# The Modular Stone-von Neumann Theorem

Lucas Hall  School of Mathematical and Statistical Sciences
Arizona State University
Tempe, Arizona 85287<EMAIL_ADDRESS>,  Leonard Huang  Department of
Mathematics & Statistics
University of Nevada, Reno, Reno, NV 89557<EMAIL_ADDRESS>and  John
Quigg  School of Mathematical and Statistical Sciences
Arizona State University
Tempe, Arizona 85287<EMAIL_ADDRESS>
###### Abstract.

In this paper, we use the tools of nonabelian duality to formulate and prove a
far-reaching generalization of the Stone-von Neumann Theorem to modular
representations of actions and coactions of locally compact groups on
elementary $C^{\ast}$-algebras. This greatly extends the Covariant Stone-von
Neumann Theorem for Actions of Abelian Groups recently proven by L. Ismert and
the second author. Our approach is based on a new result about Hilbert
$C^{\ast}$-modules that is simple to state yet is widely applicable and can be
used to streamline many previous arguments, so it represents an improvement —
in terms of both efficiency and generality — in a long line of results in this
area of mathematical physics that goes back to J. von Neumann’s proof of the
classical Stone-von Neumann Theorem.

###### Key words and phrases:

Crossed product, action, coaction, $C^{\ast}$-correspondence, Morita
equivalence, nonabelian duality, Stone-von Neumann Theorem

###### 2000 Mathematics Subject Classification:

Primary 46L55; Secondary 22D25, 22D35, 43A65, 46L06, 81R15, 81S05

## Introduction

In a groundbreaking 1925 paper ([9]), W. Heisenberg introduced the very first
version of quantum mechanics, known as “matrix mechanics”. According to matrix
mechanics, classical observables (continuous real-valued functions on a phase
space describing observable properties of classical particles) were to be
replaced by quantum observables (self-adjoint unbounded operators on a Hilbert
space describing observable properties of quantum particles), and all quantum
phenomena could be mathematically explained by the noncommutativity of certain
pairs of quantum observables, as expressed by the Heisenberg Commutation
Relation (HCR):

$\left[A,B\right]\coloneqq AB-BA=i\hbar\cdot\mathbf{1}.$

A well-known example of a pair of quantum observables that satisfies the HCR
is $\left(X,P\right)$, where $X$ and $P$ are the position and momentum
operators on the Hilbert space $L^{2}\left(\mathbb{R}\right)$, defined by

$\displaystyle\left(X\psi\right)\left(x\right)$ $\displaystyle\coloneqq
x\psi(x),$ $\displaystyle\left(P\psi\right)\left(x\right)$
$\displaystyle\coloneqq-i\hbar\psi^{\prime}\left(x\right).$

Note that $X$ and $P$ are not bounded operators on
$L^{2}\left(\mathbb{R}\right)$. Rather, they are unbounded operators on
$L^{2}\left(\mathbb{R}\right)$, where we may take the Schwartz space
$\mathcal{S}\left(\mathbb{R}\right)$ (for example) to be a suitable dense
common domain.

In 1926, E. Schrödinger published his famous equation that formed the
cornerstone of a second version of quantum mechanics, known as “wave
mechanics” ([30]). As matrix mechanics and wave mechanics made the same
experimental predictions, a mathematical equivalence between them was
immediately suspected. Schrödinger attempted to prove this equivalence in
[31], but his proof failed to meet the standards of mathematical rigor due to
its inability to address technical domain issues surrounding unbounded
operators.

To overcome the challenges faced by Schrödinger, one may take any pair
$\left(A,B\right)$ of self-adjoint unbounded operators on a Hilbert space
$\mathcal{H}$ that satisfies the HCR and use Stone’s Theorem to exponentiate
them, thus obtaining (strongly continuous) one-parameter unitary groups $U$
and $V$ on $\mathcal{H}$:

$U=(e^{ixA})_{x\in\mathbb{R}},\qquad V=(e^{iyB})_{y\in\mathbb{R}}.$

It can then be shown that the pair $\left(U,V\right)$ satisfies the so-called
“Weyl Commutation Relation (WCR)” on $\mathcal{H}$:

$\forall x,y\in\mathbb{R}:\quad V_{y}U_{x}=e^{ixy}U_{x}V_{y}.$

The Stone-von Neumann Theorem — formulated by M. Stone in [33] and rigorously
demonstrated by J. von Neumann in [22] — states that $\left(U,V\right)$ can be
completely described in terms of the pair
$\left(\lambda,\operatorname{Mod}\right)$, where $\lambda$ and
$\operatorname{Mod}$ are unitary groups on $L^{2}\left(\mathbb{R}\right)$
obtained by exponentiating (via Stone’s Theorem) the pair $\left(X,P\right)$.
These are called, respectively, the “left regular representation” and the
“phase modulation” of $\mathbb{R}$. The theorem rigorously proves the
equivalence between matrix mechanics and wave mechanics, and is a fundamental
result in the mathematical theory of nonrelativistic quantum mechanics.

The following is a rather general version of the Stone-von Neumann Theorem.

######  Theorem 0.1 (Stone-von Neumann Theorem [22]).

Let $\left(U,V\right)$ be a pair of unitary representations of
$\mathbb{R}^{n}$ on a Hilbert space $\mathcal{H}$ that satisfies the WCR, i.e,

$\forall x,y\in\mathbb{R}^{n}:\quad V_{y}U_{x}=e^{i\left\langle
x,y\right\rangle}U_{x}V_{y}.$

Let $\lambda$ and $\operatorname{Mod}$ be the unitary representations of
$\mathbb{R}^{n}$ on the Hilbert space $L^{2}\left(\mathbb{R}^{n}\right)$
defined by

$\lambda_{x}\left(f\right)\coloneqq
f\left(\cdot-x\right)\quad\text{and}\quad\operatorname{Mod}_{y}\left(f\right)\coloneqq
e^{i\left\langle\cdot,y\right\rangle}f.$

Then $\left(\lambda,\operatorname{Mod}\right)$ satisfies the WCR on
$L^{2}\left(\mathbb{R}^{n}\right)$. Moreover, there exist a set $I$ and a
unitary map $W:\mathcal{H}\to\bigoplus_{i\in
I}L^{2}\left(\mathbb{R}^{n}\right)$ such that for all $x,y\in\mathbb{R}^{n}$,

$WU_{x}W^{-1}=\bigoplus_{i\in I}\lambda_{x}\quad\text{and}\quad
WV_{y}W^{-1}=\bigoplus_{i\in I}\operatorname{Mod}_{y}.$

The Stone-von Neumann Theorem thus completely classifies — up to unitary
equivalence — all pairs of unitary Hilbert-space representations of
$\mathbb{R}^{n}$ that satisfy the WCR.

In [19], G. Mackey noted that as $\mathbb{R}^{n}$ is its own Pontryagin dual,
the WCR could be formulated for locally compact abelian groups in general. He
then proceeded to formulate and establish the Stone-von Neumann Theorem for
the class of all separable locally compact abelian groups, where the
separability hypothesis was required by the measure-theoretic machinery of his
proof. Shortly after, in [18], L.H. Loomis was able to remove this superfluous
hypothesis with a new proof that did not rely on measure theory. Finally, in
[28], M. Rieffel found a mostly algebraic proof containing rudiments of the
concept of Morita equivalence for $C^{\ast}$-algebras, which he went on to
develop more fully in [29]. It is believed by most operator algebraists that
the proper way to view the Mackey-Stone-von Neumann Theorem is as a Morita-
equivalence result.

To give an efficient formulation of the Mackey-Stone-von Neumann Theorem, let
us define Heisenberg and Schrödinger representations.

######  Definition 0.2 ([19, 28]).

Let $G$ be a locally compact abelian group. Then a _Heisenberg
$G$-representation_ is defined as a triple $\left(\mathcal{H},U,V\right)$ with
the following properties:

  1. (1)

$\mathcal{H}$ is a Hilbert space.

  2. (2)

$U$ is a unitary representation of $G$ on $\mathcal{H}$.

  3. (3)

$V$ is a unitary representation of $\widehat{G}$ on $\mathcal{H}$.

  4. (4)

$\left(U,V\right)$ satisfies the WCR on $\mathcal{H}$: For all $x\in G$ and
$\varphi\in\widehat{G}$,

$V_{\varphi}U_{x}=\varphi\left(x\right)U_{x}V_{\varphi}.$

A Heisenberg $G$-representation $\left(\mathcal{H},U,V\right)$ is said to be
_equivalent_ to another one
$\left(\mathcal{H}^{\prime},U^{\prime},V^{\prime}\right)$ if and only if there
is a unitary map $W:\mathcal{H}\to\mathcal{H}^{\prime}$ such that for all
$x\in G$ and $\varphi\in\widehat{G}$,

$WU_{x}W^{-1}=U^{\prime}_{x}\quad\text{and}\quad
WV_{\varphi}W^{-1}=V^{\prime}_{\varphi},$

in which case we write
$\left(\mathcal{H},U,V\right)\simeq\left(\mathcal{H}^{\prime},U^{\prime},V^{\prime}\right)$.
Note that $\simeq$ is an equivalence relation on the class of all Heisenberg
$G$-representations.

######  Definition 0.3 ([19, 28]).

Let $G$ be a locally compact abelian group. Then the _Schrödinger
$G$-representation_ is the triple

$(L^{2}\left(G\right),\lambda,\operatorname{Mod}),$

where $\lambda$ and $\operatorname{Mod}$ are, respectively, the unitary
representations of $G$ and $\widehat{G}$ on the Hilbert space
$L^{2}\left(G\right)$ defined by:

$\lambda_{x}\left(f\right)\coloneqq
f(x^{-1}\cdot)\quad\text{and}\quad\operatorname{Mod}_{\varphi}\left(f\right)\coloneqq\varphi
f.$

######  Theorem 0.4 ([19, 28]).

Let $G$ be a locally compact abelian group. Then the following statements are
true:

  1. (1)

The Schrödinger $G$-representation is a Heisenberg $G$-representation.

  2. (2)

Let $I$ be a set and $\left(\mathcal{H}_{i},U_{i},V_{i}\right)_{i\in I}$ an
$I$-indexed family of Heisenberg $G$-representations. Then the direct sum

$\bigoplus_{i\in
I}\left(\mathcal{H}_{i},U_{i},V_{i}\right)\coloneqq\left(\bigoplus_{i\in
I}\mathcal{H}_{i},\bigoplus_{i\in I}U_{i},\bigoplus_{i\in I}V_{i}\right)$

is also a Heisenberg $G$-representation.

######  Theorem 0.5 (Mackey-Stone-von Neumann Theorem [19, 28]).

Let $G$ be a locally compact abelian group. Then every Heisenberg
$G$-representation is a multiple of the Schrödinger $G$-representation.

Several generalizations of the Mackey-Stone-von Neumann Theorem have appeared
in the literature over time. They deal with nonregular group representations
([4]) or nonabelian duality ([11, 23]). As the theory of Hilbert
$C^{\ast}$-modules was already highly developed, it seemed natural to expect
that the theorem could also be generalized to the setting of Hilbert
$C^{\ast}$-modules, but heuristic arguments against the feasibility of such a
generalization were made in [1], owing to the lack of a spectral decomposition
theorem for Hilbert $C^{\ast}$-modules.

In [10], L. Ismert and the second author proposed an alternative way of
viewing the Mackey-Stone-von Neumann Theorem. Their main idea was that the
theorem should be viewed as a special case of a far more general result about
the uniqueness of representations of abelian $C^{\ast}$-dynamical systems on
Hilbert $C^{\ast}$-modules. For a $C^{\ast}$-dynamical system
$\left(A,G,\alpha\right)$ with $A$ an elementary $C^{\ast}$-algebra (i.e.,
isomorphic to the compacts on some Hilbert space) and $G$ abelian, they were
able to completely classify, up to unitary equivalence, all quadruples
$\left(X,\pi,U,V\right)$ with the following properties:

  1. (1)

$X$ is a Hilbert $A$-module.

  2. (2)

$\left(X,\pi,U\right)$ is a covariant representation of
$\left(A,G,\alpha\right)$.

  3. (3)

$\left(X,\pi,V\right)$ is a covariant representation of
$(A,\widehat{G},\iota)$, where $\iota$ denotes the trivial action of
$\widehat{G}$ on $A$.

  4. (4)

$\left(U,V\right)$ satisfies the WCR on $X$.

In the present paper, we further promote the paradigm shift in [10],
streamlining many of their arguments to present what may reasonably be thought
of as an abstract Stone-von Neumann Theorem, which we state in terms of
Hilbert modules over elementary $C^{\ast}$-algebras. Notably, this abstract
characterization makes no mention of groups but largely takes on the flavor of
the von Neumann Uniqueness Theorem (cf. [34]*Theorem 4.29). This abstract
characterization offers more flexibility, allowing us to consider dynamical
systems involving arbitrary locally compact groups, successfully removing the
abelian hypothesis through the tools of nonabelian duality.

