# A de Finetti theorem for quasi-invariant states

###### Abstract

We develop the theory of quasi–invariant (resp. strongly quasi–invariant)
states under the action of a group $G$ of normal $*$–automorphisms of a von
Neumann alegbra $\mathcal{A}$. We prove that these states are naturally
associated to left–$G$–$1$–cocycles. If $G$ is compact, the structure of
strongly $G$–quasi–invariant states is determined. The case of unimodular
locally compact group is also discussed showing that many of the structure
properties of strongly $G$–quasi–invariant states continue to hold in this
case. For any $G$–quasi–invariant state $\varphi$, we construct a unitary
representation associated to the triple $(\mathcal{A},G,\varphi)$. We prove
that any quantum Markov chain, with commuting, invertible and hermitean
conditional density amplitudes, on a countable tensor product of type I
factors is strongly quasi–invariant with respect to the natural action of the
group $\mathcal{S}_{\infty}$ of local permutations and we give the explicit
form of the associated cocycle. This provides a family of non–trivial examples
of strongly quasi–invariant states for locally compact groups obtained as
inductive limit of an increasing sequence of compact groups whose structure is
formally identical to the one deduced in the compact case. If also for groups
in this larger class this structure characterizes all possible strongly
quasi–invariant states, is an open problem, however we prove that for the
group $\mathcal{S}_{\infty}$, the one originally considered in de Finetti’s
paper, this characterization holds in the classical (commutative) case.

Luigi Accardi

Volterra Center, University of Roma Tor Vergata

Via Columbia 2, 00133 Roma, Italy

<EMAIL_ADDRESS>
Ameur Dhahri

Dipartimento di Matematica, Politecnico di Milano

Piazza Leonardo da Vinci 32, I-20133 Milano, Italy

<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Quasi–invariant states under a group of normal $*$-automorphisms
    1. 2.0.1 The case $\mathcal{A}=\mathcal{B}(\mathcal{H})$
  3. 3 Structure of strongly quasi–invariant states
    1. 3.1  The case of compact $G$
    2. 3.2 Normalized left multiplicative $G$–$1$–co–cycles on unimodular locally compact groups
    3. 3.3 The case of a unimodular locally compact group
    4. 3.4 Inductive limits of compact groups
  4. 4 Unitary representations associated to strongly quasi–invariant states
  5. 5 Examples of quasi–invariant states under permutations
    1. 5.1 Product states
      1. 5.1.1 Generic non–$\mathcal{S}_{\infty}$–integrability in norm of the cocycle $x_{g}$
    2. 5.2 Markov Chains with commuting conditional density amplitudes
  6. 6 A martingale convergence theorem
  7. 7 de Finetti’s Theorem for quasi–invariant states: classical case
    1. 7.1 Some general conditions
    2. 7.2 Structure of the limit conditional expectation: classical case
    3. 7.3  Algebraic extension of Moy’s theorem

##  1 Introduction

Given a normal faithful state on von Neumann algebra $\mathcal{A}$ and a group
$G$ of normal $*$-automorphisms of $\mathcal{A}$, we say that $\varphi$ is
$G$–invariant if

$\varphi(g(a))=\varphi(a),\quad\forall g\in G,\;\;a\in\mathcal{A}$ (1.1)

This class of states is of paramount importance in many branches of
mathematics and physics and it has been studied in different contexts (cf [4],
[5], [12], [13], [17], [18]…). In this paper, motivated by the analogy with
the classical case, we extend definition (1.1) as follows: we assume that for
every $g\in G$ there exists $x_{g}\in\mathcal{A}$ such that

$\varphi(g(x))=\varphi(x_{g}a)$ (1.2)

and we call $G$–quasi–invariant a state satisfying (1.2). Moreover, if in
addition $x_{g}=x_{g}^{*}$ for every $g\in G$, the state $\varphi$ is said to
be $G$–strongly quasi–invariant. In this case we prove that the Radon–Nikodym
derivatives $x_{g}$’s are strictly positive operators and satisfy the
normalized left–$G$–$1$–cocycle property

$x_{g_{2}g_{1}}=x_{g_{1}}(g_{1}^{-1}(x_{g_{2}})),\quad x_{e}=1$

Moreover, we show that the $C^{*}$-algebra generated by the Radon–Nikodym
derivatives $x_{g}$’s is commutative. We further study the case of a
unimodular locally compact group $G$. In particular, we give the structure of
a $G$–strongly quasi–invariant state $\varphi$, when $G$ is a compact group,
and we prove that it has the following form

$\varphi(a)=\varphi_{G}(\kappa_{G}^{-1}a),\quad\forall a\in\mathcal{A},$ (1.3)

for some bounded strictly positive operator $\kappa_{G}$ in the centraliser of
$\varphi$ whose structure is explicitly given (see Theorem 1). Conversely, if
$\varphi_{G}$ is a $G$–invariant state and $\kappa_{G}$ is an invertible
positive operator in the centraliser of $\varphi$, then the state $\varphi$,
defined by the right hand side of (1.3), is a $G$–quasi–invariant state with
cocycle

$x_{g}=\kappa_{G}g^{-1}(\kappa_{G}^{-1}),\quad\forall g\in G$ (1.4)

We study the inductive limit associated to a uni–modular group $G$ which is
the union of an increasing sequence of compact groups. For any $G$–strongly
quasi–invariant state $\varphi$, the unitary representation of the group $G$
is also given. We finally give some examples and we investigate the structure
of a $\mathcal{S}_{\infty}$–quasi–invariant state $\varphi$, where
$\mathcal{S}_{\infty}=\cup_{N=1}^{\infty}\mathcal{S}_{N}$ is the group of
local permutations on an infinite tensor product of a von Neumann algebra
($\mathcal{S}_{N}$ is the group of permutations on $\\{1,2,\dots,N\\}$).

Finally we prove a martingale convergence theorem. Moreover, in the
commutative case, we prove a de Finetti’s type Theorem for quasi-invariant
states - a result which seems to be new.

##  2 Quasi–invariant states under a group of normal $*$-automorphisms

In the following $\mathcal{A}$ will denote a von Neumann algebra, $\varphi$ a
faithful normal state on $\mathcal{A}$ and $G\subseteq\hbox{Aut}(\mathcal{A})$
a group of normal $*$–automorphisms of $\mathcal{A}$.

###### Proposition 1

Suppose that, for any $g\in G$, there exists a $x_{g}\in\mathcal{A}$ such
that:

$\varphi(g(a))=\varphi(x_{g}a)\ ;\qquad\forall\,a\in\mathcal{A}$ (2.1)

Then the map $g\in G\mapsto x_{g}\in\mathcal{A}$ is a normalized
multiplicative left $G$–$1$–co–cycle, i.e. it satisfies the identities

$x_{e}=1$ (2.2)
$x_{g_{2}g_{1}}=x_{g_{1}}(g_{1}^{-1}(x_{g_{2}}))\quad,\quad\forall
g_{2},g_{1}\in G$ (2.3)

In particular each $x_{g}$ is invertible and its inverse is

$x_{g}^{-1}=g^{-1}(x_{g^{-1}})\left(\iff x_{g^{-1}}=g(x_{g}^{-1})\right)$
(2.4)

Proof. One has

$\varphi(a)=\varphi(e(a))=\varphi(x_{e}a)\ ;\qquad\forall\,a\in\mathcal{A}$

Since $\varphi$ is faithful, (2.2) follows. Now for any $g_{2},g_{1}\in G$,
one has

$\varphi(g_{2}g_{1}(a))=\varphi(x_{g_{2}g_{1}}a)\\\
=\varphi(x_{g_{2}}(g_{1}(a)))\\\ =\varphi(g_{1}(g_{1}^{-1}(x_{g_{2}})a))\\\ $
$=\varphi(x_{g_{1}}(g_{1}^{-1}(x_{g_{2}})a))\\\
=\varphi((x_{g_{1}}g_{1}^{-1}(x_{g_{2}}))a)$

This is equivalent to

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\varphi(x_{g_{2}g_{1}}a)-\varphi(x_{g_{1}}(g_{1}^{-1}(x_{g_{2}}))a)$
$\displaystyle=$
$\displaystyle\varphi((x_{g_{2}g_{1}}-x_{g_{1}}(g_{1}^{-1}(x_{g_{2}})))a)$

Since $a$ is arbitrary, one can choose
$a=(x_{g_{2}g_{1}}-x_{g_{1}}(g_{1}^{-1}(x_{g_{2}})))^{*}$. This gives

$0=\varphi(|(x_{g_{2}g_{1}}-x_{g_{1}}(g_{1}^{-1}(x_{g_{2}})))^{*}|^{2})$

Since $\varphi$ is faithful, (2.3) follows. Taking $g_{2}:=g_{1}^{-1}$ in
(2.3), one finds, using $x_{e}=1$,

$1=x_{e}=x_{g_{1}}(g_{1}^{-1}(x_{g_{1}^{-1}}))$

This implies that $x_{g_{1}}$ is invertible and its inverse is

$x_{g_{1}}^{-1}=g_{1}^{-1}(x_{g_{1}^{-1}})$

since $g_{1}$ is arbitrary, the first identity in (2.4) follows. Multiplying
both sides of it by $g_{1}$, one obtains

$g_{1}\left(x_{g_{1}}^{-1}\right)=x_{g_{1}^{-1}}$

which is equivalent to the second identity in (2.4). $\qquad\square$
The following result is known. We include a simple proof of it for
completeness.

###### Lemma 1

Suppose that

$x=g(y)$

with $x$ and $y$ are selfadjoint, invertible in $\mathcal{A}$. Then

$x^{s}=g(y^{s})\quad,\quad\forall s\in\mathbb{R}$ (2.5)

Proof.

$1=g(yy^{-1})=g(y)g(y^{-1})$

Therefore $g(y)$ is invertible and

$g(y)^{-1}=g(y^{-1})=x^{-1}$ (2.6)

Thus

$x^{s}=g(y^{s})\quad,\quad\forall s\in\mathbb{Z}$

For $n\in\mathbb{N}$ one has

$g(y^{\frac{1}{n}})^{n}=g((y^{\frac{1}{n}})^{n})=g(y)=x$

Therefore

$x^{\frac{1}{n}}=g(y^{\frac{1}{n}})\quad;\quad\forall n\in\mathbb{N}$

Consequently

$x^{\frac{m}{n}}=(x^{\frac{1}{n}})^{m}=g(y^{\frac{1}{n}})^{m}=g(y^{\frac{m}{n}})$

and, because of (2.6), $x^{s}=g(y^{s})$ for all $s\in\mathbb{Q}$. By
continuity

$x^{s}=g(y^{s})\quad,\quad\forall s\in\mathbb{R}$

$\square$

###### Corollary 1

If $x_{g}=x_{g}^{*}$, then for any $s\in\mathbb{R}$,

$x^{-s}_{g}=g^{-1}(x^{s}_{g^{-1}})$ (2.7)

Proof. Replacing in (2.5) $g$ by $g^{-1},x$ by $x_{g}^{-1}$ and $y$ by
$x_{g^{-1}}$, one finds

$x_{g}^{-s}=g^{-1}(x_{g^{-1}}^{s})$

which is (2.7). $\square$

###### Definition 1

If the pair $(G,x)$ satisfies condition (2.1) of Proposition 1 the state
$\varphi$ is called $(G,x)$–quasi–invariant, or quasi–invariant with cocycle
$x$.

###### Lemma 2

If $\varphi$ is $(G,x)$–quasi–invariant, then for all $g\in G$

$a\in\mathcal{A}\ ,\ a\geq 0\Rightarrow\varphi(x_{g}a)\geq 0\quad,\quad
1=\varphi(x_{g})$ (2.8) $\varphi(x_{g}a)=\varphi(ax_{g}^{*})\
;\qquad\forall\,a\in\mathcal{A}$ (2.9)

Proof. If $a\in\mathcal{A}\ ,\ a\geq 0$ and $g\in G$, one has
$0\leq\varphi(g(a))=\varphi(x_{g}a)$ and this proves the inequality in (2.8).
The equality follows from

$1=\varphi(1)=\varphi(g(1))=\varphi(x_{g})\ ;\qquad\forall\,g\in G$

Moreover one has

$\varphi(g(a^{*}))=\varphi(x_{g}a^{*})\ ;\qquad\forall\,a\in\mathcal{A}$

On the other hand

$\varphi(g(a^{*}))=\varphi((g(a))^{*})=\overline{\varphi(g(a))}=\overline{\varphi(x_{g}a)}=\varphi((x_{g}a)^{*})=\varphi(a^{*}x_{g}^{*})$

It follows that

$\varphi(x_{g}a^{*})=\varphi(a^{*}x_{g}^{*})\ ;\qquad\forall\,a\in\mathcal{A}$

and, since $a\in\mathcal{A}$ is arbitrary, this is equivalent to (2.9).
$\qquad\square$

###### Remark 1

Recall that the  centralizer of $\varphi$, hereinafter denoted
$\hbox{Centrz}(\varphi)$, is characterized by the property

$c\in\
\hbox{Centrz}(\varphi)\Leftrightarrow\varphi(ac)=\varphi(ca)\quad;\quad\forall\,a\in\mathcal{A}$
(2.10)

###### Lemma 3

If $\varphi$ is $(G,x)$–quasi–invariant, for any $g\in G$ the following are
equivalent:

  1. (i)

$x_{g}$ is a hermitean element of $\mathcal{A}$

  2. (ii)

$x_{g}$ is in the centralizer of $\varphi$.

Proof. (i) $\Rightarrow$ (ii). If $x_{g}=x_{g}^{*}$, (ii) follows from (2.9).
(ii) $\Rightarrow$ (i). If $x_{g}$ is in the centralizer of $\varphi$ then the
same is true for $x_{g}^{*}$. Therefore (2.9) implies that for all
$a\in\mathcal{A}$,

$\varphi(x_{g}a)=\varphi(ax_{g}^{*})=\varphi(x_{g}^{*}a)\iff\varphi((x_{g}-x_{g}^{*})a)=0\overset{a:=(x_{g}-x_{g}^{*})^{*}}{\Rightarrow}\
\varphi(|x_{g}-x_{g}^{*}|^{2})=0$

and (i) follows because $\varphi$ is faithful. If either (i) or (ii) holds,
then $x_{g}=x_{g}^{*}=x_{g,+}-x_{g,-}$, with $x_{g,\pm}$ are positive such
$x_{g,+}x_{g,-}=0$. Therefore, choosing in (2.9) $a=x_{g,-}$, one finds

$0\leq\varphi(x_{g}x_{g,-})=\varphi((x_{g,+}-x_{g,-})x_{g,-})=-\varphi(x_{g,-}^{2})\iff
0=\varphi(x_{g,-}^{2})\iff 0=x_{g,-}$

Thus $x_{g}=x_{g}^{*}=x_{g,+}\geq 0$. Since, by Proposition 1, $x_{g}$ is
invertible in $\mathcal{A}$, there exist $0<s<S<+\infty$ such that (2.13)
holds. $\square$

###### Definition 2

A $(G,x)$–quasi–invariant state $\varphi$ is called $(G,x)$–strongly
quasi–invariant if, for any $g\in G$, $x_{g}$ is hermitean.

