# Coactions and Skew Products for Toplogical Quivers

Lucas Hall  School of Mathematical and Statistical Sciences
Arizona State University
Tempe, Arizona 85287<EMAIL_ADDRESS>
###### Abstract.

Given a cocycle on a topological quiver by a locally compact group, the author
constructs a skew product topological quiver, and determines conditions under
which a topological quiver can be identified as a skew product. We investigate
the relationship between the $C^{*}$-algebra of the skew product and a certain
native coaction on the $C^{*}$-algebra of the original quiver, finding that
the crossed product by the coaction is isomorphic to the skew product. As an
application, we show that the reduced crossed product by the dual action is
Morita equivalent to the $C^{*}$-algebra of the original quiver.

##  1\. Introduction

The dynamics of $C^{*}$-algebras has long been an area of interest for
operator algebraists. This theory of crossed product duality simplifies when
acting groups are abelian, but the wide array of uses left a desire for
generalizations to duality suitable for nonabelian groups. This desire was
furnished by the duality theorem of Imai and Takai, which depended on the
design of coactions to replace the role of the dual group in the nonabelian
setting. While these objects greatly generalized dynamical results, the cost
was significant. The loss of the dual group in favor of coactions in this
setting cast this new field into a challenging intersection of tensor products
and multiplier algebras.

Independently, the introduction of graph algebras in the late twentieth
century shed much new light in the field of operator algebras. Popularized as
$C^{*}$-algebras that you can “see,” graph algebras succinctly relate the
algebraic structure of the $C^{*}$-algebra to observable combinatorial data
associated to the defining (directed) graph. Their ongoing study is
simultaneously tractable and fruitful, since on one hand the hope of using
simple pictures to analyze a $C^{*}$-algebra is both clear and stimulating,
while on the other hand $C^{*}$-algebras from across the field have found
natural expressions as graph algebras. Once again, this wide array of uses has
produced an interest in generalizing the constructions of graph algebras,
which have been pursued along many different lines. Examples include higher
ranked graphs, Exel-Laca algebras, topological graphs, and the subject of this
paper, topological quivers.

One boon graph algebras offered to the theory of nonabelian duality is the
incredible leverage offered by “seeing” how the coactions work in the context
of graph algebras. The first result of this type [kqr:graph] considered
labelings of the edge set by a discrete group, introducing a new directed
graph whose vertex and edge sets are products of the originals with the
discrete group. At the same time, this labeling induced a coaction on the
original graph algebra, and the authors showed in [kqr:graph]*Theorem 2.4 that
these two constructions are isomorphic using the technology devoted to graph
algebras, namely Cuntz-Krieger families and the Gauge-Invariant Uniqueness
Theorem.

Later, Katsura codefied the data of a graph algebra into a
$C^{*}$-correspondence [kat:class], and offered the modern perspective [kat02]
that the graph algebra of a specified graph is the universal $C^{*}$-algebra
for (what we would call today) Cuntz-Pimsner covariant homomorphisms of the
associated $C^{*}$-correspondence. Here, universal means that every Cuntz-
Pimsner covariant representation of the correspondence into a $C^{*}$-algebra
induces a homomorphism from the universal algebra into the representing
algebra. Simultaneously, Katsura introduced the notion of topological graphs,
which offered the first continuous analog to directed graphs by puting
topological constraints on the data.

Equipped with the technology of Cuntz-Pimsner covariant homomorphisms into
$C^{*}$-algebras, Kaliszewski, Quigg, and Robertson introduced the notion of
Cuntz-Pimsner covariance for morphisms _between correspondences_ [kqrfunctor].
There, they showed that the passage from $C^{*}$-correspondences to the Cuntz-
Pimsner algebra was functorial between a suitible pair of categories. Shortly
after, these three authors investigated this functor for correspondences
equipped with a coaction[kqrcoact], deducing that a suitable correspondence
coaction generates a coaction on the associated Cuntz-Pimsner algebra.
Kaliszewski and Quigg then used this machinery to recover for Katsura’s
topological graphs what had been discovered for directed graphs [tgcoact]. The
skew product topological graph by a (locally compact) group is isomorphic to
the crossed product by a naturally occuring coaction, producing this time a
coaction that you can “see” in the continuous setting.