We provide a brief overview of nonabelian duality in §1 as preparation for the
parallel sections §3 and §4 involving the Covariant Stone von-Neumann Theorem
for Actions and Coactions respectively.

We thank the referee for comments that improved our paper.

##  1\. Preliminaries

We refer to [17] and [7] for Hilbert $C^{\ast}$-modules and
$C^{\ast}$-correspondences. Here we merely record our conventions. Throughout,
$A$ and $B$ denote $C^{\ast}$-algebras, and $X$ and $Y$ denote Hilbert
$C^{\ast}$-modules. A _$B$ -$A$ correspondence_ is a Hilbert $A$-module $X$
equipped with a homomorphism $\phi_{X}:B\to\mathcal{L}\left(X\right)$, which
we will always assume to be nondegenerate in the sense that

$BX=\\{\left[\phi_{X}\left(b\right)\right]\left(x\right):b\in B,x\in X\\}=X.$

We also call $\phi_{X}$ a _representation_ of $B$ on $X$. Given a $C$-$B$
correspondence $Y$ and a $B$-$A$ correspondence $X$, we write $Y\otimes_{B}X$
for the $B$-balanced tensor product, which is a $C$-$A$ correspondence.
Caution: Lance [17]*pages 39–41 would write $Y\otimes_{B}X$ for the
_algebraic_ $B$-balanced tensor product, and $Y\otimes_{\phi_{X}}X$ for the
completion; nowadays, it seems more customary to use both notations for the
completion.

A _$B$ -$A$ imprimitivity bimodule_ is a $B$-$A$ correspondence $X$ such that
$\phi_{X}$ is an isomorphism of $B$ onto $\mathcal{K}\left(X\right)$, in which
case we write $\widetilde{X}$ for the _conjugate_ $A$-$B$ imprimitivity
bimodule. We regard a Hilbert space $H$ as a
$\mathcal{K}\left(H\right)$-$\mathbb{C}$ imprimitivity bimodule (caution: for
this purpose, the inner product on $H$ is taken to be linear in the second
variable).

An _isomorphism_ of $B$-$A$ correspondences $X$ and $Y$ is a linear bijection
$\Phi:X\to Y$ such that

  1. (1)

$\left\langle\Phi\left(x\right),\Phi\left(y\right)\right\rangle_{A}=\left\langle
x,y\right\rangle_{A}$ and

  2. (2)

$\Phi\left(bx\right)=b\Phi\left(x\right)$

for all $x,y\in X$ and $b\in B$. Note that it follows from (1) that
$\Phi\left(xa\right)=\Phi\left(x\right)a$ for all $x\in X$ and $a\in A$.

Given a Hilbert $A$-module $X$ and a Hilbert $B$-module $Y$, the _external
tensor product_ $X\otimes Y$ is a Hilbert $\left(A\otimes B\right)$-module. In
particular, if $B=\mathbb{C}$, then we have a Hilbert space $Y$, so we can
regard the Hilbert $\left(A\otimes\mathbb{C}\right)$-module $X\otimes Y$ as a
Hilbert $A$-module.

We record the following standard facts for convenient reference:

######  Lemma (Calculus of $C^{\ast}$-correspondences).

Let $X$, $\left(X_{i}\right)_{i\in I}$ be Hilbert $A$-modules and $Y$ an
$A$-$C$ correspondence. Then

  1. (1)

$X\otimes_{A}A\simeq X$ as Hilbert $A$-modules;

  2. (2)

$\left(\bigoplus_{i\in I}X_{i}\right)\otimes_{A}Y\simeq\bigoplus_{i\in
I}\left(X_{i}\otimes_{A}Y\right)$ as Hilbert $C$-modules.

Similarly, if the family $X$, $\left(X_{i}\right)_{i\in I}$ consists of
$B$-$A$ correspondences, and $Z$ is a $D$-$B$ correspondence, then also

  1. (3)

$B\otimes_{B}X\simeq X$ as $B$-$A$ correspondences;

  2. (4)

$Z\otimes_{B}\left(\bigoplus_{i\in I}X_{i}\right)\simeq\bigoplus_{i\in
I}\left(Z\otimes_{B}X_{i}\right)$ as $D$-$A$ correspondences.

If $X$ is a $B$-$A$ imprimitivity bimodule (so that the conjugate
$\widetilde{X}$ exists), then also

  1. (5)

$\widetilde{X}\otimes_{B}X\simeq A$ as $A$-$A$ imprimitivity bimodules;

  2. (6)

$X\otimes_{A}\widetilde{X}\simeq B$ as $B$-$B$ imprimitivity bimodules.

###### Proof.

The proof is routine. Standard references include [17, 27, 7]. ∎

In Sections 3 and 4, we will apply the Abstract Modular Stone-von Neumann
Theorem to a couple of situations where the imprimitivity bimodule comes from
crossed-product duality. For completeness, we review the necessary background.
For the remainder of this section, $G$ is a locally compact group and $A,B$
are $C^{\ast}$-algebras. An _action_ $\left(A,G,\alpha\right)$ of $G$ is a
strongly continuous homomorphism
$\alpha:G\to\operatorname{Aut}\left(A\right)$. A _covariant representation_ of
$\left(A,G,\alpha\right)$ is a triple $\left(X,\pi,U\right)$, where $X$ is a
Hilbert $C^{\ast}$-module, $\pi:A\to\mathcal{L}\left(X\right)$ is a
nondegenerate representation, and $U:G\to\mathcal{L}\left(X\right)$ is a
strongly continuous unitary representation satisfying

$\pi\circ\alpha_{x}=\operatorname{Ad}U_{x}\circ\pi\qquad\text{for}~{}x\in G.$

If $X$ is a $C^{\ast}$-algebra $B$, regarded as a Hilbert $B$-module in the
canonical way, then $\mathcal{L}\left(X\right)=M\left(B\right)$ and the
continuity assumption on $U$ is with respect to the strict topology.

A _crossed product_ of $\left(A,G,\alpha\right)$ is a covariant representation
$\left(A\rtimes_{\alpha}G,i_{A},i_{G}\right)$, where $A\rtimes_{\alpha}G$ is a
$C^{\ast}$-algebra and the pair $\left(i_{A},i_{G}\right)$ is universal in the
sense that for every covariant representation $\left(X,\pi,U\right)$, there is
a unique nondegenerate representation $\pi\times
U:A\rtimes_{\alpha}G\to\mathcal{L}\left(X\right)$, called the _integrated
form_ of $\left(\pi,U\right)$, making the diagram

${A}$${M\left(A\rtimes_{\alpha}G\right)}$${G}$${\mathcal{L}\left(X\right)}$$\scriptstyle{i_{A}}$$\scriptstyle{\pi}$$\scriptstyle{\pi\times
U}$$\scriptstyle{!}$$\scriptstyle{i_{G}}$$\scriptstyle{U}$

commute.

We identify $A\otimes C_{0}\left(G\right)$ with $C_{0}\left(G,A\right)$ in the
usual way. The regular covariant representation of $\left(A,G,\alpha\right)$
is $\left(A\otimes L^{2}\left(G\right),\left(\operatorname{id}\otimes
M\right)\circ\widetilde{\alpha},1\otimes\lambda\right)$, where $M$ is the
representation of $C_{0}\left(G\right)$ on $L^{2}\left(G\right)$ by
multiplication operators, $\lambda$ is the left regular representation of $G$,
and $\widetilde{\alpha}:A\to M\left(A\otimes C_{0}\left(G\right)\right)$ is
given by

$\left[\widetilde{\alpha}\left(a\right)\left(b\otimes
f\right)\right]\left(x\right)=f\left(x\right)\alpha_{x^{-1}}\left(a\right)b\qquad\text{for}~{}a,b\in
A,~{}f\in C_{0}\left(G\right),~{}x\in G.$

Dually, a _coaction_ $\left(A,G,\delta\right)$ of $G$ is a homomorphism
$\delta:A\to M\left(A\otimes C^{\ast}(G)\right)$ satisfying both of the
following:

$\displaystyle\overline{\operatorname{span}}\\{\delta\left(A\right)\left(1\otimes
C^{\ast}(G)\right)\\}$ $\displaystyle=A\otimes C^{\ast}(G),$
$\displaystyle\left(\delta\otimes\operatorname{id}\right)\circ\delta$
$\displaystyle=\left(\operatorname{id}\otimes\delta_{G}\right)\circ\delta,$

where $\delta_{G}:C^{\ast}(G)\to M\left(C^{\ast}(G)\otimes C^{\ast}(G)\right)$
is the integrated form of the unitary homomorphism $s\mapsto s\otimes s$. Note
that $\delta_{G}$ is a coaction of $G$ on $C^{\ast}(G)$. A _covariant
representation_ of $\left(A,G,\delta\right)$ is a triple
$\left(X,\pi,\mu\right)$, where $X$ is a Hilbert $C^{\ast}$-module, $\pi$ and
$\mu$ are nondegenerate representations of $A$ and $C_{0}\left(G\right)$,
respectively, on $X$ satisfying

$\left(\delta\otimes\operatorname{id}\right)\circ\delta=\operatorname{Ad}\left(\mu\otimes\operatorname{id}\right)\left(w_{G}\right)\circ\delta,$

and where $w_{G}\in M\left(C_{0}\left(G\right)\otimes C^{\ast}(G)\right)$
corresponds to the canonical embedding of $G$ in $M\left(C^{\ast}(G)\right)$.
A _crossed product_ of $\left(A,G,\delta\right)$ is a covariant representation
$(A\rtimes_{\delta}G,j_{A},j_{G})$, where $A\rtimes_{\delta}G$ is a
$C^{\ast}$-algebra and the pair $\left(j_{A},j_{G}\right)$ is universal in the
sense that for every covariant representation $\left(X,\pi,\mu\right)$, there
is a unique nondegenerate representation
$\pi\times\mu:A\rtimes_{\delta}G\to\mathcal{L}\left(X\right)$, called the
_integrated form_ of $\left(\pi,\mu\right)$, making the diagram

${A}$${M\left(A\rtimes_{\delta}G\right)}$${C_{0}\left(G\right)}$${\mathcal{L}\left(X\right)}$$\scriptstyle{j_{A}}$$\scriptstyle{\pi}$$\scriptstyle{\pi\times\mu}$$\scriptstyle{!}$$\scriptstyle{j_{G}}$$\scriptstyle{\mu}$

commute.

The _regular covariant representation_ of $\left(A,G,\delta\right)$ is
$\left(A\otimes
L^{2}\left(G\right),\left(\operatorname{id}\otimes\lambda\right)\circ\delta,1\otimes
M\right)$.

Raeburn shows in [26]*Example 2.9 (1) that if
$U:G\to\mathcal{L}\left(X\right)$ is a unitary representation and
$\mu:C_{0}\left(G\right)\to\mathcal{L}\left(X\right)$ is a nondegenerate
representation, then $\left(X,U,\mu\right)$ is a covariant representation of
$\left(C^{\ast}(G),G,\delta_{G}\right)$ if and only if $\left(X,\mu,U\right)$
is a covariant representation of
$\left(C_{0}\left(G\right),G,\textnormal{lt}\right)$, where lt is the action
of $G$ on $C_{0}\left(G\right)$ given by left translation.

If $\left(A,G,\alpha\right)$ is an action, then the _dual coaction_
$\widehat{\alpha}$ of $G$ on $A\rtimes_{\alpha}G$ is determined by

$\displaystyle\widehat{\alpha}\left(i_{A}\left(a\right)\right)$
$\displaystyle=i_{A}\left(a\right)\otimes 1\qquad\text{for}~{}a\in A,$
$\displaystyle\widehat{\alpha}\left(i_{G}\left(x\right)\right)$
$\displaystyle=i_{G}\left(x\right)\otimes x\qquad\text{for}~{}x\in G.$

If $\left(A,G,\delta\right)$ is a coaction, then the _dual action_
$\widehat{\delta}$ of $G$ on $A\rtimes_{\delta}G$ is determined by

$\displaystyle\widehat{\delta}_{x}\circ j_{A}$ $\displaystyle=j_{A},$
$\displaystyle\widehat{\delta}_{x}\circ j_{G}$
$\displaystyle=j_{G}\circ\textnormal{rt}_{x}\qquad\text{for}~{}x\in G,$

where rt is the action of $G$ on $C_{0}\left(G\right)$ given by right
translation.

##  2\. The Abstract Stone-von Neumann Theorem

We now specialize the methods of the preliminaries to the case of $X$ a
nonzero Hilbert $A$-module, where $A$ is an elementary $C^{\ast}$-algebra.
Thus $A\cong\mathcal{K}\left(H\right)$ for some nonzero Hilbert space $H$,
which is necessarily unique up to isomorphism (see [5]*Corollary 4.1.6). Since
$A$ is simple, the Hilbert module $X$ is full.