###### Lemma 4

If $\varphi$ is $(G,x)$–strongly quasi–invariant, $x$ satisfies

$x_{g_{1}}(g_{1}^{-1}(x_{g_{2}}))=(g_{1}^{-1}(x_{g_{2}}))x_{g_{1}}\quad,\quad\forall
g_{2},g_{1}\in G$ (2.11)

which is equivalent to

$x_{g_{2}}x_{g_{1}}=x_{g_{1}}x_{g_{2}}\quad,\quad\forall g_{2},g_{1}\in G$
(2.12)

Moreover $x_{g}$ is strictly positive with bounded inverse, i.e. there exist
$0<s<S<+\infty$ such that

$\hbox{Spec}(x_{g})\subset[s,S]$ (2.13)

Proof. From the cocycle identity (2.3) and the hermiteanity of the $x_{g}$,
one has, for all $g_{2},g_{1}\in G$,

$x_{g_{1}}(g_{1}^{-1}(x_{g_{2}}))=x_{g_{2}g_{1}}=x_{g_{2}g_{1}}^{*}=(g_{1}^{-1}(x_{g_{2}}))x_{g_{1}}$

which is (2.11). (2.11) is equivalent to

$(g_{1}^{-1}(x_{g_{2}}))x_{g_{1}}^{-1}=x_{g_{1}}^{-1}(g_{1}^{-1}(x_{g_{2}}))\overset{\eqref{xg-1}}{\iff}(g_{1}^{-1}(x_{g_{2}}))g_{1}^{-1}(x_{g_{1}})=g_{1}^{-1}(x_{g_{1}})(g_{1}^{-1}(x_{g_{2}}))$
$\iff g_{1}^{-1}(x_{g_{2}}x_{g_{1}})=g_{1}^{-1}(x_{g_{1}}x_{g_{2}})\iff
x_{g_{2}}x_{g_{1}}=x_{g_{1}}x_{g_{2}}$

which is (2.12).
To prove (2.13) notice that the $C^{*}$–algebra $\mathcal{C}$ generated by the
$x_{g}\;(g\in G)$ is abelian because of (2.12) and contains the identity of
$\mathcal{A}$ ($1_{\mathcal{A}}=x_{e}$). Therefore, by Gelfand Theorem ((see
[16], section 1.1.9) $\mathcal{C}$ can be identified to the algebra

$\mathcal{C}_{\mathbb{C}}(S):=\\{\hbox{ continuous complex valued functions on
a compact space $S$}\\}$ (2.14)

Since $S$ is compact, in this identification, the $x_{g}$ become strictly
positive functions on a compact set. Hence for all $g\in G$ there exist
$0<s<S<+\infty$ (depending on $g$) such that (2.13) holds. $\square$

###### Lemma 5

If $\varphi$ is $(G,x)$–strongly quasi–invariant then $\forall g\in G$ and
$a\in\mathcal{A}$

$\varphi(g(x)a)=\varphi(ag(x_{g}xx_{g}^{-1}))$ (2.15)

In particular, for $\mathcal{C}$ as in (2.14)

$g(\mathcal{C})\subseteq\hbox{Centrz}(\varphi)\qquad,\quad\forall g\in G$
(2.16)

Proof. In the above notations, one has

$\varphi(g(x)a)=\varphi(g(xg^{-1}(a)))=\varphi(x_{g}xg^{-1}(a))$
$=\varphi(g^{-1}(a)x_{g}x)=\varphi(g^{-1}(ag(x_{g}x)))=\varphi(x_{g^{-1}}ag(x_{g})g(x))=\varphi(ag(x_{g})g(x)x_{g^{-1}})$
$\overset{\eqref{xg-1}}{=}\varphi(ag(x_{g})g(x)g(x_{g}^{-1}))=\varphi(ag(x_{g}xx_{g}^{-1}))$

which is (2.15). (2.16) follows from (2.15) because,
$\mathcal{C}\subseteq\hbox{Centrz}(\varphi)$ and, if $x\in\mathcal{C}$, then
it is known that $x^{-1}\in\mathcal{C}$ (in fact, if
$x\in\hbox{Centrz}(\varphi)$ is invertible,
$\varphi(x^{-1}xa)=\varphi(ax^{-1}x)=\varphi(xax^{-1})$ and, since the map
$a\in\mathcal{A}\mapsto xa$ is invertible, it follows that, if
$a\in\mathcal{A}$ is arbitrary, so it is $xa$). Therefore
$\varphi(ax_{g}xx_{g}^{-1})=\varphi(x_{g}xx_{g}^{-1}a)$ for any $g\in G$.
$\qquad\square$

####  2.0.1 The case $\mathcal{A}=\mathcal{B}(\mathcal{H})$

The following Proposition shows that, if
$\mathcal{A}=\mathcal{B}(\mathcal{H})$, there exist both
$G$–quasi–invariant states and $G$–strongly quasi–invariant states.

###### Proposition 2

If $\mathcal{A}=\mathcal{B}(\mathcal{H})$ and $\varphi=Tr(W\,\cdot\,)$ where
$W$ is a density operator, then $x_{g}$ is a solution of equation (2.9) if and
only if

$Wx_{g}=x_{g}^{*}W$ (2.17)

and the solutions of equation (2.17) are given by the set

$S_{W}:=\\{W^{-1}z\colon z\in\mathcal{B}(\mathcal{H})\quad,\quad z=z^{*}\\}$
(2.18)

Proof. In the assumptions of the Proposition, equation (2.9), i.e.

$\varphi(x_{g}a)=\varphi(ax_{g}^{*})\ ;\qquad\forall\,a\in\mathcal{A}$ (2.19)

becomes equivalent to

$Tr(Wx_{g}a)=Tr(Wax_{g}^{*})=Tr(x_{g}^{*}Wa)\quad;\quad\forall a$ $\iff
Wx_{g}=x_{g}^{*}W$

which is (2.17). If $x$ is a solution of equation (2.17), defining

$z:=Wx$

one has

$z^{*}=(Wx)^{*}=x^{*}W\overset{\eqref{(Wxg=xg*W)}}{=}Wx=z$

thus $x=W^{-1}z\in S_{W}$. Conversely, if $x\in S_{W}$, then $x=W^{-1}z$ for
some $z=z^{*}$.
Therefore

$Wx=W(W^{-1}z)=z=zW^{-1}W=(W^{-1}z)^{*}W=x^{*}W$

i.e. $x$ is a solution of (2.17). $\square$

###### Remark 2

1) If $z=z^{*}\in\mathcal{B}(\mathcal{H})$ commutes with $W$, then it commutes
with $W^{-1}$ and therefore, if $x:=W^{-1}z$, one has

$x^{*}=zW^{-1}=W^{-1}z=x$

In other words, equation (2.17) always admits hermitean solutions when
$\mathcal{A}=\mathcal{B}(\mathcal{H})$.
2) From Proposition 2, one also deduces that the family of strong
quasi–invariant states is strictly smaller than the family of quasi–invariant
states. In fact, let

$\mathcal{G}=\\{g_{0}^{n}:n\in\mathbb{N}\\}\quad;\quad g^{0}_{0}:=e\neq g_{0}$

be a group of $*$–automorphism of $\mathcal{B}(\mathcal{H})$ with a single
generator $g_{0}\neq e$.
Given $W$, fix $z_{g_{0}}=z_{g_{0}}^{*}$ subject to the only condition

$[z_{g_{0}},W]\neq 0$ (2.20)

(clearly, for any $W$, there are many $z_{g_{0}}$ satisfying (2.20)) and
define

$x_{g_{0}}:=W^{-1}z_{g_{0}}$ (2.21)

The identity

$x_{g_{0}^{n}}=x_{g_{0}}g_{0}^{-1}(x_{g_{0}})\cdots
g_{0}^{-(n-1)}(x_{g_{0}})\quad,\quad n\in\mathbb{N}$

easily verified by induction on the cocycle equation, shows that (2.21)
uniquely determines $x_{g_{0}^{n}}$ for any $n\in\mathbb{N}$, i.e. the cocycle
$x:\mathcal{G}\rightarrow\mathcal{A}$. On the other hand condition (2.20)
means that $W^{-1}z_{g_{0}}$ is not in the centralizer of
$\varphi=\hbox{Tr}(W\,\cdot\,)$. Hence it is not hermitian by Lemma 3.

##  3 Structure of strongly quasi–invariant states

###  3.1  The case of compact $G$

From now on we will assume that the action of the group $G$, $g\in G\mapsto
x_{g}\in\mathcal{C}$, is continuous.

In this section we assume that $G$ is a compact group of normal
$*$–automorphisms of $\hbox{Aut}(\mathcal{A})$ with normalized Haar measure
$dg$.

###### Theorem 1

In the above assumptions the element

$\kappa:=\int_{G}x_{g}dg\in\mathcal{C}\subseteq\hbox{Centrz}(\varphi)\subseteq\mathcal{A}$
(3.1)

$\kappa$ is an invertible contraction with bounded inverse (hence
$\kappa^{-1}$ belongs to $\mathcal{C}$).

Proof. Using the functional realization of the algebra $\mathcal{C}$, we
realize the $x^{\prime}_{g}s$ as continuous strictly positive functions on the
compact Hausdorff space $S$. Then each function $s\in S\mapsto x_{g}(s)$ is
bounded away from zero and the continuity of the map $g\in G\mapsto
x_{g}\in\mathcal{C}$

$\kappa(s)=\int_{G}x_{g}(s)dg>0\qquad,\qquad\forall s\in S$

implies that $\kappa$ is continuous and bounded away from zero. Hence
$\kappa^{-1}$ is bounded strictly positive and $\kappa^{-1}$ belongs to
$\mathcal{C}$. $\square$

###### Theorem 2

Define

$E_{G}=\int_{G}gdg$ (3.2)

Then

  1. (i)

$E_{G}$ is a $*$–map from $\mathcal{A}$ to $\mathcal{A}$.

  2. (ii)

For any $g\in G$, $E_{G}$ satisfies the following identity

$gE_{G}=E_{G}g=E_{G}$ (3.3) $E_{G}(g(a))=gE_{G}(a)=E_{G}(a)$ (3.4)

In particular

$\hbox{Range}(E_{G})=\hbox{Fix}(G):=\\{a\in\mathcal{A}:\;\;\;g(a)=a\>\>\forall
g\in G\\}$ (3.5)

and $E_{G}(\mathcal{A})$ is a $W^{*}$–sub–algebra of $\mathcal{A}$.

  3. (iii)

$E_{G}$ is a Umegaki conditional expectation from $\mathcal{A}$ onto
$E_{G}(\mathcal{A})=\hbox{Fix}(G)$ ($=:$ the fixed points of $G$).

Proof. For each $a\in\mathcal{A}$

$\mathcal{A}\ni(E_{G}(a))^{*}=\left(\int_{G}g(a)dg\right)^{*}=\int_{G}g(a)^{*}dg=\int_{G}g(a^{*})dg=E_{G}(a^{*})$

Therefore $E_{G}(\mathcal{A})$ is a $*$–sub–space of $\mathcal{A}$. This
proves (i).
Left–invariance of the Haar measure implies that, for each $a\in\mathcal{A}$
and $g\in G$,

$gE_{G}(a)=\int_{G}gh(a)dh=\int_{G}gh(a)d(gh)=\int_{G}g^{\prime}(a)dg^{\prime}=E_{G}(a)$

Therefore $\hbox{Range}(E_{G})\subseteq\hbox{Fix}(G)$. The converse inclusion
is clear because, if $a\in\hbox{Fix}(G)$, then

$a=\int_{G}g(a)dg=E_{G}(a)$

So the second identity in (3.4) holds and it implies (3.5). Similarly,
right–invariance of the Haar measure of $G$ implies that, for each
$a\in\mathcal{A}$ and $g\in G$,

$E_{G}(g(a))=\int_{G}hg(a)dh=\int_{G}hg(a)dhg=\int_{G}g^{\prime}(a)dg^{\prime}=E_{G}(a)$

Therefore the first identity in (3.4) holds. This proves (ii).
To prove (iii) notice that $E_{G}$ is completely positive being a convex
combination of automorphisms and identity preserving because
$g(1_{\mathcal{A}})=1_{\mathcal{A}}$ for any $g\in G$. Moreover, for any
$a\in\mathcal{A}$, one has

$E_{G}^{2}(a)=\int_{G}dg\,g(E_{G}(a))\overset{\eqref{EG-
commutes-G2}}{=}\int_{G}dg\,E_{G}(a)=E_{G}(a)$

due to the normalization of the Haar measure of $G$. So $E_{G}$ is a
completely positive norm–$1$ projector on $\mathcal{A}$ and, by Tomijama’s
theorem [20], $E_{G}(\mathcal{A})$ is a sub–$W^{*}$–algebra of $\mathcal{A}$
and $E_{G}$ is a Umegaki conditional expectation from $\mathcal{A}$ to
$E_{G}(\mathcal{A})$. This proves (iii). $\qquad\square$

The structure of strongly quasi–invariant states with respect to compact
groups is described by the following theorem.

###### Theorem 3

Let $G$ be a compact group of normal $*$–automorphisms of $\mathcal{A}$ and
let $\varphi$ be a $G$–strongly quasi–invariant faithful normal state on
$\mathcal{A}$. Denote $x:g\in G\to x_{g}\in\mathcal{A}$ the (continuous,
hermitean) left–$G$–$1$–cocycle associated to $\varphi$ and let $\kappa$ be as
in Theorem 1, namely

$\kappa=\int_{G}x_{g}dg$ (3.6)

Then $\varphi$ can be written in the form

$\varphi(a)=\left(\varphi\circ
E_{G}\right)(\kappa^{-1}a)=:\varphi_{G}(\kappa^{-1}a)\quad;\quad\forall\,a\in\mathcal{A}$
(3.7)

In particular $\varphi_{G}$ is a faithful $G$–invariant state and

$x_{g}=\kappa g^{-1}(\kappa^{-1})$ (3.8)

Moreover

$E_{G}(\kappa^{-1})=1$ (3.9)

Conversely, if $G$ is any group (not necessarily locally compact),
$\varphi_{G}$ is a $G$–invariant state on $\mathcal{A}$ and $X^{-1}$ is any
element of $\mathcal{A}$, invertible with inverse in $\mathcal{A}$, then

$\varphi(\,\cdot\,):=\varphi_{G}(X\,\cdot\,)$ (3.10)

is a $(G,x)$–quasi-invariant state with cocycle

$x_{g}:=X^{-1}g^{-1}(X)$ (3.11)

If $G$ is compact, $X=\kappa^{-1}$ where $\kappa$ is given by (3.6) with
$x_{g}$ given by (3.11).

Proof. If $\varphi$ is a $(G,x_{.})$ strongly–quasi–invariant state, define
$\varphi_{G}:=\varphi\circ E_{G}$. Since $E_{G}g\overset{\eqref{EG-
commutes-G}}{=}E_{G}$ for every $g\in G$, the state $\varphi_{G}$ is
$G$–invariant. From Theorem 1, one knows that $\kappa$ is an invertible
element of $\mathcal{A}$ with inverse in $\mathcal{A}$. Moreover, for any
$a\in\mathcal{A}$ one has:

$\varphi_{G}(\kappa^{-1}a)=\varphi(E_{G}(\kappa^{-1}a))=\int_{G}dg\varphi(g(\kappa^{-1}a))=\int_{G}dg\varphi(x_{g}\kappa^{-1}a)$
$=\varphi(\kappa\kappa^{-1}a)=\varphi(a)$

which is (3.7). In particular

$\varphi(x_{g}a)=\varphi(g(a))\overset{\eqref{struc-q-inv-st-
cpct}}{=}\varphi_{G}(\kappa^{-1}g(a))=\varphi_{G}(g(\kappa^{-1})a)$
$=\varphi_{G}(\kappa^{-1}\kappa g(\kappa^{-1})a)=\varphi(\kappa
g(\kappa^{-1})a)$

and (3.8) follows because $\varphi$ is faithful. $\varphi_{G}$ is also
faithful because, if $0\neq a$ is a positive element in $\mathcal{A}$ such
that $\varphi_{G}(a)=0$, then

$0\leq\varphi(a^{1/2})\overset{\eqref{struc-q-inv-st-
cpct}}{=}\varphi_{G}(\kappa^{-1}a^{1/2})\leq\varphi_{G}(\kappa^{-2})\varphi_{G}(a)=0$

against the assumption that $\varphi$ is faithful. Since the Haar measure on a
unimodular group (in particular on a compact group) is invariant by inversion,
one gets

$E_{G}(\kappa^{-1})=\int_{G}g(\kappa^{-1})dg\overset{\eqref{constr--
xg}}{=}\int_{G}\kappa^{-1}x_{g^{-1}}dg=\kappa^{-1}\int_{G}x_{g^{-1}}dg=\kappa^{-1}\kappa=1$

which is (3.9). Conversely, if $\varphi_{G}$, $\varphi$ and $X^{-1}$ are as in
the statement, then for any $a\in\mathcal{A}$ and $g\in G$, one has

$\varphi(g(a))=\varphi_{G}(Xg(a))=\varphi_{G}(g[g^{-1}(X)a])=\varphi_{G}([g^{-1}(X)a])$
$=\varphi_{G}(XX^{-1}[g^{-1}(X)a])=\varphi((X^{-1}g^{-1}(X)a)$

Therefore $\varphi$ is $(G,x)$–quasi–invariant with cocycle (3.11).
From Definition 2, $\varphi$ is $(G,x)$–strongly quasi–invariant iff each
$x_{g}$ is hermitean. Finaly, the last statement follows from the first part
of the theorem. $\square$

###  3.2 Normalized left multiplicative $G$–$1$–co–cycles on unimodular
locally compact groups

###### Lemma 6

A map of the form

$g\in G\mapsto x_{g}=\kappa g^{-1}\left(\kappa^{-1}\right)$ (3.12)

for some $\kappa\in\mathcal{A}$ is a normalized left multiplicative
$G$–$1$–co–cycle.

