# Hilbert-Schmidt and Trace Class Pseudo-differential Operators on the
Abstract Heisenberg Group

Aparajita Dasgupta  Aparajita Dasgupta Department of Mathematics Indian
Institute of Technology Delhi Hauz Khas, New Delhi - 110 016, India.
<EMAIL_ADDRESS>and  Vishvesh Kumar  Vishvesh Kumar School of
Mathematical Sciences National Institute of Science Education and Research
Bhubaneshwar, HBNI At/Po- Jatni, District- Khurda, Odisha- 7520250, India.
<EMAIL_ADDRESS>
###### Abstract.

In this paper we introduce and study pseudo-differential operators with
operator valued symbols on the abstract Heisenberg group
$\mathbb{H}(G):=G\times\widehat{G}\times\mathbb{T},$ where $G$ a locally
compact abelian group with its dual group $\widehat{G}$. We obtain a necessary
and sufficient condition on symbols for which these operators are in the class
of Hilbert-Schmidt operators. As a key step in proving this we derive a trace
formula for the trace class $j$-Weyl transform, $j\in\mathbb{Z}^{*}$ with
symbols in $L^{2}(G\times\widehat{G}).$ We go on to present a characterization
of the trace class pseudo-differential operators on $\mathbb{H}(G)$. Finally,
we also give a trace formula for these trace class operators.

###### Key words and phrases:

Pseudo-differential operators, $j$-Weyl transform, Hilbert Schmidt operators,
Trace class operators, Abstract Heisenberg group

###### 2010 Mathematics Subject Classification:

Primary 35S05, 47G30; Secondary 43A85, 43A77

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 $L^{2}$-boundedness
  4. 4 Trace of Weyl transform on a locally compact Abelian group
  5. 5 Hilbert-Schmidt pseudo-differential operators on the abstract Heisenberg groups
  6. 6 Trace class pseudo-differential operators on the abstract Heisenberg groups

##  1\. Introduction

The aim of this paper is to look at the pseudo-differential operators on the
abstract Heisenberg group $\mathbb{H}(G)$, for locally compact abelian group
$G.$ In 1964, A. Weil [17] studied certain group of unitary operators
associated with a locally compact abelian group in connection with the study
of the celebrated work of Seigel on quadratic form. In [17], the author
introduced this group where he considered $G$ to be an ad$\grave{\text{e}}$le
group or the additive group of vector space over a local field, which has
applications in number theory. Recently Radha, et al. [2, 15] studied Weyl
multipliers and shift invariant spaces on the group $\mathbb{H}(G).$ It is a
well-known fact from [18] that pseudo-differential operators on
${\mathbb{R}}^{n}$ are based on the Plancherel formula for the Fourier
transform on ${\mathbb{R}}^{n}$. The Plancherel formula gives rise to the
Fourier inversion formula, which says that the identity operator for
$L^{2}({\mathbb{R}}^{n})$ can be expressed in terms of the Fourier transform
on ${\mathbb{R}}^{n}$. The Fourier inversion formula, albeit useful in many
situations, gives a perfect symmetry, namely, the identity operator. The
classical pseudo-differential operator $T_{\sigma}$ associated to a symbol
$\sigma,$ (a measurable function on $\mathbb{R}^{n}\times\mathbb{R}^{n}$) is
defined by

(1)
$\left(T_{\sigma}\phi\right)(x)=(2\pi)^{-n/2}\int_{\mathbb{R}^{n}}e^{ix\xi}\sigma(x,\xi)\widehat{\phi}(\xi)d\xi,~{}~{}~{}x\in\mathbb{R}^{n},$

for all $\phi$ in the Schwartz space $S(\mathbb{R}^{n})$ on $\mathbb{R}^{n},$
provided the integral exists. The function $\widehat{\phi}$ in (1) is the
Fourier transform of $\phi$ defined by

$\widehat{\phi}(\xi)=(2\pi)^{-n/2}\int_{\mathbb{R}^{n}}e^{-ix\cdot\xi}\phi(x)dx,~{}~{}~{}\xi\in\mathbb{R}^{n}.$

The formation of a pseudo-differential operator as define in (1) is mainly
based on the Fourier inversion formula given by,

$\phi(x)=(2\pi)^{-n/2}\int_{\mathbb{R}^{n}}e^{ix\cdot\xi}\widehat{\phi}(\xi)d\xi,~{}~{}~{}x\in\mathbb{R}^{n},$

for all $\phi$ in $S(\mathbb{R}^{n}).$ By inserting a symbol, which is a
suitable function on the phase space ${\mathbb{R}}^{n}\times{\mathbb{R}}^{n},$
we break the symmetry and obtain the pseudo-differential operator. To extend
pseudo-differential operators to other settings, we first observe that
${\mathbb{R}}^{n}$ is a group and its dual is also $\mathbb{R}^{n}.$ It is
then natural to extend pseudo-differential operators to other groups which
have explicit dual objects and Fourier inversion formulas. Recently, such
works have been done for $\mathbb{S}^{1},~{}~{}\mathbb{Z},$ finite abelian
groups, locally compact abelian groups, affine groups, compact groups,
homogeneous spaces of compact groups, Heisenberg group and on general locally
compact type I groups [3, 5, 8, 13, 14, 10, 11, 16] among others.
A basic result in the theory of pseudo-differential operators on
${\mathbb{R}}^{n}$ is that if $\sigma\in
L^{2}({\mathbb{R}}^{n}\times{\mathbb{R}}^{n}),$ then $T_{\sigma}$ can be
extended to a bounded linear operator from $L^{2}({\mathbb{R}}^{n})$ into
$L^{2}({\mathbb{R}}^{n}).$ Moreover the resulting bounded linear operator is
in Hilbert-Schimdt class as explained in [12, 19]. Recently in [5], Dasgupta
and Wong has obtained a necessary and sufficient conditions on the symbols for
which the pseudo-differential operators on Heisenberg groups are in Hilbert-
Schmidt class. Motivated by this we wish to study the boundedness property of
the pseudo-differential operators on $\mathbb{H}(G)$ and obtain conditions on
the symbols for which these operators are in Hilbert-Schimdt class. In Section
2, we recall the basics of the abstract Heisenberg group,$\mathbb{H}(G)$, and
define the pseudo-differential operators on the group $\mathbb{H}({G}),$ where
$G$ is a locally compact abelian group. The $L^{2}$ boundedness property of
the pseudo-differential operators on $\mathbb{H}(G)$ is given in Section 3. In
Section 4 we obtained the trace formula of a trace class $j$-Weyl transform,
$j\in\mathbb{Z}^{*}$ associated to the symbol in $L^{2}(G\times\widehat{G})$ ,
where $G$ is a locally compact abelian group and this result is the key step
in obtaining the necessary and sufficient conditions for the pseudo-
differential operators on the abstract Heisenberg group, $\mathbb{H}(G)$ to be
in Hilbert-Schmidt class, which we proved in Section 5. In Section 6 trace
class pseudo-differential operators on $\mathbb{H}(G)$ are given and a trace
formula is obtained for them.

##  2\. Preliminaries

Let $G$ be a locally compact abelian group such that the map $x\mapsto jx,$
$j\in\mathbb{Z}^{\ast}=\mathbb{Z}\setminus\\{0\\},$ is an automorphism of $G$.
We denote the abstract Heisenberg group associated with $G$ by
$\mathbb{H}(G):=G\times\widehat{G}\times\mathbb{T}$ with the group operation
given by

$(x,\chi,\theta)(x^{\prime},\chi^{\prime},\theta)=(xx^{\prime},\chi\chi^{\prime},\theta\theta^{\prime}\chi^{\prime}(x)).$

The unitary dual $\widehat{\mathbb{H}(G)}$ of $\mathbb{H}(G)$ can be
identified with $\mathbb{Z}^{\ast},$ the set of non-zero integers, as follows:
for any $j\in\mathbb{Z}^{*},$ the irreducible representation of
$\mathbb{H}(G)$ on $L^{2}(G)$ is given by

$(\rho_{j}(x,\chi,\theta)\varphi)(y)=\theta^{j}(\chi(y))^{j}\varphi(xy)\,\,\,\,\,\text{for}\,\,\varphi\in
L^{2}(G).$

It is well known from Stone-von Neumann theorem that every infinite
dimensional irreducible unitary representation of $\mathbb{H}(G)$ is unitarily
equivalent to $\rho_{j},$ $j\in\mathbb{Z}^{*}.$ For further details we refer
Folland [6, 7].

For each $j\in\mathbb{Z}^{*},$ the Fourier transform, $\widehat{f}$ of $f\in
L^{1}(\mathbb{H}(G))$ at $j$ is an operator $\widehat{f}(j)$ on $L^{2}(G)$
given by

$\widehat{f}(j)\psi:=\int_{G\times\widehat{G}\times\mathbb{T}}f(x,\chi,\theta)\,\rho_{j}(x,\chi,\theta)\psi\,d\mu_{G}\,d\mu_{\widehat{G}}\,d\mu_{\mathbb{T}},$

where $d\mu_{G}\,d\mu_{\widehat{G}}\,d\mu_{\mathbb{T}}$ is a product of the
Haar measure on $G,$ $\widehat{G}$ and $\mathbb{T}.$ The above integral is a
Bochner integral taking values in the Hilbert space $L^{2}(G).$ This operator
$\widehat{f}(j)$ is a bounded operator on $L^{2}(G)$ with
$||\widehat{f}(j)||_{\mathcal{B}(L^{2}(G))}\leq||f||_{L^{1}(G)}.$ The inverse
Fourier transform, $f^{j},$ $j\in\mathbb{Z}^{\ast},$ of $f\in
L^{1}(\mathbb{H}(G))$ in the $\theta$ variable is given by

(2)
$f^{j}(x,y)=\int_{\mathbb{T}}f(x,\gamma,\theta)\theta^{j}d\mu_{\mathbb{T}}(\theta).$

Then $f^{j}\in L^{1}(G\times\widehat{G})$ and using (2), $\widehat{f}$ can be
written as

(3)
$\widehat{f}(j)\phi=\int_{G\times\widehat{G}}f^{j}(x,\gamma)\rho_{j}(x,\gamma,1)\phi
d\mu_{G}\,d\mu_{\widehat{G}},~{}~{}~{}\phi\in L^{2}(G).$

Further the Fourier transform $\widehat{f}$ of $f\in L^{2}(G)$ satisfies

$||\widehat{f}(j)||_{\mathcal{B}_{2}(L^{2}(G))}=(C_{j,G})^{-1}||f^{j}||_{L^{2}(G\times\widehat{G})}^{2},$

where $C_{j,G}$ is a constant depending on $j$ and $G$ such that
$C_{j,G}d\mu_{G}(j^{-1}x)=d\mu_{G}(x),$ for $j\in\mathbb{Z}^{*}$ and $C_{0,G}$
is taken be to zero.
Since the group is $\mathbb{H}(G)=G\times\widehat{G}\times\mathbb{T}$ so to
obtain the Plancheral theorem we need to consider the one-dimensional
representations of $\mathbb{H}(G)$ in addition to the infinite dimensional
irreducible unitary representations. Then the Plancherel formula for $f\in
L^{2}(\mathbb{H}(G))$ is given by

(4)
$\|f\|^{2}_{L^{2}(\mathbb{H}(G))}=\|\widehat{f}\|^{2}_{\ell^{2}(\mathbb{Z}^{*},S_{2},C_{j,G})}+\int_{\widehat{G}}\int_{G}|\widehat{(f^{0})}(\gamma,y)|^{2}\,d\mu_{G}(y)\,d\mu_{\widehat{G}}(\gamma),$

where
$f_{0}(x,\chi)=\int_{\mathbb{T}}f(x,\chi,\theta)\,d\mu_{\mathbb{T}}(\theta)$.
However for the case when $f-f^{0},$ $f\in L^{2}(\mathbb{H}(G))$ has mean
value zero in the central variable, the Plancheral formula reduces to the
following,

(5)
$||\widehat{f}||_{l^{2}(\mathbb{Z}\setminus\\{0\\},\mathcal{B}(L^{2}(G);C_{j,G})}=||f-f^{0}||_{L^{2}(\mathbb{H}(G))}.$

In other words, we can say there is an isometry from $L^{2}(\mathbb{H}(G))$
into $l^{2}(\mathbb{Z}\setminus\\{0\\},\mathcal{B}(L^{2}(G);C_{j,G}),$ where
$l^{2}(\mathbb{Z}\setminus\\{0\\},\mathcal{B}(L^{2}(G);C_{j,G})$ denotes the
space of all sequences indexed in $\mathbb{Z}^{\ast},$ taking values in
$\mathcal{B}(L^{2}(G)$ and square summable with weight $C_{j,G}.$ See [2, 17]
for more details.