A Hilbert space $H$ is already an $A$-$\mathbb{C}$ imprimitivity bimodule, so
its conjugate $\widetilde{H}$ is a $\mathbb{C}$-$A$ imprimitivity bimodule.
For any Hilbert space $L$, $L\otimes_{\mathbb{C}}\widetilde{H}$ is a Hilbert
$A$-module. In fact, every Hilbert $A$-module is of this form, as we show in
the following lemma.

######  Lemma 2.1.

If $X$ is any Hilbert $A$-module, then there is a Hilbert space $L$ such that
$X\cong L\otimes_{\mathbb{C}}\widetilde{H}$ as Hilbert $A$-modules. This
Hilbert space is unique up to isomorphism.

###### Proof.

We take $L=X\otimes_{A}H$, which is a Hilbert space since $H$ is a Hilbert
$\mathbb{C}$-module. In fact, $H$ is an $A$-$\mathbb{C}$ imprimitivity
bimodule, and the calculus of imprimitivity bimodules gives

$L\otimes_{\mathbb{C}}\widetilde{H}\simeq
X\otimes_{A}H\otimes_{\mathbb{C}}\widetilde{H}\simeq X\otimes_{A}A\simeq X.$

For uniqueness up to isomorphism, if $M$ is another Hilbert space such that
$L\otimes_{\mathbb{C}}\widetilde{H}\simeq M\otimes_{\mathbb{C}}\widetilde{H}$,
then similar computations show that tensoring with $H$ gives $L\simeq M$. ∎

######  Lemma 2.2.

In Lemma 2.1, identify $X$ with $L\otimes_{\mathbb{C}}\widetilde{H}$. Then the
map

$b\mapsto b\otimes_{\mathbb{C}}1$

from $\mathcal{K}\left(L\right)$ to $\mathcal{L}\left(X\right)$ is an
isomorphism onto $\mathcal{K}\left(X\right)$. In this way, Lemma 2.1 gives an
isomorphism of $\mathcal{K}\left(X\right)$-$A$ correspondences.

###### Proof.

Since $H$ is an $A$-$\mathbb{C}$ imprimitivity bimodule, the first assertion
follows from [17]*Proposition 4.7, and then the second assertion follows
immediately. ∎

######  Remark 2.3.

With the notation above, Schweizer proves in [32]*Lemma 3 that
$X\simeq\mathcal{K}\left(H,L\right)$. We can give an alternative proof of this
by combining Lemma 2.1 with the elementary fact that for any
$C^{\ast}$-algebra $D$ and any two Hilbert $D$-modules $X$ and $Y$, there is a
unique isomorphism

$\Upsilon:X\otimes_{D}\widetilde{Y}\xrightarrow{\simeq}\mathcal{K}\left(Y,X\right)$

of $\mathcal{K}\left(X\right)$-$\mathcal{K}\left(Y\right)$ correspondences
such that

$\Upsilon\left(x\otimes\widetilde{y}\right)=\theta_{x,y}\qquad\text{for}~{}x\in
X,~{}y\in Y.$

Actually, Schweizer gets the Hilbert space $L$ a different way, namely by
recognizing that $\mathcal{K}\left(X\right)$ is an elementary
$C^{\ast}$-algebra. It is easy to see that his Hilbert space is isomorphic to
our $L$. Related results on Hilbert modules over elementary
$C^{\ast}$-algebras are contained in, for example, [3, 20].

######  Definition 2.4.

For any two $B$-$A$ correspondences $X$ and $Y$, we say that $Y$ is a
_multiple_ of $X$ provided $Y\cong\bigoplus_{I}X$ for some set $I$.

######  Proposition 2.5 (Abstract Modular Stone-von Neumann Theorem).

If $A$ is elementary and $X$ a $B$-$A$ imprimitivity bimodule, then every
$B$-$A$ correspondence is a multiple of $X$.

###### Proof.

Let $Y$ be a $B$-$A$ correspondence. By Lemmas 2.1 and 2.2, without loss of
generality,

$X=L\otimes_{\mathbb{C}}\widetilde{H}\quad\text{and}\quad
Y=M\otimes_{\mathbb{C}}\widetilde{H},$

where $H$, $M$, and $L$ are Hilbert spaces, with $B$ acting irreducibly on $L$
and nondegenerately on $M$. Since $B$ is elementary, its image in
$\mathcal{B}\left(L\right)$ is the algebra of compact operators
$\mathcal{K}\left(L\right)$. Thus the nondegenerate representation of $B$ on
$M$ is a multiple of the irreducible representation on $L$. In other words,
for some set $I$, we have $M\simeq\bigoplus_{I}L$ as $B$-$\mathbb{C}$
correspondences.

Then by the calculus of $C^{\ast}$-correspondences,

$\displaystyle M\otimes_{\mathbb{C}}\widetilde{H}$
$\displaystyle\simeq\left(\bigoplus_{I}L\right)\otimes_{\mathbb{C}}\widetilde{H}$
$\displaystyle\simeq\bigoplus_{I}(L\otimes_{\mathbb{C}}\widetilde{H})$
$\displaystyle\simeq\bigoplus_{I}X$

as $B$-$A$ correspondences, and we are done. ∎

######  Remark 2.6.

Proposition 2.5 could be used to help classify Hilbert modules over elementary
$C^{\ast}$-algebras. However, in fact [14]*Remark 6.5 gives a classification
where the coefficient algebra can be any direct sum of elementary
$C^{\ast}$-algebras.

##  3\. The Covariant Stone-von Neumann Theorem for Actions

The Covariant Stone-von Neumann Theorem for Actions of Abelian Groups
formulated and proven in [10] generalizes the Mackey-Stone-von Neumann Theorem
by replacing Hilbert spaces by Hilbert $C^{\ast}$-modules as the basis of
representations. However, the theorem is more than just a statement about
representations of a locally compact abelian group $G$ and its dual
$\widehat{G}$ on a Hilbert $C^{\ast}$-module, as the WCR alone suggests. As a
Hilbert $C^{\ast}$-module is a right module $X$ over a $C^{\ast}$-algebra $A$,
one can introduce a strongly continuous action $\alpha$ of $G$ on $A$ and
attempt to classify “representations” of the $C^{\ast}$-dynamical system
$\left(A,G,\alpha\right)$ on $X$, assuming one knows precisely what these
“representations” are. The next definition seeks to realize this assumption.

######  Definition 3.1 ([10]).

Let $\left(A,G,\alpha\right)$ be a $C^{\ast}$-dynamical system with $G$
abelian. Then an _abelian Heisenberg $\left(A,G,\alpha\right)$-modular
representation_ is defined as a quadruple $\left(X,\pi,U,V\right)$ with the
following properties:

  1. (1)

$X$ is a Hilbert $A$-module.

  2. (2)

$\left(X,\pi,U\right)$ is a covariant representation of
$\left(A,G,\alpha\right)$.

  3. (3)

$\left(X,\pi,V\right)$ is a covariant representation of
$(A,\widehat{G},\iota)$, where $\iota$ denotes the trivial action of
$\widehat{G}$ on $A$.

  4. (4)

$\left(U,V\right)$ satisfies the WCR on $X$: For all $x\in G$ and
$\varphi\in\widehat{G}$,

$V_{\varphi}U_{x}=\varphi\left(x\right)U_{x}V_{\varphi}.$

An abelian Heisenberg $\left(A,G,\alpha\right)$-modular representation
$\left(X,\pi,U,V\right)$ is said to be _equivalent_ to another one
$\left(X^{\prime},\pi^{\prime},U^{\prime},V^{\prime}\right)$ if and only if
there is a unitary map $W:X\to X^{\prime}$ such that for all $a\in A$, $x\in
G$, and $\varphi\in\widehat{G}$,

$W\pi\left(a\right)W^{-1}=\pi^{\prime}\left(a\right),\quad
WU_{x}W^{-1}=U^{\prime}_{x},\quad\text{and}\quad
WV_{\varphi}W^{-1}=V^{\prime}_{\varphi},$

in which case we write
$\left(X,\pi,U,V\right)\simeq\left(X^{\prime},\pi^{\prime},U^{\prime},V^{\prime}\right)$.
Note that $\simeq$ is an equivalence relation on the class of all abelian
Heisenberg $\left(A,G,\alpha\right)$-modular representations.

Note that in the definition above, in addition to the WCR, there are two
commutation relations coming from the covariant representations in Properties
(2) and (3). These two relations are novel features of the Covariant Stone-von
Neumann Theorem for Actions of Abelian Groups that do not arise in the Mackey-
Stone-von Neumann Theorem.

######  Definition 3.2 ([10]).

Let $\left(A,G,\alpha\right)$ be a $C^{\ast}$-dynamical system with $G$
abelian. Then the _abelian Schrödinger $\left(A,G,\alpha\right)$-modular
representation_ is the quadruple

$(A\otimes L^{2}\left(G\right),\left(\operatorname{id}\otimes
M\right)\circ\widetilde{\alpha},1\otimes\lambda,1\otimes\operatorname{Mod}),$

where $(A\otimes L^{2}\left(G\right),\left(\operatorname{id}\otimes
M\right)\circ\widetilde{\alpha},1\otimes\lambda)$ is the regular covariant
representation of $\left(A,G,\alpha\right)$ (defined in the introduction) and
$\operatorname{Mod}$ is the phase modulation of $G$, as in Definition 0.3.

The following theorem is reminiscent of Theorem 0.4.

######  Theorem 3.3 ([10]).

Let $\left(A,G,\alpha\right)$ be a $C^{\ast}$-dynamical system with $G$
abelian. Then the following statements are true:

  1. (1)

The abelian Schrödinger $\left(A,G,\alpha\right)$-modular representation is an
abelian Heisenberg $\left(A,G,\alpha\right)$-modular representation.

  2. (2)

Let $I$ be a set and $\left(X_{i},\pi_{i},U_{i},V_{i}\right)_{i\in I}$ an
$I$-indexed family of abelian Heisenberg $\left(A,G,\alpha\right)$-modular
representations. Then the direct sum

$\bigoplus_{i\in I}\left(X_{i},\pi_{i},U_{i},V_{i}\right)_{i\in
I}\coloneqq\left(\bigoplus_{i\in I}X_{i},\bigoplus_{i\in
I}\pi_{i},\bigoplus_{i\in I}U_{i},\bigoplus_{i\in I}V_{i}\right)$

is also an abelian Heisenberg $\left(A,G,\alpha\right)$-modular
representation.

The Covariant Stone-von Neumann Theorem for Actions of Abelian Groups can now
be stated concisely as follows.

######  Theorem 3.4 (Covariant Stone-von Neumann Theorem for Actions of
Abelian Groups [10]).

Let $\left(A,G,\alpha\right)$ be a $C^{\ast}$-dynamical system with $A$ an
elementary $C^{\ast}$-algebra and $G$ abelian. Then every abelian Heisenberg
$\left(A,G,\alpha\right)$-modular representation is a multiple of the abelian
Schrödinger $\left(A,G,\alpha\right)$-modular representation.

In order to generalize the Covariant Stone-von Neumann Theorem for Actions of
Abelian Groups to include actions of nonabelian groups, we must first
reformulate the WCR in nonabelian terms.

Let $\left(A,G,\alpha\right)$ be a $C^{\ast}$-dynamical system with $G$
abelian, and let the following data be given:

  * •

A Hilbert $A$-module $X$.

  * •

A unitary representation $U$ of $G$ on $X$.

  * •

A unitary representation $V$ of $\widehat{G}$ on $X$.

  * •

A representation $\pi$ of $A$ by adjointable operators on $X$.

Let $\widetilde{V}$ denote the integrated form of $V$, which is a
representation of $C^{\ast}(\widehat{G})$ on $X$. Letting
$\mathcal{F}:C^{\ast}(\widehat{G})\to C_{0}\left(G\right)$ denote the
$\widehat{G}$-Fourier transform, we get a representation
$\mu\coloneqq\widetilde{V}\circ\mathcal{F}^{-1}$ of $C_{0}\left(G\right)$ on
$X$. It is a standard consequence of properties of Fourier transforms that
$\left(U,V\right)$ satisfies the WCR if and only if $\left(X,\mu,U\right)$ is
a covariant representation of
$\left(C_{0}\left(G\right),G,\textnormal{lt}\right)$ (see [34]*Lemma 4.28, for
example).

This reformulation of the WCR does not appeal to the abelian nature of $G$ and
can thus be applied even to nonabelian groups. Moreover,
$\left(X,\pi,V\right)$ is a covariant representation of
$(\widehat{G},A,\iota)$ if and only if $\pi$ and $\mu$ commute.

These observations motivate the following alternative definition of a modular
representation of a $C^{\ast}$-dynamical system, which makes sense even if the
underlying group is not abelian.

######  Definition 3.5.

Let $\left(A,G,\alpha\right)$ be a $C^{\ast}$-dynamical system. Then a
_Heisenberg $\left(A,G,\alpha\right)$-modular representation_ is a quadruple
$\left(X,\pi,U,\mu\right)$ with the following properties:

  1. (1)

$X$ is a Hilbert $A$-module.