Proof. Condition (2.2) follows from

$x_{e}=\kappa\kappa^{-1}=1$

Condition (2.3) follows from

$x_{g_{2}g_{1}}=\kappa(g_{2}g_{1})^{-1}\left(\kappa^{-1}\right)=\kappa
g_{1}^{-1}g_{2}^{-1}\left(\kappa^{-1}\right)=\kappa
g_{1}^{-1}\left(\kappa^{-1}\left(\kappa
g_{2}^{-1}\left(\kappa^{-1}\right)\right)\right)$ $=\kappa
g_{1}^{-1}(\kappa^{-1})g_{1}^{-1}\left(\kappa
g_{2}^{-1}\left(\kappa^{-1}\right)\right)=x_{g_{1}}g_{1}^{-1}(x_{g_{2}})\quad,\quad\forall
g_{2},g_{1}\in G$

$\square$

###### Definition 3

Let $G$ be a group. A multiplicative left $G$–$1$–co–cycle of the form (3.12)
is called trivial. If $G$ is a locally compact group, a multiplicative left
$G$–$1$–co–cycle is called integrable if the integral

$\int_{G}dgx_{g}$

exists where $dg$ is the right Haar measure on $G$.

###### Proposition 3

On a unimodular locally compact group $G$, any normalized integrable,
invertible, multiplicative left $G$–$1$–co–cycle $x$ is trivial.

Proof. Integrability for the right Haar measure on $G$ and (2.2) imply that,
for any $g_{1}\in G$,

$\int_{G}dg_{2}x_{g_{2}g_{1}}=\int_{G}d(g_{2}g_{1})x_{g_{2}g_{1}}=\int_{G}dgx_{g}=:\kappa$

Therefore

$\kappa\overset{\eqref{xg-1G-left-
cocy}}{=}\int_{G}dg_{2}x_{g_{1}}(g_{1}^{-1}(x_{g_{2}}))=x_{g_{1}}g_{1}^{-1}\left(\int_{G}dg_{2}x_{g_{2}}\right)=x_{g_{1}}g_{1}^{-1}\left(\kappa\right)$
(3.13)

The invertibility of $\kappa$ implies that (3.13) is equivalent to

$\kappa g_{1}^{-1}\left(\kappa^{-1}\right)=x_{g_{1}}$ (3.14)

Since $g_{1}\in G$ is arbitrary, the thesis follows. $\qquad\square$

###  3.3 The case of a unimodular locally compact group

In the following, we will assume that $G$ is unimodular locally compact group
of normal $*$–automorphisms of $\mathcal{A}$ with Haar measure denoted
$\mu_{G}$ or simply $dg$ and we will assume that $\varphi$ is $(G,x)$–strongly
quasi–invariant. Note that compact and discrete groups are particular cases of
unimodular locally compact groups.
Since the results discussed in Section 2 are valid for any group of normal
$*$–automorphisms of $\mathcal{A}$, $\varphi$ defines a cocycle $x_{g}$ ($g\in
G$) enjoying all the properties discussed in that section.

If the Haar measure of $G$ is not finite (i.e. $G$ is not compact), the
integral

$E_{G}=\int_{G}gdg$ (3.15)

with domain

$\hbox{Dom}(E_{G}):=\left\\{a\in\mathcal{A}\colon\int_{G}g(a)dg\in\mathcal{A}\right\\}$
(3.16)

is not a Umegaki conditional expectation, but an operator valued weight in the
sense of Haagerup [10], [11] with values in $\hbox{Fix}(G)_{+}$. This means
that,$E_{G}$ satisfies the following conditions:

$E_{G}(\lambda x)=\lambda E_{G}(x)\quad,\quad\lambda\geq 0\ ,\
x\in\mathcal{A}_{+}$ $E_{G}(x+y)=E_{G}(x)+E_{G}(y)\quad,\quad
x,y\in\mathcal{A}_{+}$ $E_{G}(a^{*}xa)=a^{*}E_{G}(x)a\quad,\quad
x\in\mathcal{A}_{+}\ ,\ a\in\hbox{Range}(E_{G})$ (3.17)

It is known (see [8], Lemma 1.10) that, denoting

$\hbox{Dom}^{+}(E_{G}):=\\{x\in\mathcal{A}_{+}\colon\|E_{G}(x)\|<+\infty\\}$
$\hbox{Dom}^{1/2}(E_{G}):=\\{x\in\mathcal{A}\colon\|E_{G}(x^{*}x)\|<+\infty\\}$
$=\left\\{\sum_{i=1}^{n}x_{i}^{*}y_{i}\colon
x_{i},y_{i}\in\hbox{Dom}^{1/2}(E_{G})\ ,\ n\in\mathbb{N}\right\\}$
$\hbox{Dom}(E_{G}):=\left(\hbox{Dom}^{1/2}(E_{G})\right)^{*}\hbox{Dom}^{1/2}(E_{G})$

  1. (i)

$\hbox{Dom}(E_{G})$ is linearly spanned by $\hbox{Dom}^{+}(E_{G})$.

  2. (ii)

$\hbox{Dom}(E_{G})$ and $\hbox{Dom}^{1/2}(E_{G})$ are two sided modules over
$\hbox{Range}(E_{G})$.

  3. (iii)

$E_{G}$ has a unique linear extension, still denoted
$E_{G}\colon\hbox{Dom}(E_{G})\to\hbox{Range}(E_{G})$ and it is a module map,
i.e.

$E_{G}(axb)=aE_{G}(x)b\quad,\quad a,b\in\hbox{Range}(E_{G})\ ,\
x\in\hbox{Dom}(E_{G})$ (3.18)

###### Lemma 7

Let $y:g\in G\to y_{g}\in\mathcal{A}$ be the left multiplicative
$G$–$1$–cocycle associated to $\varphi$ and suppose that $y_{g}$ is integrable
for some topology on $\mathcal{A}$ and that its integral

$\kappa:=\int_{G}y_{g}dg\in\mathcal{A}$ (3.19)

exists in $\mathcal{A}$. Then $\kappa$ is an invertible element of
$\mathcal{A}$ and

$\displaystyle y_{g}$ $\displaystyle=$ $\displaystyle\kappa
g^{-1}(\kappa^{-1})\quad,\quad\forall\,g\in G$ (3.20) $\displaystyle
E_{G}(\kappa^{-1})$ $\displaystyle=1$ (3.21)

Conversely, if $\kappa$ is any invertible element of $\mathcal{A}$, then the
map

$g\in G\mapsto y_{g}:=\kappa g^{-1}(\kappa^{-1})\in\mathcal{A}$ (3.22)

is a left multiplicative $G$–$1$–cocycle, integrable if and only if
$\kappa^{-1}\in\hbox{Dom}(E_{G})$ and hermitean if $\kappa$ is hermitean and

$\kappa g^{-1}(\kappa^{-1})=g^{-1}(\kappa^{-1})\kappa$ (3.23)

Proof. If $y$ is an integrable left multiplicative $G$–$1$–cocycle with values
in $\mathcal{A}$, $\kappa$ is well defined by (3.19) and Lemma 4 and Lemma 5
imply that it is an invertible element of the centralizer of $\mathcal{A}$.
Then, since the Haar measure on $G$ is invariant under translation, for any
$g^{\prime}\in G$ one has

$\kappa=\int_{G}y_{g}dg=\int_{G}y_{gg^{\prime}}dg\overset{(\ref{xg-1G-left-
cocy})}{=}\int_{G}y_{g^{\prime}}(g^{\prime})^{-1}(y_{g})dg$
$=y_{g^{\prime}}(g^{\prime})^{-1}\left(\int_{G}y_{g}dg\right)\overset{\eqref{xg-1}}{=}y_{g^{\prime}}(g^{\prime})^{-1}(\kappa)$
(3.24)

By assumption $\kappa$ is invertible and by (2.4) $y_{g^{\prime}}$ is
invertible. Therefore, since $(g^{\prime})^{-1}$ is an automorphism, (3.20) is
equivalent to

$y_{g^{\prime}}^{-1}=(g^{\prime})^{-1}(\kappa)\kappa^{-1}\overset{\eqref{xg-1}}{\iff}(g^{\prime})^{-1}(y_{(g^{\prime})^{-1}})=(g^{\prime})^{-1}(\kappa)\kappa^{-1}$
$\iff y_{(g^{\prime})^{-1}}=\kappa\,g^{\prime}(\kappa^{-1})$ (3.25)

Since $g^{\prime}\in G$ is arbitrary one can replace $g^{\prime}$ by $g^{-1}$
and, with this replacement, (3.25) becomes (3.20). Finally, (3.20) is
equivalent to

$\kappa^{-1}=g^{-1}(\kappa^{-1})y_{g}^{-1}\iff
g(\kappa^{-1})=\kappa^{-1}g(y_{g}^{-1})\overset{\eqref{xg-1}}{=}\kappa^{-1}y_{g^{-1}}\
,\qquad\forall\,g\in G$

Now, since $G$ is unimodular locally compact group, then the Haar measure on
$G$ is invariant by inversion. Therefore, one gets

$\displaystyle\kappa=\int_{G}y_{g}dg=\kappa\int_{G}g^{-1}(\kappa^{-1})dg=\kappa\int_{G}g(\kappa^{-1})dg^{-1}=\kappa\int_{G}g(\kappa^{-1})dg=\kappa
E_{G}(\kappa^{-1})$

This proves that $\kappa^{-1}\in\mbox{Dom}(E_{G})$ and $E_{G}(\kappa^{-1})=1.$
Conversely, if $\kappa\in\mathcal{A}$ is invertible then $y_{g}$, defined by
(3.20), satisfies

$y_{g_{2}g_{1}}=\kappa(g_{2}g_{1})^{-1}(\kappa^{-1})=\kappa
g_{1}^{-1}g_{2}^{-1}(\kappa^{-1})=\kappa g_{1}^{-1}(\kappa^{-1}\kappa
g^{-1}_{2}(\kappa^{-1}))$ $=\kappa g_{1}^{-1}(\kappa^{-1})g_{1}^{-1}(\kappa
g_{2}^{-1}(\kappa^{-1}))=y_{g_{1}}g_{1}^{-1}(y_{g_{2}})$

i.e. the map $g\in G\mapsto y_{g}:=\kappa g^{-1}(\kappa^{-1})\in\mathcal{A}$
is a left–$G$–$1$–cocycle. Finally, if $\kappa$ is hermitean and (3.22) holds,
then

$y_{g}^{*}=(\kappa g^{-1}(\kappa^{-1}))^{*}=g^{-1}(\kappa^{-1})\kappa=\kappa
g^{-1}(\kappa^{-1})=y_{g}$

$\qquad\square$

###### Remark 3

From the results of the paper [8] it follows that $E_{G}$ can be further
extended to the case in which, for a large class of positive operators
$a\in\mathcal{A}$, $E_{G}(a)$ is an unbounded positive operator affiliated to
$\mathcal{A}$.
This extension should allow to weaken the assumption of bounded invertibility
in Lemma 7.

The following result shows that many of the results of Theorem 2 remain true
if restricted from $\mathcal{A}$ to $\hbox{Dom}(E_{G})$.

###### Theorem 4

  1. (i)

$E_{G}$ is a $*$–map in the sense that, if $a\in\hbox{Dom}(E_{G})$, then
$a^{*}\in\hbox{Dom}(E_{G})$ and $E_{G}(a)^{*}=E_{G}(a^{*})$. In particular
$\hbox{Dom}(E_{G})$ is a $*$–sub–space of $\mathcal{A}$.

  2. (ii)

The identities (3.3), (3.4) must be understood in the sense that these
identities holds for any $a\in\hbox{Dom}(E_{G})$ such that $g\in G$ such that
$g(a)\in\hbox{Dom}(E_{G})$. Moreover, $\hbox{Fix}(G)$ is defined by

$\hbox{Fix}(G):=\\{a\in\hbox{Dom}(E_{G})\colon\forall g\in G\ ,\
g(a)\in\hbox{Dom}(E_{G})\hbox{ and \eqref{EG-commutes-G2}}\;\;\hbox{holds}\\}$

  3. (iii)

The identity (3.18) must be understood in the sense that
$E_{G}(a)bE_{G}(c)\in\hbox{Dom}(E_{G})$ and the equality holds.

  4. (iv)

For all $a,b\in\hbox{Dom}(E_{G})$ such that $E_{G}(a)b\in\hbox{Dom}(E_{G})$,

$E_{G}(a)E_{G}(b)=E_{G}\left(E_{G}(a)b\right)$

In particular, for any $x\in\hbox{Fix}(G)$,

$xb\in\hbox{Dom}(E_{G})\ \iff\ b\in\hbox{Dom}(E_{G})$

Proof. The proofs of (i), (ii) and (iii) are similar as in Theorem 2 using the
unimodularity of $G$. Now in order to prove (iv), let
$a,b\in\hbox{Dom}(E_{G})$ such that $E_{G}(a)b\in\hbox{Dom}(E_{G})$. Then, one
has

$E_{G}(a)E_{G}(b)=\int_{G}g(a)dg\int_{G}h(b)dh=\int_{G}\int_{G}dgdh\,g(a)h(b)$
(3.26)
$=\int_{G}\int_{G}dgdh\,h(h^{-1}g(a)b)=\int_{G}dh\,h\left(h^{-1}\Big{(}\int_{G}dg\,g(a)\Big{)}b\right)$
$=\int_{G}dh\,h\left(h^{-1}(E_{G}(a))b\right)\overset{\eqref{EG-
commutes-G}}{=}\int_{G}dh\,h\left(E_{G}(a)b\right)=E_{G}\left(E_{G}(a)b\right)\in\hbox{Range}(E_{G})$

It follows that, for any $x\in\hbox{Fix}(G)$,

$xb\in\hbox{Dom}(E_{G})\ \iff\ b\in\hbox{Dom}(E_{G})$

$\qquad\square$

###### Theorem 5

Let $G$ be a unimodular locally compact group of normal $*$–automorphisms of
$\mathcal{A}$ and let $\varphi$ be a $G$–strongly quasi–invariant faithful
normal state on $\mathcal{A}$ with cocycle $x:g\in G\to x_{g}\in\mathcal{A}$.
Suppose that $x_{g}$ satisfies all the conditions of Lemma 7. Then, in the
notations of this lemma, $\varphi$ can be written in the form

$\varphi(a)=\left(\varphi\circ
E_{G}\right)(\kappa^{-1}a)=:\varphi_{G}(\kappa^{-1}a)\quad;\quad\forall\,a\in\mathcal{A}:\;\kappa^{-1}a\in\mbox{Dom}(E_{G})$
(3.27)

In particular $\varphi_{G}$ is a $G$–invariant state.
Conversely, if $\varphi_{G}$ is a $G$–invariant state on $\mathcal{A}$ and
$\kappa$ is any element of $\mathcal{A}$, invertible with inverse in
$\mathcal{A}$, then

$\varphi(\,\cdot\,):=\varphi_{G}(\kappa^{-1}\,\cdot\,)$ (3.28)

is a $(G,x)$–quasi-invariant state with cocycle

$x_{g}:=\kappa g^{-1}(\kappa^{-1})$ (3.29)

$\varphi_{G}$ is $(G,x)$–strongly quasi-invariant if and only if $x_{g}$ is
hermitean.