The following Fourier Inversion formula for the Fourier transform on the
abstract Heisenberg group is the starting point for the analysis of the
pseudo-differential operators on the abstract Heisenberg group.
For a function with compact support, $f\in\mathcal{C}_{c}(\mathbb{H}(G)),$ the
Fourier Inversion formula is given by,

(6)
$f(x,\chi,\theta)=\sum_{j\in\mathbb{Z}^{*}}tr(\rho_{j}^{*}(x,\chi,\theta)\widehat{f}(j)),$

where $\rho_{j}^{\ast}(x,\chi,\theta)$ is the adjoint of
$\rho_{j}(x,\chi,\theta).$
Now let $\mathbb{B}(L^{2}(G))$ be the $C^{\ast}$-algebra of bounded linear
operators on $L^{2}(G).$ Then we call the mapping
$\sigma:\mathbb{H}(G)\times\mathbb{Z}^{\ast}\rightarrow B(L^{2}(G))$ an
operator valued symbol or simply a symbol.

We define a pseudo-differential operator $T_{\sigma}$ associated with the
operator-valued symbol
$\sigma:\mathbb{H}(G)\times\mathbb{Z}^{*}\rightarrow\mathcal{B}(L^{2}(G))$ as

$(T_{\sigma}f)(x,\chi,\theta):=\sum_{j\in\mathbb{Z}^{*}}tr(\rho_{j}^{*}(x,\chi,\theta)\sigma(x,\chi,\theta,j)\widehat{f}(j))\,\,\,\,\,\forall
f\in C_{c}(\mathbb{H}(G)).$

##  3\. $L^{2}$-boundedness

In this section we prove $L^{2}$\- boundedness of pseudo-differential
operators on $\mathbb{H}(G).$ We also show that if two symbols with some
conditions give arise to same pseudo-differential operator then symbol must be
same. Before stating our results of this section we would like to recall some
notation from operator theory.

Let $X$ be a complex and separable Hilbert space with the inner product
denoted by $(\cdot,\cdot)$ and let $\\{\psi_{k},k=1,2,..\\}$ be an orthonormal
basis for $X.$

An operator $T\in\mathcal{B}(X)$ is a Hilbert-Schmidt operator if for any
(hence all) orthonormal basis $\\{\psi_{k}\\}_{k=1}^{\infty}$ of $X$ we have
$\sum_{k}\|T\psi_{k}\|_{X}<\infty.$ The set of Hilbert-Schmidt operators,
denoted by $S_{2}(X),$ is two sided ideal of $\mathcal{B}(X).$ At times we
also denote $S_{2}(X)$ only by $S_{2}.$ The Hilbert-Schmidt norm on $S_{2}$ is
given by

$\|T\|_{HS}=\left(\sum_{k=1}^{\infty}\|T\psi_{k}\|_{X}^{2}\right)^{\frac{1}{2}}.$

Now, we are ready to state the following theorem on $L^{2}$-boundedness of
pseudo-differential operators on $\mathbb{H}(G).$

######  Theorem 3.1.

Let $\sigma:\mathbb{H}(G)\times\mathbb{Z}^{*}\rightarrow S_{2}$ be a operator-
valued symbol such that

$\sum_{j\in\mathbb{Z}^{*}}\int_{G\times\widehat{G}\times\mathbb{T}}C_{j,G}^{-1}\,\|\sigma(x,\chi,\theta,j)\|^{2}_{S_{2}}\,d\mu_{G}\,d\mu_{\widehat{G}}\,d\mu_{\mathbb{T}}<\infty.$

Then the pseudo-differential operator
$T_{\sigma}:L^{2}(\mathbb{H}(G))\rightarrow L^{2}(\mathbb{H}(G))$ is a bounded
operator.

###### Proof.

Using Minkowski integral inequality and the estimate
$\|\widehat{f}\|_{\ell^{2}(\mathbb{Z}^{*},S_{2},C_{j,G})}\leq\|f\|_{L^{2}(\mathbb{H}(G))}$
we have, for $f\in L^{2}(\mathbb{H}(G)),$

$\displaystyle\|T_{\sigma}f\|_{L^{2}(\mathbb{H}(G))}$
$\displaystyle=\left(\int_{G\times\widehat{G}\times\mathbb{T}}|(T_{\sigma}f)(x,\chi,\theta)|^{2}d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{\mathbb{T}}(\theta)\right)^{\frac{1}{2}}$
$\displaystyle=\left(\int_{G\times\widehat{G}\times\mathbb{T}}\left|\sum_{j\in\mathbb{Z}^{*}}tr(\rho_{j}^{*}(x,\chi,\theta)\sigma(x,\chi,\theta,j)\widehat{f}(j))\right|^{2}d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{\mathbb{T}}(\theta)\right)^{\frac{1}{2}}$
$\displaystyle\leq\sum_{j\in\mathbb{Z}^{*}}\left(\int_{G\times\widehat{G}\times\mathbb{T}}\left|tr(\rho_{j}^{*}(x,\chi,\theta)\sigma(x,\chi,\theta,j)\widehat{f}(j))\right|^{2}d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{\mathbb{T}}(\theta)\right)^{\frac{1}{2}}$
$\displaystyle\leq\sum_{j\in\mathbb{Z}^{*}}\left(\int_{G\times\widehat{G}\times\mathbb{T}}\|\sigma(x,\chi,\theta,j)\|_{S_{2}}^{2}\,\,\|\widehat{f}(j))\|_{S_{2}}^{2}d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{\mathbb{T}}(\theta)\right)^{\frac{1}{2}}$
$\displaystyle=\sum_{j\in\mathbb{Z}^{*}}\|\widehat{f}(j))\|_{S_{2}}\left(\int_{G\times\widehat{G}\times\mathbb{T}}\|\sigma(x,\chi,\theta,j)\|_{S_{2}}^{2}\,\,d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{\mathbb{T}}(\theta)\right)^{\frac{1}{2}}$
$\displaystyle\leq\left(\sum_{j\in\mathbb{Z}^{*}}C_{j,G}\|\widehat{f}(j))\|_{S_{2}}^{2}\right)^{\frac{1}{2}}\times$
$\displaystyle\left(\sum_{j\in\mathbb{Z}^{*}}\int_{G\times\widehat{G}\times\mathbb{T}}C_{j,G}^{-1}\|\sigma(x,\chi,\theta,j)\|_{S_{2}}^{2}\,\,d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{\mathbb{T}}(\theta)\right)^{\frac{1}{2}}$
$\displaystyle=\|\widehat{f}\|_{\ell^{2}(\mathbb{Z}^{*},S_{2},C_{j,G})}\left(\sum_{j\in\mathbb{Z}^{*}}\int_{G\times\widehat{G}\times\mathbb{T}}C_{j,G}^{-1}\|\sigma(x,\chi,\theta,j)\|_{S_{2}}^{2}\,\,d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{\mathbb{T}}(\theta)\right)^{\frac{1}{2}}$
$\displaystyle\leq\|f\|_{L^{2}(\mathbb{H}(G))}\left(\sum_{j\in\mathbb{Z}^{*}}\int_{G\times\widehat{G}\times\mathbb{T}}C_{j,G}^{-1}\|\sigma(x,\chi,\theta,j)\|_{S_{2}}^{2}\,\,d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{\mathbb{T}}(\theta)\right)^{\frac{1}{2}}.$

Therefore,

$\|T_{\sigma}\|_{\mathcal{B}(L^{2}(\mathbb{H}(G)))}\leq\left(\sum_{j\in\mathbb{Z}^{*}}\int_{G\times\widehat{G}\times\mathbb{T}}C_{j,G}^{-1}\|\sigma(x,\chi,\theta,j)\|_{S_{2}}^{2}\,\,d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{\mathbb{T}}(\theta)\right)^{\frac{1}{2}}<\infty.$

Hence, $T_{\sigma}$ is a bounded operator on $L^{2}(\mathbb{H}(G)).$ ∎

In the next theorem we prove that two symbols giving the same pseudo-
differential operators are equal.

######  Theorem 3.2.

Let $\sigma:\mathbb{H}(G)\times\mathbb{Z}^{*}\rightarrow S_{2}$ such that

(7)
$\sum_{j\in\mathbb{Z}^{*}}\int_{G\times\widehat{G}\times\mathbb{T}}C_{j,G}^{-1}\,\|\sigma(x,\chi,\theta,j)\|^{2}_{S_{2}}\,d\mu_{G}\,d\mu_{\widehat{G}}\,d\mu_{\mathbb{T}}<\infty.$

Furthermore, suppose that

(8)
$\sum_{j\in\mathbb{Z}^{*}}\|\sigma(x,\chi,\theta,j)\|_{S_{2}}<\infty,\,\,\,\,\,\,\,\,\,\sup_{x,\chi,\theta,j}\|\sigma(x,\chi,\theta,j)\|_{S_{2}}<\infty$

and the mapping
$(x,\chi,\theta,j)\mapsto\rho_{j}^{*}(x,\chi,\theta)\sigma(x,\chi,\theta,j)$
from $\mathbb{H}(G)\times\mathbb{Z}^{*}$ into $S_{2}$ is weakly continuous.
Then $T_{\sigma}f=0\,\,\,\forall f\in L^{2}(\mathbb{H}(G))$ if and only if
$\sigma(x,\chi,\theta,j)=0$ for almost all
$(x,\chi,\theta,j)\in\mathbb{H}(G)\times\mathbb{Z}^{*}.$

###### Proof.

If $\sigma(x,\chi,\theta,j)=0$ for almost all
$(x,\chi,\theta,j)\in\mathbb{H}(G)\times\mathbb{Z}^{*}$ then it is obvious
that $T_{\sigma}f=0$ for all $f\in f\in\mathbb{H}(G).$

It remains to show that if $T_{\sigma}f=0\,\,\,\forall f\in
L^{2}(\mathbb{H}(G))$ then $\sigma(x,\chi,\theta,j)=0$ for almost all
$(x,\chi,\theta,j)\in\mathbb{H}(G)\times\mathbb{Z}^{*}.$

For $(x,\chi,\theta)\in\mathbb{H}(G),$ define a function $f_{x,\chi,\theta}\in
L^{2}(\mathbb{H}(G))$ as

$\widehat{f}_{x,\chi,\theta}(j)=\sigma(x,\chi,\theta,j)^{*}\rho_{j}(x,\chi,\theta)\,\,\,\,j\in\mathbb{Z}^{*}.$

For all $(x^{\prime},\chi^{\prime},\theta^{\prime})\in\mathbb{H}(G),$

$(T_{\sigma}f_{x,\chi,\theta})(x^{\prime},\chi^{\prime},\theta^{\prime})=\sum_{j\in\mathbb{Z}^{*}}tr(\rho_{j}^{*}(x^{\prime},\chi^{\prime},\theta^{\prime})\sigma(x^{\prime},\chi^{\prime},\theta^{\prime},j)\sigma(x,\chi,\theta,j)^{*}\rho_{j}(x,\chi,\theta)).$

Now, let $(x_{0},\chi_{0},\theta_{0})\in\mathbb{H}(G).$ Then, using the weak
continuity of the mapping

$(x,\chi,\theta,j)\mapsto\rho_{j}^{*}(x,\chi,\theta)\sigma(x,\chi,\theta,j)$

we get that

$\displaystyle
tr(\rho_{j}^{*}(x^{\prime},\chi^{\prime},\theta^{\prime})\sigma(x^{\prime},\chi^{\prime},\theta^{\prime},j)\sigma(x,\chi,\theta,j)^{*}\rho_{j}(x,\chi,\theta))$
$\displaystyle\rightarrow
tr(\rho_{j}^{*}(x_{0},\chi_{0},\theta_{0})\sigma(x_{0},\chi_{0},\theta_{0},j)\sigma(x,\chi,\theta,j)^{*}\rho_{j}(x,\chi,\theta))$

as
$(x^{\prime},\chi^{\prime},\theta^{\prime},)\rightarrow(x_{0},\chi_{0},\theta_{0})$
in $\mathbb{H}(G).$ Moreover, by using
$\sup_{x,\chi,\theta,j}\|\sigma(x,\chi,\theta,j)\|_{S_{2}}<\infty$ there
exists a constant $K>0$ such that for all
$(x^{\prime},\chi^{\prime},\theta^{\prime},j)\in\mathbb{H}(G)\times\mathbb{Z}^{*},$

$|tr(\rho_{j}^{*}(x^{\prime},\chi^{\prime},\theta^{\prime})\sigma(x^{\prime},\chi^{\prime},\theta^{\prime},j)\sigma(x,\chi,\theta,j)^{*}\rho_{j}(x,\chi,\theta))|\leq
K\|\sigma(x,\chi,\theta,j)\|_{S_{2}}.$

Now, using the assumption that
$\sum_{j\in\mathbb{Z}^{*}}\|\sigma(x,\chi,\theta,j)\|_{S_{2}}<\infty,$ we get,
an application of Lebesgue dominated convergence theorem, that

$\displaystyle\sum_{j\in\mathbb{Z}^{*}}tr(\rho_{j}^{*}(x^{\prime},\chi^{\prime},\theta^{\prime})\sigma(x^{\prime},\chi^{\prime},\theta^{\prime},j)\sigma(x,\chi,\theta,j)^{*}\rho_{j}(x,\chi,\theta))$
$\displaystyle\rightarrow\sum_{j\in\mathbb{Z}^{*}}tr(\rho_{j}^{*}(x_{0},\chi_{0},\theta_{0})\sigma(x_{0},\chi_{0},\theta_{0},j)\sigma(x,\chi,\theta,j)^{*}\rho_{j}(x,\chi,\theta))$

as
$(x^{\prime},\chi^{\prime},\theta^{\prime},)\rightarrow(x_{0},\chi_{0},\theta_{0})$
in $\mathbb{H}(G).$ Therefore, $T_{\sigma}f_{x,\chi,\theta}$ is continuous.