  2. (2)

$\left(X,\pi,U\right)$ is a covariant representation of
$\left(A,G,\alpha\right)$.

  3. (3)

$\left(X,\mu,U\right)$ is a covariant representation of
$\left(C_{0}\left(G\right),G,\textnormal{lt}\right)$.

  4. (4)

$\pi$ and $\mu$ commute.

######  Lemma 3.6.

Let $\left(A,G,\alpha\right)$ be a $C^{\ast}$-dynamical system, and let
$\widehat{\alpha}$ denote the dual coaction of $G$ on $A\rtimes_{\alpha}G$.
Then

$\left(X,\pi,U,\mu\right)\mapsto\left(X,\pi\times U,\mu\right)$

gives a bijection from the Heisenberg $\left(A,G,\alpha\right)$-modular
representations to the covariant representations of
$\left(A\rtimes_{\alpha}G,G,\widehat{\alpha}\right)$. Consequently,

$\left(X,\pi,U,\mu\right)\mapsto\left(X,\left(\pi\times
U\right)\times\mu\right)$

gives a bijection from the Heisenberg $\left(A,G,\alpha\right)$-modular
representations to the nondegenerate representations of
$\left(A\rtimes_{\alpha}G\right)\rtimes_{\widehat{\alpha}}G$.

###### Proof.

By [7]*Proposition A.63 and the discussion preceding it (see also
[26]*Examples 2.9), $\left(X,\pi,U,\mu\right)$ is a Heisenberg
$\left(A,G,\alpha\right)$-modular representation if and only if
$\left(X,\pi\times U,\mu\right)$ is a covariant representation of
$\left(A\rtimes_{\alpha}G,G,\widehat{\alpha}\right)$. Since the latter is
equivalent to $\left(X,\left(\pi\times U\right)\times\mu\right)$ being a
nondegenerate representation of
$\left(A\rtimes_{\alpha}G\right)\rtimes_{\widehat{\alpha}}G$ by definition, we
are done. ∎

######  Definition 3.7.

A Heisenberg $\left(A,G,\alpha\right)$-modular representation
$\left(X,\pi,U,\mu\right)$ is said to be _equivalent_ to another one
$\left(X^{\prime},\pi^{\prime},U^{\prime},\mu^{\prime}\right)$ if and only if
there is a unitary map $W:X\to X^{\prime}$ such that for all $a\in A$, $x\in
G$, and $f\in C_{0}\left(G\right)$,

$W\pi\left(a\right)W^{-1}=\pi^{\prime}\left(a\right),\quad
WU_{x}W^{-1}=U^{\prime}_{x},\quad\text{and}\quad
W\mu\left(f\right)W^{-1}=\mu^{\prime}\left(f\right),$

in which case we write
$\left(X,\pi,U,\mu\right)\simeq\left(X^{\prime},\pi^{\prime},U^{\prime},\mu^{\prime}\right)$.

Note that $\simeq$ is an equivalence relation on the class of all Heisenberg
$\left(A,G,\alpha\right)$-modular representations. Heisenberg modular
representations are equivalent if and only if the associated representations
of the double crossed product are equivalent.

######  Definition 3.8.

Let $\left(A,G,\alpha\right)$ be a $C^{\ast}$-dynamical system. Then the
_Schrödinger $\left(A,G,\alpha\right)$-modular representation_ is the
quadruple

$(A\otimes L^{2}\left(G\right),\left(\operatorname{id}\otimes
M\right)\circ\widetilde{\alpha},1\otimes\lambda,1\otimes M).$

Note: If $G$ is abelian, then $M=\mu_{\operatorname{Mod}}$.

######  Proposition 3.9.

Let $\left(A,G,\alpha\right)$ be a $C^{\ast}$-dynamical system. Then the
following statements are true:

  1. (1)

The Schrödinger $\left(A,G,\alpha\right)$-modular representation is a
Heisenberg $\left(A,G,\alpha\right)$-modular representation.

  2. (2)

Let $I$ be a set and $\left(X_{i},\pi_{i},U_{i},\mu_{i}\right)_{i\in I}$ an
$I$-indexed family of Heisenberg $\left(A,G,\alpha\right)$-modular
representations. Then the direct sum

$\bigoplus_{i\in
I}\left(X_{i},\pi_{i},U_{i},\mu_{i}\right)\coloneqq\left(\bigoplus_{i\in
I}X_{i},\bigoplus_{i\in I}\pi_{i},\bigoplus_{i\in I}U_{i},\bigoplus_{i\in
I}\mu_{i}\right)$

is also a Heisenberg $\left(A,G,\alpha\right)$-modular representation.

###### Proof.

Statement (1) is true by virtue of the following facts:

  * •

The regular covariant representation of $\left(A,G,\alpha\right)$ is precisely

$(A\otimes L^{2}\left(G\right),\left(\operatorname{id}\otimes
M\right)\circ\widetilde{\alpha},1\otimes\lambda).$

  * •

$\left(L^{2}\left(G\right),M,\lambda\right)$ is a covariant representation of
$\left(C_{0}\left(G\right),G,\textnormal{lt}\right)$.

  * •

$\left(\operatorname{id}\otimes M\right)\circ\widetilde{\alpha}$ commutes with
$1\otimes M$ as $C_{0}\left(G\right)$ is commutative.

Statement (2) follows routinely from the definitions. ∎

By Lemma 3.6 and Proposition 3.9, the direct sum of Heisenberg modular
representations is associated to the direct sum of their associated
representations of the double crossed product. In particular, a Heisenberg
modular representation $\left(X,\pi,U,\mu\right)$ is a multiple of another one
$\left(X^{\prime},\pi^{\prime},U^{\prime},\mu^{\prime}\right)$ if and only if
the associated representation $\left(\pi\times U\right)\times\mu$ of the
double crossed product is a multiple of $\left(\pi^{\prime}\times
U^{\prime}\right)\times\mu^{\prime}$.

######  Proposition 3.10 (Covariant Stone-von Neumann Theorem for Actions).

Let $\left(A,G,\alpha\right)$ be a $C^{\ast}$-dynamical system with $A$ an
elementary $C^{\ast}$-algebra. Then every Heisenberg
$\left(A,G,\alpha\right)$-modular representation is a multiple of the
Schrödinger $\left(A,G,\alpha\right)$-modular representation.

###### Proof.

Let $\left(X,\pi,U,\mu\right)$ be a Heisenberg
$\left(A,G,\alpha\right)$-modular representation. Imai-Takai Duality says that
the integrated form of the Schrödinger $\left(A,G,\alpha\right)$-modular
representation

$(A\otimes L^{2}\left(G\right),\left(\operatorname{id}\otimes
M\right)\circ\widetilde{\alpha},1\otimes\lambda,1\otimes M)$

is a $C^{\ast}$-isomorphism from
$\left(A\rtimes_{\alpha}G\right)\rtimes_{\widehat{\alpha}}G$ to
$A\otimes\mathcal{K}\left(L^{2}\left(G\right)\right)$. Using this
$C^{\ast}$-isomorphism and the identification

$\mathcal{K}(A\otimes L^{2}\left(G\right))\cong
A\otimes\mathcal{K}(L^{2}\left(G\right)),$

the $\mathcal{K}\left(A\otimes L^{2}\left(G\right)\right)$-$A$ imprimitivity
bimodule $A\otimes L^{2}\left(G\right)$ becomes an
$\left(\left(A\rtimes_{\alpha}G\right)\rtimes_{\widehat{\alpha}}G\right)$-$A$
imprimitivity bimodule, and $\phi_{A\otimes L^{2}\left(G\right)}$ becomes the
integrated form of the Schrödinger $\left(A,G,\alpha\right)$-modular
representation. The Covariant Stone-von Neumann Theorem for Actions then
follows from the paragraph preceding this proposition. ∎

##  4\. The Covariant Stone-von Neumann Theorem for Coactions

In this section, we apply the Abstract Modular Stone-von Neumann Theorem again
to crossed-product duality, this time starting with a coaction rather than an
action. The development is parallel to that in Section 3, and we significantly
streamline the presentation.

######  Definition 4.1.

Given a coaction $\left(A,G,\delta\right)$, a _Heisenberg
$\left(A,G,\delta\right)$-modular representation_ is a quadruple
$\left(X,\pi,\mu,U\right)$ with the following properties:

  * •

$\left(X,\pi,\mu\right)$ is a covariant representation of
$\left(A,G,\delta\right)$.

  * •

$\left(X,\mu,U\right)$ is a covariant representation of
$\left(C_{0}\left(G\right),G,\textnormal{rt}\right)$.

  * •

$\pi$ and $U$ commute.

######  Lemma 4.2.

With the above notation, the assignments
$\left(X,\pi,\mu,U\right)\mapsto\left(X,\pi\times\mu,U\right)$ and
$\left(X,\pi,\mu,U\right)\mapsto\left(X,\left(\pi\times\mu\right)\times
U\right)$ give bijections from the Heisenberg
$\left(A,G,\delta\right)$-modular representations to the covariant
representations of $(A\rtimes_{\delta}G,G,\widehat{\delta})$ and the
nondegenerate representations of
$\left(A\rtimes_{\delta}G\right)\rtimes_{\widehat{\delta}}G$, respectively.

It seems difficult to find the above statement in the literature, but it is an
easy application of standard techniques, and for completeness, we include an
outline of the proof below.

###### Proof.

For the first part, it suffices to note that

  * •

if $\left(X,\pi,\mu,U\right)$ is a Heisenberg $\left(A,G,\delta\right)$-module
representation, then $\left(X,\pi\times\mu,U\right)$ is a covariant
representation of $(A\rtimes_{\delta}G,G,\widehat{\delta})$, and this can be
quickly checked separately on the generators $\pi\left(a\right)$ and
$\mu\left(f\right)$ for $a\in A$ and $f\in C_{0}\left(G\right)$, while, on the
other hand,

  * •

if $\left(X,\sigma,U\right)$ is a covariant representation of
$(A\rtimes_{\delta}G,G,\widehat{\delta})$, then there is a unique covariant
representation $\left(X,\pi,\mu\right)$ of $\left(A,G,\delta\right)$ such that
$\sigma=\pi\times\mu$, and, moreover, covariance of $\left(X,\sigma,U\right)$
applied to the generators $\pi\left(a\right)$ and $\mu\left(f\right)$ shows
that $\left(X,\mu,U\right)$ is covariant for
$\left(C_{0}\left(G\right),G,\textnormal{rt}\right)$ and that $\pi$ commutes
with $U$.

Then the second part follows immediately since, by definition, the covariant
representations of the action $\widehat{\delta}$ are in bijective
correspondence with the nondegenerate representations of the crossed product
by $\widehat{\delta}$. ∎

######  Definition 4.3.

The _Schrödinger $\left(A,G,\delta\right)$-modular representation_ is

$(A\otimes
L^{2}\left(G\right),\left(\operatorname{id}\otimes\lambda\right)\circ\delta,1\otimes
M,1\otimes\rho),$

where $\rho$ is the right regular representation of $G$.

######  Lemma 4.4.

The Schrödinger $\left(A,G,\delta\right)$-modular representation is a
Heisenberg modular representation.

###### Proof.

This is true because of the following facts:

  * •

$\left(A\otimes
L^{2}\left(G\right),\left(\operatorname{id}\otimes\lambda\right)\circ\delta,1\otimes
M\right)$ is the regular covariant representation of
$\left(A,G,\delta\right)$.

  * •

$\left(A\otimes L^{2}\left(G\right),M,\rho\right)$ is a covariant
representation of $\left(C_{0}\left(G\right),G,\textnormal{rt}\right)$.

  * •

$\left(\operatorname{id}\otimes\lambda\right)\circ\delta$ commutes with
$1\otimes\rho$ since $\lambda$ and $\rho$ commute. ∎

Just as we did for actions, we say that a Heisenberg modular representation is
a multiple of another one if and only if it is equivalent to a direct sum of
copies of it.

######  Theorem 4.5 (Covariant Stone-von Neumann Theorem for Coactions).

If $\delta$ is a maximal coaction of $G$ on an elementary $C^{\ast}$-algebra
$A$, then every Heisenberg $\left(A,G,\delta\right)$-modular representation is
a multiple of the Schrödinger $\left(A,G,\delta\right)$-modular
representation.

###### Proof.

The Katayama Duality Theorem says that if the coaction $\delta$ is maximal,
then the integrated form of the Schrödinger modular representation is a
$C^{\ast}$-isomorphism of
$\left(A\rtimes_{\delta}G\right)\rtimes_{\widehat{\delta}}G$ onto
$A\otimes\mathcal{K}$. Using this $C^{\ast}$-isomorphism, the
$\left(A\otimes\mathcal{K}\right)$-$A$ imprimitivity bimodule $A\otimes
L^{2}\left(G\right)$ becomes an
$\left(\left(A\rtimes_{\delta}G\right)\rtimes_{\widehat{\delta}}G\right)$-$A$
imprimitivity bimodule, and $\phi_{A\otimes L^{2}\left(G\right)}$ is the
integrated form of the Schrödinger $\left(A,G,\delta\right)$-modular
representation. Now the result follows from Proposition 2.5. ∎

######  Remark 4.6.

In its original formulation, the Katayama Duality Theorem [16]*Theorem 8 (see
also [6]*Proposition 2.2) involved the reduced crossed product by the dual
action. The version we quoted is really more of a definition than a theorem
(see [6]*Definition 3.1).