Proof. Define

$\varphi_{G}:=\varphi\circ E_{G}$

Since $E_{G}g\overset{\eqref{EG-commutes-G}}{=}E_{G}$ for every $g\in G$, the
state $\varphi_{G}$ is $G$–invariant.
Notice that by assumption $\kappa$ is an invertible element of $\mathcal{A}$
with inverse in $\mathcal{A}$.
Moreover, for any $a\in\mathcal{A}$ such that
$\kappa^{-1}a\in\hbox{Dom}(E_{G})$, one has

$\varphi_{G}(\kappa^{-1}a)=\varphi(E_{G}(\kappa^{-1}a))=\int_{G}dg\varphi(g(\kappa^{-1}a))=\int_{G}dg\varphi(x_{g}\kappa^{-1}a)$
$=\varphi(\kappa\kappa^{-1}a)=\varphi(a)$

which is (3.27). Finally, The converse is already proved in Theorem 3.

$\square$

###### Remark 4

The structure, Theorem 5, of the $(G,x)$–strongly–quasi–invariant states in
the uni–modular case is the same found in the case when the group $G$ is
compact. However, the conditions of Lemma 7 are very strong. In the following
sections we discuss the possibility to weaken these conditions.

###  3.4 Inductive limits of compact groups

Let $(G_{N})_{N\in\mathbb{N}}$ be an increasing sequence of compact groups of
normal $*$-automorphisms of a von Neumann algebra $\mathcal{A}$ and assume
that the group

$G:=\bigcup_{N\in\mathbb{N}}G_{N}$ (3.30)

is unimodular and locally compact. Denote

$E_{G_{N}}=\int_{G_{N}}gd_{N}g$ (3.31)

where $d_{N}g$ is the Haar measure on $G_{N}$.

###### Proposition 4

The family $(E_{G_{N}})_{N\in\mathbb{N}}$ is a projective family of
conditional expectations, i.e.

$E_{G_{N+1}}E_{G_{N}}=E_{G_{N+1}}\quad,\quad\forall N\in\mathbb{N}$ (3.32)

Proof. Since $G_{N}\subseteq G_{N+1}$, one has

$\hbox{Range}(E_{G_{N+1}})=\hbox{Fix}({G_{N+1}})\subseteq\hbox{Fix}({G_{N}})=\hbox{Range}(E_{G_{N}})$
(3.33)

Therefore projectivity is equivalent to (3.32). Moreover

$E_{G_{N+1}}E_{G_{N}}=E_{G_{N+1}}\int_{G_{N}}gd_{N}g=\int_{G_{N}}d_{N}gE_{G_{N+1}}g\overset{\eqref{EG-
commutes-G}}{=}\int_{G_{N}}d_{N}gE_{G_{N+1}}=E_{G_{N+1}}$

$\qquad\square$

###### Lemma 8

A state $\varphi$ on $\mathcal{A}$ is $G$–quasi–invariant if and only if it is
$G_{N}$–quasi–invariant for each $N\in\mathbb{N}$. If this is the case,
denoting for each $g\in G$

$N_{g}:=\min\\{N\in\mathbb{N}\colon g\in G_{N}\\}$ (3.34) $x_{g}:=\hbox{the
$G$--cocycle}\quad;\quad x_{G_{N};g}:=\hbox{the
$G_{N}$--cocycle}\quad(N\in\mathbb{N})$ (3.35)

one has

$x_{g}=x_{G_{N_{g}};g}\quad,\quad\forall g\in G_{N}$ (3.36)

In particular, $\varphi$ is $G$–strongly quasi–invariant if and only if it is
$G_{N}$–strongly quasi–invariant for each $N\in\mathbb{N}$.

Proof. Since any $g\in G$ belongs to some $G_{N}$, $N_{g}$ is well defined by
(3.34). Since, for each $N\in\mathbb{N}$, $G_{N}\subseteq G$, it is clear
that, if $\varphi$ is $G$–quasi–invariant, it is also $G_{N}$–quasi–invariant.
To prove that (3.36) holds notice that, for any $N\in\mathbb{N}$ such that
$g\in G_{N}$,

$\varphi(x_{g}a)=\varphi(g(a))=\varphi(x_{G_{N};g}a)$

The faithfulness of $\varphi$ then implies that $x_{g}=x_{G_{N};g}$ for any
$N\in\mathbb{N}$ such that $g\in G_{N}$. Thus in particular (3.36) holds.
Conversely, suppose that $\varphi$ is $G_{N}$–quasi–invariant for each
$N\in\mathbb{N}$ and notice that, since any $g\in G$ belongs to some
$G_{N_{0}}$. Then one has, for any $N\geq N_{g}$,

$\varphi(x_{G_{N_{g}};g}a)=\varphi(g(a))=\varphi(x_{g}a)$

The faithfulness of $\varphi$ then implies that, defining $x_{g}$ by the right
hand side of (3.36), one has

$\varphi(g(a))=\varphi(x_{g}a)\quad,\quad\forall g\in G$

i.e. that $\varphi$ is $G$–quasi–invariant with cocycle $g\in G\mapsto x_{g}$.
This proves the first statement of the theorem. Given this and (3.36), the
second statement is clear. $\qquad\square$

##  4 Unitary representations associated to strongly quasi–invariant states

Let $G\subseteq\hbox{Aut}(\mathcal{A})$ be a group of normal $*$–automorphisms
of $\mathcal{A}$. In the following the cyclic representation of
$\\{\mathcal{A},\varphi\\}$ shall be denoted
$\\{\mathcal{H}_{\varphi},\pi,\Phi\\}$.

###### Theorem 6

If $\varphi$ is $(G,x)$–strongly quasi–invariant, there exists a unique
unitary representation $U$ of $G$ on $\mathcal{A}_{\varphi}$ characterized by
the property

$U_{g}\pi(a)\Phi=\pi(g(a)x^{1/2}_{g^{-1}}))\Phi\
;\quad\forall\,a\in\mathcal{A}$ (4.1)

where $x^{1/2}_{g^{-1}}$ is the positive square root of $x_{g^{-1}}$. Moreover

$u_{g}(\pi(a)):=U^{*}_{g}\pi(a)U_{g}=\pi(g^{-1}(a))\ ;\quad\forall\,g\in G\
;\quad\forall\,a\in\mathcal{A}$ (4.2)

Proof. In the above notations, for any $a,b\in\mathcal{A}$ and $g\in G$, one
has

$\displaystyle\langle U_{g}\pi(a)\Phi,\ U_{g}\pi(b)\Phi\rangle$
$\displaystyle\overset{\eqref{df-Ug}}{=}$
$\displaystyle\langle\pi(g(a)x^{1/2}_{g^{-1}})\Phi,\pi(g(b)x^{1/2}_{g^{-1}})\Phi\rangle$
$\displaystyle=$
$\displaystyle\langle\Phi,\pi(x^{1/2}_{g^{-1}}g(a^{*}b)x^{1/2}_{g^{-1}})\Phi\rangle$
$\displaystyle=$
$\displaystyle\varphi(x^{1/2}_{g^{-1}}g(a^{*}b)x^{1/2}_{g^{-1}})$
$\displaystyle\overset{Lemma\,\ref{lm:xg-herm}}{=}$
$\displaystyle\varphi(g(a^{*}b)x_{g^{-1}})$ $\displaystyle=$
$\displaystyle\varphi\circ g(a^{*}b\cdot g^{-1}(x_{g^{-1}}))$
$\displaystyle\overset{\eqref{df-G-q-inv}}{=}$
$\displaystyle\varphi(x_{g}a^{*}bg^{-1}(x_{g^{-1}}))$
$\displaystyle\overset{Lemma\,\ref{lm:xg-herm}}{=}$
$\displaystyle\varphi(a^{*}bg^{-1}(x_{g^{-1}})x_{g})$
$\displaystyle\overset{\eqref{xg-1}}{=}$
$\displaystyle\varphi(a^{*}b)=\langle\pi(a)\Phi,\pi(b)\Phi\rangle$

Thus $U_{g}$ is isometric. Now, if $g,g^{\prime}\in G$, then

$\displaystyle U_{g}U_{g^{\prime}}\pi(a)\Phi$ $\displaystyle=$ $\displaystyle
U_{g}\pi(g^{\prime}(a)x^{1/2}_{g^{\prime-1}})\Phi$ $\displaystyle=$
$\displaystyle\pi(g[g^{\prime}(a)x^{1/2}_{g^{\prime-1}}]x^{1/2}_{g^{-1}})\Phi$
$\displaystyle=$ $\displaystyle\pi(gg^{\prime}(a)\cdot
g(x^{1/2}_{g^{\prime-1}})x^{1/2}_{g^{-1}})\Phi$

One has

$U_{g}^{*}=U_{g^{-1}}$ (4.4)

In fact, for any $a,b\in\mathcal{A}$,

$\displaystyle\langle\pi(a)\Phi\,,\,U_{g}^{*}\pi(b)\Phi\rangle$
$\displaystyle=$ $\displaystyle\langle U_{g}\pi(a)\Phi\,,\,\pi(b)\Phi\rangle$
$\displaystyle=$
$\displaystyle\langle\pi(g(a))x_{g-1}^{1/2}\Phi\,,\,\pi(b)\Phi\rangle$
$\displaystyle=$ $\displaystyle\varphi(x^{1/2}_{g-1}g(a^{*})b)$
$\displaystyle=$
$\displaystyle\varphi(g(g^{-1}(x_{g-1}^{1/2})a^{*}g^{-1}(b)))$
$\displaystyle=$
$\displaystyle\varphi(x_{g}g^{-1}(x_{g^{-1}}^{1/2})a^{*}g^{-1}(b))$
$\displaystyle\overset{\eqref{x(-s)g}}{=}$
$\displaystyle\varphi(x_{g}x_{g}^{-1/2}a^{*}g^{-1}(b))$ $\displaystyle=$
$\displaystyle\varphi(x_{g}^{1/2}a^{*}g^{-1}(b))$
$\displaystyle\overset{Lemma\,\ref{lm:xg-herm}}{=}$
$\displaystyle\varphi(a^{*}g^{-1}(b)x_{g}^{1/2})$ $\displaystyle=$
$\displaystyle\varphi^{*}(a^{*}g^{-1}(b)x_{(g^{-1})^{-1}}^{1/2})$
$\displaystyle=$
$\displaystyle\langle\pi(a)\Phi\,,\,U_{g^{-1}}\pi(b)\Phi\rangle$

which is equivalent to (4.4). Since $x_{g^{\prime}}$ and
${g^{\prime}}^{-1}(x_{g})$ are hermitean and commute because of (2.11), the
cocycle property implies that

$g(x^{1/2}_{g^{\prime-1}})x^{1/2}_{g^{-1}}=\left(g(x_{g^{\prime-1}})x_{g^{-1}}\right)^{1/2}=\left(x_{g^{-1}}g(x_{g^{\prime-1}})\right)^{1/2}\
\overset{\eqref{xg-1G-left-cocy}}{=}\
\left(x_{{g^{\prime}}^{-1}g{-1}}\right)^{1/2}=\left(x_{(gg^{\prime})^{-1}}\right)^{1/2}$

Therefore the right hand side of (4) becomes

$\pi(gg^{\prime}(a)\cdot
x_{(gg^{\prime})^{-1}}^{1/2})\Phi=U_{gg^{\prime}}\pi(a)\Phi$

Thus $U$ is a representation hence, by isometry and (4.4), a unitary
representation. Finally, for $a,b,c\in\mathcal{A}$ and $g\in G$, one has

$U^{*}_{g}\pi(a)U_{g}\pi(b)\Phi=U_{g^{-1}}\pi(a\cdot g(b)\cdot
x^{1/2}_{g^{-1}})\Phi=\pi(g^{-1}[ag(b)x^{1/2}_{g^{-1}}]\cdot x^{1/2}_{g})\Phi$
(4.5) $=\pi(g^{-1}(a)b\cdot[g^{-1}(x^{1/2}_{g^{-1}})x^{1/2}_{g}])\Phi$

and, by (2.7) with $s=1/2$, the right hand side of (4.5) is equal to

$\pi(g^{-1}(a)b)\Phi=\pi(g^{-1}(a))\pi(b)\Phi$

which implies (4.2) by the cyclicity of $\Phi$. $\square$

###### Corollary 2

The family $(u_{g})_{g\in G}$ of $*$–automorphisms of $\pi(\mathcal{A})$
defined by (4.2) extends uniquely to a representation of $G$ into the normal
(or equivalently strongly continuous, see x. ) $*$–automorphisms of
$\pi(\mathcal{A})^{\prime\prime}$, still denoted with the same symbol.
If $G=\bigcup_{N\in\mathbb{N}}G_{N}$ is the union of an increasing sequence of
compact groups, denoting for each $N\in\mathbb{N}$,

$\overline{E}_{G_{N}}:=\int_{G_{N}}u_{g}d_{N}g$ (4.6)

the family $(\overline{E}_{G_{N}})$ is a projective family of normal Umegaki
conditional expectations. Moreover, for each $N\in\mathbb{N}$,

$\overline{E}_{G_{N}}(\pi(a))=\pi(E_{G_{N}}(a))$ (4.7)

Each $\overline{E}_{N}$ can be extended by continuity to a Umegaki conditional
expectation, denoted with the same symbol $\overline{E}_{N}$, onto the weak
closure of

$\hbox{Fix}(u(G_{N})):=\hbox{weak closure of
}\hbox{Fix}(u(G_{N}))\subset\pi(\mathcal{A})^{\prime\prime}$ (4.8)

The family $(\overline{E}_{N})_{N\in\mathbb{N}}$ is a projective decreasing
family of normal Umegaki conditional expectations.

Proof. Each $\overline{E}_{N}$ is a normal map because each $u_{g}$ ($g\in
G_{N}$) is normal and this property is preserved under integration over a
compact set. By normality, it can be extended by continuity to
$\pi(\mathcal{A})^{\prime\prime}$. (4.7) holds because

$\overline{E}_{G_{N}}(\pi(a))=\int_{G_{N}}d_{G_{N}}gu_{g}(\pi(a))=\int_{G_{N}}d_{G_{N}}gU_{g}^{*}(\pi(a))U_{g}\overset{\eqref{*-auto-
ug}}{=}\int_{G_{N}}d_{G_{N}}(\pi(g^{-1}a))$
$=\int_{G_{N}}d_{G_{N}}(\pi(ga))=\pi\left(\int_{G_{N}}d_{G_{N}}ga\right)=\pi(E_{G_{N}}(a))$

Therefore $\overline{E}_{G_{N}}$ is a Umegaki conditional expectation from
$\pi(\mathcal{A})$ onto $\hbox{Fix}(G_{N})$. This implies that its extension
is a Umegaki conditional expectation from $\pi(\mathcal{A})^{\prime\prime}$
onto $\hbox{Fix}(u(G_{N}))$. Since $(E_{N})_{N\in\mathbb{N}}$ is a projective
decreasing family, the same is true for $(\overline{E}_{N})_{N\in\mathbb{N}}$.
$\qquad\square$

##  5 Examples of quasi–invariant states under permutations

Let $\mathcal{B}:=\mathcal{B}(\mathcal{H})$ for some (separable) Hilbert space
$\mathcal{H}$. Denote

$\mathcal{A}=\bigotimes_{\mathbb{N}^{*}}\mathcal{B}$ (5.1)

the algebraic tensor product of countably many copies of $\mathcal{B}$ and let
$j_{n}$ be the natural embedding of $\mathcal{B}$ onto the $n$–th factor of
$\mathcal{A}$ (we also use the notation
$j_{n}(\mathcal{B})=:\mathcal{A}_{n}\equiv\mathcal{B}\otimes\bigotimes_{\mathbb{N}^{*}\setminus\\{n\\}}1_{\mathcal{B}}$).
With these notations, introducing the local algebras

$\mathcal{A}_{[1,N]}:=\bigvee_{n\in[1,N]}\mathcal{A}_{n}:=\hbox{algebra
generated by the }\mathcal{A}_{n}\,\colon\,n\in[1,N]$

one has

$\mathcal{A}=\bigcup_{N\in\mathbb{N}^{*}}\mathcal{A}_{[1,N]}$ (5.2)