Around the time that Katsura studied his topological graphs, Muhly and Solel
introduced topological quivers to exemplify traits in the study of Morita
equivalence of tensor algebras. Muhly and Tomforde later took up the task of
investigating the associated $C^{*}$-algebras in [quiver]. Apart from the
matter of convention (and we conform to the conventions of Katsura here), the
principle difference between quivers and topological graphs is that
topological graphs insist that the source map is a local homeomorphism, while
quivers insist only that the source map is open. To make up for this freedom,
we insist that the fibres over any vertex can be appropriately measured, and
the authors follow the strategy used for measuring groupoids by a system of
measures. In this article, we take up the study of skew products for
topological quivers.

In §2, we record our conventions for bundles, topological quivers and
associated constructions, and coactions. Starting with §3 we introduce the
skew product quiver and produce a Gross-Tucker type classification for skew
products. In §4 we study the associated $C^{*}$-algebras in earnest,
culminating in the main objective Theorem 4.3. We end with a section on
applications, where we offer an open problem regarding qualities of the
coaction, and show that the reduced crossed product by the dual action is
Morita equivalent to the $C^{*}$-algebra of the original quiver.

### Acknowledgements

The author would like to thank John Quigg for numerous discussions and
insights regarding this work. Without his patient encouragement and diligent
guidance, this work would be far inferior.

##  2\. Preliminaries

###  2.1. Bundle Constructions

In the development of bundles, we follow [husemoller], but we define principal
bundles accoring to the hypothesis of [RaeWil]*Lemma 4.63. The remark
following the lemma reconciles the disparity in definitions.

######  Definition 2.1.

A _bundle_ is a triple $(E,p,B)$, where $p:E\to B$ is a continuous function.
$B$ is called the base space, $E$ the total space, $p$ the bundle projection,
and for each $b\in B$, the space $p^{-1}(b)$ is called the fibre over $b$. We
often identify a bundle with its bundle projection.

A bundle morphism is a pair of continuous maps
$(u,f):(E,p,B)\to(E^{\prime},p^{\prime},B^{\prime})$ with $p^{\prime}u=fp$,
i.e. the diagram below commutes.

${E}$${E^{\prime}}$${B}$${B^{\prime}.}$$\scriptstyle{p}$$\scriptstyle{u}$$\scriptstyle{p^{\prime}}$$\scriptstyle{f}$

An bundle _isomorphism_ is an invertible bundle morphism (which requires each
map $u,f$ to be a homeomorphism).

######  Example 2.2.

Given a bundle $(E,p,B)$ and a continuous map $f:C\to B$, the space

$E*_{f}C=\\{(x,c)\in E\times C|p(x)=f(c)\\}$

is called the pullback under $f$, and the triple $(E*_{f}C,\text{pr}_{2},C)$
is a bundle, where $\text{pr}_{2}$ is the standard projection onto the second
factor. The pair $(\text{pr}_{2},f)$ is then a bundle morphism.

######  Example 2.3.

A topological group $G$ acts on on the right a locally compact Hausdorff space
$X$ provided there is a continuous group anti-homomorphism
$\alpha:G\to\text{Homeo}(X)$ (written $\alpha(g)(x)=xg$), and we call such a
space a right $G$-space. $G$ acts _freely_ on $X$ provided all stabilizers are
trivial. $G$ acts _properly_ on $X$ if the translation map
$(x,g)\mapsto(x,xg)$ is a proper map. The action is called _principal_ if $G$
acts both freely and properly. If $G$ acts on the right of $X$, then the
triple $(X,q,X/G)$ is a bundle, where $X/G$ (the set of orbits of $G$) is
given the quotient topology induced by the quotient map $q$. This is called
the quotient bundle for the $G$ action. When $G$ acts principally on $X$, then
the space $X/G$ is also locally compact and Hausdorff.

For $g\in G$, the pair $(\text{rt}_{g},\text{id})$ is a bundle morphism of
$q$, where $\text{rt}_{g}(x)=xg$ is right translation by $g$, and this
morphism is invertible.

######  Definition 2.4.

Given a locally compact group $G$, a bundle $p$ is a _$G$ -bundle_ provided
the total space $E$ is a $G$-space and there is a bundle isomorphism
$(\text{id},f):(E,p,B)\to(E,q,E\backslash G)$. A $G$-bundle $p$ is called
_principal_ if $G$ acts freely and properly on the total space. In this case,
the fibre $p^{-1}(b)$ over any point is isomorphic to $G$.