## References

  * [1] B.V. Baht, M. Skeide, Pure semigroups of isometries on Hilbert $C^{\ast}$-modules, J. Funct. Anal., 269(2015), no. 5, 1539–1562.
  * [2] D. Bakić, B. Guljaš, On a class of module maps of Hilbert $C^{\ast}$-modules, Math. Commun., 7(2002), no. 2, 177–192.
  * [3] D. Bakić, B. Guljaš, Hilbert $C^{\ast}$-modules over $C^{\ast}$-algebras of compact operators, Acta Sci. Math. (Szeged), 68(2002), no. 1-2, 249–269.
  * [4] S. Cavallaro, G. Morchio, F. Strocchi, A generalization of the Stone-von Neumann theorem to nonregular representations of the CCR-algebra, Lett. Math. Phys., 47(1999), no. 4, 307–320.
  * [5] J. Dixmier, $C^{\ast}$-algebras, Translated from the French by Francis Jellett; North-Holland Mathematical Library, Vol. 15. North-Holland Publishing Co., Amsterdam-New York-Oxford (1977).
  * [6] S. Echterhoff, S. Kaliszewski, J. Quigg, Maximal coactions, Internat. J. Math., 15(2004), no. 1, 47–61.
  * [7] S. Echterhoff, S. Kaliszewski, J. Quigg, I. Raeburn, A categorical approach to imprimitivity theorems for $C^{\ast}$-dynamical systems, Mem. Amer. Math. Soc., 180(2006), no. 850.
  * [8] P. Green, The local structure of twisted covariance algebras, Acta Math., 140(1978), no. 3-4, 191–250.
  * [9] W. Heisenberg, Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Z. f. Physik, 33(1925), no. 1, 879–893.
  * [10] L. Huang, L. Ismert, The covariant Stone-von Neumann theorem for actions of abelian groups on $C^{\ast}$-algebras of compact operators, Comm. Math. Phys., 378(2020), no. 1, 117–147.
  * [11] A. an Huef, S. Kaliszewski, I. Raeburn, Covariant representations of Hecke algebras and imprimitivity for crossed products by homogeneous spaces, J. Pure Appl. Algebra, 212(2008), no. 10, 2344–2357.
  * [12] A. an Huef, I. Raeburn, D. P. Williams, Properties preserved under Morita equivalence of $C^{\ast}$-algebras, Proc. Amer. Math. Soc., 135(2007), no. 5, 1495–1503.
  * [13] S. Imai, H. Takai, On a duality for $C^{\ast}$-crossed products by a locally compact group, J. Math. Soc. Japan, 30(1978), no. 3, 495–504.
  * [14] S. Kaliszewski, N. S. Larsen, J. Quigg, Subgroup correspondences, Proc. Edinb. Math. Soc. (2), 61(2018), no. 4, 1127–1154.
  * [15] S. Kaliszewski, J. Quigg, Categorical Landstad duality for actions, Indiana Univ. Math. J., 58(2009), no. 1, 415–441.
  * [16] Y. Katayama, Takesaki’s duality for a nondegenerate co-action, Math. Scand., 55(1984), no. 1, 141–151.
  * [17] E. C. Lance, Hilbert $C^{\ast}$-modules: a toolkit for operator algebraists, London Mathematical Society Lecture Note Series, Vol. 210. Cambridge University Press, Cambridge (1995).
  * [18] L. H. Loomis, Note on a theorem of Mackey, Duke Math. J., 19(1952), no. 4, 641–645.
  * [19] G. Mackey, A theorem of Stone and von Neumann, Duke Math. J., 16(1949), no. 2, 313–326.
  * [20] B. Magajna, Hilbert $C^{\ast}$-modules in which all closed submodules are complemented, Proc. Amer. Math. Soc., 125(1997), no. 3, 849–852.
  * [21] G. Murphy, $C^{\ast}$-algebras and operator theory, Academic Press, Inc., Boston, MA (1990).
  * [22] J. von Neumann, Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann., 104(1931), no. 1, 570–578.
  * [23] R. Palma, Crossed products by Hecke pairs, Mem. Amer. Math. Soc., 1204(2018).
  * [24] J. Quigg, Full $C^{\ast}$-crossed product duality, J. Austral. Math. Soc. Ser. A, 50(1991), no. 1, 34–52.
  * [25] I. Raeburn, A duality theorem for crossed products by nonabelian groups, Miniconference on harmonic analysis and operator algebras (Canberra, 1987), Proc. Centre Math. Anal. Austral. Nat. Univ., Vol 15, Austral. Nat. Univ., Canberra, 1987, 214–227.
  * [26] I. Raeburn, On crossed products by coactions and their representation theory, Proc. London Math. Soc., 64(1992), no. 3, 625–652.
  * [27] I. Raeburn, D. P. Williams, Morita equivalence and continuous-trace $C^{\ast}$-algebras, Math. Surveys Monogr., Vol. 60, American Mathematical Society, Providence, RI (1998).
  * [28] M. Rieffel, On the uniqueness of the Heisenberg commutation relations, Duke Math. J., 39(1972), no. 4, 745–752.
  * [29] M. Rieffel, Induced representations of $C^{\ast}$-algebras, Advances in Math., 13(1974), no. 2, 176–257.
  * [30] E. Schrödinger, Quantisierung als Eigenwertproblem, Ann. der Physik, 384(1926), no. 4, 361–377.
  * [31] E. Schrödinger, Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinen, Ann. der Physik, 79(1926), no. 4, 734–756.
  * [32] Schweizer, J., A description of Hilbert $C^{\ast}$-modules in which all closed submodules are orthogonally closed, Proc. Amer. Math. Soc., 127(1999), no. 7, 2123–2125.
  * [33] M. Stone, Linear transformations in Hilbert space, III: Operational methods and group theory, Proc. Nat. Acad. Sci. USA, 16(1930), no. 2, 172–175.
  * [34] D. P. Williams, Crossed products of $C^{\ast}$-algebras, Math. Surveys Monogr., Vol. 134, American Mathematical Society, Providence, RI (2007).


#  Some characterizations of $\omega$-balanced topological groups with a
$q$-point

Deng-Bin Chen<EMAIL_ADDRESS>Hai-Hua Lin<EMAIL_ADDRESS>Li-Hong Xie
<EMAIL_ADDRESS>School of Mathematics and Computational Science, Wuyi
University, Jiangmen, Guangdong 529000, P.R. China

###### Abstract

In this paper, we study some characterizations of $q$-spaces, strict
$q$-spaces and strong $q$-spaces under $\omega$-balanced topological groups as
follows:

  1. (1)

A topological group $G$ is $\omega$-balanced and a $q$-space if and only if
for each open neighborhood $O$ of the identity in $G$, there is a countably
compact invariant subgroup $H$ which is of countable character in $G$, such
that $H\subseteq O$ and the canonical quotient mapping $p:G\rightarrow G/H$ is
quasi-perfect and the quotient group $G/H$ is metrizable.

  2. (2)

A topological group $G$ is $\omega$-balanced and a strict $q$-space if and
only if for each open neighborhood $O$ of the identity in $G$, there is a
closed sequentially compact invariant subgroup $H$ which is of countable
character in $G$, such that $H\subseteq O$ and the canonical quotient mapping
$p:G\rightarrow G/H$ is sequential-perfect and the quotient group $G/H$ is
metrizable.

  3. (3)

A topological group $G$ is $\omega$-balanced and a strong $q$-space if and
only if for each open neighborhood $O$ of the identity in $G$, there is a
closed sequentially compact invariant subgroup $H$ of countable character
$\\{V_{n}:n\in\omega\\}$, such that $H\subseteq O$ and
$\\{V_{n}:n\in\omega\\}$ is a strong $q$-sequence at each $y\in H$, in $G$
such that the canonical quotient mapping $p:G\rightarrow G/H$ is strongly
sequential-perfect and the quotient group $G/H$ is metrizable.

###### keywords:

Strict $q$-space; strong $q$-space; $q$-space; topological group; invariant
subgroup.

###### MSC:

[2020]Primary 54A20, secondary 54H11, 54E18, 54E35.

##  1 Introduction

A group $G$ endowed with a topology $\tau$ is called a paratopological group
if multiplication in $G$ is continuous as a mapping of $G\times G$ to $G$,
where $G\times G$ carries the usual product topology. A topological group is a
paratopological group with continuous inversion. All topological groups and
spaces in this paper are assumed to be Hausdorff. Given a topological group
$G$ and its subgroup $H$, the quotient space $G/H$ means that the left coset
with the quotient topology.

Recall that a point $x$ in a space $X$ is called an accumulation point of a
family $\gamma$ of subsets in $X$, if every neighborhood $U$ of $x$ joints
with infinitely many elements of $\gamma$. A space $X$ is called sequentially
compact if every sequence of $X$ has a convergent subsequence [3]. Next we
give the concepts of $q$-space, strict $q$-space and strong $q$-space.

  1. $\bullet$

A point $x\in X$ is called a $q$-point [2, p.389] of a space $X$ if there
exists a sequence $\\{U_{n}:n\in\omega\\}$ of open neighborhoods of $x$ in $X$
such that every sequence $\\{x_{n}\\}_{n\in\omega}$ of points in $X$ such that
$x_{n}\in U_{n}$ for each $n\in\omega$ has a point of accumulation in $X$. We
also call $\\{U_{n}:n\in\omega\\}$ a $q$-sequence at $x$. A space $X$ is said
to be a $q$-space if every point of it is a $q$-point.

  2. $\bullet$

A point $x\in X$ is called a strict $q$-point of a space $X$ if there exists a
$q$-sequence $\\{U_{n}:n\in\omega\\}$ at $x$ such that
$\bigcap_{n\in\omega}U_{n}$ is a sequentially compact set. We also call
$\\{U_{n}:n\in\omega\\}$ a strict $q$-sequence at $x$. A space $X$ is said to
be a strict $q$-space if every point in $X$ is a strict $q$-point.

  3. $\bullet$

A point $x\in X$ is called a strong $q$-point of a space $X$ if there exists a
sequence $\\{U_{n}:n\in\omega\\}$ of open neighborhoods of $x$ in $X$ such
that every sequence $\\{x_{n}\\}_{n\in\omega}$ of points in $X$ such that
$x_{n}\in U_{n}$ for each $n\in\omega$ has a convergent subsequence. We also
call $\\{U_{n}:n\in\omega\\}$ a strong $q$-sequence at $x$. A space $X$ is
said to be a strong $q$-space if every point in $X$ is a strong $q$-point [8].

Recall that a continuous mapping $f:X\rightarrow Y$ is perfect if $f$ is a
closed onto mapping and all fibers $f^{-1}(y)$ are compact subsets of $X$[3,
p.182]. Recall that a continuous mapping $f:X\rightarrow Y$ is quasi-perfect
[7, Definition 2.1.3] if $f$ is closed such that $f^{-1}(y)$ is countably
compact for each $y\in Y$.

Peng and Liu gave a characterization of $q$-space.

######  Theorem 1.1.

[8, Theorem 7] A topological group $G$ is a $q$-space if and only if there is
a closed countably compact subgroup $H$ of $G$ such that the quotient group
$G/H$ is metrizable and the canonical quotient mapping $p:G\rightarrow G/H$ is
quasi-perfect.

Recall that a continuous mapping $f:X\rightarrow Y$ is called sequential-
perfect if $f$ is closed and $f^{-1}(y)$ is a sequentially compact set for
each $y\in Y$. Recall that a continuous mapping $f:X\rightarrow Y$ is strongly
sequential-perfect if $f$ is closed and $f^{-1}(F)$ is a sequentially compact
set for each sequentially compact set $F\subseteq Y$. The next two theorems
appear in [6].

######  Theorem 1.2.

[6, Theorem 2.4] A topological group $G$ is a strict $q$-space if and only if
for each open neighborhood $U$ of the identity in $G$ there is a closed
sequentially compact subgroup $H\subseteq U$ such that the quotient space
$G/H$ is metrizable and the canonical quotient mapping $p:G\rightarrow G/H$ is
sequential-perfect.

######  Theorem 1.3.

[6, Theorem 2.6] A topological group $G$ is a strong $q$-space if and only if
for each open neighborhood $U$ of the identity in $G$, there is a closed
sequentially compact subgroup $H$ of countable character, which is a strong
$q$-sequence at each $x\in H$, in $G$ such that the quotient space $G/H$ is
metrizable and the canonical quotient mapping $p:G\rightarrow G/H$ is strongly
sequential-perfect.