Next, by letting $(x_{0},\chi_{0},\theta_{0})=(x,\chi,\theta)$ we get

$\displaystyle T_{\sigma}f_{x,\chi,\theta}(x,\chi,\theta)$
$\displaystyle=\sum_{j\in\mathbb{Z}^{*}}tr(\rho_{j}^{*}(x,\chi,\theta)\sigma(x,\chi,\theta,j)\sigma(x,\chi,\theta,j)^{*}\rho_{j}(x,\chi,\theta))$
$\displaystyle=\sum_{j\in\mathbb{Z}^{*}}tr(\sigma(x,\chi,\theta,j)\sigma(x,\chi,\theta,j)^{*})$
$\displaystyle=\sum_{j\in\mathbb{Z}^{*}}\|\sigma(x,\chi,\theta,j)\|_{S_{2}}^{2}=0$

So, $\|\sigma(x,\chi,\theta,j)\|_{S_{2}}=0$ for almost every
$j\in\mathbb{Z}^{*}$ and hence $\sigma(x,\chi,\theta,j)=0$ for almost all
$(x,\chi,\theta,j)\in\mathbb{H}(G)\times\mathbb{Z}^{*}.$ ∎

##  4\. Trace of Weyl transform on a locally compact Abelian group

Let $X$ be a complex and separable Hilbert space in which the inner product is
denoted by $(,)$ and let $A:X\rightarrow X$ be a compact operator. If we
denote by $A^{\ast}:X\rightarrow X$ the adjoint of $A:X\rightarrow X$ then the
linear operator $(A^{\ast}A)^{1/2}:X\rightarrow X$ is positive and compact.
Let $\\{\psi_{k},k=1,2,..\\}$ be an orthonormal basis for $X$ consisting of
eigenvalues of $(A^{\ast}A)^{1/2}:X\rightarrow X$ and let $s_{k}(A)$ be the
eigenvalue corresponding to the eigenvector $\psi_{k},k=1,2,3,...$. Then
$s_{k}(A)$ $k=1,2,3,...$, are the singular values of $A:X\rightarrow X$. If

$\sum\limits_{k=1}^{\infty}s_{k}(A)<\infty,$

then the linear operator $A:X\rightarrow X$ is said to be in the trace class
$S_{1}.$ It can be shown that $S_{1}$ is a Banach space in which the norm
$||\cdot||_{S_{1}}$ is given by

$||A||_{S_{1}}=\sum\limits_{k=1}^{\infty}s_{k}(A),~{}~{}~{}A\in S_{1}.$

Let $A:X\rightarrow X$ be a linear operator in $S_{1}$ and let
$\\{\phi_{k}:k=1,2,3,...\\}$ be any orthonormal basis for $X$. Then from [12],
the series $\sum\limits_{k=1}^{\infty}(A\phi_{k},\phi_{k})$ is absolutely
convergent and the sum is independent of the choice of the orthonormal basis
$\\{\phi_{k}:k=1,2,3,...\\}.$ Thus the trace of any linear operator
$A:X\rightarrow X$ in $S_{1}$ is defined by

$\text{tr}(A)=\sum\limits_{k=1}^{\infty}(A\phi_{k},\phi_{k}),$

where $\\{\phi_{k},k=1,2,3...\\}$ is any orthonormal basis of $X$.

The following well-known theorem describe a relation between trace class
operator and Hilbert-Schmidt operators.

######  Theorem 4.1.

Let $T\in\mathcal{B}(\mathcal{H}).$ Then $T$ is a trace class operator if and
only if there exist two Hilbert-Schmidt operators $U$ and $V$ on $\mathcal{H}$
such that $T=UV.$

In this section we have obtained the trace formula for the trace class Weyl
transform associated to a symbol in $L^{2}(G\times\widehat{G}),$ where $G$ is
a locally compact abelian group.
Here first we recall that the Schrödinger representation $\rho_{j},$
$j\in\mathbb{Z}^{*}$ of the abstract Heisenberg group $\mathbb{H}(G)$ on
$L^{2}(G)$ is defined by

$(\rho_{j}(x,\chi,\theta)\varphi)(y)=\theta^{j}\chi(y)^{j}\varphi(xy)\,\,\,\,\,\,\varphi\in
L^{2}(G).$

Using $\rho_{j},$ we define the $j$-Weyl transform  $W^{j}:\sigma\mapsto
W^{j}_{\sigma}$ from $C_{c}(G\times\widehat{G})$ into $\mathcal{B}(L^{2}(G))$
as

$W_{\sigma}^{j}(\varphi)(y)=\int_{G\times\widehat{G}}\sigma(x,\chi)(\rho_{j}(x,\chi,1)\varphi)(y)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{G}(x).$

Which is further can be written as

$W^{j}_{\sigma}(\varphi)(y)=\int_{G\times\widehat{G}}\sigma(x,\chi)\chi(y)^{j}\varphi(xy)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{G}(x)=\int_{G}K^{j}_{\sigma}(x,y)\,d\mu_{G}(x),$

where

(9)
$K^{j}_{\sigma}(x,y)=\int_{\widehat{G}}\sigma(xy^{-1},\chi)\chi(y)^{j}\,d\mu_{\widehat{G}}(\chi).$

Therefore, $W_{\sigma}^{j}$ is an integral operator with kernel
$K^{j}_{\sigma}.$ Note that for $\sigma\in L^{2}(G\times\widehat{G}),$ the
$j$-Weyl transform $W_{\sigma}^{j}$ is a Hilbert Schmidt operator. In fact,

(10)
$\|W_{\sigma}^{j}\|_{S_{2}}=\|K_{\sigma}^{j}\|_{L^{2}(G\times\widehat{G})}=(C_{j,G})^{-1}\|\sigma\|_{L^{2}(G\times\widehat{G})}$

and more generally, $\langle
W_{f}^{j},W_{g}^{j}\rangle_{S_{2}}=(C_{j,G})^{-1}\langle
f,g\rangle_{L^{2}(G\times\widehat{G})}.$

For two function $f$ and $g$ in $L^{2}(G\times\widehat{G}),$ the $j$-twisted
convolution $f\times_{j}g,$ $j\in\mathbb{Z}^{*}$ of $f$ and $g$ is defined by

$f\times_{j}g(x,\chi)=\int_{G}\int_{\widehat{G}}f(x^{\prime},\chi^{\prime})g(xx^{\prime-1},\chi\chi^{\prime-1})\overline{\chi(x^{\prime})}^{j}\,d\mu_{G}(x^{\prime})\,d\mu_{\widehat{G}}(\chi^{\prime}).$

Note for $j=1,$ $W^{j}_{\sigma}$ turns out to be the well known Weyl transform
$W_{\sigma}$ and $j$\- twisted convolution is nothing but twisted convolution
studied, see [2, 15]. In the same line of [15] it can be shown that
$(L^{2}(G\times\widehat{G}),\times_{j})$ is a Banach algebra. Further, the
$j$-Weyl transform is a Banach algebra isomorphism from
$L^{2}(G\times\widehat{G})$ onto the space of all Hilbert-Schmidt operators on
$L^{2}(G)$ denoted by $S_{2}(L^{2}(G))$ or $S_{2}.$ Therefore, for any $A\in
S_{2}(L^{2}(G)),$ there exist a unique $\sigma\in L^{2}(G\times\widehat{G})$
such that $A=W_{\sigma}^{j}.$ Also,
$W_{\sigma}^{j}W_{\tau}^{j}=W_{\sigma\times_{j}\tau}^{j}.$

Denote the subset of all $\lambda\in L^{2}(G\times\widehat{G})$ such that
there exist functions $\sigma,\tau\in L^{2}(G\times\widehat{G})$ such that
$\lambda=\sigma\times_{j}\tau$ by $W_{j}.$

Now we present the following theorem on the characterization of trace class
$j$-Weyl transform.

######  Theorem 4.2.

Let $W_{\sigma}^{j}$ be the $j$-Weyl transform on $G$ associated with symbol
$\sigma\in L^{2}(G\times\widehat{G}).$ Then $W_{\sigma}^{j}$ is a trace class
operator if and only if $\sigma\in W_{j}.$

###### Proof.

Let $\sigma\in W_{j}.$ Then there exist $\lambda$ and $\tau$ in
$L^{2}(G\times\widehat{G})$ such that $\sigma=\lambda\times_{j}\tau.$ So,

$W_{\sigma}^{j}=W_{\lambda\times_{j}\tau}^{j}=W_{\lambda}^{j}W_{\tau}^{j}.$

Since $\tau$ and $\lambda$ are in $L^{2}(G\times\widehat{G}).$ It follows that
$W_{\lambda}$ and $W_{\tau}$ are Hilbert Schmidt operators. Therefore,
$W_{\sigma}$ is a trace class operator being product of two Hilbert Schmidt
operators.

Conversely, suppose that $W_{\sigma}^{j}$ is a trace class operator. Then, it
follows that $W_{\sigma}=AB$ for some Hilbert Schmidt operators $A$ and $B$ on
$L^{2}(G).$ Since $j$-Weyl transform is an algebra isomorphism from
$L^{2}(G\times\widehat{G})$ onto $S_{2},$ it follows that there exist
$\lambda,\tau\in L^{2}(G\times\widehat{G})$ such that $A=W_{\lambda}^{j}$ and
$B=W_{\tau}^{j}$ and therefore,
$W_{\sigma}^{j}=W_{\lambda}^{j}W_{\tau}^{j}=W_{\lambda\times_{j}\tau}^{j}.$
Hence, $\sigma\in W_{j}.$ ∎

The following corollary of Theorem 4.2 is immediate once we recall that
$\sigma_{j}\mapsto W_{\sigma}^{j}$ is a Banach algebra isomorphism from
$(L^{2}(G\times\widehat{G}),\times_{j})$ onto $S_{2}(L^{2}(G)).$

######  Corollary 4.3.

The $W_{j}$ is a subspace of $L^{2}(G\times\widehat{G}).$

In addition to this corollary, we have the following result the space $W_{j}.$

######  Theorem 4.4.

$W_{j}$ is a dense subspace of $L^{2}(G\times\widehat{G}).$

###### Proof.

In view of Corollary 4.3, we only need to prove that $W_{j}$ is dense in
$L^{2}(G\times\widehat{G}).$ Let $F$ be the set of all functions $\sigma$ on
$G\times\widehat{G}$ such that $W_{\sigma}^{j}$ is a finite rank operator on
$L^{2}(G)$. Since every element in $S_{2}(L^{2}(G))$ is the limit in
$S_{2}(L^{2}(G))$ of a sequence of finite rank operators on $L^{2}(G)$ and
$W_{\sigma}^{j}$ is in $S_{2}(L^{2}(G))$ if and only if $\sigma\in
L^{2}(G\times\widehat{G}),$ it follows that $F$ is a dense subspace of
$L^{2}(G\times\widehat{G})$. Obviously, $F$ is a subspace of $W_{j}$.
Therefore $W_{j}$ is dense in $L^{2}(G\times\widehat{G}).$ ∎

Now we calculate the trace of trace of $j$-Weyl transform $W_{\sigma}^{j}$
associated with symbol $\sigma.$

######  Theorem 4.5.

Let $\sigma\in L^{2}(G\times\widehat{G})$ such that the $j$-Weyl transform
$W_{\sigma}^{j}$ is a trace class operator. Then

$tr(W_{\sigma}^{j})=\int_{G}\int_{\widehat{G}}\sigma(x,\chi)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{G}(x).$

###### Proof.

Since $\sigma\in L^{2}(G\times\widehat{G}),$ it follows that $W_{\sigma}^{j}$
is a Hilbert Schmidt operator with kernel $K_{\sigma}^{j}(x,y)$ given by

$K_{\sigma}^{j}(x,y)=\int_{\widehat{G}}\sigma(xy^{-1},\chi)\chi(y)^{j}\,d\mu_{\widehat{G}}(\chi).$

Now, as by assumption $W_{\sigma}^{j}$ is a trace class operator so
$\int_{G}K_{\sigma}^{j}(x,x)d\mu_{G}(x)$ exists and

$\displaystyle tr(W_{\sigma}^{j})$
$\displaystyle=\int_{G}K_{\sigma}^{j}(x,x)\,d\mu_{G}(x)$
$\displaystyle=\int_{G}\int_{\widehat{G}}\sigma(e,\chi)\,\chi(x)^{j}\,d\mu_{\widehat{G}}(\chi)\,d\mu_{G}(x)$
$\displaystyle=\int_{G}\int_{\widehat{G}}\sigma(x^{-1},\chi)\,\chi(e)^{j}\,d\mu_{\widehat{G}}(\chi)\,d\mu_{G}(x)$
$\displaystyle=\int_{G}\int_{\widehat{G}}\sigma(x,\chi)\,\chi(e)^{j}\,d\mu_{\widehat{G}}(\chi)\,d\mu_{G}(x)$

where $e$ denotes the identity element of $G.$ Since $\chi(e)=1,$ we get

$tr(W_{\sigma}^{j})=\int_{G}\int_{\widehat{G}}\sigma(x,\chi)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{G}(x).$

∎

We will give another formula for the trace of a trace class Weyl transform
which will be useful for our study in next section.