A _$G$ -bundle morphism_ is a bundle morphism
$(u,f):(E,p,B)\to(E^{\prime},p^{\prime},B^{\prime})$ whose map of total spaces
is equivariant for the $G$-action, that is $u(xg)=u(x)g$.

######  Example 2.5.

If $(E,p,B)$ is a $G$-bundle and $f:C\to B$ is continuous, then there is a
$G$-action on $C*_{f}E$ given by $(c,e)g=(c,eg)$ making the pullback into a
$G$-bundle over $C$. The pullback bundle inherits any structure enjoyed by the
original bundle.

The following theorem will be foundational for us. For details, see
[husemoller, Theorem 4.4.2] and the discussion surrounding [palais,
Proposition 1.3.4]. This result also holds for groupoids, see [hkqwstab] and
the references therein.

######  Theorem 2.6.

Suppose we have principal $G$-bundles $p:E\to B$ and $p^{\prime}:E^{\prime}\to
B^{\prime}$, together with a $G$-bundle morphism
$(u,f):(E,p,B)\to(E^{\prime},p^{\prime},B^{\prime})$. Then the assignment

$\displaystyle E$ $\displaystyle\to B*_{f}E^{\prime}$ $\displaystyle e$
$\displaystyle\mapsto(p(e),u(e)),$

is a $G$-equivariant homeomorphism between $E^{\prime}$ and
$B*_{f}E^{\prime}$. In particular, this map produces a $G$-bundle isomorphism
from $E$ to $B*_{f}E^{\prime}$ over $B$.

######  Definition 2.7 ([wil:toolkit]*§3.2).

Given a bundle $\pi:E\to B$, a system of measures over $\pi$, or _$\pi$
-system_, is a family of positive Radon measures $\lambda=\\{\lambda_{b}:b\in
B\\}$ defined on $E$ and satisfying two conditions

  1. (1)

the support of $\lambda_{b}$ is a subset of the fibre over $b$, and

  2. (2)

for any $f\in C_{c}(E)$, the assignment

$b\mapsto\lambda(f)(b):=\int fd\lambda_{b}$

is a continuous function.

Condition 2 is called continuity for the system. A $\pi$-system is called
_full_ provided $\text{supp}(\lambda_{b})=\pi^{-1}(b)$, and _proper_ provided
$\lambda_{b}\neq 0$ for every $b\in B$. We will confine our attention to full
and proper $\pi$-systems.

Suppose the spaces $E,B$ admit (right) actions by a group $G$ and $\pi$ is
equivariant for the action. A $\pi$-system is called _equivariant_ provided

$\int_{E}f(eg)d\lambda_{b}(e)=\int_{E}f(e)d\lambda_{bg}(e)$

for every $f\in C_{c}(E)$, $b\in B$, and $g\in G$.

If $G$ acts principally on each of $E,B$ and if $\pi$ is equivariant and
surjective, then there is a canonical map $\dot{\pi}$ making the diagram

${E}$${B}$${E\backslash G}$${B\backslash
G}$$\scriptstyle{q_{E}}$$\scriptstyle{\pi}$$\scriptstyle{q_{B}}$$\scriptstyle{\dot{\pi}}$

commute. The orbit map for $E$ induces a homeomorphism
$\pi^{-1}(b)\to\dot{\pi}^{-1}(bG)$, so that equivariance of $\lambda$ induces
for each orbit $bG\in B\backslash G$ a measure $\mu_{bG}$ with support
$\dot{\pi}(bG)$. For $\phi\in C_{c}(E\backslash G)$, this measure is given by

$\int\phi d\mu_{bG}.$

In this circumstance, we have the following theorem relating $G$\- equivariant
$\pi$-systems to those of their orbits.

######  Theorem 2.8 ([wil:toolkit]*Proposition 3.14).

Let $G$ be a group, $E$ and $B$ principal right $G$-spaces, and $\pi:E\to B$ a
continuous open surjection. Let $\dot{\pi}$ be the induced map for the orbit
spaces of $E$ and $B$.

  1. (a)

If $\lambda$ is an invariant $\pi$-system, then the induced family of measures
$\mu=\\{\mu_{bG}\\}_{bG\in B\backslash G}$ is a $\dot{\pi}$-system.

  2. (b)

If $\mu$ is a $\dot{\pi}$-system, then there is a $\pi$-system $\lambda$ whose
induced $\dot{\pi}$-system agrees with $\mu$.