Recall that a subset $A$ of a space $X$ is of countable character if there is
a countable family $\\{U_{n}:n\in\omega\\}$ of open neighborhoods of $A$ such
that for each open neighborhood $U$ of $A$ there is a $U_{n}$ such that
$U_{n}\subseteq U$.

Feathered topological groups are very important in topological algebra [2].
Recall that a topological group $G$ is feathered [2, p.235] if $G$ has a non-
empty compact set $K$ of countable character in $G$.

Pasynkov gave characterizations of feathered topological group.

######  Theorem 1.4.

[2, Theorem 4.3.20] A topological group G is feathered if and only if it
contains a compact subgroup H such that the quotient space G/H is metrizable.

Recall that a paratopological group $G$ is called $\omega$-balanced if for
every neighbourhood $U$ of the identity $e$ in $G$, there exists a countable
family $\gamma$ of open neighbourhoods of $e$ in $G$ such that for each $x\in
G$ one can find $V\in\gamma$ satisfying $xVx^{-1}\subseteq U$.

In 2022, Xie and Yan proved the following result:

######  Theorem 1.5.

[9, Proposition 2.3] A topological group $G$ is $\omega$-balanced and
feathered if and only if for each open neighbourhood $O$ of the identity in
$G$ there is a compact invariant subgroup $H$ such that $H\subseteq O$ and the
quotient group $G/H$ is metrizable.

In view of Theorem 1.5, it is natural to ask how to characterize $q$-space,
strict $q$-space and strong $q$-space under $\omega$-balanced topological
groups.

This paper is organized as follows. First, it is proved that a topological
group $G$ is $\omega$-balanced and a $q$-space if and only if for each open
neighborhood $O$ of the identity in $G$, there is a countably compact
invariant subgroup $H$ which is of countable character in $G$, such that
$H\subseteq O$ and the canonical quotient mapping $p:G\rightarrow G/H$ is
quasi-perfect and the quotient group $G/H$ is metrizable(see Theorem 2.3). As
a corollary, we show that a topological group $G$ is $\omega$-narrow and a
$q$-space if and only if for each open neighborhood $O$ of the identity in
$G$, there is a countably compact invariant subgroup $H$ which is of countable
character in $G$, such that $H\subseteq O$ and the canonical quotient mapping
$p:G\rightarrow G/H$ is quasi-perfect and the quotient group $G/H$ is
separable and metrizable(see Corollary 2.7). Next, we prove that a topological
group $G$ is $\omega$-balanced and a strict $q$-space if and only if for each
open neighborhood $O$ of the identity in $G$, there is a closed sequentially
compact invariant subgroup $H$ which is of countable character in $G$, such
that $H\subseteq O$ and the canonical quotient mapping $p:G\rightarrow G/H$ is
sequential-perfect and the quotient group $G/H$ is metrizable(see Theorem
2.8). As a corollary, we show that a topological group $G$ is $\omega$-narrow
and a strict $q$-space if and only if for each open neighborhood $O$ of the
identity in $G$, there is a closed sequentially compact invariant subgroup $H$
which is of countable character in $G$, such that $H\subseteq O$ and the
canonical quotient mapping $p:G\rightarrow G/H$ is sequential-perfect and the
quotient group $G/H$ is separable and metrizable(see Corollary 2.9). Finally,
we show that a topological group $G$ is $\omega$-balanced and a strong
$q$-space if and only if for each open neighborhood $O$ of the identity in
$G$, there is a closed sequentially compact invariant subgroup $H$ of
countable character $\\{V_{n}:n\in\omega\\}$, such that $H\subseteq O$ and
$\\{V_{n}:n\in\omega\\}$ is a strong $q$-sequence at each $y\in H$, in $G$
such that the canonical quotient mapping $p:G\rightarrow G/H$ is strongly
sequential-perfect and the quotient group $G/H$ is metrizable(see Theorem
2.10). As a corollary, we show that a topological group $G$ is $\omega$-narrow
and a strong $q$-space if and only if for each open neighborhood $O$ of the
identity in $G$, there is a closed sequentially compact invariant subgroup $H$
of countable character $\\{V_{n}:n\in\omega\\}$, such that $H\subseteq O$ and
$\\{V_{n}:n\in\omega\\}$ is a strong $q$-sequence at each $y\in H$, in $G$
such that the canonical quotient mapping $p:G\rightarrow G/H$ is strongly
sequential-perfect and the quotient group $G/H$ is separable and
metrizable(see Corollary 2.11).

##  2 Characterizations of generalized countably compact topological groups

In view of Theorems 1.4 and 1.5, it is naturally to ask how to give ”internal”
characterizations of $q$-spaces, strict $q$-spaces and strong $q$-spaces under
$\omega$-balanced topological groups. To study this question, we firstly
establish some very useful lemmas.

A subgroup $H$ of a topological group $G$ is called neutral if for every open
neighborhood $U$ of the identity $e$ in $G$, there exists an open neighborhood
$V$ of $e$ such that $HV\subseteq UH$ (equivalently, $VH\subseteq HU$) [5].

######  Lemma 2.1.

[4, Theorem 3.1] If $H$ is a closed neutral subgroup of a topological group
$G$ such that $G/H$ is first-countable, then $G/H$ is metrizable.

######  Lemma 2.2.

[7, Lemma 1.7.6] Let $\\{U_{n}:n\in\omega\\}$ be a decreasing sequence of open
subsets in a topological space $X$ such that
$\bigcap_{n\in\omega}U_{n}=\bigcap_{n\in\omega}\overline{U_{n}}$. Put
$C=\bigcap_{n\in\omega}U_{n}$. Then the following statements are equivalent:

  1. (1)

$\\{U_{n}:n\in\omega\\}$ is a $q$-sequence in $X$;

  2. (2)

the family $\\{U_{n}:n\in\omega\\}$ is a base at $C$ in $X$ and $C$ is
countably compact in $X$.

A topological group $G$ is called range-metrizable if for every open
neighbourhood $U$ of the identity $e$ in $G$, there exists a continuous
homomorphism $p$ of $G$ onto a metrizable group $H$ such that
$p^{-1}(V)\subseteq U$, for some open neighbourhood $V$ of the identity in
$H$.

Now we give an ”internal” characterization of $q$-space in an
$\omega$-balanced topological group.

######  Theorem 2.3.

A topological group $G$ is $\omega$-balanced and a $q$-space if and only if
for each open neighborhood $O$ of the identity in $G$, there is a countably
compact invariant subgroup $H$ which is of countable character in $G$, such
that $H\subseteq O$ and the canonical quotient mapping $p:G\rightarrow G/H$ is
quasi-perfect and the quotient group $G/H$ is metrizable.

###### Proof.

Necessity. Suppose that for each open neighborhood $O$ of the identity in $G$,
there is a countably compact invariant subgroup $H$ which is of countable
character in $G$, such that $H\subseteq O$ and the canonical quotient mapping
$p:G\rightarrow G/H$ is quasi-perfect and the quotient group $G/H$ is
metrizable. Then, by Theorem 1.1, $G$ is a $q$-space. Put $V=(G/H)\setminus
p(G\setminus O)$. Then one can easily show that $V$ is an open neighbourhood
of the identity in $G/H$ such that

$\displaystyle\begin{split}p^{-1}(V)&=p^{-1}(G/H)\setminus p^{-1}(p(G\setminus
O))\\\ &\subseteq G\setminus(G\setminus O)\\\ &=O\\\ \end{split}$

This implies that $G$ is range-metrizable. Thus from the fact that a
topological group is $\omega$-balanced if it is range-metrizable [2, Theorem
3.4.22] it follows that $G$ is $\omega$-balanced.

Sufficiency. (1) Let $e$ be the identity of $G$. Since topological group $G$
is regular and a $q$-space, there is a $q$-sequence $\\{U_{n}:n\in\omega\\}$
at $e$ such that
$\bigcap_{n\in\omega}U_{n}=\bigcap_{n\in\omega}\overline{U_{n}}$. Let
$\\{U_{n}:n\in\omega\\}$ be a decreasing sequence. Put
$C=\bigcap_{n\in\omega}U_{n}$. Then by Lemma 2.2, $\\{U_{n}:n\in\omega\\}$ is
a base at $C$ in $G$ and $C$ is countably compact. Take any open neighbourhood
$O$ of $e$. Then one can find an open symmetric neighbourhood $W$ at $e$ such
that $W\subseteq U_{0}\cap O$. Let $\gamma_{0}=\\{W\\}$. Suppose that for some
$n\in\omega\ $ we have defined families
$\gamma_{0},\gamma_{1},\dots,\gamma_{k}$ of open symmetric neighbourhoods at
$e$ satisfying the following conditions for each $1\leqslant k\leqslant n$:

  1. (i)

$|\gamma_{k-1}|\leqslant\omega$ and $W\subseteq U_{k-1}\cap O$ for each
$W\in\gamma_{k-1}$;

  2. (ii)

$\gamma_{k-1}\subseteq\gamma_{k}$;

  3. (iii)

for every $W\in\gamma_{k-1}$, there exists $V\in\gamma_{k}$ such that
$V^{2}\subseteq W$;

  4. (iv)

for every $W\in\gamma_{k-1}$ and $x\in G$, there exists $V\in\gamma_{k}$ such
that $xVx^{-1}\subseteq W$.

Since $\gamma_{n}$ is countable and $G$ is a topological group, we can find a
countable family $\lambda_{n,1}$ of open symmetric neighbourhoods at $e$ such
that every element of $\gamma_{n}$ contains the square of some element
$V\in\lambda_{n,1}$. In addition, the group $G$ is $\omega$-balanced, so there
exists a countable family $\lambda_{n,2}$ of open symmetric neighbourhoods at
$e$ such that for each $W\in\gamma_{n}$ and each $x\in G$ there is
$V\in\lambda_{n,2}$ satisfying $xVx^{-1}\subseteq W$. We can also assume that
each element $V\in\lambda_{n,1}\cup\lambda_{n,2}$ is contained in $U_{n+1}\cap
O$. Let the family
$\gamma_{n+1}=\gamma_{n}\cup\lambda_{n,1}\cup\lambda_{n,2}$. Then the families
$\gamma_{0},\gamma_{1},\dots,\gamma_{n+1}$ of symmetric open neighbourhoods at
$e$ satisfying (i)-(iv).

It is easy to see that the family $\gamma=\bigcup_{n\in\omega}\gamma_{n}$ is
countable. Put $H=\bigcap\gamma$. Then we claim that the following $(\ast)$
formula is valid:

$(\ast)$ $H$ is a countably compact invariant subgroup of countable character
in $G$ such that $H\subseteq O$.

(a) Firstly, we shall show that $H$ is an invariant subgroup. Since every
element in $\gamma$ is symmetric, we have $H^{-1}=H$. Let us verify that
$HH\subseteq H$. Take any points $a,b\in H$. By (iii), then for each
$W\in\gamma$ there is $V\in\gamma$ such that $V^{2}\subseteq W$. Thus $ab\in
V^{2}\subseteq W$ for each $W\in\gamma$. Then
$ab\in{\bigcap_{W\in\gamma}}W=H$. This implies that $HH\subseteq H$. Thus $H$
is a subgroup. Next we show that $H$ is invariant. In fact, take any $x\in G$.
By (iv), then for each $W\in\gamma$ there is $V\in\gamma$ such that
$xVx^{-1}\subseteq W$. This implies $xHx^{-1}\subseteq xVx^{-1}\subseteq W$
for every $W\in\gamma$. Thus $xHx^{-1}\subseteq{\bigcap_{W\in\gamma}}W=H$,
which implies that $H$ is invariant.

(b) Secondly, we shall show that $H$ is countably compact. By (i),
$H=\bigcap\gamma\subseteq{\bigcap_{n\in\omega}U_{n}}=C$ is obvious. Since C is
countably compact, it is enough to show that $H$ is closed. Take any $y\notin
H=\bigcap\gamma$. Then there is $V\in\gamma_{n_{0}}$ such that $y\notin V$. By
(iii), there is $U\in\gamma_{n_{0}+1}$ such that $U^{2}\subseteq V$. Thus
$y\notin U^{2}$. Since every element in $\gamma$ is symmetric and open, we
have $yU\cap U=\emptyset$. Thus $yU\cap H=\emptyset$ because of $H\subseteq
U$. Since $yU$ is an open neighbourhood of $y$, $H$ is closed.