######  Theorem 4.6.

Let $\sigma=\lambda\times_{j}\tau$ for some $\lambda,\tau\in
L^{2}(G\times\widehat{G})$ such that the $j$-Weyl transform $W_{\sigma}^{j}$
is a trace class operator. Then

(11) $\displaystyle
tr(W_{\sigma}^{j})=(C_{j,G})^{-1}\int_{G\times\widehat{G}}\tau(x,\chi)\,\lambda(x^{-1},\chi)\,\overline{\chi(x)}\,d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi).$

###### Proof.

First note that

(12)
$W_{\sigma}^{j}=W_{\lambda}^{j}W_{\tau}^{j}=W_{\lambda\times_{j}\tau}^{j}$

Since $W_{\sigma}^{j}$ is a trace class operator and hence a Hilbert Schmidt
operator on $L^{2}(G)$. Let $\\{\varphi_{k}:k\in\mathbb{N}\\}$ is an
orthonormal basis for $L^{2}(G).$ Then, by using the fact that
$(W^{j}_{\lambda})^{*}=W_{\tilde{\lambda}},$ where
$\tilde{\lambda}(x,\chi)=\chi(x)\,\overline{\lambda(x^{-1},\chi)},$ we have

$\displaystyle tr(W_{\sigma}^{j})$ $\displaystyle=\sum_{k\in\mathbb{N}}\langle
W_{\sigma}^{j}\varphi_{k},\varphi_{k}\rangle=\sum_{k\in\mathbb{N}}\langle
W_{\lambda}^{j}W_{\tau}^{j}\varphi_{k},\varphi_{k}\rangle$
$\displaystyle=\sum_{k\in\mathbb{N}}\langle
W_{\tau}^{j}\varphi_{k},(W^{j}_{\lambda})^{*}\varphi_{k}\rangle=\sum_{k\in\mathbb{N}}\langle
W_{\tau}^{j}\varphi_{k},W_{\tilde{\lambda}}^{j}\varphi_{k}\rangle=\langle
W_{\tau}^{j},W_{\tilde{\lambda}}^{j}\rangle_{S_{2}}.$

By Using the relation $\langle
W_{f}^{j},W_{g}^{j}\rangle_{S_{2}}=(C_{j,G})^{-1}\langle
f,g\rangle_{L^{2}(G\times\widehat{G})},$ we get

(13) $\displaystyle tr(W_{\sigma}^{j})$
$\displaystyle=(C_{j,G})^{-1}\langle\tau,\tilde{\lambda}\rangle_{L^{2}(G\times\widehat{G})}$
(14)
$\displaystyle=(C_{j,G})^{-1}\int_{G\times\widehat{G}}\overline{\chi(x)}\,\tau(x,\chi)\,\lambda(x^{-1},\chi)\,d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi).$

∎

##  5\. Hilbert-Schmidt pseudo-differential operators on the abstract
Heisenberg groups

In this section, we characterize the Hilbert-Schmidt pseudo-differential
operators in terms of their corresponding symbols. We begin this section with
following observation.

Let $f\in L^{2}(\mathbb{H}(G)).$ For $j\in\mathbb{Z}^{*},$ we know that
$f^{j}$ is defined as

$f^{j}(x,\chi)=\int_{\mathbb{T}}f(x,\chi,\theta)\,\theta^{j}\,d\theta,~{}~{}(x,\chi,\theta)\in
G\times\widehat{G}\times\mathbb{T}.$

Note that $f^{j}$ is the inverse Fourier transform of $f$ in $\theta$ variable
or Fourier transform of $f$ with respect to the center of $\mathbb{H}(G).$
Therefore, it is convenient to write $f^{j}$ is the following form:

$f^{j}(x,\chi)=(\mathcal{F}_{c}^{-1}f)(x,\chi,j)=(\mathcal{F}_{c}f)(x,\chi,-j),$

where $\mathcal{F}_{c}$ denote the Fourier transform with respect to center of
$\mathbb{H}(G).$

Before stating our main theorem of this section, we would like to note
$\widehat{f}(j)\varphi=W_{f^{j}}^{j}(\varphi)$ for all $\varphi\in
L^{2}(\mathbb{H}(G)).$ In fact,

$\displaystyle\widehat{f}(j)\varphi(y)$
$\displaystyle=\int_{G\times\widehat{G}\times\mathbb{T}}f(x,\chi,\theta)\,(\rho_{j}(x,\chi,\theta)\varphi)(y)d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{\mathbb{T}}(\theta)$
$\displaystyle=\int_{G\times\widehat{G}\times\mathbb{T}}f(x,\chi,\theta)\,\theta^{j}(\rho_{j}(x,\chi,1)\varphi)(y)d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{\mathbb{T}}(\theta)$
$\displaystyle=\int_{G\times\widehat{G}}f^{j}(x,\chi)\,\chi(x)^{j}\varphi(xy)d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)$
$\displaystyle=(W_{f^{j}}^{j}\varphi)(y).$

The following theorem is the main theorem of this section which characterizes
Hilbert-Schmidt pseudo-differential operators on $L^{2}(\mathbb{H}(G)).$

######  Theorem 5.1.

Let $\sigma:\mathbb{H}(G)\times\mathbb{Z}^{*}\rightarrow S_{2}$ be a symbol
such that the hypothesis of Theorem 3.2 is satisfied. Then the corresponding
pseudo-differential operator
$T_{\sigma}:L^{2}(\mathbb{H}(G))\rightarrow\mathbb{H}(G)$ is Hilbert-Schmidt
operator if and only if

$\sigma(x,\chi,\theta,j)=C_{j,G}\,\rho_{j}(x,\chi,\theta)W^{j}_{\alpha(x,\chi,\theta)^{-j}}\,\,\,\,\,(x,\chi,\theta)\in\mathbb{H}(G),\,j\in\mathbb{Z}^{*},$

where $\alpha:\mathbb{H}(G)\rightarrow L^{2}(\mathbb{H}(G))$ is a weakly
continuous mapping for which

$\int_{G\times\widehat{G}\times\mathbb{T}}\|\alpha(x,\chi,\theta)\|_{L^{2}(\mathbb{H}(G))}^{2}d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{\mathbb{T}}(\theta)<\infty,$
$\sup_{(x,\chi,\theta,j)\in\mathbb{H}(G)\times\mathbb{Z}^{*}}\|\mathcal{F}_{c}\alpha(x,\chi,\theta)(\cdot,\cdot,j)\|_{L^{2}(G\times\widehat{G})}<\infty$

and

$\sum_{j\in\mathbb{Z}^{*}}\|(\mathcal{F}_{c}\alpha(x,\chi,\theta))(\cdot,\cdot,j)\|_{L^{2}(G\times\widehat{G})}<\infty.$

###### Proof.

We first show the sufficiency part. Let $f\in C_{c}(\mathbb{H}(G)).$ Then for
all $(x,\chi,\theta)\in\mathbb{H}(G),$

$(T_{\sigma}f)(x,\chi,\theta)=\sum_{j\in\mathbb{Z}^{*}}tr(\rho_{j}^{*}(x,\chi,\theta)\sigma(x,\chi,\theta,j)\widehat{f}(j)).$

Using the expression of $\sigma$ and the fact that
$\widehat{f}(j)=W^{j}_{f^{j}},$ we have

$\displaystyle(T_{\sigma}f)(x,\chi,\theta)$
$\displaystyle=\sum_{j\in\mathbb{Z}^{*}}C_{j,G}tr(\rho_{j}^{*}(x,\chi,\theta)\rho_{j}(x,\chi,\theta)W^{j}_{\alpha(x,\chi,\theta)^{-j}}W^{j}_{f^{j}})$
$\displaystyle=\sum_{j\in\mathbb{Z}^{*}}C_{j,G}tr(W^{j}_{\alpha(x,\chi,\theta)^{-j}}W^{j}_{f^{j}})$
$\displaystyle=$
$\displaystyle\sum_{j\in\mathbb{Z}^{*}}tr(W^{j}_{\alpha(x,\chi,\theta)^{-j}\times_{j}C_{j,G}f^{j}}).$

By (11), we get

$\displaystyle(T_{\sigma}f)(x,\chi,\theta)$
$\displaystyle=\sum_{j\in\mathbb{Z}^{*}}\int_{G\times\widehat{G}}(\alpha(x,\chi,\theta)^{-j})(x^{\prime-1},\chi^{\prime})\,f^{j}(x^{\prime},\chi^{\prime})\,\chi^{\prime}(x^{\prime-1})\,d\mu_{G}(x^{\prime})\,d\mu_{\widehat{G}}(\chi^{\prime})$
$\displaystyle=\sum_{j\in\mathbb{Z}^{*}}\int_{G\times\widehat{G}}(\alpha(x,\chi,\theta)^{-j})(x^{\prime-1},\chi^{\prime})\,f^{j}(x^{\prime},\chi^{\prime})\,\chi^{\prime}(x^{\prime-1})\,d\mu_{G}(x^{\prime})\,d\mu_{\widehat{G}}(\chi^{\prime})$
$\displaystyle=\sum_{j\in\mathbb{Z}^{*}}\int_{G\times\widehat{G}}(\mathcal{F}_{c}\alpha(x,\chi,\theta))(x^{\prime-1},\chi^{\prime},j)\,(\mathcal{F}_{c}f)(x^{\prime},\chi^{\prime},j)\chi^{\prime}(x^{\prime-1})\,d\mu_{G}(x^{\prime})\,d\mu_{\widehat{G}}(\chi^{\prime})$
$\displaystyle=\int_{G\times\widehat{G}\times\mathbb{T}}(\alpha(x,\chi,\theta))(x^{\prime-1},\chi^{\prime},\theta^{\prime})\,f(x^{\prime},\chi^{\prime},\theta^{\prime})\chi^{\prime}(x^{\prime-1})d\mu_{G}(x^{\prime})\,d\mu_{\widehat{G}}(\chi^{\prime})d\mu_{\mathbb{T}}(\theta^{\prime}).$

Thus, the kernel of the operator $T_{\sigma}$ is a function $k$ on
$(G\times\widehat{G}\times\mathbb{T})\times(G\times\widehat{G}\times\mathbb{T})$
given as

(15)
$k((x,\chi,\theta),(x^{\prime},\chi^{\prime},\theta^{\prime}))=\chi^{\prime}(x^{\prime-1})\alpha(x,\chi,\theta)(x^{\prime-1},\chi^{\prime},\theta^{\prime}).$

Now, using Fubini theorem

$\displaystyle\int_{\mathbb{H}(G)\times\mathbb{H}(G)}|k((x,\chi,\theta),(x^{\prime},\chi^{\prime},\theta^{\prime}))|^{2}d\mu_{G}(x^{\prime})\,d\mu_{\widehat{G}}(\chi^{\prime})d\mu_{\mathbb{T}}(\theta^{\prime})d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)d\mu_{\mathbb{T}}(\theta)$
$\displaystyle=\int_{\mathbb{H}(G)\times\mathbb{H}(G)}|\chi^{\prime}(x^{\prime-1})\alpha(x,\chi,\theta)(x^{\prime-1},\chi^{\prime},\theta^{\prime})|^{2}d\mu_{G}(x^{\prime})\,d\mu_{\widehat{G}}(\chi^{\prime})d\mu_{\mathbb{T}}(\theta^{\prime})d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)d\mu_{\mathbb{T}}(\theta)$
$\displaystyle=\int_{G\times\widehat{G}\times\mathbb{T}}\|\alpha(x,\chi,\theta)\|_{L^{2}(\mathbb{H}(G))}^{2}d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{\mathbb{T}}(\theta)<\infty.$

Therefore, $T_{\sigma}:L^{2}(\mathbb{H}(G))\rightarrow L^{2}(\mathbb{H}(G))$
is a Hilbert-Schmidt operator.