Denote by $\mathcal{S}_{N}$ the permutation group on $\\{1,\dots,N\\}$. Define
$\mathcal{S}_{\infty}=\cup_{N\in\mathbb{N}^{*}}\mathcal{S}_{N}$.
$\mathcal{S}_{\infty}$ has a natural action on $\mathcal{A}$ by
$*$–automorphisms given by $g\circ j_{n}:=j_{gn}$ where we use the same symbol
for the actions of $\mathcal{S}_{\infty}$ on $\mathbb{N}^{*}$ and on
$\mathcal{A}$. With this identification, $\mathcal{S}_{\infty}$ is called the
group of local permutations on $\mathcal{A}$. Recall that, for each
$g\in\mathcal{S}_{\infty}$, the support of $g$ is defined by

$\Lambda_{g}:=\hbox{supp}(g):=\\{n\in\mathbb{N}^{*}\colon gn\neq
n\\}=\hbox{supp}(g^{-1})$

Denoting

$\mathcal{S}_{N}:=\\{g\in\mathcal{S}_{\infty}\colon\hbox{supp}(g)\subseteq[1,N]\\}$

one has

$\mathcal{S}_{\infty}=\bigcup_{N\in\mathbb{N}^{*}}\mathcal{S}_{N}$ (5.3)

###  5.1 Product states

###### Proposition 5

Any product state on $\mathcal{A}$

$\varphi=\bigotimes_{n\in\mathbb{N}^{*}}\varphi_{n}=\bigotimes_{n\in\mathbb{N}^{*}}\rm{Tr}(W_{n}\,\cdot\,)=:\rm{Tr}_{\mathbb{N}^{*}}\left(\bigotimes_{n\in\mathbb{N}^{*}}W_{n}\,\cdot\,\right)=\rm{Tr}_{\mathbb{N}^{*}}\left(\prod_{n\in\mathbb{N}^{*}}j_{n}(W_{n})\,\cdot\,\right)$
(5.4)

with $W_{n}$ invertible for each $n\in\mathbb{N}^{*}$, is
$\mathcal{S}_{\infty}$–quasi–invariant with cocycle

$x_{g}:=\prod_{n\in\Lambda_{g}}j_{n}(W^{-1}_{n})\prod_{n\in\Lambda_{g}}j_{g^{-1}n}(W_{n})=\prod_{n\in\Lambda_{g}}j_{n}(W^{-1}_{n}W_{gn})\quad,\quad\forall
g\in\mathcal{S}_{\infty}$ (5.5)

Moreover, $\varphi$ is $\mathcal{S}_{\infty}$–strongly quasi–invariant iff
$[W_{n},\ W_{m}]=0$ for all $m,n\in\mathbb{N}^{*}$ and,

$x_{g}=\prod_{n\in\Lambda_{g}}j_{n}(W^{-1}_{n})j_{g^{-1}n}(W_{n})\quad,\quad\forall
g\in\mathcal{S}_{\infty}$ (5.6)

Proof. Let $g\in\mathcal{S}_{\infty}$ with support $\Lambda_{g}$. Note that
$g(\Lambda_{g})=\Lambda_{g}$. Therefore, for any $n\in\Lambda_{g}$ and
$a\in\mathcal{A}$ one has

$\displaystyle\varphi(g(a))$ $\displaystyle=$
$\displaystyle\rm{Tr}_{\mathbb{N}^{*}}\left(\prod_{n\in\mathbb{N}^{*}}j_{n}(W_{n})g(a)\right)$
$\displaystyle=$
$\displaystyle\rm{Tr}_{\mathbb{N}^{*}}\left(g\left(g^{-1}\left(\prod_{n\in\mathbb{N}^{*}}j_{n}(W_{n})\right)a\right)\right)$
$\displaystyle=$
$\displaystyle\rm{Tr}_{\mathbb{N}^{*}}\left(g^{-1}\left(\prod_{n\in\mathbb{N}^{*}}j_{n}(W_{n})\right)a\right)$
$\displaystyle=$
$\displaystyle\rm{Tr}_{\mathbb{N}^{*}}\left(\prod_{n\in\mathbb{N}^{*}}j_{g^{-1}n}(W_{n})a\right)$
$\displaystyle=$
$\displaystyle\rm{Tr}_{\mathbb{N}^{*}}\left(\left(\prod_{n\in\Lambda_{g}^{c}}j_{n}(W_{n})\right)\left(\prod_{n\in\Lambda_{g}}j_{g^{-1}n}(W_{n})\right)a\right)$
$\displaystyle=$
$\displaystyle\rm{Tr}_{\mathbb{N}^{*}}\left(\left(\prod_{n\in\mathbb{N}^{*}}j_{n}(W_{n})\right)\left(\prod_{n\in\Lambda_{g}}j_{n}(W^{-1}_{n})\right)\left(\prod_{n\in\Lambda_{g}}j_{g^{-1}n}(W_{n})\right)a\right)$
$\displaystyle=$
$\displaystyle\varphi\left(\left(\prod_{n\in\Lambda_{g}}j_{n}(W^{-1}_{n})\right)\left(\prod_{n\in\Lambda_{g}}j_{g^{-1}n}(W_{n})\right)a\right)$

Thus, if for each $g\in\mathcal{S}_{\infty}$, $x_{g}$ is defined by (5.5), one
obtains

$\varphi(g(a))=\varphi\left(x_{g}a\right)$

i.e. $\varphi$ is $\mathcal{S}_{\infty}$–quasi–invariant with cocycle given by
(5.5). Finally, if $[W_{n},\ W_{m}]=0$ for all $m,n\in\mathbb{N}^{*}$, it is
clear that the cocycle $x_{g}$ is given by (5.6) and it is hermitean.
Therefore $\varphi$ is $\mathcal{S}_{\infty}$–strongly quasi–invariant.
$\qquad\square$

###### Theorem 7

Let $\varphi$ be a product state on $\mathcal{A}$ of the form (5.4) and let
$\psi_{0}=\hbox{Tr}(W_{\infty}\,\cdot)\,$ be an arbitrary faithful state on
$\mathcal{B}$. Define, for all $n\in\mathbb{N}^{*}$,

$x_{n}:=W_{\infty}^{-1}W_{n}\quad;\quad
x_{[1,n]}:=\bigotimes_{k\in[1,n]}W_{\infty}^{-1}W_{k}=\prod_{k\in[1,n]}j_{k}(W_{\infty}^{-1}W_{k})$
(5.7)
$\psi:=\bigotimes_{\mathbb{N}^{*}}\psi_{0}=\bigotimes_{\mathbb{N}^{*}}\rm{Tr}_{\mathbb{N}^{*}}(W_{\infty}\,\cdot\,)$
(5.8)

Then $\varphi$ is $\mathcal{S}_{\infty}$–quasi invariant with cocycle

$x_{g}:=\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1})\prod_{n\in\Lambda_{g}}j_{g^{-1}n}(x_{n})=\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1}x_{gn})$
(5.9)

and $\varphi$ is $\mathcal{S}_{\infty}$–strongly quasi–invariant iff all
$W_{\infty}$ and $W_{n}$ commute. Moreover if we denote by
$(\mathcal{H}_{\psi},\pi_{\psi},\Psi)$ be the cyclic representation of $\psi$,
then the weak limit

$\lim_{n\to+\infty}\pi_{\psi}(x_{[1,n]})=:X_{\psi,\infty}$ (5.10)

exists in $\mathcal{H}_{\psi}$ and takes place in the strongly finite sense on
$\pi_{\psi}(\mathcal{A})\cdot\Psi$. Moreover $X_{\psi,\infty}\geq 0$ and,
denoting $\hat{\psi}$ the linear functional on
$\pi_{\psi}(\mathcal{A})^{\prime\prime}$ defined by

$\hat{\psi}(\bar{a}):=\langle\Psi\,,\,\bar{a}\Psi\rangle\quad,\quad\bar{a}\in\pi_{\psi}(\mathcal{A})^{\prime\prime}$
(5.11)

the restriction of $\hat{\psi}$ on $\pi_{\psi}(\mathcal{A})$ induces an
exchangeable state on $\mathcal{A}$ and one has:

$\varphi(a)=\hat{\psi}(X_{\psi,\infty}\pi_{\psi}(a))\quad,\quad\forall
a\in\mathcal{A}$ (5.12)

Remark. The identity (5.12) is the first example of a decomposition of the
form (3.7) for a locally compact but non–compact group.

Proof. For any $N\in\mathbb{N}^{*}$ and any element $a\in\mathcal{A}_{[1,N]}$,
one has

$\varphi(a)=\lim_{n\to+\infty}\psi(x_{[1,n]}a)=\psi(x_{[1,N]}a)$ (5.13)

Notice that $\psi$ is exchangeable by construction. Now fix
$N\in\mathbb{N}^{*}$ and notice that, for each $a\in\mathcal{A}$, there exists
$N_{a}\in\mathbb{N}^{*}$ such that $a\in\mathcal{A}_{[1,N^{\prime}]}$ for each
$N^{\prime}\geq N_{a}\geq N+1$. Therefore, since each $x_{n}$ is invertible,
for each $g\in\mathcal{S}_{N}$ and for each $N^{\prime}\geq N_{a}$, one has

$\displaystyle\varphi(g(a))$ $\displaystyle=$
$\displaystyle\psi(x_{[1,N^{\prime}]}g(a))$ $\displaystyle=$
$\displaystyle\psi\left(x_{[1,N^{\prime}]}g(a)\right)$
$\displaystyle\overset{\psi\circ g=\psi}{=}$
$\displaystyle\psi\left(g^{-1}\left(\left(x_{[1,N^{\prime}]}\right)g\left(a\right)\right)\right)$
$\displaystyle=$
$\displaystyle\psi\left(g^{-1}\left(x_{[1,N^{\prime}]}\right)a\right)$
$\displaystyle=$
$\displaystyle\psi\left(x_{[1,N^{\prime}]}x_{[1,N^{\prime}]}^{-1}g^{-1}\left(x_{[1,N^{\prime}]}\right)a\right)$
$\displaystyle=$
$\displaystyle\varphi\left(x_{[1,N^{\prime}]}^{-1}g^{-1}\left(x_{[1,N^{\prime}]}\right)a\right)$

Since $g\in\mathcal{S}_{N}$, for each $N^{\prime}\geq N_{a}$, one has

$\displaystyle x_{[0,N^{\prime}]}^{-1}g^{-1}(x_{[0,N^{\prime}]})$
$\displaystyle\overset{\eqref{df-xn-fi}}{=}$
$\displaystyle(x_{[0,N]}^{-1}\otimes
x_{[N+1,N_{a}]}^{-1})g^{-1}(x_{[0,N]}\otimes x_{[N+1,N_{a}]})$
$\displaystyle\overset{g\in\mathcal{S}_{N}}{=}$
$\displaystyle(x_{[0,N]}^{-1}\otimes
x_{[N+1,N_{a}]}^{-1})(g^{-1}(x_{[0,N]})\otimes x_{[N+1,N_{a}]})$
$\displaystyle=$ $\displaystyle x_{[0,N]}^{-1}g^{-1}(x_{[0,N]})\otimes
1_{[N+1,+\infty)})$ $\displaystyle\equiv$ $\displaystyle
x_{[0,N]}^{-1}g^{-1}(x_{[0,N]})$ $\displaystyle\overset{\eqref{df-xn-fi}}{=}$
$\displaystyle\prod_{n\in[0,N]}j_{n}(x_{n}^{-1})\prod_{n\in[0,N]}j_{g^{-1}n}(x_{n})$
$\displaystyle=$
$\displaystyle\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1})\prod_{n\in\Lambda_{g}}j_{g^{-1}n}(x_{n})$

which depends only on $g$. This proves that $\varphi$ is
$\mathcal{S}_{\infty}$–quasi invariant with cocycle given by (5.9). Finally,
if all $W_{\infty}$ and $W_{n}$ commute, (5.9) implies that $x_{g}$ is
hermitean, i.e. $\varphi$ is $\mathcal{S}_{\infty}$–strongly quasi–invariant.

To prove the existence of the weak limit (5.10), it is sufficient to prove its
existence on the dense sub–space $\pi_{\psi}(\mathcal{A})\cdot\Psi$ and, since
any element of $\pi_{\psi}(\mathcal{A})\cdot\Psi$ is a linear combination of
elements of the form $\prod_{k=1}^{M}j_{k}(a_{k})\cdot\Psi$ with
$M\in\mathbb{N}$ and $a_{1},\dots,a_{M}\in\mathcal{A}$, it is sufficient to
restrict one’s attention to these elements. Let
$\prod_{k=1}^{M}j_{k}(a_{k})\cdot\Psi$ and
$\prod_{k=1}^{M}j_{k}(b_{k})\cdot\Psi$ be two elements of this form (one can
always suppose that $M$ is the same for both). Then for any
$M<N\in\mathbb{N}^{*}$, one has

$\displaystyle\langle\prod_{k=1}^{M}\pi_{\psi}(j_{k}(a_{k}))\cdot\Psi\,,\,\prod_{h=1}^{N}\pi_{\psi}(j_{h}(x_{h}))\prod_{k=1}^{M}\pi_{\psi}(j_{k}(b_{k}))\cdot\Psi\rangle$
(5.15)
$\displaystyle=\langle\prod_{k=1}^{M}\pi_{\psi}(j_{k}(a_{k}))\cdot\Psi\,,\,\prod_{h=1}^{M}\pi_{\psi}(j_{h}(x_{h}))\prod_{h=M+1}^{N}\pi_{\psi}(j_{h}(x_{h}))\prod_{k=1}^{M}\pi_{\psi}(j_{k}(b_{k}))\cdot\Psi\rangle$
$\displaystyle=\langle\prod_{k=1}^{M}\pi_{\psi}(j_{k}(a_{k}))\cdot\Psi\,,\,\prod_{h=1}^{M}\pi_{\psi}(j_{h}(x_{h}))\prod_{k=1}^{M}\pi_{\psi}(j_{k}(b_{k}))\cdot\Psi\rangle\;\langle\Psi\,,\,\prod_{h=M+1}^{N}\pi_{\psi}(j_{h}(x_{h}))\cdot\Psi\rangle$

Since

$\langle\Psi\,,\,\prod_{h=M+1}^{N}\pi_{\psi}(j_{h}(x_{h}))\cdot\Psi\rangle=\prod_{h=M+1}^{N}\psi_{0}(x_{h})\overset{\eqref{df-
xn-fi}}{=}\prod_{h=M+1}^{N}\hbox{Tr}(W_{\infty}\cdot W_{\infty}^{-1}W_{h})$
$=\prod_{h=M+1}^{N}\hbox{Tr}(W_{h})=1$

it follows that, for any $N>M$

$\displaystyle\langle\prod_{k=1}^{M}\pi_{\psi}(j_{k}(a_{k}))\cdot\Psi\,,\,\prod_{h=1}^{N}\pi_{\psi}(j_{h}(x_{h}))\prod_{k=1}^{M}\pi_{\psi}(j_{k}(b_{k}))\cdot\Psi\rangle$
$\displaystyle=\langle\prod_{k=1}^{M}\pi_{\psi}(j_{k}(a_{k}))\cdot\Psi\,,\,\prod_{h=1}^{M}\pi_{\psi}(j_{h}(x_{h}))\prod_{k=1}^{M}\pi_{\psi}(j_{k}(b_{k}))\cdot\Psi\rangle$
$\displaystyle=\prod_{k=1}^{M}\psi(a_{k}^{*}x_{h}b_{k})$ (5.16)

Therefore the sequence $(\prod_{h=1}^{N}\pi_{\psi}(j_{h}(x_{h})))_{N}$
converges weakly on $\pi_{\psi}(\mathcal{A})\cdot\Psi$ in the strongly finite
sense, hence it converges weakly on $\mathcal{H}_{\psi}$. This proves (5.10).
Taking the limit of (5.16) for $N\to\infty$ with all the $a_{k}=1$, one
obtains