###  $C^{*}$-Constructions for Topological Quivers

We refer to [quiver] for topological quivers, but adapt the definitions to the
conventions of [rae:graphalg, kat:class], which we use along with [enchilada].

######  Definition 2.9.

A Topological Quiver is an ensemble
$\mathcal{Q}=\\{E^{0},E^{1},r,s,\lambda\\}$ comprising the following data:

  * •

$E^{1}$ and $E^{0}$ are locally compact Hausdorff spaces.

  * •

$r$ and $s$ are continuous from $E^{1}$ to $E^{0}$.

  * •

$s$ is open.

  * •

$\lambda$ is a full $s$-system

######  Remarks 2.10.

$r$ and $s$ are called the range map and source map, elements of $E^{1}$ are
typically called edges, and elements of $E^{0}$ vertices. These names are
anticipated by the analogy of quivers with directed graphs [rae:graphalg],
which are essentially quivers whose edge and vertex spaces are discrete and
countable, and the family $\lambda$ (typically suppressed in this context)
consists of counting measures. A vertex which is in the image of the source
map but not the range map is called a source, and a vertex which is in the
image of the range map but not the source map is called a sink. Using the
technique of adding tails to quivers (cf. [quiver, Section 4]), one can always
restrict their attention to quivers without sinks. We will often briefly call
a topological quiver a quiver.

Given a topological quiver $\mathcal{Q}$, [quiver] defines a quiver
$C^{*}$-correspondence as follows: Let $A=C_{0}(E^{0})$. For
$X_{0}=C_{c}(E^{1})$, there is an $A$-valued inner product
$\langle\cdot,\cdot\rangle:X_{0}\times X_{0}\to A$ by

$\langle\eta,\xi\rangle(v)=\int\overline{\eta(e)}\xi(e)\,d\lambda_{v}(e).$

We then get left and right actions of $f\in A$ on $\xi\in X_{0}$ by

$\displaystyle f\xi(e)$ $\displaystyle=f(r(e))\xi(e)\text{ and}$
$\displaystyle\xi f(e)$ $\displaystyle=\xi(e)f(s(e)),$

which encode in the module $X_{0}$ the structure of the topological quiver.
Denoting by $X$ the completion of $X_{0}$ with respect to this inner product,
$X$ is an $A-A$ correspondence. In fact, for $A^{\prime}=C_{0}(E^{1})$, $X$ is
also an $A^{\prime}-A$ correspondence with the left action given by pointwise
multiplication. Notice that, whether we regard $X$ as an $A-A$ correspondence
or an $A^{\prime}-A$ correspondence, $X$ is nondegenerate, by which we mean
that $\overline{\text{span}}(A\cdot X)=\overline{\text{span}}(A^{\prime}\cdot
X)=X$. In fact, the Cohen-Hewitt Factorization Theorem guarantees that $A\cdot
X=X$, so authors will often take this to be the definition of nondegeneracy
for $C^{*}$-correspondences.

######  Remark 2.11.

When we declare a left action of a $C^{*}$-algebra $A$ on $X$, what we really
mean is a homomorphism $\phi_{A}:A\to\mathcal{L}(X)$. In particular, that the
left action of $A^{\prime}$ on $X$ is implimented by pointwise multiplication,
and the left action by $A$ is induced by the range map by assigning $f\mapsto
f\circ r$ and then acting by pointwise multiplication. In our context, this
homomorphism $\phi_{A}$ is faithful when the set of sources has empty
interior.

Given the $C^{*}$-correspondence of a topological quiver, we are interested in
constructing a $C^{*}$-algebra which reflects the quiver’s data. We will also
find value in representing the correspondence of one quiver in the
correspondence of another quiver in a way which preserves the “essential”
structure of interest to us. In what follows, recall that for an $A-A$
correspondence $X$, the multiplier correspondence is
$M(X)=\mathcal{L}_{A}(A,X),$ the space of all adjointable $A$-linear operators
from $A$ (regarded as the standard correspondence over itself) into $X$.
$M(X)$ is an $M(A)-M(A)$ correspondence. Given a pair $(X,A)$, $(Y,B)$ of
correspondences, a _morphism_ of correspondences is a pair
$(\psi,\pi):(X,A)\to(M(Y),M(B))$ of maps consisting of a homomorphism
$\pi:A\to M(B)$ and a linear map $\psi:X\to M(Y)$ which preserves the
correspondence operations. A _representation_ of $(X,A)$ in a $C^{*}$-algebra
$D$ is a correspondence homomorphism $(\psi,\pi):(X,A)\to(M(D),M(D))$ into the
multiplier of the standard correspondence associated to $D$. We are interested
in a particular class of correspondence homomorphisms.

