq f_{1},\ldots,f_{n}\in C(Q_{0})$ and
$g_{1}\prec_{S}\cdots\prec_{S}g_{n}$. The same calculations as in the proof of
Theorem 3.18 now show that $\mu$ is invariant under $G$ and $\theta$. ∎

######  Remark 3.23.

In the standard case, it is interesting to take a closer look at the structure
of the measure space $(Q,\Sigma,\mu)$. Then $\mathcal{H}\cong
L^{2}(M,{\mathfrak{S}},\nu)$, and for $L^{\infty}(M,{\mathfrak{S}},\nu)\cong
C(Q_{0})$ we obtain on the compact space $Q:=Q_{0}^{G}$ a measure which is
determined by the corresponding cylinder measures:

$\displaystyle\int_{Q}f_{g_{1}}\cdots
f_{g_{n}}\,d\mu=\int_{M}M_{f_{1}}P_{g_{1}^{-1}g_{2}}\cdots
M_{f_{n-1}}P_{g_{n-1}^{-1}g_{n}}M_{f_{n}}1\,d\nu$ (3.5)

for $0\leq f_{1},\ldots,f_{n}\in C(Q_{0})$ and
$g_{1}\prec_{S}\cdots\prec_{S}g_{n}$.

For every $g\in G$, we then have a continuous projection $q_{g}\colon Q\to
Q_{0}$, so that we obtain a homomorphism $q_{g}^{*}\colon
L^{\infty}(M,{\mathfrak{S}},\nu)\cong C(Q_{0})\to C(Q)$ of $C^{*}$-algebras.
This leads to a morphism ${\mathfrak{S}}/J_{\nu}\to\Sigma/J_{\mu}$ of Boolean
$\sigma$-algebras. At this point it is a natural question whether this
morphism can be realized by a measurable function $f_{g}\colon Q\to M$. In
[66, Theorem 1.6] this is answered in the affirmative for
$M={\mathbb{R}}^{n}$, but this implies the corresponding result for completely
metrizable separable spaces, because these spaces can be realized as closed
subsets of some ${\mathbb{R}}^{\mathbb{N}}$. Once we have the measurable maps
$f_{g}\colon Q\to M$, they combine to a stochastic process $(f_{g})_{g\in G}$
with state space $(M,{\mathfrak{S}})$, so that we obtain an isomorphism of
measure spaces
$f\colon(Q,\Sigma,\mu)\to(M^{G},{\mathfrak{S}}^{G},\mu^{\prime})$.

If $(M,{\mathfrak{S}},\nu)$ is a polish space, i.e., $M$ carries a topology
for which it is completely metrizable and separable and ${\mathfrak{S}}$ is
the $\sigma$-algebra of Borel sets, then [7, Corollary 35.4] directly implies
the existence of a Borel measure $\mu^{\prime}$ on the measurable space
$(M^{G},{\mathfrak{S}}^{M})$ with the correct projections onto finite
products:

$\displaystyle\int_{M^{G}}f_{1}(\omega(g_{1}))\cdots
f_{n}(\omega(g_{n}))\,d\mu^{\prime}(\omega)=\int_{Q}M_{f_{1}}P_{g_{1}^{-1}g_{2}}\cdots
M_{f_{n-1}}P_{g_{n-1}^{-1}g_{n}}M_{f_{n}}1\,d\nu$

for $0\leq f_{1},\ldots,f_{n}\in C(Q_{0})$ and
$g_{1}\prec_{S}\cdots\prec_{S}g_{n}$. Therefore we obtain a realization of our
$(G,S,\tau)$-measure space on $(M^{G},{\mathfrak{S}}^{G},\mu^{\prime})$.

######  Remark 3.24.

(a) Consider the case $(G,S,\tau)=({\mathbb{R}},{\mathbb{R}}_{+},-\mathop{{\rm
id}}\nolimits_{\mathbb{R}})$ and assume that the operators $P_{t}f$ are
obtained from Markov kernels $P_{t}\colon M\times{\mathfrak{S}}\to[0,\infty]$
in the sense that

$\displaystyle(P_{t}f)(x)=\int_{M}P_{t}(x,dy)f(y)$

(cf. Appendix B). In view of Remark B.3, the measures $P_{\bf{t}}^{\nu}$ can
be written as

$\displaystyle P_{\bf{t}}^{\nu}=\nu P_{\bf{t}},\qquad\mbox{i.e.,}\qquad
P_{\bf{t}}^{\nu}(A)=\int_{M}\nu(dx)P_{\bf{t}}(x,A)$

for the kernel

$\displaystyle
P_{\bf{t}}(x_{0},B)=\int_{M^{n}}\chi_{B}(x_{1},\ldots,x_{n})P_{t_{1}}(x_{0},dx_{1})P_{t_{2}-t_{1}}(x_{1},dx_{2})\cdots
P_{t_{n}-t_{n-1}}(x_{n-1},dx_{n})$

on $M\times{\mathfrak{S}}^{n}$. In particular, we have for each $x\in M$ a
probability measure $P_{x}$ on

$\displaystyle\big{\\{}\omega\in
M^{{\mathbb{R}}_{+}}\colon\omega(0)=x\big{\\}},$

defining a kernel $P\colon
M\times{\mathfrak{S}}^{{\mathbb{R}}_{+}}\to[0,\infty]$, such that

$\displaystyle P^{\nu}=\nu P,\qquad\mbox{i.e.,}\qquad
P^{\nu}(A)=\int_{M}\nu(dx)P(x,A).$

For any $t\geq 0$, we then have

$\displaystyle
P_{t}(x_{0},B)=\int_{M^{{\mathbb{R}}_{+}}}\chi_{B}(\omega(t))\,P(x_{0},d\omega)=P(x_{0},\\{\omega(t)\in
B\\}),$

which leads to

$\displaystyle(P_{t}f)(x)=\int_{M}P_{t}(x,dy)f(y)=\int_{M^{{\mathbb{R}}_{+}}}P(x,d\omega)f(\omega(t)).$
(3.6)

This is an abstract version of the Feynman–Kac–Nelson formula that expresses
the value of $P_{t}f$ in $x\in M$ as an integral over all paths $[0,t]\to M$
starting in $x$ with respect to the probability measure $P_{x}$.

We also note that, for $t>0$, we have the relation

$\displaystyle\int_{M^{\mathbb{R}}}f(\omega(t))\,dP^{\nu}(\omega)=\int_{Q}f\,d\nu\quad\mbox{for}\quad
t\in{\mathbb{R}},$

and, for $t<s$,

$\displaystyle\int_{M}\int_{M}f_{1}(x)\nu(dx)f_{2}(y)P_{s-t}(x,dy)=\int_{M}f_{1}(x)(P_{s-t}f_{2})(x)\,d\nu(x)$
$\displaystyle\hphantom{\int_{M}\int_{M}f_{1}(x)\nu(dx)f_{2}(y)P_{s-t}(x,dy)}{}=\int_{M^{\mathbb{R}}}f_{1}(\omega(t))f_{2}(\omega(s))\,dP^{\nu}(\omega).$

(b) In the special case where $M=G$ is a topological group and
$P_{t}f=f*\mu_{t}$, we have

$\displaystyle(P_{t}f)(x)=\int_{G}f(xy)\,d\mu_{t}(y)=\int_{G}P_{t}(x,dy)f(y)\qquad\mbox{for}\quad
P_{t}(x,A)=\mu_{t}(x^{-1}A).$

Therefore the operators $P_{t}$ are given by Markov kernels. Here, the
measurability of the functions

$\displaystyle x\mapsto\mu_{t}(x^{-1}A)=\int_{G}\chi_{A}(xy)\,d\mu_{t}(y)$

follows from the measurability of the function $(x,y)\mapsto\chi_{A}(xy)$.

Let $P(G)$ be the path group $G^{{\mathbb{R}}}$ and let $P_{*}(G)$ be the
subgroup of pinned paths

$\displaystyle P_{*}(G)=\\{\omega\in P(G)\colon\omega(0)=\mathbf{1}\\}.$

We have the relations

$\displaystyle\int_{P(G)}f_{1}(\omega(0))f_{2}(\omega(t))\,d\mu(\omega)=\int_{G}\int_{G}f_{1}(g_{1})f_{2}(g_{1}g_{2})\,d\nu(g_{1})d\mu_{t}(g_{2})\qquad\mbox{for}\quad
t>0$

and

$\displaystyle\int_{P_{*}(G)}f(\omega(t))\,dP^{\nu}(\omega)=\int_{G}f(g)\,d\mu_{t}(g).$

This leads for $f\in L^{2}(G,\nu)$ and $t\geq 0$ to

$\displaystyle(P_{t}f)(x)=(f*\mu_{t})(x)=\int_{G}f(xg)\,d\mu_{t}(g)=\int_{G^{{\mathbb{R}}_{+}}}f(x\omega(t))\,dP(\omega).$

This is a group version of the Feynman–Kac–Nelson formula (3.6) which
expresses $(P_{t}f)(x)$ as an integral over all paths $\omega\colon[0,t]\to G$
with $\omega(0)=x$.