$\langle\Psi\,,\,X_{\infty}\pi_{\psi}\left(\prod_{k=1}^{M}j_{k}(b_{k})\right)\cdot\Psi\rangle=\prod_{k=1}^{M}\psi(x_{h}b_{k})\overset{\eqref{df-
xn-fi}}{=}\prod_{k=1}^{M}\hbox{Tr}(W_{\infty}\cdot W_{\infty}^{-1}W_{k}b_{k})$
$=\prod_{k=1}^{M}\hbox{Tr}(W_{k}b_{k})=\varphi\left(\prod_{k=1}^{M}b_{k}\right)$

and this is equivalent to (5.12). $\qquad\square$

Remark. Formally, denoting

$x_{n}:=W_{\infty}^{-1}W_{n}\quad,\quad n\in\mathbb{N}^{*}$ (5.17)

one can write $\varphi$ in the form

$\displaystyle\varphi$ $\displaystyle=$
$\displaystyle\bigotimes_{\mathbb{N}^{*}}\rm{Tr}(W_{\infty}W_{\infty}^{-1}W_{n}\,\cdot\,)=\psi\left(\bigotimes_{\mathbb{N^{*}}}W_{\infty}^{-1}W_{n}\,\cdot\,\right)$
$\displaystyle=$
$\displaystyle\psi\left(\bigotimes_{n\in\mathbb{N}^{*}}x_{n}\,\cdot\,\right)=\psi\left(\prod_{n\in\mathbb{N}^{*}}j_{n}(x_{n})\,\cdot\,\right)$

so, defining

$X_{\infty}:=\bigotimes_{\mathbb{N}^{*}}W_{\infty}^{-1}W_{n}\equiv\prod_{n\in\mathbb{N}^{*}}j_{n}(x_{n})$
(5.18)

one has

$\varphi=\psi(X_{\infty}\,\cdot\,)$ (5.19)

For generic sequences $(W_{n})$, the right hand side of (5.18) is a formal
expression because the product does not converge in any reasonable topology on
$\mathcal{A}$. But the formal expression gives the right intuition because,
for any $g\in\mathcal{S}_{\infty}$, $a\in\mathcal{A}_{[1,N_{a}]}$, one has

$\displaystyle\varphi(g(a))$ $\displaystyle=$
$\displaystyle\psi(X_{\infty}g(a))$ $\displaystyle=$
$\displaystyle\psi\left(\bigotimes_{n\in\mathbb{N}^{*}}x_{n}g(a)\right)$
$\displaystyle=$
$\displaystyle\psi\left(g^{-1}\left(\bigotimes_{n\in\mathbb{N}^{*}}x_{n}\right)a\right)$
$\displaystyle=$
$\displaystyle\psi\left(\left(\prod_{n\in\mathbb{N}}j_{g^{-1}n}(x_{n})\right)\right)$
$\displaystyle=$
$\displaystyle\psi\left(\left(\prod_{n\in\Lambda_{g}}j_{g^{-1}n}(x_{n})\right)\left(\prod_{n\in\Lambda_{g}^{c}}j_{n}(x_{n})\right)a\right)$
$\displaystyle=$
$\displaystyle\psi\left(\left(\prod_{n\in\mathbb{N}}j_{n}(x_{n})\right)\left(\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1})\prod_{n\in\Lambda_{g}}j_{g^{-1}n}(x_{n})\right)a\right)$
$\displaystyle=$
$\displaystyle\varphi\left(\left(\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1})\prod_{n\in\Lambda_{g}}j_{g^{-1}n}(x_{n})\right)a\right)$
$\displaystyle=$
$\displaystyle\varphi\left(\left(\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1}x_{gn})\right)a\right)$

which is (5.9).

####  5.1.1 Generic non–$\mathcal{S}_{\infty}$–integrability in norm of the
cocycle $x_{g}$

Notice that the product (5.18), i.e. $X_{\infty}$, in generic conditions will
not be defined as an element of $\mathcal{A}$ for any reasonable topology.
However the following result shows that the cocycle
$x_{g}\overset{\eqref{cocy-X}}{=}X_{\infty}^{-1}g^{-1}(X_{\infty})$ (formal
equality) is well defined as a norm limit.

###### Lemma 9

For any $g\in\mathcal{S}_{\infty}$, one has

$x_{g}:=\hbox{norm--}\lim_{N\to+\infty}X_{N}^{-1}g^{-1}(X_{N})=\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1})\prod_{n\in\Lambda_{g}}j_{g^{-1}n}(x_{n})=\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1}x_{gn})$
(5.20)

and the limit takes place in the strongly finite sense (i.e. it is constant
for all $N$ satisfying $\Lambda_{g}\subseteq[1,N]$).

Proof. Since any element of $g\in S_{\infty}$ has finite support
$\Lambda_{g}$, one has (all the calculations being meant in the sense of
norm–limits)

$\displaystyle X_{\infty}^{-1}g^{-1}(X_{\infty})$
$\displaystyle\overset{\eqref{df-X-infty}}{=}$
$\displaystyle\prod_{n\in\mathbb{N}^{*}}j_{n}(x_{n})^{-1}g^{-1}(\prod_{n\in\mathbb{N}^{*}}j_{n}(x_{n}))$
$\displaystyle=$
$\displaystyle\prod_{n\in\mathbb{N}^{*}}j_{n}(x_{n})^{-1}(\prod_{n\in\mathbb{N}^{*}}j_{g^{-1}n}(x_{n}))$
$\displaystyle=$
$\displaystyle\prod_{n\in\Lambda_{g}}j_{n}(x_{n})^{-1}\prod_{n\in\Lambda_{g}^{c}}j_{n}(x_{n})^{-1}\left(\prod_{n\in\Lambda_{g}}j_{g^{-1}n}(x_{n})\prod_{n\in\Lambda_{g}^{c}}j_{g^{-1}n}(x_{n})\right)$
$\displaystyle=$
$\displaystyle\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1})\prod_{n\in\Lambda_{g}^{c}}j_{n}(x_{n})^{-1}\left(\prod_{n\in\Lambda_{g}}j_{g^{-1}n}(x_{n})\prod_{n\in\Lambda_{g}^{c}}j_{n}(x_{n})\right)$
$\displaystyle=$
$\displaystyle\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1})\prod_{n\in\Lambda_{g}}j_{g^{-1}n}(x_{n})$
$\displaystyle=$
$\displaystyle\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1})\prod_{n\in\Lambda_{g}}j_{n}(x_{gn})$
$\displaystyle=$ $\displaystyle\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1}x_{gn})$

because we have seen that $n\in\Lambda_{g}\iff gn\in\Lambda_{g}$. This proves
(5.20). $\qquad\square$

The following proposition shows however that the cocycle cannot be integrable
for any reasonable topology.

###### Proposition 6

Suppose that the state $\varphi\equiv(W_{n})$ is strongly
$\mathcal{S}_{\infty}$–quasi–invariant and that there exists a finite set
$F\subset_{fin}\mathbb{N}^{*}$ such that

$W_{n}=W_{\infty}\quad,\quad\forall n\in F^{c}$ (5.21)

Then the $\mathcal{S}_{\infty}$–cocycle (5.20) is not integrable for any
non–trivial topology on $\mathcal{A}$.

Proof. The integrability of the cocycle (5.20) means (see Lemma 7) that the
series

$\sum_{g\in\mathcal{S}_{\infty}}x_{g}=\sum_{g\in\mathcal{S}_{\infty}}\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1}x_{gn})$

is convergent. On the other hand (5.17) and (5.20) imply that $x_{n}=1$ for
all $n\in F^{c}$. Denote

$\mathcal{S}_{\infty;F}:=\\{g\in\mathcal{S}_{\infty}\colon\Lambda_{g}\cap
F\neq\emptyset\\}$

Then, since $F$ is a finite set, $|\mathcal{S}_{\infty;F}^{c}|=+\infty$.
Strongly $\mathcal{S}_{\infty}$–quasi–invariance implies the commutativity of
the $W_{n}$ with $W_{\infty}$ (see Proposition 7) and this implies that
$x_{n}^{-1}x_{gn}\geq 0$ for each $n$. Therefore

$\sum_{g\in\mathcal{S}_{\infty}}x_{g}=\sum_{g\in\mathcal{S}_{\infty};F}\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1}x_{gn})+\sum_{g\in\mathcal{S}_{\infty;F}^{c}}\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1}x_{gn})$
$\geq\sum_{g\in\mathcal{S}_{\infty;F}^{c}}\prod_{n\in\Lambda_{g}}j_{n}(x_{n}^{-1}x_{gn})=\sum_{g\in\mathcal{S}_{\infty;F}^{c}}\prod_{n\in\Lambda_{g}}\
1=+\infty$

$\square$

###  5.2 Markov Chains with commuting conditional density amplitudes

Recall that, by the $C^{*}$–version of de Finetti’s Theorem, all local
sub–algebras of $\mathcal{A}=\bigotimes_{n\in\mathbb{N}^{*}}\mathcal{B}$ are
$\psi$–expected for any exchangeable state $\psi$ on $\mathcal{A}$. For
$I\subseteq\mathbb{N}^{*}$, the sub–algebra of $\mathcal{A}$ localized in $I$
is denoted $\mathcal{A}_{I}$. We fix an exchangeable state $\psi$ on
$\mathcal{A}$, we denote $E^{\psi}_{I}$ the (unique) Umegaki conditional
expectation from $\mathcal{A}$ onto $\mathcal{A}_{I}$ satisfying

$\psi\circ E^{\psi}_{I}=\psi\quad(\mathcal{A}_{I}\hbox{ $\psi$--expected })$

Recall that an $E^{\psi}_{I}$–conditionally density amplitude (CDA) localized
in $\mathcal{A}_{[n,n+1]}=\mathcal{A}_{n}\otimes\mathcal{A}_{n+1}$
($n\in\mathbb{N}^{*}$) is an operator $K_{n,n+1}\in\mathcal{A}_{[n,n+1]}$
satisfying

$E^{\psi}_{\\{n\\}}(j_{n}(K_{n,n+1}^{*}K_{n,n+1}))=\left(E^{\psi}_{n]}(j_{n}(K_{n,n+1}^{*}K_{n,n+1}))\right)=1_{\mathcal{A}_{n}}$

and that the family $(K_{n,n+1})_{n\in\mathbb{N}^{*}}$ of CDA is called
commutative if the $C^{*}$–algebra generated by it is commutative. Let
$j_{(n,n+1)}:=j_{n}\otimes j_{n+1}$ (we also use the notation
$j_{(n,n+1)}(\mathcal{B}\otimes\mathcal{B})=:\mathcal{A}_{n}\otimes\mathcal{A}_{n+1}$).
So with this notation, for any $g\in\mathcal{S}_{\infty}$, one has
$gj_{(n,n+1)}(a\otimes b):=j_{[gn,g(n+1)]}(a\otimes b)$ which is equal to
$j_{gn}(a)\otimes j_{g(n+1)}(b)$ if $g(n+1)>gn$ and $j_{g(n+1)}(a)\otimes
j_{gn}(b)$ if $gn>g(n+1)$.

###### Remark 5

In the following proposition we use the formal language introduced in the
remark after Proposition 5. More explicitly: the right hand side of (5.22) is
a formal expression and its precise meaning is given by the general theory of
quantum Markov chains using arguments similar to those used in the proof of
Proposition 5. On the other hand, the use of formal expressions clearly
indicates that, dealing with quantum Markov chains, the natural notion of
quasi–invariant state is more general than the one discussed in the present
paper. This more general notion will be discussed in a paper in preparation.

###### Proposition 7

In the above notations, the state $\varphi$ on $\mathcal{A}$ defined by

$\varphi(a):=\psi\left(\left(\prod_{n\in\mathbb{N}^{*}}j_{(n,n+1)}(K_{n,n+1}^{*})\right)a\left(\prod_{n\in\mathbb{N}^{*}}j_{(n,n+1)}(K_{n,n+1})\right)\right)$
(5.22)

satisfies

$\displaystyle\varphi(g(a))$ $\displaystyle=$
$\displaystyle=\varphi\left(\left(\prod_{n\in\Lambda_{g}}j_{(n,n+1)}((K_{n,n+1}^{*})^{-1})j_{[g^{-1}n,g^{-1}(n+1)]}(K_{n,n+1}^{*})\right)a\right.$
(5.23)
$\displaystyle\left.\left(\prod_{n\in\Lambda_{g}}j_{(n,n+1)}((K_{n,n+1})^{-1})j_{[g^{-1}n,g^{-1}(n+1)]}(K_{n,n+1})\right)\right)$

where $\Lambda_{g}$ is the support of $g$. In particular, if each $K_{n,n+1}$
is in the centralizer of $\psi$, then $\varphi$ is a
$\mathcal{S}_{\infty}$–strongly quasi–invariant state with

$x_{g}=\prod_{n\in\Lambda_{g}}j_{(n,n+1)}(|K_{n,n+1}|^{2})^{-1})j_{[g^{-1}n,g^{-1}(n+1)]}(|K_{n,n+1}|^{2}),\;\;\;\forall
g\in\mathcal{S}_{\infty}$ (5.24)

Proof. Let $g\in\mathcal{S}_{\infty}$. Then, one has (see (5.22))

$\varphi(g(a))=\psi\left(\left(\prod_{n\in\mathbb{N}^{*}}j_{(n,n+1)}(K_{n,n+1}^{*})\right)g(a)\left(\prod_{n\in\mathbb{N}^{*}}j_{(n,n+1)}(K_{n,n+1})\right)\right)\\\
$
$=\psi\left(g\left(\left(\prod_{n\in\mathbb{N}^{*}}g^{-1}(j_{(n,n+1)}(K_{n,n+1}^{*}))\right)a\left(\prod_{n\in\mathbb{N}^{*}}g^{-1}\left(j_{(n,n+1)}(K_{n,n+1})\right)\right)\right)\right)\\\
$
$=\psi\left(\left(\prod_{n\in\mathbb{N}^{*}}j_{[g^{-1}n,g^{-1}(n+1)]}(K_{n,n+1}^{*})\right)a\left(\prod_{n\in\mathbb{N}^{*}}j_{[g^{-1}n,g^{-1}(n+1)]}\left(K_{n,n+1}\right)\right)\right)\\\
$
$=\psi\left(\left(\prod_{n\in\Lambda^{c}_{g}}j_{(n,n+1)}(K_{n,n+1}^{*})\right)\left(\prod_{n\in\Lambda_{g}}j_{[g^{-1}n,g^{-1}(n+1)]}(K_{n,n+1}^{*})\right)a\right.$
$\left.\left(\prod_{n\in\Lambda_{g}}j_{[g^{-1}n,g^{-1}(n+1)]}(K_{n,n+1})\right)\left(\prod_{n\in\Lambda^{c}_{g}}j_{(n,n+1)}(K_{n,n+1})\right)\right)\\\
$
$=\psi\left(\left(\prod_{n\in\mathbb{N}^{*}}j_{(n,n+1)}(K_{n,n+1}^{*})\right)\left(\prod_{n\in\Lambda_{g}}j_{(n,n+1)}((K_{n,n+1}^{*})^{-1})j_{[g^{-1}n,g^{-1}(n+1)]}(K_{n,n+1}^{*})\right)a\right.$
$\left.\left(\prod_{n\in\Lambda_{g}}j_{(n,n+1)}((K_{n,n+1})^{-1})j_{[g^{-1}n,g^{-1}(n+1)]}(K_{n,n+1})\right)\left(\prod_{n\in\mathbb{N}^{*}}j_{(n,n+1)}(K_{n,n+1})\right)\right)\\\
$
$=\varphi\left(\left(\prod_{n\in\Lambda_{g}}j_{(n,n+1)}((K_{n,n+1}^{*})^{-1})j_{[g^{-1}n,g^{-1}(n+1)]}(K_{n,n+1}^{*})\right)a\right.$
$\left.\left(\prod_{n\in\Lambda_{g}}j_{(n,n+1)}((K_{n,n+1})^{-1})j_{[g^{-1}n,g^{-1}(n+1)]}(K_{n,n+1})\right)\right)$

In the last equality we used the facts that the terms in the products commute
because, for any $n\in\Lambda_{g}$, $gn\neq n$ and the $K_{n,n+1}$ mutually
commute.
If each $K_{n,n+1}$ is in the centralizer of $\psi$, then (5.23) becomes

$\displaystyle\varphi(g(a))\\!\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\varphi\left(\prod_{n\in\Lambda_{g}}j_{(n,n+1)}((K_{n,n+1})^{-1}(K_{n,n+1}^{*})^{-1})j_{[g^{-1}n,g^{-1}(n+1)]}(K_{n,n+1}K_{n,n+1}^{*})a\right)$
$\displaystyle=$
$\displaystyle\varphi\left(\prod_{n\in\Lambda_{g}}j_{(n,n+1)}(|K_{n,n+1}|^{2})^{-1})j_{[g^{-1}n,g^{-1}(n+1)]}(|K_{n,n+1}|^{2})a\right)$
$\displaystyle=$ $\displaystyle\varphi(x_{g}a)$

Therefore $\varphi$ is $\mathcal{S}_{\infty}$–strongly quasi–invariant state
with cocycle given by (5.24). $\qquad\square$
Remark. Notice that all the examples of $\mathcal{S}_{\infty}$–strongly
quasi–invariant states constructed in this section, have the form
$\varphi=\psi(\rho\,\cdot\,)$ where $\psi$ is a
$\mathcal{S}_{\infty}$–invariant state and $\rho$ is a positive element of
$\mathcal{A}$ satisfying $\psi(\rho)=1$. Conversely, if $\varphi$ has this
form and $\rho$ is invertible, then $\varphi$ is
$\mathcal{S}_{\infty}$–strongly quasi–invariant with cocycle
$x_{g}=\rho^{-1}g^{-1}(\rho)$, which is the same structure as in (1.4) with
$\kappa_{G}^{-1})$ replaced by $\rho$.