######  Definition 2.12 ([kqrfunctor]*Definition 3.1).

A correspondence homomorphism is _Cuntz-Pimsner covariant_ provided

  1. (i)

$\psi(X)\subset M_{B}(Y)$

  2. (ii)

$\pi:A\to M(B)$ is nondegenerate

  3. (iii)

$\pi(J_{X})\subset M(B;J_{Y})$

  4. (iv)

The following diagram commutes

${J_{X}}$${M(B;J_{Y})}$${\mathbb{K}(X)}$${M_{B}(\mathbb{K}(Y)).}$$\scriptstyle{\pi|}$$\scriptstyle{\phi_{A}|}$$\scriptstyle{\overline{\phi_{B}}|}$$\scriptstyle{\psi^{(1)}}$

######  Remark 2.13.

Above, $J_{X}$ is the largest ideal mapping $A$ injectively into
$\mathbb{K}(X)$ (likewise for $J_{Y}$), $\psi^{(1)}$ is the naturally
occurring map from $\mathbb{K}(X)$ to $M(\mathbb{K}(Y))$, and

$M(B;J_{Y}):=\\{m\in M(B)|\,mB\cup Bm\subset J_{Y}\\}.$

For topological quivers, [quiver]*Proposition 3.15 provides a satisfactory
description of $J_{X}$ in terms of the _regular verticies_ of the topological
quiver, themselves defined in [quiver]*Definition 3.14. The regular verticies
are characterized as

$E^{0}_{\text{reg}}=E^{0}_{\text{fin}}\cap\text{Int}(\overline{r(E^{1})}),$

where

$\displaystyle E^{0}_{\text{fin}}=\\{v\in E^{0}:\text{there exists a
precompact neighborhood $V$ of $v$ such that }r^{-1}(\overline{V})$
$\displaystyle\text{is compact and }s|_{r^{-1}(V)}\text{ is a local
homeomorphism}\\}.\ $

According to definition 2.12, [kqrfunctor]*Theorem 3.7 defines a functor from
the category of $C^{*}$-correspondences with Cuntz-Pimsner covariant
homomorphisms to the category of $C^{*}$-algebras with nondegenerate
homomorphisms. Through the remainder of this article, we denote by $(X,A)$ the
$A-A$ correspondence associated to $\mathcal{Q}$ and
$(C^{*}(\mathcal{Q}),k_{X},k_{A})$ the universal Cuntz-Pimsner covariant
representation of $(X,A)$. This universal $C^{*}$-algebra is the definition of
the $C^{*}$-algebra associated to $\mathcal{Q}$.

### Coaction Constructions

For a detailed exposition on coactions, one can consult the appendix to
[enchilada]. Antecedents for this article are due to [kqrcoact].

######  Definition 2.14.

Given a locally compact group $G$ and a $C^{*}$-algebra $A$, a coaction
$(A,\delta)$ of $G$ is an injective nondegenerate homomorphism

$\delta:A\to M(A\otimes C^{*}(G))$

which satisfies the _coassociative condition,_

$(\delta\otimes\text{id})\circ\delta=(\text{id}\otimes\delta_{G})\circ\delta$

together with _coaction nondegeneracy_

$\overline{\text{span}}\\{\delta(A)\cdot(1\otimes C^{*}(G))\\}=A\otimes
C^{*}(G).$

######  Remark 2.15.

The theory of coactions makes emphatic use of the group $C^{*}$-algebra
together with tensor products, and these structures are deeply bound to the
theory. Consider, for example, that for a locally compact group $G$, we can
associate with the canonical homomorphism $\iota:G\hookrightarrow M(C^{*}(G))$
a distinguished unitary $w_{G}\in UM(C_{0}(G)\otimes C^{*}(G))$ as follows. We
identify $M(C_{0}(G)\otimes C^{*}(G))$ with $C_{b}(G,M^{\beta}(C^{*}(G)))$,
the bounded continuous functions from $G$ into $M(C^{*}(G))$ endowed with the
strict topology, and then define $w_{G}$ by $w_{G}(g)=\iota(g)\in
M(C^{*}(G))$. This unitary casts the embedding $\iota$ into the theory of
tensor products, which allows us to describe covariant representations of
coactions in what follows.