Conversely, assume that $T_{\sigma}:L^{2}(\mathbb{H}(G))\rightarrow
L^{2}(\mathbb{H}(G))$ is a Hilbert-Schmidt operator. Then there exist a
function $\alpha\in L^{2}(\mathbb{H}(G)\times\mathbb{H}(G))$ such that for
$f\in L^{2}(\mathbb{H}(G))$

$(T_{\sigma}f)(x,\chi,\theta)=\int_{\mathbb{H}(G)\times\mathbb{H}(G)}\alpha((x,\chi,\theta),(x^{\prime},\chi^{\prime},\theta^{\prime}))f(x^{\prime},\chi^{\prime},\theta^{\prime})d\mu_{G}(x^{\prime})\,d\mu_{\widehat{G}}(\chi^{\prime})d\mu_{\mathbb{T}}(\theta^{\prime}).$

Define $\alpha:\mathbb{H}(G)\rightarrow L^{2}(\mathbb{H}(G))$ as

$\alpha(x,\chi,\theta)(x^{\prime},\chi^{\prime},\theta^{\prime})=\alpha((x,\chi,\theta),(x^{\prime},\chi^{\prime},\theta^{\prime})),\,\,\,\,\,(x,\chi^{\prime},\theta),\,(x^{\prime},\chi^{\prime},\theta^{\prime})\in\mathbb{H}(G).$

Then using (10), we see

$\|\sigma(x,\chi,\theta,j)\|_{S_{2}}=\|(\mathcal{F}_{c}\alpha(x,\chi,\theta))(\cdot,\cdot,j)\|_{L^{2}(G\times\widehat{G})}\,\,\,\,\,\,(x,\chi,\theta,j)\in\mathbb{H}(G)\times\mathbb{Z}^{*}.$

Then reversing the argument in the proof of sufficiency and using Theorem 3.2,
the converse is proved. ∎

Now we present following corollary on a trace class pseudo-differential
operator on $\mathbb{H}(G)$ and its trace formula.

######  Corollary 5.2.

Let $\alpha\in L^{2}(\mathbb{H}(G)\times\mathbb{H}(G)$ be such that

$\int_{G\times\widehat{G}\times\mathbb{T}}|\alpha((x,\chi,\theta),(x,\chi,\theta))|\,d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)d\mu_{\mathbb{T}}(\theta).$

Let $\sigma:\mathbb{H}(G)\times\mathbb{Z}^{*}\rightarrow\mathcal{B}(L^{2}(G))$
be a symbol as in the Theorem 5.1. Then
$T_{\sigma}:L^{2}(\mathbb{H}(G))\rightarrow L^{2}(\mathbb{H}(G))$ is a trace
class operator and

$tr(T_{\sigma})=\int_{\mathbb{H}(G)}\overline{\chi(x)}\alpha((x,\chi,\theta),(x^{-1},\chi,\theta))\,\,d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)d\mu_{\mathbb{T}}(\theta).$

###### Proof.

The proof of the corollary follows immediately from the formula 15 on the
kernel of pseudo-differential operator in Theorem 5.1. ∎

##  6\. Trace class pseudo-differential operators on the abstract Heisenberg
groups

######  Theorem 6.1.

Let $\sigma:\mathbb{H}(G)\times\rightarrow S_{2}$ be a symbol satisfying the
hypothesis of Theorem 3.2. Then the psuedo-differential operator
$T_{\sigma}:L^{2}(\mathbb{H}(G))\rightarrow L^{2}(\mathbb{H}(G))$ is a trace
class operator if and only if

$\sigma(x,\chi,\theta,j)=C_{j,G}\,\rho_{j}(x,\chi,\theta)W^{j}_{\alpha(x,\chi,\theta)^{-j}}\,\,\,\,\,(x,\chi,\theta)\in\mathbb{H}(G),\,j\in\mathbb{Z}^{*},$

where $\alpha:\mathbb{H}(G)\rightarrow L^{2}(\mathbb{H}(G))$ is a mapping such
that the conditions of Theorem 5.1 are satisfied and

$\alpha(x,\chi,\theta)(x^{\prime},\chi^{\prime},\theta^{\prime})=\int_{G\times\widehat{G}\times\mathbb{T}}\alpha_{1}(x,\chi,\theta)(x^{\prime\prime},\chi^{\prime\prime},\theta^{\prime\prime})\alpha_{2}(x^{\prime\prime},\chi^{\prime\prime},\theta^{\prime\prime})(x^{\prime},\chi^{\prime},\theta^{\prime})\,d\mu_{G}(x^{\prime\prime})d\mu_{\widehat{G}}(\chi^{\prime\prime})d\mu_{\mathbb{T}}(\theta^{\prime\prime})$

for all
$(x,\chi,\theta),\,(x^{\prime},\chi^{\prime},\theta^{\prime})\in\mathbb{H}(G);$
here $\alpha_{1}:\mathbb{H}(G)\rightarrow L^{2}(\mathbb{H}(G))$ and
$\alpha_{2}:\mathbb{H}(G)\rightarrow L^{2}(\mathbb{H}(G))$ are such that

$\int_{G\times\widehat{G}\times\mathbb{T}}\|\alpha_{i}(x,\chi,\theta)\|_{L^{2}(\mathbb{H}(G))}<\infty\,\,\,\,i=1,2.$

Moreover, if $T_{\sigma}:L^{2}(\mathbb{H}(G))\rightarrow L^{2}(\mathbb{H}(G))$
is a trace class operator then

$\displaystyle tr(T_{\sigma})$
$\displaystyle=\int_{G\times\widehat{G}\times\mathbb{T}}\alpha(x,\chi,\theta)(x,\chi,\theta)\,d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{\mathbb{T}}(\theta)$
$\displaystyle=\int_{\mathbb{H}(G)}\int_{\mathbb{H}(G)}\alpha_{1}(x,\chi,\theta)(x^{\prime},\chi^{\prime},\theta^{\prime})\,\alpha_{2}(x^{\prime},\chi^{\prime},\theta^{\prime})(x,\chi,\theta)d\mu_{G}(x^{\prime})\,d\mu_{\widehat{G}}(\chi^{\prime})\,d\mu_{\mathbb{T}}(\theta^{\prime})\,d\mu_{G}(x)\,d\mu_{\widehat{G}}(\chi)\,d\mu_{\mathbb{T}}(\theta).$

###### Proof.

The proof of this theorem follows from Theorem 5.1 and the fact the every
trace class operator is the product of two Hilbert-Schmidt operators. ∎

## Acknowledgment

Vishvesh Kumar wants to thank Prof. V. Muruganandam for his support and
encouragement. He also thanks the National Institute of Science Education and
Research, the Department of Atomic Energy, Government of India, for providing
excellent research facility.

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# Constructing the space of valuations of a quasi-Polish space as a space of
ideals

Matthew de Brecht This work was supported by JSPS KAKENHI Grant Number
18K11166. We thank the reviewers for carefully reading this paper and
providing feedback.

###### Abstract

We construct the space of valuations on a quasi-Polish space in terms of the
characterization of quasi-Polish spaces as spaces of ideals of a countable
transitive relation. Our construction is closely related to domain theoretical
work on the probabilistic powerdomain, and helps illustrate the connections
between domain theory and quasi-Polish spaces. Our approach is consistent with
previous work on computable measures, and can be formalized within weak formal
systems, such as subsystems of second order arithmetic.

##  1 Introduction

Quasi-Polish spaces [2] are a class of well-behaved countably based sober
spaces that includes Polish spaces, $\omega$-continuous domains, and countably
based spectral spaces. They can be interpreted via Stone-duality as the spaces
of models of countably axiomatized propositional geometric theories [12, 1].
In [7] another characterization of quasi-Polish spaces was presented that is a
natural generalization of the notion of an _abstract basis_ for
$\omega$-continuous domains [8]. In this paper we use this latter
characterization to extend domain theoretical work on probabilistic
powerdomains to the study of valuations on quasi-Polish spaces.

Valuations are a substitute for Borel measures which are used in the
denotational semantics of probabilistic programming languages [14] and in
computable approaches to measure theory, probability theory, and randomness
[19, 13, 18]. See R. Heckmann’s excellent paper [11] for more on the theory of
valuations, spaces of valuations, and integration111The valuations in this
note correspond to the _Scott-continuous valuations_ in [11].. Every valuation
on a quasi-Polish space can be extended to a Borel measure [5], and this
extension is unique if the valuation is locally finite [3]. Conversely, it is
easy to see that the restriction of a Borel measure to the open sets is a
valuation. Thus, in particular, there is a bijection between probabilistic
valuations and probabilistic Borel measures on quasi-Polish spaces.

The main result in this paper is a construction of the space of valuations on
a quasi-Polish space as a space of ideals of a transitive relation on a
countable set (Theorem 13). Our construction is closely related to domain
theoretical work on the probabilistic powerdomain (see [14] and [8, Section
IV-9]). Along with the constructions of the upper and lower powerspaces of
quasi-Polish spaces as spaces of ideals given in [4], our results demonstrate
how some domain theoretic results generalize well to quasi-Polish spaces (see
also [6] for more on the upper and lower powerspaces of quasi-Polish spaces).

An immediate corollary of our construction is that the space of valuations on
a quasi-Polish space is again a quasi-Polish space, although this already
follows from well-known results. A locale theoretic proof easily follows from
S. Vickers’ geometricity result in [20, Proposition 5] by using R. Heckmann’s
characterization of quasi-Polish spaces as countably presented locales [12]. A
proof based on quasi-metrics, at least for the case of subprobabilistic
valuations, follows from J. Goubault-Larrecq’s work on continuous Yoneda-
complete quasi-metric spaces in [9, Section 11] and his characterization of
quasi-Polish spaces in [10, Theorem 8.18]. Independently, the first proof we
found (which we presented at the Domains XII conference in August 2015) was
largely based on M. Schröder’s work in [19] on the space of (probabilistic)
measures within the cartesian closed category $\mathsf{QCB_{0}}$. That proof
starts with the observation that the $\mathsf{QCB_{0}}$ exponential
${\mathbb{S}}^{{\mathbb{S}}^{X}}$ is quasi-Polish whenever $X$ is222See [6]
for a proof. The ${\mathbb{S}}$ here is the Sierpinski space, and the space
${\mathrm{\mathbf{O}}}({\mathrm{\mathbf{O}}}(X))$ defined in [6] is
homeomorphic to the $\mathsf{QCB_{0}}$ exponential object
${\mathbb{S}}^{{\mathbb{S}}^{X}}$ when $X$ is quasi-Polish., then uses the
cartesian closed structure of $\mathsf{QCB_{0}}$ to show that
$Y^{{\mathbb{S}}^{X}}$ is quasi-Polish whenever $X$ and $Y$ are, and finally
observes that M. Schröder’s construction of the space of valuations on $X$ can
be obtained as the equalizer of the continuous functions
$\ell,r\colon\overline{\mathbb{R}}_{+}^{{\mathbb{S}}^{X}}\to\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}^{{\mathbb{S}}^{X}\times{\mathbb{S}}^{X}}$
defined as:

$\displaystyle\ell(\nu)$ $\displaystyle=$
$\displaystyle\big{\langle}\nu(\emptyset),\lambda\langle
U,V\rangle.\nu(U)+\nu(V)\big{\rangle},\text{ and}$ $\displaystyle r(\nu)$
$\displaystyle=$ $\displaystyle\big{\langle}0,\lambda\langle
U,V\rangle.\nu(U\cup V)+\nu(U\cap V)\big{\rangle}.$

It follows that the space of valuations is quasi-Polish because the space of
extended reals $\overline{\mathbb{R}}_{+}$ is quasi-Polish and the category of
quasi-Polish spaces is closed under countable limits.

A nice characteristic of the construction we give in this paper is that it can
be formalized within relatively weak formal systems. For example, our approach
is related to C. Mummert’s formalization of general topology within subsystems
of second order arithmetic [15, 16, 17]333Note that C. Mummert’s _MF-spaces_
are in general $\mathbf{\Pi}^{1}_{1}$-complete spaces, whereas quasi-Polish
spaces correspond to the $\mathbf{\Pi}^{0}_{2}$-level of the Borel hierarchy.
This explains why ${\mathbf{\Pi}^{1}_{1}-\mathsf{CA_{0}}}$ is required to
prove MF-spaces are closed under $G_{\delta}$-subsets, whereas our
construction of $\mathbf{\Pi}^{0}_{2}$-subspaces of quasi-Polish spaces in
Theorem 3 of [4] can be done within $\mathsf{ACA_{0}}$..

##  2 Main result

We let $\overline{\mathbb{R}}_{+}$ denote the positive extended reals (i.e.,
$[0,\infty]$) with the Scott-topology induced by the usual order. Given a
topological space $X$, we let ${\mathrm{\mathbf{O}}}(X)$ denote the lattice of
open subsets of $X$ with the Scott-topology.

######  Definition 1 (Valuations)

Let $X$ be a topological space. A _valuation_ on $X$ is a continuous function
$\nu\colon{\mathrm{\mathbf{O}}}(X)\to\overline{\mathbb{R}}_{+}$ satisfying:

  1. 1.

$\nu(\emptyset)=0$, and (_strictness_)

  2. 2.