######  Remark 3.25.

The assumption that $G=S\cup S^{-1}$ is very restrictive (see the discussion
in Subsection 3.5). We mainly use it to ensure that the order $\prec_{S}$ on
$G$ is total, which is a crucial ingredient in the construction. However, if
$S\subseteq G$ is a subsemigroup not satisfying the totality condition, i.e.,
$S\cup S^{-1}\not=G$, then one may still consider subsets
$\mathcal{C}\subseteq G$ on which the order $\prec_{S}$ is total, so-called
$\mathcal{C}$-chains. Typical examples of such $\prec_{S}$-chains are curves
$\mathcal{C}=\gamma(I)$, $I\subseteq{\mathbb{R}}$, with
$\gamma(t_{2})\in\gamma(t_{1})S$ for $t_{1}<t_{2}$ in $I$. For any such chain
$\mathcal{C}$, one can still construct a stochastic process indexed by
$\mathcal{C}$, but we then don’t have the symmetries implemented by a
$G$-action on the corresponding probability space.

######  Remark 3.26.

The Abel–Klein reconstruction theorem leads in particular to realizations of
dilations of the corresponding standard positive semigroup structures
$(P_{t})_{t\geq 0}$ of the following kind:

  * (a)

Positivity preserving semigroups obtained by integrating a measure preserving
action of $G$ on some measure space $(X,{\mathfrak{S}},\nu)$ to operators

$\displaystyle(P_{t}f)(x):=\int_{G}(g.f)(x)\,d\mu_{t}(g),$

where the measures $\mu_{t}$ on $G$ are symmetric. If $(X,{\mathfrak{S}})$ is
polish, the corresponding measure space can be realized on the path space
$X^{\mathbb{R}}$. Here the requirement that $G$ acts continuously on
$L^{2}(X,{\mathfrak{S}},\nu)$ implies that $(P_{t})_{t\geq 0}$ is strongly
continuous on $L^{2}(X,{\mathfrak{S}},\nu)$ (see (b) below for a more general
argument) and therefore continuous in measure (Remark 3.6(c)).

  * (b)

Semigroups of the type $(\Gamma(\pi(\mu_{t})))_{t\geq 0}$, where
$(\pi,\mathcal{H})$ is a continuous unitary representation of $G$ on
$\mathcal{H}$ and $\Gamma$ is the functor of second quantization (see Section
5.1 for more information about second quantization). Here we use that, for
every continuous orthogonal representation $(\pi,\mathcal{H})$ of $G$ and a
Borel probability measure $\nu$ on $G$, we can define

$\displaystyle\pi(\nu):=\int_{G}\pi(g)\,d\nu(g)$

as the operator representing the hermitian form

$\displaystyle q(v,w):=\int_{G}\langle\pi(g)v,w\rangle\,d\nu(g)$

which satisfies $|q(v,w)|\leq\|v\|\|w\|$ for $v,w\in\mathcal{H}$. For any one-
parameter semigroup $(\mu_{t})_{t\geq 0}$ of symmetric probability measures,
we thus obtain a hermitian one-parameter semigroup of contractions
$(\pi(\mu_{t}))_{t\geq 0}$. The continuity of this one-parameter semigroup
follows from the continuity of the functions

$\displaystyle
t\mapsto\langle\pi(\mu_{t})v,w\rangle=\int_{G}\langle\pi(g)v,w\rangle\,d\mu_{t}(g),$

which is a consequence of the weak convergence $\mu_{t}\to\delta_{\mathbf{1}}$
for $t\to 0$ and the boundedness of the matrix coefficients of $\pi$.

###  3.4 Klein’s characterization of the Markov case

The following theorem characterizes the positive semigroup structures arising
from Markov path spaces as those for which $\Omega$ is a cyclic vector for
$\mathcal{A}$, which is considerably stronger than condition (PS4)(b).

######  Theorem 3.27 ([41, Theorem 3.1]).

Let $((Q,\Sigma,\mu),\Sigma_{0},U,\theta)$ be a $(G,S,\tau)$-probability space
and let $(\widehat{\mathcal{E}},\widehat{U},\mathcal{A},\Omega)$ be its
associated positive semigroup structure. Then
$((Q,\Sigma,\mu),\Sigma_{0},U,\theta)$ is Markov if and only if $\Omega$ is
cyclic for $\mathcal{A}$.

###### Proof.

The Markov property is equivalent to
$q(\mathcal{E}_{0})=\widehat{\mathcal{E}}$ (Proposition 3.3). Since
$\mathcal{E}_{0}=\mathcal{A}\cdot 1$, this condition is equivalent to
$\Omega=q(1)$ being $\mathcal{A}$-cyclic in $\widehat{\mathcal{E}}$. ∎

######  Remark 3.28 (cf. [41, Remark 3.3]).

Suppose that $(\mathcal{H},P,\mathcal{A},\Omega)$ is a positive semigroup
structure for which $\Omega$ is $\mathcal{A}$-cyclic, so that Proposition 3.7
implies that $\mathcal{H}\cong L^{2}(M)$ for a probability space $M$. Then
condition (PS4)(c) follows from the seemingly weaker condition

$\displaystyle A,B\in\mathcal{A}_{+}:=\\{C\in\mathcal{A}\colon 0\leq
C\\},\quad s\in S\qquad\Rightarrow\qquad\langle
A\Omega,P_{s}B\Omega\rangle\geq 0.$

In fact, in this case $\mathcal{A}_{+}\Omega$ is dense in the closed cone
$L^{2}(Q,\Sigma_{0},\mu)_{+}$ of non-negative functions, so that $\langle
A\Omega,P_{s}B\Omega\rangle\geq 0$ for all $A,B\in\mathcal{A}_{+}$ implies
that the operator $P_{s}$ on $L^{2}(Q,\Sigma_{0},\mu)$ is positivity
preserving. Then

$\displaystyle P_{s_{1}}A_{1}\cdots P_{s_{n}}A_{n}\Omega\geq 0$

for $A_{j}\in\mathcal{A}_{+}$ and $s_{j}\in S$, and the assertion follows.

######  Remark 3.29.

(a) We consider the Markov case where $\mathcal{A}\Omega$ is dense in
$\mathcal{H}$, so that $\mathcal{E}_{0}\cong L^{2}(Q,\Sigma_{0},\mu)$ is a
$G$-cyclic subspace of $\mathcal{E}$. Then $\varphi(g)=E_{0}U_{g}E_{0}$
satisfies $\varphi(s)=\widehat{U_{s}}=P_{s}$ for $s\in S$. Therefore the
$G$-invariant subspace of $\mathcal{E}$ generated by $\mathcal{E}_{0}$ carries
a representation which is a unitary dilation of the representation
$(P,\mathcal{H})$ of $S$ (cf. Remark 2.8).

(b) The set of finite products $f_{g_{1}}\cdots f_{g_{n}}$, $g_{j}\in G$,
$f_{j}\in\mathcal{E}_{0}\cong L^{\infty}(M,{\mathfrak{S}},\nu)\cap
L^{1}(M,{\mathfrak{S}},\nu)$ spans a dense subspace of $\mathcal{E}$.
Accordingly, the restriction $g_{j}\in S$ leads to a dense subspace of
$\mathcal{E}_{+}$. This permits us to give sufficient conditions for
$\mathcal{E}_{+}$ to be cyclic in $\mathcal{E}$. Since

$\displaystyle U_{g}f_{g_{1}}\cdots f_{g_{n}}=f_{gg_{1}}\cdots f_{gg_{n}},$

this is the case if, for every finite subset $F\subseteq G$, there exists a
$g\in G$ with $gF\subseteq S$, i.e., $g^{-1}\prec_{S}F$. This means that the
order $\prec_{S}$ on $G$ is filtered, i.e., for $g_{1},g_{2}\in G$, there
exists a $g_{3}\in G$ with $g_{3}\prec_{S}g_{1},g_{2}$. This property is
easily seen to be equivalent to $G=S^{-1}S$ because $g_{1}^{-1}g_{2}\in
S^{-1}S$ is equivalent to the existence of $s_{1},s_{2}\in S$ with
$g_{3}:=g_{2}s_{2}^{-1}=g_{1}s_{1}^{-1}\prec_{S}g_{1},g_{2}$ (cf. [25]). We
conclude that $\mathcal{E}_{+}$ is $G$-cyclic in $\mathcal{E}$ if $G=S^{-1}S$.
Note that this is in particular the case if $G=S\cup S^{-1}$.

###  3.5 Total subsemigroups of Lie groups

In this section we briefly discuss total subsemigroups. Our standard
references are [25, Chapter 6] and [43]. For Lie groups $G$, subsemigroups $S$
with dense interior satisfying $G=S\cup S^{-1}$ are rare. Typical examples
are:

  * (a)

$S=[0,\infty[={\mathbb{R}}_{+}\subseteq G={\mathbb{R}}$.

  * (b)

$S={\mathbb{R}}_{+}\rtimes{\mathbb{R}}^{\times}_{+}\subseteq
G={\mathbb{R}}\rtimes{\mathbb{R}}^{\times}_{+}$ (the $ax+b$-group).