##  6 A martingale convergence theorem

Let $\mathcal{B}$ be a given von Neumann algebra,
$\mathcal{A}=\bigotimes_{\mathbb{N}^{*}}\mathcal{B}$ be defined by (5.2), and
$\varphi$ be a normal faithful state on $\mathcal{A}$. Denote by
$\\{\mathcal{H}_{\varphi},\pi,\Phi\\}$ the cyclic representation of
$\\{\mathcal{A},\varphi\\}$. We will use the notations of section 4, where
$G=\mathcal{S}_{\infty}=\cup_{\mathbb{N}^{*}}\mathcal{S}_{N}$. Denote by

$\pi(j_{n}(b))=:\hat{j}_{n}(b)\in\pi(\mathcal{A})\quad,\quad b\in\mathcal{B}$

###### Definition 4

The state $\varphi$ is called strongly ergodic if for every $b\in\mathcal{B}$,
the limit

$\lim_{N\to\infty}\left(\frac{1}{N}\sum_{h=1}^{N}\hat{j}_{h}(b)\right)$

exists in the strong topology on $\mathcal{H}_{\varphi}$.

###### Theorem 8

Suppose that $\varphi$ is strongly ergodic and denote

$\overline{E}_{\infty}\colon\pi(\mathcal{A})^{\prime\prime}\to\mathcal{A}_{\infty}:=\bigcap_{k\in\mathbb{N}}\pi(\mathcal{A}_{[k,+\infty)})^{\prime\prime}$

the Umegaki conditional expectation defined by

$\overline{E}_{\infty}(\hat{j}_{0}(b)):=\hbox{s--}\lim_{N\to\infty}\left(\frac{1}{N}\sum_{h=1}^{N}\hat{j}_{h}(b)\right)\quad,\quad
b\in\mathcal{B}$ (6.1)

Then, for each $m\in\mathbb{N}$ and $b_{1},\ldots,b_{m}\in\mathcal{B}$,

$\hbox{weak--}\lim_{N\to\infty}\left(\overline{E}_{N}\bigl{(}\hat{j}_{n_{1}}(b_{1})\cdots\hat{j}_{n_{m}}(b_{m})\bigr{)}\right)=\overline{E}_{\infty}(\hat{j}_{0}(b_{1}))\cdots\overline{E}_{\infty}(\hat{j}_{0}(b_{m}))$
(6.2)

Proof. From Lemma $1$ of [2] one knows that

$E_{N}\circ j_{k}=\frac{1}{N}\sum_{h=1}^{N}j_{h}\qquad,\qquad\forall k\leq N$
(6.3)

Therefore, for $m=1$, the existence of the limit (6.2) is equivalent to the
strong convergence assumption (6.1). Suppose by induction that the limit (6.2)
exists for $k\leq m-1$. From Lemma $1$ and Lemma $2$ of [2] one knows that for
each $m,N\in\mathbb{N}$ with $m<N$ and $b_{1},\ldots,b_{m}\in\mathcal{B}$, one
has

$\|\overline{E}_{N}\bigl{(}\hat{j}_{n_{1}}(b_{1})\cdots\hat{j}_{n_{m}}(b_{m})\bigr{)}-\overline{E}_{N}\bigl{(}\hat{j}_{n_{1}}(b_{1})\bigr{)}\cdots\overline{E}_{N}\bigl{(}\hat{j}_{n_{m}}(b_{m})\bigr{)}\|$
$=\left\|\overline{E}_{N}\bigl{(}\hat{j}_{n_{1}}(b_{1})\cdots\hat{j}_{n_{m}}(b_{m})\bigr{)}-\left(\frac{1}{N}\sum_{h_{1}=1}^{N}\hat{j}_{h_{1}}(b_{1})\right)\cdots\left(\frac{1}{N}\sum_{h_{m}=1}^{N}\hat{j}_{h_{m}}(b_{m})\right)\right\|$
$\leq{m^{2}\over N}\parallel b_{1}\parallel\cdot\ldots\cdot\parallel
b_{m}\parallel$ (6.4)

Therefore

$\overline{E}_{N}\bigl{(}\hat{j}_{n_{1}}(b_{1})\cdots\hat{j}_{n_{m}}(b_{m})\bigr{)}$
$=\left(\frac{1}{N}\sum_{h_{1}=1}^{N}\hat{j}_{h_{1}}(b_{1})\right)\cdots\left(\frac{1}{N}\sum_{h_{m}=1}^{N}\hat{j}_{h_{m}}(b_{m})\right)+O\left(\frac{1}{N}\right)$

where $\|O\left(\frac{1}{N}\right)\|\to 0$ as $N\to+\infty$. Denote

$M_{N}(b):=\frac{1}{N}\sum_{h=1}^{N}\hat{j}_{h}(b)$

and notice that $\|M_{N}(b)\|\leq\|b\|$. Then, for
$\xi,\eta\in\mathcal{H}_{\varphi}$,

$\displaystyle\langle\xi,M_{N}(b_{1})\cdots
M_{N}(b_{m-1})M_{N}(b_{m})\eta\rangle$ $\displaystyle=\langle
M_{N}(b_{1})^{*}\cdots\xi,M_{N}(b_{2})\cdots M_{N}(b_{m})\eta\rangle$
$\displaystyle=\langle\overline{E}_{\infty}(\hat{j}_{0}(b_{1}^{*}))\xi,M_{N}(b_{2})\cdots
M_{N}(b_{m})\eta\rangle$
$\displaystyle+\langle(M_{N}(b_{1}^{*})-\overline{E}_{\infty}(\hat{j}_{0}(b^{*})))\xi,M_{N}(b_{1})^{*}\cdots
M_{N}(b_{m})\eta\rangle$

Since

$|\varepsilon_{1,N}|:=|\langle(M_{N}(b_{1}^{*})-\overline{E}_{\infty}(\hat{j}_{0}(b^{*})))\xi,M_{N}(b_{1})^{*}\cdots
M_{N}(b_{m})\eta\rangle|$
$\leq\|(M_{N}(b_{1}^{*})-\overline{E}_{\infty}(\hat{j}_{0}(b^{*})))\xi\|\,\|b_{1}\|\,\cdots\,\|b_{m}\|\,\|\eta\|$

$\varepsilon_{1,N}\to 0$ as $N\to+\infty$ by the first step in the induction
argument while, by the induction assumption,

$\displaystyle\lim_{N\to\infty}\langle\overline{E}_{\infty}(\hat{j}_{0}(b_{1}^{*}))\xi,M_{N}(b_{2})\cdots
M_{N}(b_{m})\eta\rangle$
$\displaystyle=\langle\overline{E}_{\infty}(\hat{j}_{0}(b_{1}^{*}))\xi,\overline{E}_{\infty}(\hat{j}_{0}(b_{2}))\cdots\overline{E}_{\infty}(\hat{j}_{0}(b_{m}))\eta\rangle$
$\displaystyle=\langle\xi,\overline{E}_{\infty}(\hat{j}_{0}(b_{1}))\overline{E}_{\infty}(\hat{j}_{0}(b_{2}))\cdots\overline{E}_{\infty}(\hat{j}_{0}(b_{m}))\eta\rangle$

$\square$

##  7 de Finetti’s Theorem for quasi–invariant states: classical case

###  7.1 Some general conditions

Let $G=\bigcup_{N\in\mathbb{N}G_{N}}$ be a unimodular locally compact group as
in Subsection 3.4 acting on the $*$–algebra
$\mathcal{A}=\bigotimes_{\mathbb{N}}\mathcal{B}$ given by (5.2) and let
$\varphi$ be a $G$–strongly quasi–invariant state on $\mathcal{A}$ with
cocycle $x_{g}$.
By Proposition 4, for each $N\in\mathbb{N}$, $\varphi$ is a $G_{N}$–strongly
quasi–invariant state with cocycle given by the restriction of $x$ to $G_{N}$
and Umegaki conditional expectation given by (3.31)

$\kappa_{N}=\int_{G_{N}}x_{g}d_{N}g\in\mathcal{C}\subseteq\hbox{Centrz}(\varphi)\subseteq\mathcal{A}$
(7.1)

###### Proposition 8

For all $N\in\mathbb{N}$, define

$\widetilde{E}_{N}(\,\cdot\,):=E_{N}(\kappa_{N}^{-1}\,\cdot\,)$ (7.2)

$\widetilde{E}_{N}$ is a projection onto $\hbox{Fix}(G_{N})$ satisfying

$\widetilde{E}_{N}(1)=1$ (7.3)
$\widetilde{E}_{N}(ab)=\widetilde{E}_{N}(a)b\quad,\quad a\in\mathcal{A}\ ,\
b\in\hbox{Fix}(\mathcal{S}_{N})$ (7.4)
$\varphi=\varphi\circ\widetilde{E}_{N}=(\varphi\circ
E_{N})(\kappa_{N}^{-1}\,\cdot\,)$ (7.5)

Moreover the family $(\widetilde{E}_{N})_{N}$ is projective, i.e.

$\widetilde{E}_{M}\widetilde{E}_{N}=\widetilde{E}_{N}\widetilde{E}_{M}=\widetilde{E}_{N}\quad,\quad\forall
M\leq N$ (7.6)

Proof. Since the map $a\in\mathcal{A}\mapsto\kappa_{N}^{-1}a\in\mathcal{A}$ is
invertible, it is surjective, hence the range of $\widetilde{E}_{N}$ coincides
with that of $E_{N}$, namely $\hbox{Fix}(G_{N})$. Furthermore

$\widetilde{E}_{N}(1)=E_{N}(\kappa_{N}^{-1})\overset{\eqref{EG(kappa-1)=1-cpct}}{=}1$

which is (7.3). For all $a\in\mathcal{A}$ and
$b\in\hbox{Fix}(\mathcal{S}_{N})$

$\widetilde{E}_{N}(ab)=E_{N}(\kappa_{N}^{-1}ab)=E_{N}(\kappa_{N}^{-1}a)b=\widetilde{E}_{N}(a)b$

which is (7.4). (7.5) follows from

$\varphi(a)\overset{\eqref{struc-q-inv-st-cpct}}{=}(\varphi\circ
E_{N})(\kappa_{N}^{-1}a)=\varphi(\widetilde{E}_{N}(a))\quad,\quad\forall
a\in\mathcal{A}$

That $\widetilde{E}_{N}$ is a projector follows from

$\displaystyle\widetilde{E}_{N}^{2}(a)=E_{N}(\kappa_{N}^{-1}E_{N}(\kappa_{N}^{-1}a))\overset{\eqref{tilde-
E-sim-
UCE}}{=}E_{N}(\kappa_{N}^{-1})E_{N}(\kappa_{N}^{-1}a)\overset{\eqref{EG(kappa-1)=1-LC}}{=}E_{N}(\kappa_{N}^{-1}a)=\widetilde{E}_{N}(a)$

To prove projectivity observe that, since for $N>M$,
$\hbox{Fix}(\mathcal{S}_{N})\subseteq\hbox{Fix}(\mathcal{S}_{M})$, it follows
that

$\widetilde{E}_{M}\widetilde{E}_{N}(a)=E_{M}(\kappa_{M}^{-1}E_{N}(\kappa_{N}^{-1}a))=E_{M}(\kappa_{M}^{-1})E_{N}(\kappa_{N}^{-1}a)=\widetilde{E}_{N}(a)$

To prove the first equality in (7.6), notice that for all
$a_{N}\in\mbox{Fix}(G_{N})=\mbox{Range}(E_{N})$

$\varphi(aa_{N})\overset{\eqref{varphi-tilde-EN-
invar}}{=}\varphi(E_{M}(\kappa_{M}^{-1}aa_{N}))=\varphi(E_{M}(\kappa_{M}^{-1}a)a_{N})\overset{\eqref{varphi-
tilde-EN-
invar}}{=}\varphi(E_{N}(\kappa_{N}^{-1}E_{M}(\kappa_{M}^{-1}a)a_{N}))$
$=\varphi(E_{N}(\kappa_{N}^{-1}E_{M}(\kappa_{M}^{-1}a))a_{N})=\varphi(\widetilde{E}_{N}\widetilde{E}_{M}(a)a_{N}))$

On the other hand

$\displaystyle\varphi(aa_{N})=\varphi(E_{N}(\kappa_{N}^{-1}aa_{N}))=\varphi(E_{N}(\kappa_{N}^{-1}a)a_{N})=\varphi(\widetilde{E}_{N}(a)a_{N})$

Since $\varphi$ is faithful and $a_{N}\in\mbox{Fix}(G_{N})$ is arbitrary, one
gets

$\displaystyle\widetilde{E}_{N}\widetilde{E}_{M}(a)=\widetilde{E}_{N}(a)\quad,\quad\forall
a\in\mathcal{A}$

which is equivalent to (7.6) because
$\hbox{Range}(\widetilde{E}_{N})\subseteq\hbox{Range}(\widetilde{E}_{M}(a))$.
$\qquad\square$

###### Remark 6

In general $\widetilde{E}_{N}$ is not a $*$–map, hence it is not a Umegaki
conditional expectation. Defining the centralizer of $E_{N}$ as

$\hbox{Centrz}(E_{N}):=\\{c\in\mathcal{A}\colon
E_{N}(ca)=E_{N}(ac)\,,\,\forall a\in\mathcal{A}\\}$

from (7.2) it is clear that, if $\kappa_{N}$ is in the centralizer of $E_{N}$
(in particular if $\mathcal{A}$ is abelian), then it is a $*$–map hence, by
Proposition 8, a Umegaki conditional expectation.