$\nu(U)+\nu(V)=\nu(U\cup V)+\nu(U\cap V)$. (_modularity_)

The _space of valuations_ on $X$ is the set ${\mathrm{\mathbf{V}}}(X)$ of all
valuations on $X$ with the _weak topology_ , which is generated by subbasic
opens of the form

$\langle U,q\rangle:=\\{\nu\in{\mathrm{\mathbf{V}}}(X)\mid\nu(U)>q\\}$

with $U\in{\mathrm{\mathbf{O}}}(X)$ and
$q\in\overline{\mathbb{R}}_{+}\setminus\\{\infty\\}$.  $\sqcap$$\sqcup$

In this paper we will only consider the whole space of valuations
${\mathrm{\mathbf{V}}}(X)$, but it is straightforward to modify our results
for the subspaces of ${\mathrm{\mathbf{V}}}(X)$ consisting of _probabilistic
valuations_ (i.e., valuations satisfying $\nu(X)=1$) and _sub-probabilistic
valuations_ (i.e., valuations satisfying $\nu(X)\leq 1$).

Quasi-Polish spaces were introduced in [2]. In this paper we will define them
using the following equivalent characterization from [7] (see also [4]).

###### Definition 2

Let $\prec$ be a transitive relation on ${\mathbb{N}}$. A subset
$I\subseteq{\mathbb{N}}$ is an _ideal_ (with respect to $\prec$) if and only
if:

  1. 1.

$I\not=\emptyset$, (_$I$ is non-empty_)

  2. 2.

$(\forall a\in I)(\forall b\in{\mathbb{N}})\,(b\prec a\Rightarrow b\in I)$,
(_$I$ is a lower set_)

  3. 3.

$(\forall a,b\in I)(\exists c\in I)\,(a\prec c\,\&\,b\prec c)$. (_$I$ is
directed_)

The collection ${{\mathrm{\mathbf{I}}}(\prec)}$ of all ideals has the topology
generated by basic open sets of the form
$[n]_{\prec}=\\{I\in{{\mathrm{\mathbf{I}}}(\prec)}\mid n\in I\\}$. A space is
_quasi-Polish_ if and only if it is homeomorphic to
${{\mathrm{\mathbf{I}}}(\prec)}$ for some transitive relation $\prec$ on
${\mathbb{N}}$.  $\sqcap$$\sqcup$

We often apply the above definition to other countable sets with the implicit
assumption that it has been suitably encoded as a subset of ${\mathbb{N}}$.

Fix a transitive relation $\prec$ on ${\mathbb{N}}$ for the rest of this
section. Let ${\mathcal{B}}$ be the (countable) set of all partial functions
$r:\subseteq{\mathbb{N}}\to{\mathbb{Q}}_{>0}$ such that $dom(r)$ is finite,
where ${\mathbb{Q}}_{>0}$ is the set of rational numbers strictly larger than
zero.

###### Definition 3

Define the transitive relation $\prec_{V}$ on ${\mathcal{B}}$ as $r\prec_{V}s$
if and only if

$\sum_{b\in F}r(b)<\sum_{c\in{\uparrow}F\cap dom(s)}s(c)$

for every non-empty $F\subseteq dom(r)$, where
${\uparrow}F=\\{c\in{\mathbb{N}}\mid(\exists b\in F)\,b\prec c\\}$.
$\sqcap$$\sqcup$

Transitivity of $\prec_{V}$ follows from the transitivity of $\prec$. Note
that if $dom(r)=\emptyset$ then $r\prec_{V}s$ for every $s\in{\mathcal{B}}$.
We will sometimes use the fact that if $r\prec_{V}s$ and $b\in dom(r)$ then
there is $c\in dom(s)$ with $b\prec c$.

###### Definition 4

Define
$f_{V}\colon{\mathrm{\mathbf{V}}}({{\mathrm{\mathbf{I}}}(\prec)})\to{{\mathrm{\mathbf{I}}}(\prec_{V})}$
and
$g_{V}\colon{{\mathrm{\mathbf{I}}}(\prec_{V})}\to{\mathrm{\mathbf{V}}}({{\mathrm{\mathbf{I}}}(\prec)})$
as

$\displaystyle f_{V}(\nu)$ $\displaystyle=$
$\displaystyle\left\\{r\in{\mathcal{B}}\,\middle|\,\sum_{b\in
F}r(b)<\nu(\bigcup_{b\in F}[b]_{\prec})\text{ for every non-empty $F\subseteq
dom(r)$}\right\\},$ $\displaystyle g_{V}(I)$ $\displaystyle=$
$\displaystyle\lambda U.\bigvee\left\\{\sum_{b\in dom(r)}r(b)\,\middle|\,r\in
I\text{ and }\bigcup_{b\in dom(r)}[b]_{\prec}\subseteq U\right\\}.$

$\sqcap$$\sqcup$

We next prove a few lemmas which will be used to show that $f_{V}$ and $g_{V}$
are continuous inverses of each other.

###### Lemma 5

If $I\in{{\mathrm{\mathbf{I}}}(\prec_{V})}$, $r\in I$, and $A\subseteq
dom(r)$, then $r|_{A}\in I$, where $r|_{A}$ is the partial function obtained
by restricting the domain of $r$ to $A$.

Proof: Since $I$ is directed there is $s\in I$ with $r\prec_{V}s$. Then
clearly $r|_{A}\prec_{V}s$ hence $r|_{A}\in I$ because $I$ is a lower set.
$\sqcap$$\sqcup$

###### Definition 6

Define the transitive binary relation $\prec_{U}$ on
${\mathcal{P}}_{\mathrm{fin}}({\mathbb{N}})$ (the set of finite subsets of
${\mathbb{N}}$) as $F\prec_{U}G$ if and only if $(\forall n\in G)\,(\exists
m\in F)\,m\prec n$.  $\sqcap$$\sqcup$

We write ${\mathrm{\mathbf{K}}}(X)$ for the space of saturated compact subsets
of $X$ (see [6]).

######  Lemma 7 (Lemma 9 & Theorem 10 of [4])

Given $J\in{{\mathrm{\mathbf{I}}}(\prec_{U})}$, the set

$g_{U}(J)=\\{I\in{{\mathrm{\mathbf{I}}}(\prec)}\mid(\forall F\in J)(\exists
m\in I)\,m\in F\\}$

is in ${\mathrm{\mathbf{K}}}({{\mathrm{\mathbf{I}}}(\prec)})$. Furthermore,
for any $S\subseteq{\mathbb{N}}$, $g_{U}(J)\subseteq\bigcup_{b\in
S}[b]_{\prec}$ if and only if there is finite $F\subseteq S$ with $F\in J$.
$\sqcap$$\sqcup$

###### Lemma 8

If $I\in{{\mathrm{\mathbf{I}}}(\prec_{V})}$ and $r\in I$, then there exists
$s\in I$ with $r\prec_{V}s$ and $dom(r)\prec_{U}dom(s)$.

Proof: Choose any $t\in I$ with $r\prec_{V}t$. Let $s$ be the restriction of
$t$ to have $dom(s)=\\{c\in dom(t)\mid(\exists b\in dom(r))\,b\prec c\\}$.
Clearly $r\prec_{V}s$ and $dom(r)\prec_{U}dom(s)$, and Lemma 5 implies $s\in
I$.  $\sqcap$$\sqcup$

###### Lemma 9

Assume $I\in{{\mathrm{\mathbf{I}}}(\prec_{V})}$ and $r\in I$. Then there
exists $K\in{\mathrm{\mathbf{K}}}({{\mathrm{\mathbf{I}}}(\prec)})$ such that

  * •

$K\subseteq\bigcup_{b\in dom(r)}[b]_{\prec}$, and

  * •

For any finite $F\subseteq{\mathbb{N}}$, if $K\subseteq\bigcup_{b\in
F}[b]_{\prec}$, then there is $s\in I$ with $r\prec_{V}s$ and
$F\prec_{U}dom(s)$ and $K\subseteq\bigcup_{c\in
dom(s)}[c]_{\prec}\subseteq\bigcup_{b\in F}[b]_{\prec}$.

Proof: Fix $I\in{{\mathrm{\mathbf{I}}}(\prec_{V})}$ and $r\in I$. Using Lemma
8, we can find a $\prec_{V}$-ascending sequence $(r_{i})_{i\in{\mathbb{N}}}$
in $I$ with $r=r_{0}$ and $dom(r_{i})\prec_{U}dom(r_{i+1})$ for each
$i\in{\mathbb{N}}$. Then
$J=\\{F\in{\mathcal{P}}_{\mathrm{fin}}({\mathbb{N}})\mid(\exists
i\in{\mathbb{N}})\,F\prec_{U}dom(r_{i})\\}$ is in
${{\mathrm{\mathbf{I}}}(\prec_{U})}$, hence
$K=g_{U}(J)\in{\mathrm{\mathbf{K}}}({{\mathrm{\mathbf{I}}}(\prec)})$ and
$K\subseteq\bigcup_{b\in dom(r)}[b]_{\prec}$ by Lemma 7 and the fact that
$dom(r)\in J$. Assume $F\subseteq{\mathbb{N}}$ is finite and
$K\subseteq\bigcup_{b\in F}[b]_{\prec}$. Then $F\in J$ by Lemma 7, hence
$F\prec_{U}dom(r_{i})$ for some $i\in{\mathbb{N}}$. Since $\prec_{U}$ is
transitive, we can assume without loss of generality that $i>0$. Setting
$s=r_{i}$, we have $s\in I$ and $r\prec_{V}s$ and $F\prec_{U}dom(s)$, and
since $dom(s)\in J$ it follows from Lemma 7 that $K\subseteq\bigcup_{c\in
dom(s)}[c]_{\prec}$. The claim $\bigcup_{c\in
dom(s)}[c]_{\prec}\subseteq\bigcup_{b\in F}[b]_{\prec}$ follows from
$F\prec_{U}dom(s)$.  $\sqcap$$\sqcup$

###### Lemma 10

Let $D\subseteq{\mathbb{N}}$ be finite, and let ${\mathcal{P}}_{+}(D)$ be the
set of non-empty subsets of $D$. Define

$\displaystyle U_{G}$ $\displaystyle=$ $\displaystyle\bigcap_{b\in
G}[b]_{\prec}$ $\displaystyle V_{G}$ $\displaystyle=$ $\displaystyle
U_{G}\cap\bigcup_{b\in D\setminus G}[b]_{\prec}$

for each $G\in\cal{P}_{+}(D)$. Let $P\subseteq{\mathcal{P}}_{+}(D)$ be an
upper set (i.e., if $F\in P$ and $F\subseteq G\subseteq D$ then $G\in P$). If
$\nu\in{\mathrm{\mathbf{V}}}({{\mathrm{\mathbf{I}}}(\prec)})$ and
$\nu(U_{G})<\infty$ for each $G\in P$, then

$\sum_{G\in P}(\nu(U_{G})-\nu(V_{G}))=\nu\left(\bigcup_{G\in P}U_{G}\right).$

Proof: The proof is by induction on the size of $P$. It is trivial when
$P=\emptyset$, so assume $P$ is a non-empty upper set and that the lemma holds
for all upper sets of size strictly less than $P$. If $F$ is any minimal
element of $P$, then

$\displaystyle V_{F}$ $\displaystyle=$ $\displaystyle\bigcup_{b\in D\setminus
F}U_{F\cup\\{b\\}}$ $\displaystyle=$ $\displaystyle\bigcup_{G\in
P\setminus\\{F\\}}U_{F\cup G}$ $\displaystyle=$ $\displaystyle
U_{F}\cap\bigcup_{G\in P\setminus\\{F\\}}U_{G},$

so the induction hypothesis and modularity yields

$\displaystyle\sum_{G\in P}(\nu(U_{G})-\nu(V_{G}))$ $\displaystyle=$
$\displaystyle\nu(U_{F})-\nu(V_{F})+\sum_{G\in
P\setminus\\{F\\}}(\nu(U_{G})-\nu(V_{G}))$ $\displaystyle=$
$\displaystyle\nu(U_{F})-\nu\left(U_{F}\cap\bigcup_{G\in
P\setminus\\{F\\}}U_{G}\right)+\nu\left(\bigcup_{G\in
P\setminus\\{F\\}}U_{G}\right)$ $\displaystyle=$
$\displaystyle\nu\left(\bigcup_{G\in P}U_{G}\right).$

$\sqcap$$\sqcup$

###### Lemma 11

$f_{V}$ is well-defined and continuous.

Proof: We first show that $f_{V}(\nu)\in{{\mathrm{\mathbf{I}}}(\prec_{V})}$
for each $\nu\in{\mathrm{\mathbf{V}}}({{\mathrm{\mathbf{I}}}(\prec)})$.

  1. 1.

_( $f_{V}(\nu)$ is non-empty)_. The partial function with empty domain is in
$f_{V}(\nu)$.

  2. 2.