(c) Finally, to show that $H$ is of countable character in $G$ it is enough to
show that $\gamma$ is a base for $G$ at $H$. Let $W$ be an open neighbourhood
of $H$ in $G$. Since
$C=\bigcap_{n\in\omega}U_{n}=\bigcap_{n\in\omega}\overline{U_{n}}$, $C$ is
closed and countably compact. Put $K=C\setminus W$. Then $K$ is a countably
compact subset and $K\cap H=\emptyset$. Since every element in $\gamma$ is
open and symmetric, one can easily show that
$H=\bigcap_{V\in\gamma}\overline{V}$ by (iii). We claim that $\overline{V}\cap
K=\emptyset$ for some $V\in\gamma_{i}$. In fact, if $\overline{V}\cap
K\neq\emptyset$ for each $V\in\gamma$, then by the countable compactness of
$K$, the set $K\cap(\bigcap_{V\in\gamma}\overline{V})=K\cap H\neq\emptyset$,
which is a contradiction. By (iii), take $U\in\gamma_{i+1}$ such that
$U^{2}\subseteq V$. It is clear that $C\subseteq K\cup W\subseteq KU\cup W$.
Since $KU\cup W$ is an open neighbourhood of $C$ and sequence
{$U_{n}:n\in\omega$} is a base for $G$ at $C$, there is
$U_{n_{0}}\in\\{U_{n}:n\in\omega\\}$ such that $C\subseteq U_{n_{0}}\subseteq
KU\cup W$. Put $l=\text{max}\\{i+1,n_{0}\\}$. Then there is
$U_{1}\in\gamma_{l+1}$ such that $U_{1}\subseteq U$ and $U_{1}\subseteq
U_{n_{0}}$. Since $U^{2}\cap K=\emptyset$, it implies that $U\cap
KU=\emptyset$. Thus $H\subseteq U_{1}\subseteq U_{n_{0}}\subseteq W$ by
$U_{1}\cap KU=\emptyset$. We prove that $\gamma$ is a base for $G$ at $H$. By
(i), $H\subseteq O$ is obvious, so we complete the proof of $(\ast)$ formula.

(2) Next we shall show that the canonical quotient mapping $p:G\rightarrow
G/H$ is a quasi-perfect mapping. Since $H$ is countably compact,
$p^{-1}(p(y))=yH$ is also a countably compact subset of $G$ for every $y\in
G$. Therefore, we only to show that $p$ is a closed mapping. Take a closed set
$F$ in $G$. To show that $p(F)$ is closed in $G/H$ it is enough to show that
$FH$ is closed in $G$, because $p$ is a quotient mapping and
$FH=p^{-1}(p(F))$. Take any $x\notin FH$. Then $F^{-1}x$ is a closed set in
$G$ such that $F^{-1}x\cap H=\emptyset$. This implies that $H\subseteq
G\setminus F^{-1}x$ and $G\setminus F^{-1}x$ is an open set in $G$. Since
$\gamma$ is a base at $H$ in $G$, there is a $V\in\gamma$ such that
$H\subseteq V^{2}\subseteq G\setminus F^{-1}x$. Thus $F^{-1}x\cap
V^{2}=\emptyset$. Since every element in $\gamma$ is open and symmetric, we
have $xV\cap FV=\emptyset$. This implies that $xV\cap FH=\emptyset$, because
$H\subseteq V$. Thus we find an open neighborhood $xV$ of $x$ which is
disjoint with $FH$. This implies that $FH$ is closed in $G$.

Finally, we shall show that $G/H$ is metrizable. To show that $G/H$ is
metrizable it is only to show that $H$ is a neutral subgroup in $G$ and $G/H$
is first-countable by Lemma 2.1. Since $H$ is of countable character in $G$
and $p$ is an open and closed mapping, one can easily show that $G/H$ is
first-countable. In fact, the family $\\{p(V):V\in\gamma\\}$ is a base at
$p(e)$. Take any open neighbourhood $W$ of $p(e)$. Then
$H=p^{-1}(p(e))\subseteq p^{-1}(W)$. Since $W$ is open in $G/H$ and $p$ is a
continuous mapping, then $p^{-1}(W)$ is open in $G$. Since $\gamma$ is a base
at $H$ in $G$, there is a $V\in\gamma$ such that $H\subseteq V\subseteq
p^{-1}(W)$. This implies that $p(V)\subseteq p(p^{-1}(W))=W$. Thus the family
$\\{p(V):V\in\gamma\\}$ is a base at $p(e)$. From the fact that $G/H$ is a
homogeneous space [2, Theorem 1.5.1] it follows that $G/H$ is first-countable.
Now we shall prove that $H$ is a neutral subgroup in $G$. Take an open
neighborhood $U$ of the identity in $G$. Then $HU$ is an open neighborhood of
$H$. Since $\gamma$ is a base at $H$ in $G$, there is $V\in\gamma$ such that
$V^{2}\subseteq HU$. This implies that $VH\subseteq HU$. Thus we have proved
that $H$ is a neutral subgroup.

∎

We recall that a topological group $G$ is called $\omega$-narrow[2] if for
every open neighbourhood $V$ of the neutral element in $G$, there exists a
countable subset $A$ of $G$ such that $AV=G$.

######  Theorem 2.4.

[2, Theorem 3.4.23] A topological group $G$ is topologically isomorphic to a
subgroup of the topological product of some family of second-countable groups
if and only if $G$ is $\omega$-narrow.

######  Proposition 2.5.

[2, Proposition 3.4.2] If a topological group $H$ is a continuous homomorphic
image of an $\omega$-narrow topological group $G$, then $H$ is also
$\omega$-narrow.

######  Proposition 2.6.

[2, Proposition 3.4.5] Every first-countable $\omega$-narrow topological group
has a countable base.

By Theorem 2.3, we have the following result.

######  Corollary 2.7.

A topological group $G$ is $\omega$-narrow and a $q$-space if and only if for
each open neighborhood $O$ of the identity in $G$, there is a countably
compact invariant subgroup $H$ which is of countable character in $G$, such
that $H\subseteq O$ and the canonical quotient mapping $p:G\rightarrow G/H$ is
quasi-perfect and the quotient group $G/H$ is separable and metrizable.

###### Proof.

Necessity. Suppose that for each open neighborhood $O$ of the identity in $G$,
there is a countably compact invariant subgroup $H$ which is of countable
character in $G$, such that $H\subseteq O$ and the canonical quotient mapping
$p:G\rightarrow G/H$ is quasi-perfect and the quotient group $G/H$ is
separable and metrizable. Then, by Theorem 1.1, $G$ is a $q$-space. Thus we
only to show that topological group $G$ is $\omega$-narrow. For each open
neighborhood $O$ of the identity in $G$, there is a countably compact
invariant subgroup $H_{O}$, such that $H_{O}\subseteq O$ and the canonical
quotient mapping $p_{O}:G\rightarrow G/H_{O}$ is quasi-perfect and every
quotient group $G/H_{O}$ is separable and metrizable. Thus every topological
group $G/H_{O}$ is second-countable. From the proof process of the necessity
of Theorem 2.3, it can be concluded that for each open neighborhood $O$ of the
identity in $G$, there is an open neighbourhood $V_{O}$ of the identity in
$G/H_{O}$ such that $p_{O}^{-1}(V_{O})\subseteq O$. Let $\mathcal{B}$ be the
family of all open neighbourhoods of the neutral element in $G$. Then the
diagonal product $h$ of the family {$p_{O}$:$O\in\mathcal{B}$ } is a
topological isomorphism of $G$ onto a topological subgroup of the topological
product of the family {$G/H_{O}$:$O\in\mathcal{B}$}. Thus by Theorem 2.4, $G$
is $\omega$-narrow.

Sufficiency. It is well known that every $\omega$-narrow topological group is
$\omega$-balanced. Thus topological group $G$ is $\omega$-balanced and a
$q$-space. Then, by Theorem 2.3, for each open neighborhood $O$ of the
identity in $G$, there is a countably compact invariant subgroup $H$ which is
of countable character in $G$, such that $H\subseteq O$ and the canonical
quotient mapping $p:G\rightarrow G/H$ is quasi-perfect and the quotient group
$G/H$ is metrizable. Thus we only to show that $G/H$ is separable. Since
topological group $G/H$ is a continuous homomorphic image of an
$\omega$-narrow topological group $G$, by Proposition 2.5, $G/H$ is
$\omega$-narrow. $G/H$ is first-countable because $G/H$ is metrizable. Then
$G/H$ is a first-countable $\omega$-narrow topological group. Thus $G/H$ has a
countable base by Proposition 2.6. This implies that $G/H$ is second-
countable. Since every second-countable topological group is separable, we
have $G/H$ is separable. ∎

Now we give an ”internal” characterization of strict $q$-space in an
$\omega$-balanced topological group.

######  Theorem 2.8.

A topological group $G$ is $\omega$-balanced and a strict $q$-space if and
only if for each open neighborhood $O$ of the identity in $G$, there is a
closed sequentially compact invariant subgroup $H$ which is of countable
character in $G$, such that $H\subseteq O$ and the canonical quotient mapping
$p:G\rightarrow G/H$ is sequential-perfect and the quotient group $G/H$ is
metrizable.

###### Proof.

Necessity. Suppose that for each open neighborhood $O$ of the identity in $G$,
there is a closed sequentially compact invariant subgroup $H$ which is of
countable character in $G$, such that $H\subseteq O$ and the canonical
quotient mapping $p:G\rightarrow G/H$ is sequential-perfect and the quotient
group $G/H$ is metrizable. Then by Theorem 1.2, $G$ is a strict $q$-space. Put
$V=(G/H)\setminus p(G\setminus O)$. Then one can easily show that $V$ is an
open neighbourhood of the identity in $G/H$ such that

$\displaystyle\begin{split}p^{-1}(V)&=p^{-1}(G/H)\setminus p^{-1}(p(G\setminus
O))\\\ &\subseteq G\setminus(G\setminus O)\\\ &=O\\\ \end{split}$

This implies that $G$ is range-metrizable. Thus from the fact that a
topological group is $\omega$-balanced if it is range-metrizable [2, Theorem
3.4.22] it follows that $G$ is $\omega$-balanced.

Sufficiency. (1) Let $e$ be the identity of $G$. Since topological group $G$
is regular and a strict $q$-space, there is a strict $q$-sequence
$\\{U_{n}:n\in\omega\\}$ at $e$ such that
$\bigcap_{n\in\omega}U_{n}=\bigcap_{n\in\omega}\overline{U_{n}}$. Let
$\\{U_{n}:n\in\omega\\}$ be a decreasing sequence. Put
$C=\bigcap_{n\in\omega}U_{n}$. Then by Lemma 2.2, $\\{U_{n}:n\in\omega\\}$ is
a base at $C$ in $G$ and $C$ is sequentially compact in $G$. We continue to
use the notation of Theorem 2.3. The following proof is completely similar to
Theorem 2.3, and we can obtain that the inclusion of $H$ is a countably
compact invariant subgroup of countable character in $G$ such that $H\subseteq
O$ already holds. Next we only to prove that $H$ is a sequentially compact
subgroup in $G$. Since $H=\bigcap\gamma\subseteq{\bigcap_{n\in\omega}U_{n}}=C$
and $C$ is sequentially compact in $G$ and $H$ is closed, then $H$ is a
sequentially compact subgroup in $G$.

(2) Next we shall show that the canonical quotient mapping $p:G\rightarrow
G/H$ is a sequential-perfect mapping. Since $H$ is sequentially compact,
$p^{-1}(p(y))=yH$ is also a sequentially compact subset of $G$ for every $y\in
G$. Therefore, we only to prove that $p$ is a closed mapping. The proof of
closed mapping is completely similar to Theorem 2.3, and it will not be
repeated here! Finally, we shall show that $G/H$ is metrizable. Following the
proof of Theorem 2.3, one can easily show that G/H is metrizable. ∎

By Theorem 2.8, we have the following result.

######  Corollary 2.9.

A topological group $G$ is $\omega$-narrow and a strict $q$-space if and only
if for each open neighborhood $O$ of the identity in $G$, there is a closed
sequentially compact invariant subgroup $H$ which is of countable character in
$G$, such that $H\subseteq O$ and the canonical quotient mapping
$p:G\rightarrow G/H$ is sequential-perfect and the quotient group $G/H$ is
separable and metrizable.

###### Proof.

Necessity. Suppose that for each open neighborhood $O$ of the identity in $G$,
there is a closed sequentially compact invariant subgroup $H$ which is of
countable character in $G$, such that $H\subseteq O$ and the canonical
quotient mapping $p:G\rightarrow G/H$ is sequential-perfect and the quotient
group $G/H$ is separable and metrizable. Then, by Theorem 1.2, $G$ is a strict
$q$-space. Thus we only to show that topological group $G$ is $\omega$-narrow.
For each open neighborhood $O$ of the identity in $G$, there is a closed
sequentially compact invariant subgroup $H_{O}$, such that $H_{O}\subseteq O$
and every canonical quotient mapping $p_{O}:G\rightarrow G/H_{O}$ is
sequential-perfect and every quotient group $G/H_{O}$ is separable and
metrizable. Thus every topological group $G/H_{O}$ is second-countable. From
the proof process of the necessity of Theorem 2.3, it can be concluded that
for each open neighborhood $O$ of the identity in $G$, there is an open
neighbourhood $V_{O}$ of the identity in $G/H_{O}$ such that
$p_{O}^{-1}(V_{O})\subseteq O$. Let $\mathcal{B}$ be the family of all open
neighbourhoods of the neutral element in $G$. Then the diagonal product h of
the family {$p_{O}$:$O\in\mathcal{B}$} is a topological isomorphism of $G$
onto a topological subgroup of the topological product of the family
{$G/H_{O}$:$O\in\mathcal{B}$}. Thus by Theorem 2.4, $G$ is $\omega$-narrow.