###### Lemma 10

If

$[\kappa_{N}^{-1},a]=0\quad,\quad\forall
a\in\mbox{Range}(E_{N})\overset{\eqref{Range-EG-in-
Fix(G)}}{=}\hbox{Fix}(G_{N})$ (7.7)

then for any $x\in\mathcal{A}$ and $a,b\in\hbox{Range}(\widetilde{E}_{N})$

$\widetilde{E}_{N}(axb)=a\widetilde{E}_{N}(x)b$ (7.8)

Proof. From Proposition 8 we know that $\widetilde{E}_{N}$ is a projection.
Let $x\in\mathcal{A}$ and
$a,b\in\hbox{Range}(\widetilde{E}_{N})=\hbox{Range}(E_{N})$. Then, since
$E_{N}$ is a Umegaki conditional expectation, one has

$\widetilde{E}_{N}(axb)=E_{N}(\kappa_{N}^{-1}axb)\overset{\eqref{UCE}}{=}E_{N}(a\kappa_{N}^{-1}xb)=aE_{N}(\kappa_{N}^{-1}x)b=a\widetilde{E}_{N}(x)b$

$\qquad\square$

###  7.2 Structure of the limit conditional expectation: classical case

In this section we use all the notations introduced in sections 4 and 5. In
particular we denote $\\{\mathcal{H}_{\varphi},\pi,\Phi\\}$ the cyclic
representation of $\\{\mathcal{A},\varphi\\}$, where $\varphi$ is the normal
faithful state induced by $\mu$ on
$\pi(\mathcal{A})^{\prime\prime}=\mathcal{A}_{\mu}=L^{\infty}(\Omega,\mathcal{F},\mu)$.
These notions are particularized here to the classical case described as
follows.
Let $(S,\mathcal{B}_{S})$ be a measurable space and define

$(\Omega,\mathcal{F}):=\prod_{\mathbb{N}}(S,\mathcal{B}_{S})$

with the $\sigma$–algebra $\mathcal{F}$ generated by the canonical projections
$\xi_{n}\colon\omega=(s_{k})_{k\in\mathbb{N}}\in\Omega\to\xi_{n}(\omega):=s_{n}$.
Given a probability measure $\mu$ on $(\Omega,\mathcal{F})$, for
$n\in\mathbb{N}$, denote

$j_{n}\colon f\in L^{\infty}(S,\mathcal{B}_{S})\to j_{n}(f):=f\circ\xi_{n}\in
L^{\infty}(\Omega,\mathcal{F},\mu)$

the $n$–th natural embedding. In this section we fix

$\mathcal{B}:=L^{\infty}(S,\mathcal{B}_{S})\quad;\quad\mathcal{A}:=\bigotimes_{\mathbb{N}}L^{\infty}(S,\mathcal{B}_{S})\equiv\bigvee_{n\in\mathbb{N}}j_{n}(L^{\infty}(S,\mathcal{B}_{S}))=:\bigvee_{n\in\mathbb{N}}\mathcal{A}_{n}=:\mathcal{A}$

$\mathcal{A}$ is a dense $*$–sub–algebra of
$L^{\infty}(\Omega,\mathcal{F},\mu)$.

###### Theorem 9

In the notations above, any $\mathcal{S}_{\infty}$–strongly quasi–invariant
state on $\mathcal{A}$ has the form

$\varphi(a)=\psi(K\pi(a))\quad,\quad a\in\mathcal{A}$ (7.9)

where $\psi$ is an $\mathcal{S}_{\infty}$–invariant state on
$\pi(\mathcal{A})^{\prime\prime}=L^{\infty}(\Omega,\mathcal{F},\mu)$ and $K$
is a $\mu$–a.e. positive function in $L^{1}(\Omega,\mathcal{F},\mu)$.

Remark. Given the decomposition (7.9), the associated cocyle is constructed as
in Theorem 5.

Proof. Since $\mathcal{A}$ is commutative, the family
$(\tilde{E}_{N})_{N\in\mathbb{N}}$ is a projective decreasing family of
classical conditional expectations. We use the same symbol $\tilde{E}_{N}$ to
denote the extension of the classical conditional expectation
$a\mapsto\pi(\tilde{E}_{N}(a))$ to $\pi(\mathcal{A})^{\prime\prime}$. By Doob
martingale theorem the limit

$\lim_{N\to+\infty}\tilde{E}_{N}=:\tilde{E}_{\infty}$ (7.10)

exists point–wise $\mu$–a.e. and, since each $\tilde{E}_{N}$ has norm $1$, the
limit also exists strongly in $\mathcal{H}_{\varphi}\equiv
L^{2}(\Omega,\mathcal{F},\mu)$. $\tilde{E}_{\infty}$ is therefore point–wise
strongly continuous as strong limit of a sequence of uniformly norm–bounded
point–wise strongly continuous maps. By Corollary 2 the range of
$\tilde{E}_{\infty}$ is the weak closure of
$\hbox{Fix}(u(\mathcal{S}_{\infty})):=\bigcap_{N\in\mathbb{N}}\hbox{Fix}_{\pi}(\mathcal{S}_{N})$.
Denote $\mathcal{F}(\mathcal{S}_{\infty})$ the $\sigma$–algebra generated by
$\hbox{Fix}_{\pi}(\mathcal{S}_{\infty})$ and $E^{\mu}_{\infty}$ the
conditional expectation of $\mu$ onto $\mathcal{F}(\mathcal{S}_{\infty})$. By
Moy’s theorem [15] (see section 7.3 below) there exists a $\mu$–a.e. positive
function $K\in L^{1}(\Omega,\mathcal{F},\mu)$ such that

$\tilde{E}_{\infty}=E^{\mu}_{\infty}(K\,\cdot\,)$

Since

$\varphi\overset{\eqref{varphi-tilde-EN-
invar}}{=}\varphi\circ\widetilde{E}_{N}\quad,\quad\forall N\in\mathbb{N}$
(7.11)

it follows by normality and (7.10) that

$\varphi=\lim_{N\to+\infty}\varphi\circ\widetilde{E}_{N}=\varphi\circ\tilde{E}_{\infty}=\varphi\circ
E^{\mu}_{\infty}(K\,\cdot\,)$ (7.12)

$\square$

###  7.3  Algebraic extension of Moy’s theorem

###### Theorem 10

Let $\mathcal{A}=L^{\infty}(\Omega,\mathcal{F},\mu^{0})$ be a commutative Von
Neumann algebra, $\varphi$ a normal state on $\mathcal{A}$ and
$\mathcal{A}_{0}=L^{\infty}(\Omega,\mathcal{F}_{0},\mu^{0}_{\mathcal{F}_{0}})$
a sub–algebra of $\mathcal{A}$.
In the following $\mu^{0}_{\mathcal{F}_{0}}$ denotes restriction
$\mu^{0}\big{|}_{\mathcal{F}_{0}}$ and similarly
$\varphi_{\mathcal{A}_{0}}:=\varphi\big{|}_{\mathcal{A}_{0}}$.
Denoting $E^{\varphi}_{\mathcal{A}_{0}}$ (resp. $E^{0}_{\mathcal{A}_{0}}$) the
$\varphi$–conditional expectation (resp. $\mu^{0}$–conditional expectation)
from $\mathcal{A}$ onto $\mathcal{A}_{0}$, one has

$E^{\varphi}_{\mathcal{A}_{0}}(a)=E_{\mathcal{A}_{0}}^{0}(Ka)\quad;\quad\forall
a\in\mathcal{A}$ (7.13)

where, denoting $\frac{d\varphi}{d\mu^{0}}$ (resp.
$\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}}^{0}}$) the
Radon–Nikodyn derivative of $\varphi$ with respect to $\mu^{0}$ (resp. of
$\varphi_{\mathcal{A}_{0}}$ with respect to $\mu_{\mathcal{A}_{0}}^{0}$),

$K:=\left(\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu^{0}_{\mathcal{A}_{0}}}\right)^{-1}\frac{d\varphi}{d\mu}\
\in\ L_{+}^{1}(\Omega,\mathcal{F},\mu^{0})$ (7.14)

and the inverse in (7.14) is understood on the support of
$d\varphi_{\mathcal{A}_{0}}/d\mu_{\mathcal{A}_{0}}$, i.e. for
$\mu^{0}_{\mathcal{F}_{0}}$–almost all $\omega\in\Omega$,

$\left(\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}}}\right)^{-1}(\omega):=\begin{cases}0,&\mbox{if
}\ \frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}}}(\omega)=0\\\
\left(\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}}}(\omega)\right)^{-1},&\mbox{otherwise}\end{cases}$

In particular

$E^{\varphi}_{\mathcal{A}_{0}}(1_{\mathcal{A}})=\chi_{supp(d\varphi_{\mathcal{A}_{0}}/d\mu_{\mathcal{A}_{0}})}$

Proof. By definition of conditional expectation one has, for each
$a_{0}\in\mathcal{A}_{0}$ and $a\in\mathcal{A}$,

$\varphi(a_{0}a)=\varphi_{\mathcal{A}_{0}}(a_{0}E^{\varphi}_{\mathcal{A}_{0}}(a))$
$=\mu^{0}\left(\frac{d\varphi}{d\mu^{0}}a_{0}a\right)=\mu^{0}\left(a_{0}a\frac{d\varphi}{d\mu^{0}}\right)=\mu_{\mathcal{A}_{0}}^{0}\left(a_{0}E_{\mathcal{A}_{0}}^{0}\left(a\frac{d\varphi}{d\mu^{0}}\right)\right)$
$=\mu_{\mathcal{A}_{0}}^{0}\left(a_{0}E_{\mathcal{A}_{0}}^{0}\left(a\frac{d\varphi}{d\mu^{0}}\left(\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}^{0}}}\right)^{-1}\right)\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}}^{0}}\right)$
$=\mu_{\mathcal{A}_{0}}^{0}\left(\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}}^{0}}a_{0}E_{\mathcal{A}_{0}}^{0}\left(a\frac{d\varphi}{d\mu^{0}}\left(\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}^{0}}}\right)^{-1}\right)\right)$
$=\varphi_{\mathcal{A}_{0}}\left(a_{0}E_{\mathcal{A}_{0}}^{0}\left(\left(\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}}^{0}}\right)^{-1}\frac{d\varphi}{d\mu^{0}}a\right)\right)$
(7.15)

From the uniqueness of conditional expectation it follows that:

$E_{\mathcal{A}_{0}}^{\varphi}(a)=E_{\mathcal{A}_{0}}^{0}\left(\left(\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}}^{0}}\right)^{-1}\frac{d\varphi}{d\mu^{0}}a\right)\quad,\quad\forall
a\in\mathcal{A}$

which, given (7.14), is equivalent to (7.13). Finally, putting $a=1$ in
(7.15), one finds

$\varphi(a_{0})=\mu^{0}\left(\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}}^{0}}a_{0}\right)=\mu_{\mathcal{A}_{0}}^{0}\left(E_{\mathcal{A}_{0}}^{0}\left(\frac{d\varphi}{d\mu^{0}}\right)a_{0}\right)$

which implies

$\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}}^{0}}=E_{\mathcal{A}_{0}}^{0}\left(\frac{d\varphi}{d\mu^{0}}\right)$

Therefore

$E_{\mathcal{A}_{0}}^{0}(K)\overset{\eqref{struct-K}}{=}E_{\mathcal{A}_{0}}^{0}\left(\left(\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}}}\right)^{-1}\frac{d\varphi}{d\mu}\right)=\left(\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}}}\right)^{-1}E_{\mathcal{A}_{0}}^{0}\left(\frac{d\varphi}{d\mu}\right)$
$=\left(\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}}}\right)^{-1}\left(\frac{d\varphi_{\mathcal{A}_{0}}}{d\mu_{\mathcal{A}_{0}}}\right)=\chi_{supp(d\varphi_{\mathcal{A}_{0}}/d\mu_{\mathcal{A}_{0}})}$

In particular $K\in\ L_{+}^{1}(\Omega,\mathcal{F},\mu^{0})$. $\qquad\square$

###### Corollary 3

In the notations of Theorem 10, a surjective map
$E_{\mathcal{A}_{0}}:\mathcal{A}\to\mathcal{A}_{0}$ is a normal conditional
expectation if and only if it has the form

$E_{\mathcal{A}_{0}}=E_{\mathcal{A}_{0}}^{0}(K\ \cdot\ )$ (7.16)

where $K\in L_{+}^{1}(\Omega,\mathcal{F},\mu^{0})$ and

$\hbox{supp}(K)=E_{\mathcal{A}_{0}}(1_{\mathcal{A}})$

Proof. The commutativity of $\mathcal{A}$ implies that any map of the form
(7.16) is a conditional expectation. Conversely, if $E_{\mathcal{A}_{0}}$ is a
normal conditional expectation, the state on $\mathcal{A}$ given by

$\varphi:=\mu^{0}\circ E_{\mathcal{A}_{0}}$ (7.17)

is normal and the restrictions of $\varphi$ and $\mu^{0}$ on
$E_{\mathcal{A}_{0}}$ coincide. Then Theorem (10) implies that the
$\varphi$–conditional expectation
$E^{\varphi}_{\mathcal{A}_{0}}:\mathcal{A}\to\mathcal{A}_{0}$ has the form
(7.13). Since

$\varphi\circ E_{\mathcal{A}_{0}}=\mu^{0}\circ
E_{\mathcal{A}_{0}}^{2}=\mu^{0}\circ E_{\mathcal{A}_{0}}=\varphi$

the uniqueness of the conditional expectation implies that

$E^{\varphi}_{\mathcal{A}_{0}}=E_{\mathcal{A}_{0}}$ (7.18)

Thus $E_{\mathcal{A}_{0}}$ has the form (7.16). $\qquad\square$

## References

  * [1] Accardi L., Pistone G.: De Finetti’s theorem, Sufficiency, and Dobrushin’s theory, In: Exchangeability in probability and statistics (1982) 125–156, G. Koch and F. Spizzichino (eds.) North Holland.
  * [2] Accardi L., Lu Y.G.: A continuous versions of De Finetti’s theorem, Ann. of Prob. 21 (3) (1993) 1478-1493, Volterra preprint N. 59 (1990).
  * [3] Accardi L., Ben Ghorbal A., Crismale V., Lu Y. G.: Singleton condition and quantum De Finetti’s theorem, Infin. Dimens. Anal. Quantum Probab. Relat. Top. (IDA-QP) 11 (4) (2008) 639-660.
  * [4] Bratteli O., Robinson D. W.: Operator Algebras and Quantum Statistical Mechanics I, Springer-Verlag Berlin Heidelberg 1979, Second Edition 1987.
  * [5] J.P. Conze J.P., N. Dang-Ngoc N.: Ergodie Theorems for Noncommutative Dynamical Systems, Inventiones mathematicae 46 (1978) 1–15.
  * [6] Dang-Ngoc N.: Pointwise convergence of martingales in von Neumann algebras, Israel Journal of Mathematics 34 (4) (1979) 273–280.
  * [7] Diaconis P., Freedman D.: De Finetti’s Theorem for Markov Chains, The Annals of Probability 8 (1) (1980) 115–130.
  * [8] Falcone T., Takesaki M.: Operator valued weights without structure theory, Trans. Amer. Math. Soc. 351 (1) (1999) 323–341.
  * [9] Fannes M., Lewis J.T., Verbeure A.: Symmetric states of composite systems, Lett. Math. Phys. 15 (1988) 255–260.
  * [10] Haagerup U.: Operator valued weights in von Neumann algebras I, J. Funct. Anal. 32 (1979) 175–206.
  * [11] Haagerup U.: Operator valued weights in von Neumann algebras II, J. Funct. Anal. 33 (1979) 339–361.
  * [12] Herman R. H., Takesaki M.: States and Automorphism Groups of Operator Algebras, Comm. Math. Phys. 19 (1970) 142–160.
  * [13] Høegh-Krohn, R., Landstad M. B., Størmer E.: Compact ergodic groups of automorphisms, Annals of Mathematics, 114 (1981), 75–86.
  * [14] Lance, E. Martingale convergence in von Neumann algebras. Mathematical Proceedings of the Cambridge Philosophical Society, 84 (1) (1978) 47–56 doi:10.1017/S0305004100054864
  * [15] Moy S-T. C.: Characterization of conditional expectations as a transformation on function spaces. Pac. J. Math. 4 (1954) 47.
  * [16] Sakai S.: $C^{*}$–Algebras and $W^{*}$–algebras, Springer (1971).
  * [17] Størmer E.: States and Invariant Maps of Operator Algebras, J. Funct. Anal. 5 (1970) 44–65.
  * [18] Størmer E.: Automorphisms and invariant states of operator algebras, Acta Mathematica 127 (1971) 1–9.
  * [19] Takesaki M.: Conditional Expectations in von Neumann Algebras, J. Funct. Anal. 9 (1972) 306–321.
  * [20] J. Tomijama: On the projection of norm one in $W^{*}$–algebras, Proc. Japan Acad. 33 (1957) 608–612.