_( $f_{V}(\nu)$ is a lower set)_. Assume $r\prec_{V}s\in f_{V}(\nu)$. Let
$F\subseteq dom(r)$ be non-empty, and define $G={\uparrow}F\cap dom(s)$. Since
$b\prec c$ implies $[c]_{\prec}\subseteq[b]_{\prec}$ it follows that
$\bigcup_{c\in G}[c]_{\prec}\subseteq\bigcup_{b\in F}[b]_{\prec}$. Then

$\displaystyle\sum_{b\in F}r(b)$ $\displaystyle<$ $\displaystyle\sum_{c\in
G}s(c)\qquad\text{\hfill(because $r\prec_{V}s$)}$ $\displaystyle<$
$\displaystyle\nu(\bigcup_{c\in G}[c]_{\prec})\qquad\text{\hfill(because $s\in
f_{V}(\nu)$)}$ $\displaystyle\leq$ $\displaystyle\nu(\bigcup_{b\in
F}[b]_{\prec})\qquad\text{\hfill(because $\nu$ is monotonic)},$

hence $r\in f_{V}(\nu)$.

  3. 3.

_( $f_{V}(\nu)$ is directed)_. Our proof is related to the series of lemmas
leading up to Theorem IV-9.16 in [8]. Assume $r_{0},r_{1}\in f_{V}(\nu)$. For
each $i\in\\{0,1\\}$ and non-empty $F\subseteq dom(r_{i})$ fix some real
number $\beta^{i}_{F}$ satisfying

$\sum_{b\in F}r_{i}(b)<\beta^{i}_{F}<\nu\left(\bigcup_{b\in
F}[b]_{\prec}\right),$

and set

$\beta=\min\left\\{\frac{\beta^{i}_{F}-\sum_{b\in F}r_{i}(b)}{\sum_{b\in
F}r_{i}(b)}\,\middle|\,i\in\\{0,1\\}\text{ \& }\emptyset\not=F\subseteq
dom(r_{i})\right\\}.$

Then $\alpha=1/\left(1+\beta/2\right)$ satisfies $0<\alpha<1$ and is such that

$\sum_{b\in F}r_{i}(b)<\alpha\nu\left(\bigcup_{b\in F}[b]_{\prec}\right)$

for each $i\in\\{0,1\\}$ and non-empty $F\subseteq dom(r_{i})$ (see Lemma
IV-9.11 (iii) of [8]). Set $M=1+\sum_{b\in dom(r_{0})}r_{0}(b)+\sum_{b\in
dom(r_{1})}r_{1}(b)$, and $D=dom(r_{0})\cup dom(r_{1})$. Let $U_{G}$ and
$V_{G}$ be defined as in Lemma 10 for each non-empty $G\subseteq D$.

We define a finite set $h(G)\subseteq{\mathbb{N}}$ and a function $s_{G}\colon
h(G)\to{\mathbb{Q}}_{>}$ for each non-empty $G\subseteq D$ as follows. If
$\nu(U_{G})=\nu(V_{G})$ then let $h(G)=\emptyset$ and let $s_{G}$ be the empty
function. Otherwise, the set

$C=\\{c\in{\mathbb{N}}\mid(\forall b\in D)\,[b\prec c\iff b\in G]\\}$

is non-empty because $\nu(U_{G})>\nu(V_{G})$ implies there is some ideal
containing $G$ which is not in $V_{G}$. If there is some $c\in C$ with
$\nu([c]_{\prec})=\infty$, then set $h(G)=\\{c\\}$ and define $s_{G}\colon
h(G)\to{\mathbb{Q}}_{>}$ as $s_{G}(c)=M$. If no such $c\in C$ exists, then let
$(c_{i})_{i\in{\mathbb{N}}}$ be an enumeration of $C$ and define

$p_{i}=\nu([c_{i}]_{\prec})-\nu\left([c_{i}]_{\prec}\cap\left(\bigcup_{k<i}[c_{k}]_{\prec}\cup
V_{G}\right)\right).$

Using modularity and a simple inductive argument, we have

$\displaystyle\sum_{i\leq n}p_{i}$ $\displaystyle=$
$\displaystyle\nu(\bigcup_{i\leq n}[c_{i}]_{\prec})-\nu\left(\bigcup_{i\leq
n}[c_{i}]_{\prec}\cap V_{G}\right)$ $\displaystyle=$
$\displaystyle\nu(\bigcup_{i\leq n}[c_{i}]_{\prec}\cup V_{G})-\nu(V_{G})$

for each $n\in{\mathbb{N}}$. Since
$U_{G}=\bigcup_{i\in{\mathbb{N}}}[c_{i}]_{\prec}\cup V_{G}$ and $\nu$ is
Scott-continuous, there is $n_{0}\in{\mathbb{N}}$ with

$\left(\frac{1+\alpha}{2}\right)\sum_{i\leq
n_{0}}p_{i}\geq\alpha(\nu(U_{G})-\nu(V_{G}))$

if $\nu(U_{G})<\infty$, and

$\left(\frac{1+\alpha}{2}\right)\sum_{i\leq n_{0}}p_{i}\geq M$

if $\nu(U_{G})=\infty$. Define

$h(G)=\\{c_{i}\mid i\leq n_{0}\,\&\,p_{i}>0\\}$

and define $s_{G}\colon h(G)\to{\mathbb{Q}}_{>}$ so that $s_{G}(c_{i})$ is a
positive rational satisfying

$\left(\frac{1+\alpha}{2}\right)p_{i}\leq s_{G}(c_{i})<p_{i}.$

Since $h(G)\cap h(G^{\prime})\not=\emptyset$ implies $G=G^{\prime}$, there is
$s\in{\mathcal{B}}$ with

$dom(s)=\bigcup\\{h(G)\mid G\subseteq D\\}$

satisfying $s(c)=s_{G}(c)$ for the unique $G\subseteq D$ with $c\in h(G)$.
From the construction of $s$, if $F\subseteq h(G)$ is non-empty then

$\displaystyle\sum_{c\in F}s(c)<\nu(\bigcup_{c\in
F}[c]_{\prec})-\nu(\bigcup_{c\in F}[c]_{\prec}\cap V_{G}).$ (1)

Furthermore, if $h(G)\not=\emptyset$, then $\nu(U_{G})<\infty$ implies

$\displaystyle\alpha\left(\nu(U_{G})-\nu(V_{G})\right)\leq\sum_{c\in
h(G)}s(c),$ (2)

and $\nu(U_{G})=\infty$ implies

$\displaystyle M\leq\sum_{c\in h(G)}s(c).$ (3)

To show $s\in f_{V}(\nu)$, we must prove $\sum_{c\in F}s(c)<\nu(\bigcup_{c\in
F}[c]_{\prec})$ for each non-empty $F\subseteq dom(s)$. This clearly holds
when $F=\\{c\\}$ is a singleton. Next, assume it holds for all sets of size
less than or equal to $n$, and let $F$ be a set of size $n+1$. We can assume
$\nu(\bigcup_{c\in F}[c]_{\prec})<\infty$, since otherwise the claim is
trivial. Let $G\subseteq D$ be a set of minimal size satisfying $F\cap
h(G)\not=\emptyset$. This implies that either $F\setminus h(G)$ is empty or
else it satisfies the induction hypothesis. Furthermore, for any $c\in
F\setminus h(G)$ there is $G^{\prime}\subseteq D$ with $c\in h(G^{\prime})$,
and since the minimality of $G$ implies $G^{\prime}\not\subseteq G$, there is
$b\in G^{\prime}\setminus G$ with $b\prec c$, which implies
$U_{G}\cap[c]_{\prec}\subseteq V_{G}$. Therefore,

$\displaystyle\sum_{c\in F}s(c)$ $\displaystyle=$ $\displaystyle\sum_{c\in
F\cap h(G)}s_{G}(c)+\sum_{c\in F\setminus h(G)}s(c)$ $\displaystyle<$
$\displaystyle\nu(\bigcup_{c\in F\cap h(G)}[c]_{\prec})-\nu(\bigcup_{c\in
F\cap h(G)}[c]_{\prec}\cap V_{G})+\nu(\bigcup_{c\in F\setminus
h(G)}[c]_{\prec})$ (by (1) and the induction hypothesis) $\displaystyle\leq$
$\displaystyle\nu(\bigcup_{c\in F\cap h(G)}[c]_{\prec})-\nu(\bigcup_{c\in
F\cap h(G)}[c]_{\prec}\cap\bigcup_{c\in F\setminus h(G)}[c]_{\prec})$
$\displaystyle+\nu(\bigcup_{c\in F\setminus h(G)}[c]_{\prec})$ (because
$U_{G}\cap[c]_{\prec}\subseteq V_{G}$ for each $c\in F\setminus h(G)$)
$\displaystyle=$ $\displaystyle\nu(\bigcup_{c\in F}[c]_{\prec}),$

which proves $s\in f_{V}(\nu)$.

Finally, we must show $r_{0}\prec_{V}s$ and $r_{1}\prec_{V}s$. Fix
$i\in\\{0,1\\}$ and non-empty $F\subseteq dom(r_{i})$. Set $P=\\{G\subseteq
D\mid G\cap F\not=\emptyset\\}$ and note that ${\uparrow}F\cap
dom(s)=\bigcup_{G\in P}h(G)$. If $\nu(U_{G})<\infty$ for each $G\in P$, then
using (2) and the fact that $G\not=G^{\prime}$ implies $h(G)\cap
h(G^{\prime})=\emptyset$, we have

$\displaystyle\sum_{c\in{\uparrow}F\cap dom(s)}s(c)$ $\displaystyle\geq$
$\displaystyle\sum_{G\in P}\alpha(\nu(U_{G})-\nu(V_{G}))$ $\displaystyle=$
$\displaystyle\alpha\nu\left(\bigcup_{G\in P}U_{G}\right)\qquad\text{ (by
Lemma~{}\ref{lem:sum_of_differences})}$ $\displaystyle=$
$\displaystyle\alpha\nu\left(\bigcup_{b\in F}[b]_{\prec}\right)$
$\displaystyle>$ $\displaystyle\sum_{b\in F}r_{i}(b).$

Otherwise, there is $G\in P$ with $\nu(U_{G})=\infty$, so (3) implies

$\sum_{c\in{\uparrow}F\cap dom(s)}s(c)\geq M>\sum_{b\in F}r_{i}(b).$

This completes the proof that $f_{V}(\nu)$ is directed.

It only remains to show that $f_{V}$ is continuous. Fix $r\in{\mathcal{B}}$.
For each $F\subseteq dom(r)$ define $W_{F}=\bigcup_{b\in F}[b]_{\prec}$ and
$q_{F}=\sum_{b\in F}r(b)$, and set $D=\\{F\subseteq dom(r)\mid
F\not=\emptyset\\}$. Then $f_{V}(\nu)\in[r]_{\prec_{V}}$ if and only if

$\nu\in\bigcap_{F\in D}\langle W_{F},q_{F}\rangle,$

hence $f_{V}$ is continuous.  $\sqcap$$\sqcup$

###### Lemma 12

$g_{V}$ is well-defined and continuous.

Proof: We first show that $\nu=g_{V}(I)$ is a valuation for each
$I\in{{\mathrm{\mathbf{I}}}(\prec_{V})}$.

  1. 1.

$\nu(\emptyset)=0$: Assume
$U\in{\mathrm{\mathbf{O}}}({{\mathrm{\mathbf{I}}}(\prec)})$ and $\nu(U)>0$.
Then there is $r_{0}\in I$ and $b_{0}\in dom(r_{0})$ such that
$[b_{0}]_{\prec}\subseteq U$ and $0<r_{0}(b_{0})$. Since $I$ is directed,
there is an infinite sequence $r_{0}\prec_{V}r_{1}\prec_{V}\cdots$ in $I$.
Since $b_{0}\in dom(r_{0})$ and $r_{0}\prec_{V}r_{1}$, there is $b_{1}\in
dom(r_{1})$ with $b_{0}\prec b_{1}$. Similarly, there must be $b_{2}\in
dom(r_{2})$ with $b_{1}\prec b_{2}$. This yields an infinite sequence
$b_{0}\prec b_{1}\prec\cdots$, hence $\\{c\in{\mathbb{N}}\mid(\exists
i\in{\mathbb{N}})\,c\prec b_{i}\\}$ is an element of $[b_{0}]_{\prec}\subseteq
U$. Therefore, $U\not=\emptyset$.

  2. 2.