  * (c)

A subsemigroup $S\subseteq G=\widetilde{\mathop{{\rm
SL}}}\nolimits_{2}({\mathbb{R}})$ (the simply connected covering group of
$\mathop{{\rm SL}}\nolimits_{2}({\mathbb{R}})$), whose boundary is a
$2$-dimensional subgroup isomorphic to the $ax+b$-group. Actually $G$ is the
simply connected covering of the projective group $\mathop{{\rm
PSL}}\nolimits_{2}({\mathbb{R}})$ acting by orientation preserving projective
maps on the projective line
${\mathbb{P}}({\mathbb{R}}^{2})\cong{\mathbb{S}}^{1}$. Identifying
$\widetilde{\mathbb{S}}^{1}$ with ${\mathbb{R}}$, we have that

$\displaystyle S=\\{g\in G\colon g(0)\geq 0\\}=\\{g\in G\colon
g({\mathbb{R}}_{+})\subseteq{\mathbb{R}}_{+}\\}.$

In terms of an Iwasawa decomposition $G=KAN$ with $K\cong{\mathbb{R}}$ and
$K_{+}:={\mathbb{R}}_{+}$, we have $S=K_{+}AN=ANK_{+}$ and ${\mathbb{R}}\cong
K\cong G/AN$.

It is easy to see that $S$ is contained in a maximal subsemigroup of $S$ and
if ${\mathfrak{g}}/\mathop{{\rm rad}}\nolimits({\mathfrak{g}})$ is a compact
Lie algebra, then the classification of maximal subsemigroups $M$ [25, Theorem
6.18] implies that $\partial M$ is a codimension $1$ subgroup of $G$.
According to the classification of hyperplane subalgebras of finite-
dimensional Lie algebras [27, 28], this means that, if $N\subseteq\partial M$
is the largest connected normal subgroup of $G$ contained in $\partial M$,
then $M/N\subseteq G/N$ is one of the three types described above. The
examples under (a)–(c) are also semigroups whose boundary $\partial S$ is a
subgroup, so that $S^{0}$ is well bounded in the sense of [16, 59]. Such a
semigroup, for which $\partial S$ contains no non-trivial connected normal
subgroup, is isomorphic to (a), (b) or (c) [16, 59].

From [50] we know that, for the first two types (a) and (b), there are non-
trivial reflection positive representations for $(G,S,\tau)$, where $\tau$ is
such that $G^{\tau}=\partial S$. On
$\mathop{{\mathfrak{sl}}}\nolimits_{2}({\mathbb{R}})$, the fixed point set of
any involutive automorphism is $1$-dimensional, hence cannot be a hyperplane.
Suppose that, for $G=\widetilde{\mathop{{\rm
SL}}}\nolimits_{2}({\mathbb{R}})$, $G^{\tau}\subseteq\partial S$. Then $S$ is
invariant under $s\mapsto s^{\sharp}=\tau(s)^{-1}$, but one can show that all
involutive continuous representations of $(S,\sharp)$ are trivial. This
follows from the Lüscher–Mack theorem [46] and the fact that all involutive
contractive representations of the group $(\mathop{{\rm
Af\/f}}\nolimits({\mathbb{R}}),\sharp)$ must be trivial on the subgroup
$\\{0\\}\rtimes{\mathbb{R}}^{\times}_{+}$. This is a consequence of the fact
that if, for a selfadjoint operator $A$, all operators $e^{tA}$ are
contractions, we have ${\rm Spec}(A)=\\{0\\}$, hence $A=0$.

######  Example 3.30.

$S={\mathbb{R}}^{d}_{+}\subseteq G={\mathbb{R}}^{d}$ satisfies $G=S\cup
S^{-1}$. Since this subsemigroup is invariant under $\mathop{\rm
O{}}\nolimits_{d-1}({\mathbb{R}})$, we likewise obtain $G_{1}=S_{1}\cup
S_{1}^{-1}$ for $G_{1}:={\mathbb{R}}^{d}\rtimes\mathop{\rm
O{}}\nolimits_{d-1}({\mathbb{R}})$ and
$S_{1}:={\mathbb{R}}^{d}_{+}\rtimes\mathop{\rm
O{}}\nolimits_{d-1}({\mathbb{R}})$. Note that $(G_{1},\tau_{1})$ with
$\tau_{1}(b,A):=(\theta(b),A)$ is a symmetric Lie group with one-dimensional
${\mathfrak{q}}$.

###  3.6 The discrete case

Our reconstruction theorem applies in particular to
$(G,S,\tau)=({\mathbb{Z}},{\mathbb{N}}_{0},-\mathop{{\rm
id}}\nolimits_{\mathbb{Z}})$. Here we may start with a Markov operator $P$ on
a polish probability space $(M,{\mathfrak{S}},\nu)$, so that $P$ is positivity
preserving and satisfies $P1=1$ (cf. Appendix B). If $P$ defines a symmetric
operator on $L^{2}(M,{\mathfrak{S}},\nu)$, then we obtain a unitary operator
$U$ acting by translation on $Q:=M^{\mathbb{Z}}$ such that $E_{0}UE_{0}=P$
holds for the conditional expectation defined by $\Sigma_{0}:=\mathop{{\rm
ev}}\nolimits_{0}^{*}({\mathfrak{S}})\subseteq\Sigma$.

An interesting class of examples arises from finite probability spaces
$M=\\{1,\ldots,n\\}$ with the $\sigma$-algebra $2^{M}$. Then a Markov kernel
$P$ on $M$ is given by the matrix $P_{ij}:=P(i,\\{j\\})$ which is stochastic
in the sense that $\sum_{j}P_{ij}=1$ for every $i$. If $\mu$ is a measure on
$M$, encoded in the row vector $p=(p_{j})_{1\leq j\leq n}$ with
$p_{j}:=\mu(\\{j\\})$, then $\mu P$ is represented by the row vector $pP$. In
particular, the invariance condition $\nu P=\nu$ translates into
$\sum_{i}p_{i}P_{ij}=p_{j}$ for every $j$. So $P$ acts on measures on $M$
(identified with row vectors) by right multiplication and on functions on $M$
(represented by column vectors) by left matrix multiplication.

If $P$ acts as a symmetric operator on $L^{2}(M,\nu)$, then a Markov process
$(X_{n})_{n\in{\mathbb{Z}}}$ can now be obtained from the projections
$X_{n}\colon Q=M^{\mathbb{Z}}\to M$ and the shift invariant probability
measure $\mu$ on $Q$ is determined by

$\displaystyle\mu(X_{0}=i)=p_{i},\qquad\mu(X_{1}=i|X_{0}=j)=P_{ij}.$ (3.7)

In this context, we have for integers $t_{1}\leq\cdots\leq t_{n}$:

$\displaystyle\int_{Q}f_{1}(\omega(t_{1}))\cdots
f_{m}(\omega(t_{m}))\,d\mu(\omega)$
$\displaystyle=\big{\langle}M_{f_{1}}P^{t_{2}-t_{1}}\cdots
M_{f_{m-1}}P^{t_{m}-t_{m-1}}M_{f_{m}}1,1\big{\rangle}_{L^{2}(M,\nu)}$
$\displaystyle=\int_{M}M_{f_{1}}P^{t_{2}-t_{1}}\cdots
M_{f_{m-1}}P^{t_{m}-t_{m-1}}f_{m}\,d\nu(m)$

(Theorem 3.22). Applying this relation to $\delta$-functions
$f_{j}=\delta_{i_{j}}$, we obtain

$\displaystyle\mu(X_{t_{1}}=i_{1},\ldots,X_{t_{n}}=i_{n})=p_{i_{1}}\big{(}P^{t_{2}-t_{1}}\big{)}_{i_{1}i_{2}}\big{(}P^{t_{3}-t_{2}}\big{)}_{i_{2}i_{3}}\cdot\big{(}P^{t_{n}-t_{n-1}}\big{)}_{i_{n-1}i_{n}}.$

For $n=1$, we obtain in particular

$\displaystyle\mu(X_{t}=i)=p_{i}=\mu(X_{0}=i).$

For $n=2$ and $t\in{\mathbb{N}}_{0}$, we obtain

$\displaystyle\mu(X_{0}=i,X_{t}=j)=p_{i}\big{(}P^{t}\big{)}_{ij}.$

For $t=1$, this leads to

$\displaystyle\mu(X_{0}=i,\,X_{1}=j)=p_{i}P_{ij}\qquad\mbox{and}\qquad\mu(X_{1}=j\,|\,X_{0}=i)=P_{ij},$

which is (3.7).

##  4 Measures on path spaces for topological groups

In this section we build a bridge between Markov processes and the
reconstruction results from the preceding section. To this end, we return to
the classical setting, where the symmetric Lie group is
$({\mathbb{R}},{\mathbb{R}}_{+},-\mathop{{\rm id}}\nolimits)$, but the measure
space is the path space $P(Q)=Q^{\mathbb{R}}$ of a polish topological group
$Q$.