Sufficiency: It is well known that every $\omega$-narrow topological group is
$\omega$-balanced. Thus topological group $G$ is $\omega$-balanced and a
strict $q$-space. Then, by Theorem 2.8, for each open neighborhood $O$ of the
identity in $G$, there is a closed sequentially compact invariant subgroup $H$
which is of countable character in $G$, such that $H\subseteq O$ and the
canonical quotient mapping $p:G\rightarrow G/H$ is sequential-perfect and the
quotient group $G/H$ is metrizable. Thus we only to show that $G/H$ is
separable. Since topological group $G/H$ is a continuous homomorphic image of
an $\omega$-narrow topological group $G$, by Proposition 2.5, $G/H$ is
$\omega$-narrow. $G/H$ is first-countable because $G/H$ is metrizable. Then
$G/H$ is a first-countable $\omega$-narrow topological group. Thus $G/H$ has a
countable base by Proposition 2.6. This implies that $G/H$ is second-
countable. Since every second-countable topological group is separable, we
have $G/H$ is separable. ∎

Now we give an ”internal” characterization of strong $q$-space in an
$\omega$-balanced topological group.

######  Theorem 2.10.

A topological group $G$ is $\omega$-balanced and a strong $q$-space if and
only if for each open neighborhood $O$ of the identity in $G$, there is a
closed sequentially compact invariant subgroup $H$ of countable character
$\\{V_{n}:n\in\omega\\}$, such that $H\subseteq O$ and
$\\{V_{n}:n\in\omega\\}$ is a strong $q$-sequence at each $y\in H$, in $G$
such that the canonical quotient mapping $p:G\rightarrow G/H$ is strongly
sequential-perfect and the quotient group $G/H$ is metrizable.

###### Proof.

Necessity. Suppose that for each open neighborhood $O$ of the identity in $G$,
there is a closed sequentially compact invariant subgroup $H$ of countable
character $\\{V_{n}:n\in\omega\\}$, such that $H\subseteq O$ and
$\\{V_{n}:n\in\omega\\}$ is a strong $q$-sequence at each $y\in H$, in $G$
such that the canonical quotient mapping $p:G\rightarrow G/H$ is strongly
sequential-perfect and the quotient group $G/H$ is metrizable. Then by Theorem
1.3, $G$ is a strong $q$-space. Put $V=(G/H)\setminus p(G\setminus O)$. Then
one can easily show that $V$ is an open neighbourhood of the identity in $G/H$
such that

$\displaystyle\begin{split}p^{-1}(V)&=p^{-1}(G/H)\setminus p^{-1}(p(G\setminus
O))\\\ &\subseteq G\setminus(G\setminus O)\\\ &=O\\\ \end{split}$

This implies that $G$ is range-metrizable. Thus from the fact that a
topological group is $\omega$-balanced if it is range-metrizable [2, Theorem
3.4.22] it follows that $G$ is $\omega$-balanced.

Sufficiency. (1) Let $e$ be the identity of $G$. Since topological group $G$
is regular and a strong $q$-space, there is a strong $q$-sequence
$\\{U_{n}:n\in\omega\\}$ at $e$ such that
$\bigcap_{n\in\omega}U_{n}=\bigcap_{n\in\omega}\overline{U_{n}}$. Let
$\\{U_{n}:n\in\omega\\}$ be a decreasing sequence. Put
$C=\bigcap_{n\in\omega}U_{n}$. Then by Lemma 2.2, $\\{U_{n}:n\in\omega\\}$ is
a base at $C$ in $G$ and $C$ is sequentially compact in $G$. We continue to
use the notation of Theorem 2.3. The following proof is completely similar to
Theorem 2.3 and Theorem 2.8, and we can obtain that the inclusion of $H$ is a
closed sequentially compact invariant subgroup of countable character
$\gamma=\\{V_{n}:n\in\omega\\}$ in $G$ such that $H\subseteq O$ already holds.
Thus we only need to prove that $\gamma=\\{V_{n}:n\in\omega\\}$ is a strong
$q$-sequence at each $y\in H$. In fact, $V_{n}\subseteq U_{n}$ for every
$n\in\omega$ and $\\{U_{n}:n\in\omega\\}$ is a strong $q$-sequence at each
$x\in G$. This implies that $\\{V_{n}:n\in\omega\\}$ is a strong $q$-sequence
at each $y\in H$.

(2) Next we shall show that $G/H$ is metrizable. Following the proof of
Theorem 2.3, one can easily show that G/H is measurable. Finally, we shall
show that the canonical quotient mapping $p:G\rightarrow G/H$ is a strongly
sequential-perfect mapping. Following the proof of Theorem 2.3, one can easily
show that $p$ is a closed mapping. Take arbitrary sequentially compact set $F$
in $G/H$. It is only to show that $p^{-1}(F)$ is a sequentially compact set in
$G$. Take a sequence $\\{x_{n}\\}_{n\in\omega}$ of $p^{-1}(F)$. Then there is
a convergent subsequence of $\\{p(x_{n})\\}_{n\in\omega}$ in $F$ by the
sequential compactness of $F$. Without loss of generality, we assume that
$\\{p(x_{n})\\}_{n\in\omega}$ converges to some point $p(x)$ in $F$, where
$x\in G$. Let $\\{V_{n}:n\in\omega\\}$ be an open neighborhood base at $H$ in
$G$ which is a strong $q$-sequence at each $y\in H$. Since $p$ is closed and
$H=p^{-1}(p(e))$, we can find an open neighborhood $U_{n}$ of $p(e)$ such that
$p^{-1}(U_{n})\subseteq V_{n}$ for each $n\in\omega$. Clearly, the
$\\{p^{-1}(U_{n}):n\in\omega\\}$ is also an open neighborhood base at $H$ in
$G$ which is a strong $q$-sequence at each $x\in H$. Thus, without loss of
generality, we can assume that $V_{n}=p^{-1}(p(V_{n}))$ holds for each
$n\in\omega$. Therefore, one can easily show that $\\{xV_{n}:n\in\omega\\}$ is
an open neighborhood base at $xH$ in $G$ which is a strong $q$-sequence at
each $z\in xH$. Since $xV_{n}=xp^{-1}(p(V_{n}))=xV_{n}H$ holds for each
$n\in\omega$, we have that $xV_{n}=p^{-1}(p(xV_{n}))$ holds for each
$n\in\omega$. For each $n\in\omega$ we claim that the $xV_{n}$ contains
infinite elements of the sequence $\\{x_{n}\\}_{n\in\omega}$. Indeed, if there
is an $xV_{n_{0}}$ such that it contains finite elements of the sequence
$\\{x_{n}\\}_{n\in\omega}$, then $p(xV_{n_{0}})$ also contains finite elements
of the sequence $\\{p(x_{n})\\}_{n\in\omega}$, because
$xV_{n_{0}}=p^{-1}(p(xV_{n_{0}}))$. This is a contradiction with
$\\{p(x_{n})\\}_{n\in\omega}$ converging to $p(x)$, because $p(xV_{n_{0}})$ is
an open neighborhood of $p(x)$ by $p$ is open. Thus we can find a subsequence
$\\{x_{n_{i}}\\}_{i\in\omega}$ of $\\{x_{n}\\}_{n\in\omega}$ such that
$x_{n_{i}}\in xV_{i}$ for each $i\in\omega$. Since $\\{xV_{i}:i\in\omega\\}$
is a strong $q$-sequence at each $z\in xH$ in $G$,
$\\{x_{n_{i}}\\}_{i\in\omega}$ has a convergent subsequence. Thus
$\\{x_{n}\\}_{n\in\omega}$ has a convergent subsequence
$\\{x_{n_{j}}\\}_{j\in\omega}$. Since $F$ is a sequentially compact set $F$ in
the metrizable space $G/H$, $p^{-1}(F)$ is closed in $G$ by continuity of $p$
and compactness of $F$. This implies that the subsequence
$\\{x_{n_{j}}\\}_{j\in\omega}$ is convergent in $p^{-1}(F)$. ∎

By Theorem 2.10, we have the following result.

######  Corollary 2.11.

A topological group $G$ is $\omega$-narrow and a strong $q$-space if and only
if for each open neighborhood $O$ of the identity in $G$, there is a closed
sequentially compact invariant subgroup $H$ of countable character
$\\{V_{n}:n\in\omega\\}$, such that $H\subseteq O$ and
$\\{V_{n}:n\in\omega\\}$ is a strong $q$-sequence at each $y\in H$, in $G$
such that the canonical quotient mapping $p:G\rightarrow G/H$ is strongly
sequential-perfect and the quotient group $G/H$ is separable and metrizable.

###### Proof.

Necessity. Suppose that for each open neighborhood $O$ of the identity in $G$,
there is a closed sequentially compact invariant subgroup $H$ of countable
character $\\{V_{n}:n\in\omega\\}$ in $G$, such that $H\subseteq O$ and
$\\{V_{n}:n\in\omega\\}$ is a strong $q$-sequence at each $y\in H$, in $G$
such that the canonical quotient mapping $p:G\rightarrow G/H$ is strongly
sequential-perfect and the quotient group $G/H$ is separable and metrizable.
Then, by Theorem 1.3, $G$ is a strong $q$-space. Thus we only to show that
topological group $G$ is $\omega$-narrow. For each open neighborhood $O$ of
the identity in $G$, there is a closed sequentially compact invariant subgroup
$H_{O}$, such that $H_{O}\subseteq O$ and every canonical quotient mapping
$p_{O}:G\rightarrow G/H_{O}$ is strongly sequential-perfect and the quotient
group $G/H_{O}$ is separable and metrizable. Thus every topological group
$G/H_{O}$ is second-countable. From the proof process of the necessity of
Theorem 2.3, it can be concluded that for each open neighborhood $O$ of the
identity in $G$, there is an open neighbourhood $V_{O}$ of the identity in
$G/H_{O}$ such that $p_{O}^{-1}(V_{O})\subseteq O$. Let $\mathcal{B}$ be the
family of all open neighbourhoods of the neutral element in $G$. Then the
diagonal product $h$ of the family {$p_{O}$:$O\in\mathcal{B}$} is a
topological isomorphism of $G$ onto a topological subgroup of the topological
product of the family {$G/H_{O}$:$O\in\mathcal{B}$}. Then by Theorem 2.4, $G$
is $\omega$-narrow.

Sufficiency: It is well known that every $\omega$-narrow topological group is
$\omega$-balanced. Thus topological group $G$ is $\omega$-balanced and a
strong $q$-space. Then, by Theorem 2.10, for each open neighborhood $O$ of the
identity in $G$, there is a closed sequentially compact invariant subgroup $H$
of countable character $\\{V_{n}:n\in\omega\\}$, such that $H\subseteq O$ and
$\\{V_{n}:n\in\omega\\}$ is a strong $q$-sequence at each $y\in H$, in $G$
such that the canonical quotient mapping $p:G\rightarrow G/H$ is strongly
sequential-perfect and the quotient group $G/H$ is metrizable. Thus we only to
show that $G/H$ is separable. Since topological group $G/H$ is a continuous
homomorphic image of an $\omega$-narrow topological group $G$, by Proposition
2.5, $G/H$ is $\omega$-narrow. $G/H$ is first-countable because $G/H$ is
metrizable. Then $G/H$ is a first-countable $\omega$-narrow topological group.
Thus $G/H$ has a countable base by Proposition 2.6. This implies that $G/H$ is
second-countable. Since every second-countable topological group is separable,
we have $G/H$ is separable.

∎

## References

  * [1]
  * [2] A.V. Arhangel’skii, M.G. Tkachenko,  Topological Groups and Related Structures,  Atlantis Studies in Mathematics, vol. 1, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.
  * [3] R. Engelking,  General Topology,  Sigma Series in Pure Mathematics, Heldermann, Berlin, 1989.
  * [4] M. Fernández, I. Sánchez, M. Tkachenko,  Coset spaces and cardinal invariants,  Acta Math. Hung. 2 (159) (2019) 486-502.
  * [5] J. Poncet,  Une classe despaces homogenes possedant une mesure invariante,  C.R.Acad. Sci. Paris, 238(1954) 553-554.
  * [6] S. Lin, L.H. Xie, D.B. Chen,  Some generalized countably compact properties in topological groups,  Topol. Appl. (2023), 108705.
  * [7] S. Lin, Z. Yun,  Generalized Metric Spaces and Mappings,  Atlantis Studies in Mathematics, vol. 6, Atlantis Press, Paris, 2016.
  * [8] L.X. Peng, Y. Liu,  Topological groups with a (strongly) $q$-point,  Houston J. Math. 2 (47) (2021) 499-516.
  * [9] L.H. Xie, P.F. Yan,  $P\mathcal{M}$-factorizable topological groups,  Topol. Appl. (2022), 108302.