$\nu(U)+\nu(V)=\nu(U\cup V)+\nu(U\cap V)$: We first show
$\nu(U)+\nu(V)\leq\nu(U\cup V)+\nu(U\cap V)$. Let $r,s\in I$ be such that
$(\forall b\in dom(r))\,[b]_{\prec}\subseteq U$ and $(\forall b\in
dom(s))\,[b]_{\prec}\subseteq V$. Set

$p_{r}=\sum_{b\in dom(r)}r(b),\qquad p_{s}=\sum_{b\in dom(s)}s(b).$

Let $t\in I$ be a $\prec_{V}$-upper bound of $r$ and $s$. Let

$\displaystyle D_{r}$ $\displaystyle=$ $\displaystyle\\{c\in
dom(t)\mid(\exists b\in dom(r))\,b\prec c\\},$ $\displaystyle D_{s}$
$\displaystyle=$ $\displaystyle\\{c\in dom(t)\mid(\exists b\in dom(s))\,b\prec
c\\}.$

Note that $c\in D_{r}\cap D_{s}$ implies $[c]_{\prec}\subseteq U\cap V$. Set

$q_{0}=\sum_{c\in D_{r}\setminus D_{s}}t(c),\qquad q_{1}=\sum_{c\in
D_{s}\setminus D_{r}}t(c),\qquad q_{2}=\sum_{c\in D_{r}\cap D_{s}}t(c).$

Then $r\prec_{V}t$ implies $p_{r}\leq q_{0}+q_{2}$ and $s\prec_{V}t$ implies
$p_{s}\leq q_{1}+q_{2}$. Furthermore, using the fact $t\in I$, Lemma 5, and
the definition of $\nu$, we obtain $\nu(U\cup V)\geq q_{0}+q_{1}+q_{2}$ and
$\nu(U\cap V)\geq q_{2}$, hence $p_{r}+p_{s}\leq\nu(U\cup V)+\nu(U\cap V)$. It
follows that $\nu(U)+\nu(V)\leq\nu(U\cup V)+\nu(U\cap V)$.

Next we show $\nu(U\cup V)+\nu(U\cap V)\leq\nu(U)+\nu(V)$. Let $r,s\in I$ be
such that $(\forall b\in dom(r))\,[b]_{\prec}\subseteq U\cup V$ and $(\forall
b\in dom(s))\,[b]_{\prec}\subseteq U\cap V$. Let $K\subseteq\bigcup_{b\in
dom(r)}[b]_{\prec}$ be as in Lemma 9. Since $K$ is compact and $K\subseteq
U\cup V$, there exists a finite set $F\subseteq{\mathbb{N}}$ with
$K\subseteq\bigcup_{b\in F}[b]_{\prec}$ and such that each $b\in F$ satisfies
$[b]_{\prec}\subseteq U$ or $[b]_{\prec}\subseteq V$. Apply Lemma 9 to get
$t\in I$ with $r\prec_{V}t$ and $F\prec_{U}dom(t)$ and
$K\subseteq\bigcup_{c\in dom(t)}[c]_{\prec}\subseteq\bigcup_{b\in
dom(r)}[b]_{\prec}$. Next let $u\in I$ be a $\prec_{V}$-upper bound of $t$ and
$s$. By restricting the domain of $u$ if necessary, we can assume that
$(dom(t)\cup dom(s))\prec_{U}dom(u)$, hence every $c\in dom(u)$ satisfies
$[c]_{\prec}\subseteq U$ or $[c]_{\prec}\subseteq V$. Let $u_{0}$ be the
restriction of $u$ to have domain $dom(u_{0})=\\{b\in
dom(u)\mid[b]_{\prec}\subseteq U\\}$, and let $u_{1}$ be the restriction of
$u$ to have domain $dom(u_{1})=\\{b\in dom(u)\mid[b]_{\prec}\subseteq V\\}$.
Note that $u_{0}$ and $u_{1}$ are both in $I$ by Lemma 5, and that
$dom(u)=dom(u_{0})\cup dom(u_{1})$. Then using the fact that $r\prec_{V}u$ and
$s\prec_{V}u$, we have

$\displaystyle\sum_{b\in dom(r)}r(b)+\sum_{b\in dom(s)}s(b)$
$\displaystyle\leq$ $\displaystyle\sum_{c\in dom(u)}u(c)+\sum_{c\in
dom(u_{0})\cap dom(u_{1})}u(c)$ $\displaystyle=$ $\displaystyle\sum_{c\in
dom(u_{0})}u_{0}(c)+\sum_{c\in dom(u_{1})}u_{1}(c)$ $\displaystyle\leq$
$\displaystyle\nu(U)+\nu(V).$

Therefore, $\nu(U\cup V)+\nu(U\cap V)\leq\nu(U)+\nu(V)$.

  3. 3.

$\nu$ is a continuous function: Assume
$U\in{\mathrm{\mathbf{O}}}({{\mathrm{\mathbf{I}}}(\prec)})$ and
$q\in{\mathbb{Q}}_{>0}$ and $\nu(U)>q$. Since ${{\mathrm{\mathbf{I}}}(\prec)}$
is consonant (see [6]), it suffices to find
$K\in{\mathrm{\mathbf{K}}}({{\mathrm{\mathbf{I}}}(\prec)})$ such that
$K\subseteq U$ and $\nu(W)>q$ whenever $W$ is an open set containing $K$. By
definition of $g_{V}(I)$, there must be $r\in I$ such that $(\forall b\in
dom(r))\,[b]_{\prec}\subseteq U$ and $\sum_{b\in dom(r)}r(b)>q$. Now let
$K\in{\mathrm{\mathbf{K}}}({{\mathrm{\mathbf{I}}}(\prec)})$ be as in Lemma 9.
Then $K\subseteq U$, and if $K\subseteq W$ then there is $s\in I$ with
$r\prec_{V}s$ and $K\subseteq\bigcup_{c\in dom(s)}[c]_{\prec}\subseteq W$,
hence $q<\sum_{c\in dom(s)}s(c)\leq\nu(W)$.

It only remains to show that $g_{V}$ is continuous. Assume $g_{V}(I)\in\langle
U,q\rangle$. Then there is $r\in I$ satisfying $(\forall b\in
dom(r))\,[b]_{\prec}\subseteq U$ and $q<\sum_{b\in dom(r)}r(b)$. Then
$I\in[r]_{\prec_{V}}\subseteq g_{V}^{-1}(\langle U,q\rangle)$, hence $g_{V}$
is continuous.  $\sqcap$$\sqcup$

###### Theorem 13

${\mathrm{\mathbf{V}}}({{\mathrm{\mathbf{I}}}(\prec)})$ and
${{\mathrm{\mathbf{I}}}(\prec_{V})}$ are homeomorphic (via $f_{V}$ and
$g_{V}$).

Proof: It only remains to show that $f_{V}$ and $g_{V}$ are inverses of each
other.

To show that $g_{V}\circ f_{V}$ is the identity function, it suffices to show
that $g_{V}(f_{V}(\nu))\in\langle U,q\rangle$ if and only if $\nu\in\langle
U,q\rangle$ for each
$\nu\in{\mathrm{\mathbf{V}}}({{\mathrm{\mathbf{I}}}(\prec)})$ and each
subbasic open $\langle U,q\rangle$. If $g_{V}(f_{V}(\nu))\in\langle
U,q\rangle$, then there must be $r\in f_{V}(\nu)$ with $q<\sum_{b\in
dom(r)}r(b)$ and $\bigcup_{b\in dom(r)}[b]_{\prec}\subseteq U$. This implies
that $dom(r)\not=\emptyset$, and using the definition of $f_{V}$ we obtain
$q<\sum_{b\in dom(r)}r(b)<\nu(\bigcup_{b\in dom(r)}[b]_{\prec})\leq\nu(U)$,
hence $\nu\in\langle U,q\rangle$. Conversely, if $\nu\in\langle U,q\rangle$
then since $\nu$ is continuous there exist $b_{0},\ldots,b_{n}\in{\mathbb{N}}$
such that $\bigcup_{i\leq n}[b_{i}]_{\prec}\subseteq U$ and
$q<\nu(\bigcup_{i\leq n}[b_{i}]_{\prec})$. If $\nu([b_{i}]_{\prec})=\infty$
for some $i\leq n$, then the partial function $r$ defined as
$dom(r)=\\{b_{i}\\}$ and $r(b_{i})=q+1$ is in $f_{V}(\nu)$, which implies
$g_{V}(f_{V}(\nu))\in\langle U,q\rangle$. Otherwise
$\nu([b_{i}]_{\prec})<\infty$ for each $i\leq n$, so define

$m_{i}=\nu([b_{i}]_{\prec})-\nu\big{(}[b_{i}]_{\prec}\cap\bigcup_{j<i}[b_{j}]_{\prec}\big{)}.$

Note that the modularity of $\nu$ implies $m_{i}=\nu(\bigcup_{j\leq
i}[b_{j}]_{\prec})-\nu(\bigcup_{j<i}[b_{j}]_{\prec})$, hence a simple
inductive argument yields $\sum_{i\leq n}m_{i}=\nu(\bigcup_{i\leq
n}[b_{i}]_{\prec})$, which is strictly larger than $q$. Let $G=\\{i\mid
m_{i}>0\\}$. Then there exists $r\in{\mathcal{B}}$ with $dom(r)=\\{b_{i}\mid
i\in G\\}$ and $(\forall i\in G)\,r(b_{i})<m_{i}$ and $q<\sum_{b\in
dom(r)}r(b)$. If $F\subseteq G$ is non-empty, then

$\displaystyle\sum_{i\in F}r(b_{i})<\sum_{i\in F}m_{i}$ $\displaystyle=$
$\displaystyle\sum_{i\in
F}\left(\nu([b_{i}]_{\prec})-\nu\big{(}[b_{i}]_{\prec}\cap\bigcup_{j<i}[b_{j}]_{\prec}\big{)}\right)$
$\displaystyle\leq$ $\displaystyle\sum_{i\in
F}\left(\nu([b_{i}]_{\prec})-\nu\big{(}[b_{i}]_{\prec}\cap\bigcup_{\begin{subarray}{c}j<i\\\
j\in F\end{subarray}}[b_{j}]_{\prec}\big{)}\right)$ $\displaystyle=$
$\displaystyle\nu\left(\bigcup_{i\in F}[b_{i}]_{\prec}\right).$

Thus, $r\in f_{V}(\nu)$ and $q<\sum_{b\in dom(r)}r(b)$, hence
$g_{V}(f_{V}(\nu))\in\langle U,q\rangle$.

Next we show that $f_{V}(g_{V}(I))=I$ for each
$I\in{{\mathrm{\mathbf{I}}}(\prec_{V})}$. By unwinding the definitions of
$f_{V}$ and $g_{V}$, we have $r\in f_{V}(g_{V}(I))$ if and only if for every
non-empty $F\subseteq dom(r)$ there is $s\in I$ such that $\bigcup_{c\in
dom(s)}[c]_{\prec}\subseteq\bigcup_{b\in F}[b]_{\prec}$ and $\sum_{b\in
F}r(b)<\sum_{c\in dom(s)}s(c)$. Thus, given any $r\in I$, by Lemma 8 there is
$s\in I$ with $r\prec_{V}s$ and $dom(r)\prec_{U}dom(s)$, hence $\bigcup_{c\in
dom(s)}[c]_{\prec}\subseteq\bigcup_{b\in F}[b]_{\prec}$ and $\sum_{b\in
F}r(b)<\sum_{c\in dom(s)}s(c)$, which implies $r\in f_{V}(g_{V}(I))$.
Therefore, $I\subseteq f_{V}(g_{V}(I))$.

To prove $f_{V}(g_{V}(I))\subseteq I$, fix any $r\in f_{V}(g_{V}(I))$. Then
for every non-empty $F\subseteq dom(r)$ there is $s_{F}\in I$ such that
$\bigcup_{c\in dom(s_{F})}[c]_{\prec}\subseteq\bigcup_{b\in F}[b]_{\prec}$ and
$\sum_{b\in F}r(b)<\sum_{c\in dom(s_{F})}s_{F}(c)$. Using Lemma 9, we can
assume that $F\prec_{U}dom(s_{F})$. Let $s\in I$ be a $\prec_{V}$-upper bound
of all of the $s_{F}$. Then for any non-empty $F\subseteq dom(r)$, we have

$\displaystyle\sum_{b\in F}r(b)$ $\displaystyle<$
$\displaystyle\sum_{c\in{\uparrow}F\cap dom(s_{F})}s_{F}(c)\qquad\text{ (by
choice of $s_{F}$)}$ $\displaystyle<$ $\displaystyle\sum_{c\in{\uparrow}F\cap
dom(s)}s(c)\qquad\text{ (because $s_{F}\prec_{V}s$ and $\prec$ is
transitive).}$

Therefore $r\prec_{V}s$, hence $r\in I$ because $I$ is a lower-set. It follows
that $f_{V}(g_{V}(I))\subseteq I$, which completes the proof that
$f_{V}(g_{V}(I))=I$.  $\sqcap$$\sqcup$

We remark that the homeomorphisms $f_{V}$ and $g_{V}$ are computable in the
sense of TTE [21] when $\prec$ is computably enumerable, and therefore our
approach is consistent with previous work on computable measures in [19, 13,
18]. The computability of $f_{V}$ is obvious. For $g_{V}$, note that for any
$U\in{\mathrm{\mathbf{O}}}({{\mathrm{\mathbf{I}}}(\prec)})$ and any
$A\subseteq{\mathbb{N}}$ satisfying $U=\bigcup_{a\in A}[a]_{\prec}$, Lemma 9
implies

$g_{V}(I)(U)=\bigvee\left\\{\sum_{c\in dom(s)}s(c)\,\middle|\,s\in
I\,\&\,(\forall c\in dom(s))(\exists a\in A)\,a\prec c\right\\},$

which shows that $g_{V}$ is computable.

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