If $\nu$ is a measure on a locally compact group $Q$ and $(\mu_{t})_{t\geq 0}$
is a convolution semigroup of symmetric probability measures on $Q$ satisfying
$\nu*\mu_{t}=\nu$ for every $t>0$, then the Klein–Landau reconstruction
theorem applies. Since the measure $\nu$ need not be finite, $\nu$ may in
particular be the right invariant Haar measure $\mu_{Q}$ on $Q$. The operators
$P_{t}f=f*\mu_{t}$ define a positive semigroup structure on $L^{2}(Q,\nu)$,
and the reconstruction process leads to a reflection positive unitary
representation of ${\mathbb{R}}$ on
$L^{2}(P(Q),{\mathfrak{B}}^{\mathbb{R}},\nu)$ by translation (Theorem 4.9). If
$\mu_{t}=\gamma_{t}$ is the gaussian semigroup on ${\mathbb{R}}^{d}$, this
construction leads to an infinite measure on $P({\mathbb{R}}^{d})$ which
realizes in particular the dilation of the heat semigroup as a
subrepresentation (cf. [50, 54]).

Here an interesting point is that the map $Q\times P_{*}(Q)\to
P(Q),(q,\omega)\mapsto q\omega$ yields a product decomposition of $P(Q)$ into
the subgroup of constant paths and the subgroup $P_{*}(Q)$ of pinned paths.
The measures constructed on $P(Q)$ are actually product measures in these
coordinates, where $Q$ carries the measure $\nu$ and the measure on $P_{*}(Q)$
is a probability measure determined by the $\mu_{t}$. There is a natural
measure preserving ${\mathbb{R}}$-action on $P_{*}(Q)$ by

$\displaystyle(V_{t}\omega)(s):=\omega(-t)^{-1}\omega(s-t).$

The measure in $P_{*}(Q)$ defines a $Q$-valued stochastic process
$(X_{t})_{t\in{\mathbb{R}}}$ with $X_{\mathbf{1}}=\mathbf{1}$ (the constant
function). For the special case, where $\mu_{t}=\gamma_{t}$ is the gaussian
semigroup on ${\mathbb{R}}^{d}$, this construction leads to the Wiener measure
on $P_{*}({\mathbb{R}}^{d})$. Presently, we do not know how to obtain a
similar factorization if $({\mathbb{R}},{\mathbb{R}}_{+},-\mathop{{\rm
id}}\nolimits_{\mathbb{R}})$ is replaced by some $(G,S,\tau)$,444This is
caused by the ambiguities related to the discrepancy between finite subsets
$\\{g_{1},\ldots,g_{n}\\}$ of $G$ and increasing tuples $(g_{1},\ldots,g_{n})$
with respect to $\prec_{S}$. so we discuss the one-dimensional case in
Subsection 4.1.

###  4.1 One-parameter convolution semigroups of measures on polish groups

In this subsection $Q$ is a polish topological group. We write
${\mathfrak{B}}$ for the $\sigma$-algebra of Borel subsets of $Q$. A kernel
$K\colon Q\times{\mathfrak{B}}\to[0,\infty]$ on $(Q,{\mathfrak{B}})$ (cf.
Appendix B) is said to be left invariant if

$\displaystyle K(gh,gA)=K(h,A)\qquad\mbox{for all}\quad g,h\in
Q,A\in{\mathfrak{B}}.$

Then $\mu(A):=K(\mathbf{1},A)$ is a positive Borel measure on $Q$ such that
$K(g,A)=\mu(g^{-1}A)$. If, conversely, $\mu$ is a $\sigma$-finite Borel
measure on $Q$, then $K(g,A):=\mu(g^{-1}A)$ is a left invariant kernel on
$Q\times{\mathfrak{B}}$ because, for every $A\in{\mathfrak{B}}$, the function
$g\mapsto\mu(g^{-1}A)$ is measurable. In fact, $\widetilde{A}:=\\{(x,y)\in
Q^{2}\colon xy\in A\\}$ is a Borel subset of $Q\times Q$, and
$\widetilde{A}\cap(\\{g\\}\times Q)=\\{g\\}\times g^{-1}A$. We thus obtain a
one-to-one correspondence between left invariant Markov kernels $K$ on $Q$ and
Borel probability measures $\mu$ on $Q$. Since the product of two kernels
corresponds to the convolution product of the corresponding measures, this
leads to a one-to-one correspondence between convolution semigroups
$(\mu_{t})_{t\geq 0}$ of Borel probability measures on $Q$ and left invariant
Markov semigroups $(P_{t})_{t\geq 0}$ on $(Q,{\mathfrak{B}})$. We recall from
Remark 3.24(b) that the kernel $P_{t}$ acts on functions by right convolution

$\displaystyle P_{t}f=f*\mu_{t}\qquad\mbox{for}\quad t\geq 0,$

where $f*\mu_{t}(x)=\int_{G}f(xg)\,d\mu_{t}(g)$.

Consider the path group $P(Q)=Q^{\mathbb{R}}$ and the subgroup

$\displaystyle P_{*}(Q):=\\{\omega\in P(Q)\colon\omega(0)=\mathbf{1}\\}.$

Starting with a convolution semigroup $(\mu_{t})_{t\geq 0}$, we obtain for the
initial distribution $\delta_{\mathbf{1}}$ the Markov process $(X_{t})_{t\geq
0}$ with independent stationary increments [7, Corollary 35.4]. The
distribution of this process is a probability measure $P^{\mu,+}$ on

$\displaystyle P(Q)^{+}=\big{\\{}\omega\in Q^{\mathbb{R}}\colon(\forall\,t\leq
0)\,\,\omega(t)=\mathbf{1}\big{\\}}\cong Q^{{\mathbb{R}}^{\times}_{+}}.$

For the finite distributions on $Q^{n}$, described by the kernels
$P_{\bf{t}}$, ${\bf{t}}=(t_{1},\ldots,t_{n})$, $t_{1}<\dots<t_{n}$, we obtain
with Remark B.3

$\displaystyle P_{\bf{t}}(B)$
$\displaystyle=\int_{Q^{n}}\chi_{B}(x_{1},\ldots,x_{n})\mu_{t_{1}}(dx_{1})\mu_{t_{2}-t_{1}}(x_{1}^{-1}dx_{2})\cdots\mu_{t_{n}-t_{n-1}}(x_{n-1}^{-1}dx_{n})$
$\displaystyle=\int_{Q^{n}}\chi_{B}(x_{1},x_{1}x_{2},\ldots,x_{1}\cdots
x_{n})\mu_{t_{1}}(dx_{1})\mu_{t_{2}-t_{1}}(dx_{2})\cdots\mu_{t_{n}-t_{n-1}}(dx_{n}).$
(4.1)

This means that

$\displaystyle
P_{\bf{t}}=(\psi_{n})_{*}(\mu_{t_{1}}\otimes\mu_{t_{2}-t_{1}}\otimes\cdots\otimes\mu_{t_{n}-t_{n-1}})$

for the map

$\displaystyle\psi_{n}(g_{1},\ldots,g_{n})=(g_{1},g_{1}g_{2},\dots,g_{1}\cdots
g_{n}).$ (4.2)

######  Remark 4.1.

Since the inverse of the map $\psi_{n}$ in (4.2) is given by

$\displaystyle\psi_{n}^{-1}(h_{1},\ldots,h_{n})=\big{(}h_{1},h_{1}^{-1}h_{2},\cdots,h_{n-1}^{-1}h_{n}\big{)},$

it follows that the random variables

$\displaystyle
X_{t_{1}},X_{t_{1}}^{-1}X_{t_{2}},\dots,X_{t_{n-1}}^{-1}X_{t_{n}}$

are independent. Since the distribution of $X_{t}^{-1}X_{t+s}$ is $\mu_{s}$
for every $t,s\geq 0$, it follows that $(X_{t})_{t\geq 0}$ has independent
stationary increments (cf. also [7, Corollary 35.4]).

For a curve $\omega\colon{\mathbb{R}}\to Q$, we define
$(\theta\omega)(t):=\omega(-t)$. We write $P(Q)^{\pm}$ for the subgroup
consisting of paths whose value is $\mathbf{1}$ on ${\mathbb{R}}_{-}$, resp.,
${\mathbb{R}}_{+}$ and $Q\subseteq P(Q)$ with the subgroup of constant paths.
For the following proposition, we recall that the involution on the space
$\mathcal{M}(Q)$ of finite real Borel measures on $Q$ is defined by

$\displaystyle\int_{Q}f(g)\,d\mu^{*}(g)=\int_{Q}f\big{(}g^{-1}\big{)}\,d\mu(g).$

######  Proposition 4.2.

Let $P^{\mu,-}:=\theta_{*}P^{\mu,+}$ and $P^{\mu}$ be the image of the measure
$P^{\mu,-}\otimes P^{\mu,+}$ on $P_{*}(Q)$ under the bijective product map

$\displaystyle\Phi\colon\ P(Q)^{-}\times P(Q)^{+}\to
P_{*}(Q),\qquad(\omega_{-},\omega_{+})\mapsto\omega_{-}\cdot\omega_{+}.$

If $\mu_{t}^{*}=\mu_{t}$ holds for every $t\geq 0$, then this measure is
invariant under $\theta$ and the one-parameter group of Borel isomorphisms,
given by

$\displaystyle(V_{t}\omega)(s):=\omega(-t)^{-1}\omega(s-t),\qquad
t\in{\mathbb{R}}.$

###### Proof.

The $\theta$-invariance of $P$ follows immediately from its construction and

$\displaystyle\theta\Phi(\omega_{-},\omega_{+})=\theta(\omega_{-}\cdot\omega_{+})=\theta(\omega_{-})\cdot\theta(\omega_{+})=\theta(\omega_{+})\cdot\theta(\omega_{-})=\Phi(\theta(\omega_{+}),\theta(\omega_{-})).$

We also note that $\theta V_{t}\theta=V_{-t}$ follows from

$\displaystyle(\theta
V_{t}\theta\omega)(s)=(V_{t}\theta\omega)(-s)=(\theta\omega)(-t)^{-1}(\theta\omega)(-s-t)=\omega(t)^{-1}\omega(s+t)=(V_{-t}\omega)(s).$
(4.3)

Since $P$ is uniquely determined by its images under evaluation in finite
tuples ${\bf{t}}=(t_{1},\ldots,t_{n})$, $t_{i}<t_{i+1}$, it suffices to show
that the corresponding distributions $P_{\bf{t}}$ on $Q^{n}$ satisfy
$V_{t}P_{\bf{t}}=P_{(t_{1}+t,\ldots,t_{n}+t)}$. In view of (4.3), we may
assume that $t>0$. We may further assume that $t_{k}=0$ for some $k$ and that
$t_{m}=-t$ for some $m<k$.

First we note that, for measurable functions
$f_{1},\ldots,f_{k-1},f_{k+1},\ldots,f_{n}$ on $Q$, we have

$\displaystyle\int_{P_{*}(Q)}\prod_{j\not=k}f_{j}(\omega(t_{j}))\,dP^{\mu}(\omega)$
$\displaystyle\qquad{}=\int_{P(Q)^{+}}\int_{P(Q)^{+}}\prod_{j=1}^{k-1}f_{j}(\omega_{-}(-t_{j}))\prod_{j=k+1}^{n}f_{j}(\omega_{+}(t_{j}))\,dP^{\mu,-}(\omega_{-})dP^{\mu,+}(\omega_{+})$
$\displaystyle\qquad{}\buildrel\eqref{eq:density}\over{=}\int_{Q^{n-1}}f_{1}(g_{k-1}\cdots
g_{1})\cdots f_{k-1}(g_{k-1})f_{k+1}(g_{k+1})\cdots f_{n}(g_{k+1}\cdots
g_{n})$ $\displaystyle\qquad\quad{}\times d\mu_{t_{2}-t_{1}}(g_{1})\cdots
d\mu_{t_{k-1}-t_{k-2}}(g_{k-2})d\mu_{-t_{k-1}}(g_{k-1})$
$\displaystyle\qquad\quad{}\times
d\mu_{t_{k+1}}(g_{k+1})d\mu_{t_{k+2}-t_{k+1}}(g_{k+2})\cdots
d\mu_{t_{n}-t_{n-1}}(g_{n}).$

For $F(\omega)=\prod\limits_{j=1}^{n}f_{j}(\omega(t_{j}))$, this leads with
$t_{m}=-t$ and $t_{k}=0$ to

$\displaystyle\int_{P(Q)}(V_{t}F)(\omega)\,dP^{\mu}(\omega)=\int_{P(Q)}F\big{(}\omega(t)^{-1}\omega(\cdot+t)\big{)}\,dP^{\mu}(\omega)$
$\displaystyle=\int_{P(Q)^{+}}\int_{Q^{m-1}}f_{1}\big{(}\omega(t)^{-1}g_{m-1}\cdots
g_{1}\big{)}\cdots
f_{m-1}\big{(}\omega(t)^{-1}g_{m-1}\big{)}f_{m}\big{(}\omega(t)^{-1}\big{)}$
$\displaystyle\qquad\quad{}\times
f_{m+1}\big{(}\omega(t)^{-1}\omega(t+t_{m+1})\cdots
f_{n}\big{(}\omega(t)^{-1}\omega(t+t_{n})\big{)}\big{)}$
$\displaystyle\qquad\quad{}\times
d\mu_{-t_{m-1}-t}(g_{m-1})d\mu_{t_{m-1}-t_{m-2}}(g_{m-2})\cdots
d\mu_{t_{2}-t_{1}}(g_{1})dP^{\mu,+}(\omega)$
$\displaystyle=\int_{Q^{n-1}}f_{1}\big{(}(g_{m}\cdots
g_{k-1})^{-1}g_{m-1}\cdots g_{1}\big{)}\cdots f_{m-1}\big{(}(g_{m}\cdots
g_{k-1})^{-1}g_{m-1}\big{)}f_{m}\big{(}(g_{m}\cdots g_{k-1})^{-1}\big{)}$
$\displaystyle\qquad\quad{}\times f_{m+1}\big{(}(g_{m}\cdots
g_{k-1})^{-1}g_{m}\big{)}\cdots f_{k-1}((g_{m}\cdots g_{k-1})^{-1}g_{m}\cdots
g_{k-2})$ $\displaystyle\qquad\quad{}\times f_{k+1}(g_{k+1})\cdots
f_{n}(g_{k+1}\cdots
g_{n})\,d\mu_{-t_{m-1}-t}(g_{m-1})d\mu_{t_{m-1}-t_{m-2}}(g_{m-2})\cdots
d\mu_{t_{2}-t_{1}}(g_{1})$ $\displaystyle\qquad\quad{}\times
d\mu_{t_{m+1}+t}(g_{m})d\mu_{t_{m+2}-t_{m+1}}(g_{m+1})\cdots$
$\displaystyle\qquad\quad{}\times
d\mu_{t_{k}-t_{k-1}}(g_{k-1})d\mu_{t_{k+1}-t_{k}}(g_{k+1})\cdots
d\mu_{t_{n}-t_{n-1}}(g_{n})$
$\displaystyle=\int_{Q^{n-1}}f_{1}\big{(}g_{k-1}^{-1}\cdots
g_{m}^{-1}g_{m-1}\cdots g_{1}\big{)}\cdots f_{m-1}\big{(}g_{k-1}^{-1}\cdots
g_{m}^{-1}g_{m-1}\big{)}f_{m}\big{(}g_{k-1}^{-1}\cdots g_{m}^{-1}\big{)}$
$\displaystyle\qquad\quad{}\times f_{m+1}\big{(}g_{k-1}^{-1}\cdots
g_{m+1}^{-1}\big{)}\cdots
f_{k-1}\big{(}g_{k-1}^{-1}\big{)}f_{k+1}(g_{k+1})\cdots f_{n}(g_{k+1}\cdots
g_{n})$ $\displaystyle\qquad\quad{}\times d\mu_{t_{2}-t_{1}}(g_{1})\cdots
d\mu_{t_{m-1}-t_{m-2}}(g_{m-2})d\mu_{t_{m}-t_{m-1}}(g_{m-1})d\mu_{t_{m+1}-t_{m}}(g_{m})$
$\displaystyle\qquad\quad{}\times d\mu_{t_{m+2}-t_{m+1}}(g_{m+1})\cdots
d\mu_{t_{k}-t_{k-1}}(g_{k-1})d\mu_{t_{k+1}-t_{k}}(g_{k+1})\cdots
d\mu_{t_{n}-t_{n-1}}(g_{n})$
$\displaystyle\buildrel(\dagger)\over{=}\int_{Q^{n-1}}f_{1}(g_{k-1}\cdots
g_{1})\cdots f_{m-1}(g_{k-1}\cdots g_{m-1})f_{m}(g_{k-1}\cdots g_{m})$
$\displaystyle\qquad\quad{}\times f_{m+1}(g_{k-1}\cdots g_{m+1})\cdots
f_{k-1}(g_{k-1})f_{k+1}(g_{k+1})\cdots f_{n}(g_{k+1}\cdots g_{n})$
$\displaystyle\qquad\quad{}\times d\mu_{t_{2}-t_{1}}(g_{1})\cdots
d\mu_{t_{m}-t_{m-1}}(g_{m-1})d\mu_{t_{m+1}-t_{m}}(g_{m})d\mu_{t_{m+2}-t_{m+1}}(g_{m+1})\cdots$
$\displaystyle\qquad\quad{}\times
d\mu_{t_{k}-t_{k-1}}(g_{k-1})d\mu_{t_{k+1}-t_{k}}(g_{k+1})\cdots
d\mu_{t_{n}-t_{n-1}}(g_{n})$ $\displaystyle=\int_{Q^{n-1}}f_{1}(g_{k-1}\cdots
g_{1})\cdots f_{k-1}(g_{k-1})f_{k+1}(g_{k+1})\cdots f_{n}(g_{k+1}\cdots
g_{n})$ $\displaystyle\qquad\quad{}\times d\mu_{t_{2}-t_{1}}(g_{1})\cdots
d\mu_{t_{k}-t_{k-1}}(g_{k-1})d\mu_{t_{k+1}-t_{k}}(g_{k+1})\cdots
d\mu_{t_{n}-t_{n-1}}(g_{n}).$

Here we have used in $(\dagger)$ the symmetry of the measures $\mu_{t}$ to get
rid of the inverses. The preceding calculation shows that the transformations
$(V_{t})_{t\in{\mathbb{R}}}$ on $P_{*}(Q)$ leave the probability measure
$P^{\mu}$ invariant. ∎

######  Remark 4.3.

Note that the transformations $(V_{t})_{t\in{\mathbb{R}}}$ on $P_{*}(Q)$
satisfy

$\displaystyle
V_{t}X_{s}=X_{t}^{-1}X_{s+t}\qquad\mbox{because}\quad(V_{t}X_{s})(\omega)=(V_{-t}\omega)(s)=\omega(t)^{-1}\omega(s+t).$

On functions on $P_{*}(Q)$, we obtain the action

$\displaystyle(V_{t}F)(\omega)=F(\omega(t)^{-1}\omega(\cdot+t))$

which is isometric on
$L^{2}(P_{*}(Q),{\mathfrak{B}}^{{\mathbb{R}}^{\times}},P^{\mu})$.

######  Example 4.4.

(a) (Poisson semigroups) For every $g\in K$, we obtain a convolution semigroup
of measures by

$\displaystyle\mu_{t}=e^{-t}\sum_{k=0}^{\infty}\frac{t^{k}}{k!}\delta_{g^{k}}=e^{t(\delta_{g}-\delta_{0})}\qquad\mbox{for}\quad
t\geq 0.$

This semigroup consists of symmetric measures if and only if $\delta_{g}$ is
symmetric, i.e., $g=g^{-1}$. In this case $g^{2n}=\mathbf{1}$ and $g^{2n+1}=g$
for all $n$, so that

$\displaystyle\mu_{t}=\frac{1+e^{-2t}}{2}\delta_{\mathbf{1}}+\frac{1-e^{-2t}}{2}\delta_{g}.$

Note that the limit for $t\to\infty$ is the measure
${\textstyle{\frac{1}{2}}}(\delta_{\mathbf{1}}+\delta_{g})$. This particular
example can also be considered as a one-parameter subsemigroup of the
semigroup of Markov matrices on ${\mathbb{R}}^{2}$, resp., Markov operators on
the two-element set, where $g$ is the transposition of the two elements (see
also Example 3.10).

(b) (Velocity processes) If $K$ is a Lie group and $X\in{\mathfrak{g}}$, then
we obtain a semigroup of measures by

$\displaystyle\mu_{t}=\delta_{\exp tX}\qquad\mbox{for}\quad t\geq 0.$

For these semigroups the condition $\mu_{t}^{*}=\mu_{t}$ for every $t\geq 0$
is satisfied only if $X=0$, i.e., $\mu_{t}=\delta_{\mathbf{1}}$ for every $t$.

######  Example 4.5.

The class of polish groups is quite large.

(a) Among locally compact groups, the second countable ones carry a complete
left invariant metric, turning them in a polish group. This covers in
particular all finite-dimensional Lie groups with at most countably many
connected components.

(b) Separable Fréchet–Lie groups are also polish, and this class contains in
particular all connected Banach–Lie groups whose Lie algebra is separable,
gauge groups and groups of diffeomorphisms of compact manifolds.

(c) Another important example of a polish group is the unitary group
$\mathop{\rm U{}}\nolimits(\mathcal{H})_{s}$ of a separable Hilbert space
$\mathcal{H}$, endowed with the strong operator topology (cf. [47, Proposition
II.1] or [62, Theorem II.1, p. 93]).

######  Remark 4.6.

In recent years, many interesting one-parameter semigroups of measures have
been studied on infinite-dimensional groups.

(a) On the group $\mathop{{\rm Dif\/f}}\nolimits({\mathbb{S}}^{1})$ of
diffeomorphisms of the circle with respect to the $H^{3/2}$-metric, Brownian
motion has been constructed by P. Malliavin. A different approach also
exhibiting the invariance under rigid rotations is described in [18] (cf. also
[21, 22]).

(b) In [10] Brownian motion on compact groups is studied. This is defined to
be a stochastic process $(X_{t})_{t\geq 0}$ with values in $K$ with
$X_{0}=\mathbf{1}$, independent stationary increments and continuous sample
paths. In addition, it is assumed to be symmetric, biinvariant and non-
degenerate, i.e., $X_{t}$ visits every open subset with positive probability.
For the corresponding convolution semigroup $(\mu_{t})_{t\geq 0}$ this implies
that

$\displaystyle\lim_{t\to 0}t^{-1}\mu_{t}\big{(}V^{c}\big{)}=0$ (4.4)

for every open $\mathbf{1}$-neighborhood $V$ in $K$. Conversely, [10, Theorem
1.2] characterizes the convolution semigroups $(\mu_{t})_{t\geq 0}$
corresponding to Brownian motions as those which satisfy, in addition to
(4.4), that $\mu_{t}\to\delta_{\mathbf{1}}$ weakly on $C(K)$,
$\mu_{t}^{*}=\mu_{t}$, $\mu_{t}$ is conjugation invariant (also called
central), and $\mathop{{\rm supp}}\nolimits(\mu_{t})=K$ for every $t>0$.

(c) In [17], Driver studies Brownian motion on the infinite-dimensional
Banach–Lie group

$\displaystyle W(K):=C_{*}([0,1],K)=\\{\omega\in
C([0,1],K)\colon\omega(0)=\mathbf{1}\\},$

where $K$ is a connected Lie group with compact Lie algebra, i.e., $K\cong
C\times{\mathbb{R}}^{d}$ for some $d\in{\mathbb{N}}_{0}$ and a compact Lie
group $C$. Here one has to construct a semigroup $(\mu_{t})_{t\geq 0}$ of heat
kernel measures on the space $W({\mathbb{R}}^{d})$. On $W({\mathbb{R}}^{d})$
one has to get hold of the smoothness properties of the heat kernel measures
$\mu_{t}$ corresponding to the Wiener measure. The corresponding Hilbert space
$H({\mathfrak{g}})$ is the space of finite energy paths with values in the Lie
algebra ${\mathfrak{g}}$ and the measures $\mu_{t}$ on $W(K)$ are quasi-
invariant under left and right multiplication with elements of the
corresponding Cameron–Martin group $H(K)$ [17, Theorem 7.7]. The unitary
representations of $H(K)$ by left, resp., right multiplications has been
identified recently with the so-called energy representation of this group
[1]. With similar techniques, the existence of heat kernel measures is also
obtained for the space of pinned paths with values in a compact Riemannian
manifold $M$.

###  4.2 Standard path space structures for locally compact groups

In this section we assume that the group $Q$ is locally compact and that the
convolution semigroup $(\mu_{t})_{t\geq 0}$ of probability measures on $Q$ is
strongly continuous in the sense that $\lim\limits_{t\to
0}\mu_{t}=\delta_{\mathbf{1}}=\mu_{0}$ weakly on the space $C_{b}(Q)$ of
bounded continuous functions on $Q$. We further assume that $\nu$ is a measure
on $Q$ satisfying $\nu*\mu_{t}=\nu$ for every $t>0$, and, in addition, that
the operators

$\displaystyle P_{t}f:=f*\mu_{t}$

on $L^{2}(Q,\nu)$ are symmetric. If $\nu$ is a right Haar measure, then the
symmetry of the operators $P_{t}$ is equivalent to $\mu_{t}^{*}=\mu_{t}$.

######  Remark 4.7.

Let $\mu_{Q}$ be a right Haar measure on $Q$ and assume that all the measures
$\mu_{t}$ are symmetric. Integrating the right regular representation
$(\pi^{r}(g)f)(x)=f(xg)$ of $Q$ on $L^{2}(Q,\mu_{Q})$, we obtain a
$*$-representation of the convolution algebra $\mathcal{M}(Q)$ on $L^{2}(Q)$
by

$\displaystyle\pi^{r}(\mu):=\int_{Q}\pi^{r}(g)\,d\mu(g),\qquad\big{(}\pi^{r}(\mu)f\big{)}(x)=\int_{Q}f(xg)\,d\mu(g)=(f*\mu)(x).$

Then $P_{t}:=(\pi^{r}(\mu_{t}))_{t\geq 0}$ is a strongly continuous semigroup
of hermitian contractions on $L^{2}(Q)$ (here we use $\mu_{t}^{*}=\mu_{t}$)
which are Markov operators. This is a positive semigroup structure because the
continuity in measure (SPS4) follows from the continuity of the action of $Q$
(cf. Remark 3.26). Hence the Klein–Landau reconstruction Theorem 3.22 provides
a path space model for the corresponding dilation representation of
${\mathbb{R}}$ [50].

For $P_{t}$ as above, formula (3.5)

$\displaystyle\int_{P(Q)}f_{t_{1}}\otimes\cdots\otimes
f_{t_{n}}\,dP^{\mu}_{\bf{t}}=\int_{Q}f_{1}P_{t_{2}-t_{1}}f_{2}\cdots
P_{t_{n}-t_{n-1}}f_{n}\,d\nu$

in Remark 3.23 specializes for $t_{1}\leq\cdots\leq t_{n}$ and $0\leq f_{j}$
to

$\displaystyle\int_{P(Q)}f_{1}\otimes\cdots\otimes f_{n}\,dP^{\mu}_{\bf{t}}$
$\displaystyle\qquad{}=\int_{Q^{n}}f_{1}(g_{1})f_{2}(g_{1}g_{2})\cdots
f_{n}(g_{1}\cdots g_{n})\,d\nu(g_{1})d\mu_{t_{2}-t_{1}}(g_{2})\cdots
d\mu_{t_{n}-t_{n-1}}(g_{n})$
$\displaystyle\qquad{}=\int_{Q^{n}}f_{1}(g_{1})f_{2}(g_{2})\cdots
f_{n}(g_{n})\,d\psi_{*}(\nu\otimes\mu_{t_{2}-t_{1}}\otimes\cdots\otimes\mu_{t_{n}-t_{n-1}}),$

where $\psi(g_{1},\ldots,g_{n})=(g_{1},g_{1}g_{2},\dots,g_{1}\cdots g_{n})$.
We conclude that

$\displaystyle
P^{\mu}_{\bf{t}}=\psi_{*}(\nu\otimes\mu_{t_{2}-t_{1}}\otimes\cdots\otimes\mu_{t_{n}-t_{n-1}}).$

######  Lemma 4.8.

Let $\nu$ be a measure on $Q$ satisfying $\nu*\mu_{t}=\nu$ for every $t>0$.
Then we obtain for $t_{1}\leq\cdots\leq t_{n}$ and
${\bf{t}}:=(t_{1},\ldots,t_{n})$ on $Q^{n}$ a consistent family of measures

$\displaystyle
P^{\mu}_{\bf{t}}:=(\psi_{n})_{*}(\nu\otimes\mu_{t_{2}-t_{1}}\otimes\cdots\otimes\mu_{t_{n}-t_{n-1}}).$

If $Q$ is a polish group, this leads to a unique measure $P^{\mu}$ on $P(Q)$
with $(\mathop{{\rm ev}}\nolimits_{\bf{t}})_{*}P^{\mu}=P^{\mu}_{\bf{t}}$ for
$t_{1}<\ldots<t_{n}$.

###### Proof.

This follows from Remark B.3. For the sake of clarity, we give a direct
argument for the consistency of the measures $P^{\mu}_{\bf{t}}$. Pick
$j\in\\{1,\ldots,n\\}$ and let $p\colon Q^{n}\to Q^{n-1}$ denote the
projection omitting the $j$th component. Then

$\displaystyle(p\circ\psi_{n})(g_{1},\ldots,g_{n})=\psi_{n-1}(g_{1},\ldots,g_{j-1},g_{j}g_{j+1},g_{j+2},\ldots,g_{n})$

implies for $j\geq 2$ that

$\displaystyle
p_{*}(\psi_{n})_{*}(\nu\otimes\mu_{t_{2}-t_{1}}\otimes\cdots\otimes\mu_{t_{n}-t_{n-1}})$
$\displaystyle\qquad{}=(\psi_{n-1})_{*}(\nu\otimes\cdots\otimes\mu_{t_{j}-t_{j-1}}*\mu_{t_{j+1}-t_{j}}\otimes\cdots\otimes\mu_{t_{n}-t_{n-1}})$
$\displaystyle\qquad{}=(\psi_{n-1})_{*}(\mu_{t_{1}}\otimes\cdots\otimes\mu_{t_{j+1}-t_{j-1}}\otimes\cdots\otimes\mu_{t_{n}-t_{n-1}}).$

For $j=1$, we obtain

$\displaystyle
p_{*}(\psi_{n})_{*}(\nu\otimes\mu_{t_{2}-t_{1}}\otimes\cdots\otimes\mu_{t_{n}-t_{n-1}})$
$\displaystyle=(\psi_{n-1})_{*}(\nu*\mu_{t_{2}-t_{1}}\otimes\mu_{t_{3}-t_{2}}\otimes\cdots\otimes\mu_{t_{n}-t_{n-1}})$
$\displaystyle=(\psi_{n-1})_{*}(\nu\otimes\mu_{t_{3}-t_{2}}\otimes\cdots\otimes\cdots\otimes\mu_{t_{n}-t_{n-1}}).$

This implies consistency of the measures $P^{\mu}_{\bf{t}}$ on $Q^{n}$. The
consistency condition implies the existence of a measure $P^{\mu}$ on
$Q^{{\mathbb{R}}}$ with $(\mathop{{\rm
ev}}\nolimits_{\bf{t}})_{*}\mu=P^{\mu}_{\bf{t}}$ for $t_{1}\leq\cdots\leq
t_{n}$ (cf. Definition 3.11). ∎

From the Klein–Landau reconstruction theorem we immediately obtain the
following specialization. We refer to [50] for other constructions of this
dilation.

######  Theorem 4.9.

Suppose that $Q$ is a second countable locally compact group. Let $\mu$ be the
measure on $Q^{\mathbb{R}}$ corresponding to the symmetric convolution
semigroup $(\mu_{t})_{t\geq 0}$ of probability measures on $Q$ and the measure
$\nu$ on $Q$ for which the operators $P_{t}f=f*\mu_{t}$ define a positive
semigroup structure on $L^{2}(Q,\nu)$. Then the translation action
$(U_{t}\omega)(s):=\omega(s-t)$ on $P(Q)=Q^{\mathbb{R}}$ is measure preserving
and $\mu$ is invariant under $(\theta\omega)(t):=\omega(-t)$. We thus obtain a
reflection positive one-parameter group of Markov type on
$\mathcal{E}:=L^{2}(P(Q),{\mathfrak{B}}^{\mathbb{R}},\mu)$ with respect to
$\mathcal{E}_{+}:=L^{2}(P(Q),{\mathfrak{B}}^{{\mathbb{R}}_{+}},\mu)$, for
which $\mathcal{E}_{0}:=\mathop{{\rm ev}}\nolimits_{0}^{*}(L^{2}(Q,\nu))\cong
L^{2}(Q,\nu)$ and $\widehat{\mathcal{E}}\cong L^{2}(Q,\nu)$ with $q(F)=E_{0}F$
for $F\in\mathcal{E}_{+}$. We further have

$\displaystyle E_{0}U_{t}E_{0}=P_{t}\qquad\mbox{holds for}\quad
P_{t}f=f*\mu_{t},$

so that the $U$-cyclic subrepresentation generated by $\mathcal{E}_{0}$ is a
unitary dilation of the hermitian one-parameter semigroup $(P_{t})_{t\geq 0}$
on $L^{2}(Q,\nu)$.

######  Example 4.10.

(a) For $Q={\mathbb{R}}^{d}$, the heat semigroup is given on
$L^{2}({\mathbb{R}}^{d})$ by

$\displaystyle e^{t\Delta}f=f*\gamma_{t}\qquad\mbox{where}\quad
d\gamma_{t}(x)=\frac{1}{(2\pi
t)^{d/2}}e^{-\frac{1}{2}\frac{\|x\|^{2}}{t}}\,dx.$

We call the corresponding measure on $Q^{\mathbb{R}}$ the Lebesgue–Wiener
measure (cf. Theorem 4.9).

(b) If $Q$ is a finite-dimensional Lie group and $X_{1},\ldots,X_{n}$ is a
basis of the Lie algebra, then we obtain a left invariant Laplacian by
$\Delta:=\sum\limits_{j=1}^{n}L_{X_{j}}^{2},$ where $L_{X_{j}}$ denotes the
right invariant vector field with $L_{X_{j}}(\mathbf{1})=X_{j}$. Then there
also exists a semigroup $(\mu_{t})_{t\geq 0}$ of probability measures on $Q$
such that [53, Section 8]

$\displaystyle e^{t\Delta}f=f*\mu_{t}\qquad\mbox{for}\quad t\geq 0.$

Accordingly, we obtain a Haar–Wiener measure on the path space
$Q^{\mathbb{R}}$.

##  5 Gaussian $\boldsymbol{(G,S,\tau)}$-probability spaces

In this section we discuss the second quantization functor and its connection
to gaussian $(G,S,\tau)$-probability spaces. We then discuss equivalence of
gaussian measures for reproducing kernel Hilbert spaces. The main results of
this section are contained in Subsection 5.4, where we discuss gaussian
measures on the space of distributions on a Lie group. Here the distribution
vectors of unitary representations play an important role.

###  5.1 Second quantization and gaussian processes

######  Definition 5.1.

Let $\mathcal{H}$ be a real Hilbert space. A gaussian random process indexed
by $\mathcal{H}$ is a random process $(\varphi(v))_{v\in\mathcal{H}}$ on a
probability space $(Q,\Sigma,\mu)$ such that

  * (GP1)

$\\{\varphi(v)\colon v\in\mathcal{H}\\}$ is full, i.e., these random variables
generate the $\sigma$-algebra $\Sigma$ modulo zero sets.

  * (GP2)

Each $\varphi(v)$ is a gaussian random variable of mean zero.

  * (GP3)

$\langle\varphi(v),\varphi(w)\rangle=\langle v,w\rangle$ is the inner product
on $\mathcal{H}$.

According to [64, Theorems I.6, I.9], gaussian random processes indexed by
$\mathcal{H}$ exist and are unique up to isomorphisms of probability spaces.
This means that, if $(Q,\Sigma,\mu)$ is the corresponding probability space,
then the algebra $L^{\infty}(Q,\mu)$ with its state given by $\mu$ is uniquely
determined by $\mathcal{H}$ (cf. [64, Section 1.1]). Its projections
correspond to $\Sigma/J_{\mu}$, where $J_{\mu}$ is the ideal of zero sets555A
short proof for the uniqueness can be derived from reproducing kernel
techniques. If $\varphi_{j}\colon\mathcal{H}\to L^{2}(Q_{j},\mu_{j})$,
$j=1,2$, are two realizations, then the corresponding covariance kernels on
$\mathcal{H}$ coincide, so that there exists a unique unitary operator
$\Phi\colon L^{2}(Q_{1},\mu_{1})\to L^{2}(Q_{2},\mu_{2})$ such that
$\Phi\circ\varphi_{1}=\varphi_{2}$. Accordingly,
$\Phi\circ\mathcal{A}_{1}\circ\Phi^{-1}=\mathcal{A}_{2}$ for the von Neumann
algebras $\mathcal{A}_{j}:=\\{e^{i\varphi_{j}(v)}\colon
v\in\mathcal{H}\\}^{\prime\prime}\cong L^{\infty}(Q_{j},\mu_{j})$..

######  Theorem 5.2.

Let $V$ be a real vector space, $V^{*}$ be its algebraic dual and
${\mathfrak{B}}^{*}$ be the smallest $\sigma$-algebra for which all evaluation
functions $V^{*}\to V,\alpha\mapsto\alpha(v)$, are measurable. Then a function
$\chi\colon V\to{\mathbb{C}}$ is the Fourier transform $\chi=\widehat{\mu}$ of
a measure $\mu$ on $(V^{*},{\mathfrak{B}}^{*})$ if and only if $\chi$ is
positive definite and continuous on every finite-dimensional subspace. In this
case $\mu$ is uniquely determined.

###### Proof.

Since the Fourier transform $\widehat{\mu}$ is positive definite and
sequentially continuous, it is in particular continuous on every finite-
dimensional subspace of $V$. The converse follows from [68, Theorem 16.2]. ∎

######  Example 5.3.

Theorem 5.2 implies that the gaussian process indexed by $\mathcal{H}$ may be
realized by the probability measure $\gamma_{\mathcal{H}}$ on
$(\mathcal{H}^{a},{\mathfrak{B}}^{*})$ (the algebraic dual) whose Fourier
transform is

$\displaystyle\widehat{\gamma}_{\mathcal{H}}(v)=e^{-\frac{\|v\|^{2}}{2}}.$

Here $\varphi(v)(\alpha)=\alpha(v)$, and

$\displaystyle\int_{\mathcal{H}^{a}}e^{i\alpha(v)}\,d\gamma_{\mathcal{H}}(\alpha)=e^{-\frac{\|v\|^{2}}{2}}\qquad\mbox{implies}\quad\int_{\mathcal{H}^{a}}\alpha(v)^{2}\,d\gamma_{\mathcal{H}}(\alpha)=\|v\|^{2}.$

This leads to

$\displaystyle\big{\langle}e^{i\varphi(v)},e^{i\varphi(w)}\big{\rangle}=e^{-\frac{\|v-w\|^{2}}{2}}=e^{-\frac{\|v\|^{2}}{2}}e^{-\frac{\|w\|^{2}}{2}}e^{\langle
v,w\rangle},$

so that $K_{v}:=e^{i\varphi(v)}e^{\frac{\|v\|^{2}}{2}}$ satisfies

$\displaystyle\langle K_{v},K_{w}\rangle=e^{\langle v,w\rangle}.$

######  Example 5.4.

If $\mathcal{H}$ is finite-dimensional, so that
$\mathcal{H}\cong{\mathbb{R}}^{d}$ for some $d\in{\mathbb{N}}_{0}$, then
$\mathcal{H}^{a}\cong{\mathbb{R}}^{d}$ and
$d\gamma_{\mathcal{H}}(x)=\frac{1}{(2\pi)^{d/2}}e^{-\frac{\|x\|^{2}}{2}}\,dx$
is a gaussian measure.

We now introduce the second quantization. Conceptually the easiest way to
define second quantization as a functor is to associate to a (real or complex)
Hilbert space $\mathcal{H}$ its Fock space

$\displaystyle\mathcal{F}(\mathcal{H}):=\widehat{\bigoplus}_{n\in{\mathbb{N}}_{0}}S^{n}(\mathcal{H}),$

where $S^{n}(\mathcal{H}):=(\mathcal{H}^{\otimes n})^{S_{n}}$ is the closed
subspace of $S_{n}$-invariant vectors in the $n$-fold tensor power of
$\mathcal{H}$. For $v_{1},\ldots,v_{n}\in\mathcal{H}$, we define the symmetric
product by

$\displaystyle v_{1}\vee\cdots\vee v_{n}:=P_{+}(v_{1}\otimes\cdots\otimes
v_{n}),$

where $P_{+}\colon\mathcal{H}^{\otimes n}\to S^{n}(\mathcal{H})$ is the
orthogonal projection. The inner products of such elements are given by

$\displaystyle\langle v_{1}\vee\cdots\vee v_{n},w_{1}\vee\cdots\vee
w_{n}\rangle$ $\displaystyle=\langle v_{1}\vee\cdots\vee
v_{n},w_{1}\otimes\cdots\otimes w_{n}\rangle$
$\displaystyle=\frac{1}{n!}\sum_{\sigma\in S_{n}}\langle
v_{\sigma(1)},w_{1}\rangle\cdots\langle v_{\sigma(n)},w_{m}\rangle.$

In particular, with $v^{n}=\underbrace{v\vee\cdots\vee v}_{n\text{-times}}$,
we have

$\displaystyle\langle v^{n},w^{n}\rangle=\langle
v,w\rangle^{n}\qquad\mbox{and}\qquad\|v^{n}\|=\|v\|^{n}.$ (5.1)

Clearly, every contraction $A\colon\mathcal{H}\to\mathcal{K}$ defines a
contraction
$\Gamma(A)\colon\mathcal{F}(\mathcal{H})\to\mathcal{F}(\mathcal{K})$ by

$\displaystyle\Gamma(A)(v_{1}\vee\cdots\vee v_{n}):=Av_{1}\vee\cdots\vee
Av_{n}$

and it is clear that $\Gamma(AB)=\Gamma(A)\Gamma(B)$ and
$\Gamma(A^{*})=\Gamma(A)^{*}$. In particular, we obtain a representation of
the involutive semigroup of contractions on $\mathcal{H}$ in the Fock space
$\mathcal{F}(\mathcal{H})$ and $\mathcal{F}$ defines an endofunctor from the
category of Hilbert spaces whose morphism are contractions into itself. The
problem with this approach is that it completely ignores positivity issues.

If $\mathcal{H}$ is a Hilbert space, then any $v\in\mathcal{H}$ defines a
function $\langle v,\cdot\rangle$, $u\mapsto\langle v,u\rangle$, which is
linear if $\mathcal{H}$ is real and antilinear if $\mathcal{H}$ is complex.
For

$\displaystyle\mathop{{\rm
Exp}}\nolimits(v):=\sum_{n=0}^{\infty}\frac{1}{\sqrt{n!}}v^{n},$

we now derive from (5.1) that $\mathop{{\rm
Exp}}\nolimits(v)\in\mathcal{F}(\mathcal{H})$ and

$\displaystyle\langle\mathop{{\rm Exp}}\nolimits(v),\mathop{{\rm
Exp}}\nolimits(w)\rangle=\sum_{n=0}^{\infty}\frac{1}{n!}\langle
v,w\rangle^{n}=e^{\langle v,w\rangle}.$

This leads to an embedding

$\displaystyle\Phi\colon\
\mathcal{F}(\mathcal{H})\to{\mathbb{C}}^{\mathcal{H}},\qquad\Phi(\xi)(v):=\langle\xi,\mathop{{\rm
Exp}}\nolimits(v)\rangle,$

where

$\displaystyle\Phi(v_{1}\vee\cdots\vee v_{n})(v)=\frac{1}{\sqrt{n!}}\langle
v_{1}\vee\cdots\vee
v_{n},v^{n}\rangle=\frac{1}{\sqrt{n!}}\prod_{j=1}^{n}\langle v_{j},v\rangle.$

The image of $\Phi$ is the reproducing kernel space $\mathcal{F}_{\rm
RK}(\mathcal{H})$ with kernel

$\displaystyle K(v,w)=\langle\mathop{{\rm Exp}}\nolimits(w),\mathop{{\rm
Exp}}\nolimits(v)\rangle=e^{\langle w,v\rangle}.$

We may thus identify $\mathcal{F}(\mathcal{H})$ with the reproducing kernel
Hilbert space $\mathcal{F}_{\rm RK}(\mathcal{H})$.

For a contraction $A\colon\mathcal{H}\to\mathcal{K}$, we have