One special example of a coaction is implimented by a homomorphism
$\psi:C_{0}(G)\to M(A)$, according to the assignment

$\displaystyle\delta^{\psi}:A$ $\displaystyle\to M(A\otimes C^{*}(G))$
$\displaystyle a$
$\displaystyle\mapsto\text{Ad}(\psi\otimes\text{id}(w_{G}))(a\otimes 1)$

and such a coaction is termed _inner_. Much as in the theory of actions, we
define a covariant representation of $(A,\delta)$ as a pair of nondegenerate
homomorphisms $(\pi,\mu)$ such that the diagram

${A}$${M(A\otimes C^{*}(G))}$${C_{0}(G)}$${M(B)}$${M(B\otimes
C^{*}(G))}$$\scriptstyle{\delta}$$\scriptstyle{\pi}$$\scriptstyle{\pi\otimes\text{id}}$$\scriptstyle{\mu}$$\scriptstyle{\delta^{\mu}}$

commutes. Loosely speaking, a covariant representation provides a setting
where the coaction $\delta$ as an inner coaction.

######  Remark 2.16.

In the full generality of locally compact groups, it is necessary for us to
work with multiplier algebras frequently. The use of multiplier algebras
affords us the power to consider strict extentions of homomorphisms, which
under nondegeneracy conditions are unique. There are natural questions about
the compatibility of strict extentions under composition, which in practice
cause us no concern. Consequently, by abuse of notation we commonly will
identify a homomorphism with its strict extension. One can follow
[boiler]*appendix for more details on the general theory.

######  Definition 2.17.

A crossed product for a coaction is a $C^{*}$-algebra $A\rtimes_{\delta}G$
together with a covariant representation $(j_{A},j_{G}):(A,\delta)\to
M(A\rtimes_{\delta}G)$ that is universal for covariant representations. That
is, if $(\pi,\mu):(A,\delta)\to M(B)$ is any covariant representation of the
coaction, there is a unique nondegenerate homomorphism
$\pi\times\mu:A\rtimes_{\delta}G\to M(B)$ with the property that

$\displaystyle\pi=(\pi\times\mu)\circ j_{A}$
$\displaystyle\mu=(\pi\times\mu)\circ j_{G}.$

$\pi\times\mu$ is called the integrated form of the representation
$(\pi,\mu)$.

######  Remarks 2.18.

Let $(A,\delta,G)$ be a coaction, and consider the multiplication
representation $M:C_{0}(G)\to M(\mathbb{K}(L^{2}(G)))$. Then one covariant
representation for $(A,\delta)$ is implimeneted by the pair

$((\text{id}\otimes\lambda)\circ\delta,1\otimes M):(A,\delta)\to
M(A\otimes\mathbb{K}(L^{2}(G)))$

which is called the _regular_ representation of the coaction. The integrated
form of this representation is always faithful, indicating a departure from
the theory of actions on $C^{*}$-algebras. Instead, the notion of
“intermediate” completions for $C^{*}$-dynamical systems is recovered in this
dual category by different classes of coactions. We will shortly introduce the
two most important classes.

Fix $s\in G$. Another example of a representation for $(A,\delta)$ is given by
the pair $(j_{A},j_{G}\circ\text{rt}_{s}):(A,\delta)\to M(A\times_{\delta}G)$,
which is an automorphism of $A\times_{\delta}G$. The family of integrated
forms indexed by the elements of $G$ produces an action of $G$ on the crossed
product, called the _dual_ action to $\delta$, and denoted by $\hat{\delta}$.

######  Definition 2.19 (cf. [boiler], [maximal]).

A coaction $(A,\delta)$ is called _normal_ provided the canonical homomorphism
$j_{A}:A\to A\times_{\delta}G$ is faithful.

A coaction $(A,\delta)$ is called _maximal_ provided there is an isomorphism

$A\times_{\delta}G\times_{\hat{\delta}}G\cong A\otimes\mathbb{K}(L^{2}(G))$

given by a certain canonical map.

A given coaction $(A,\delta)$ always has both a _normalization_
$(A^{n},\delta^{n})$ and a _maximalization_ $(A^{m},\delta^{m})$, and there
are natural surjections $A\to A^{n}$ and $A^{m}\to A$. It is possible for a
coaction to be both maximal and normal. This has strong implications for the
dual action, as we see below.

######  Proposition 2.20.

Suppose $(A,\delta)$ is any coaction which is both normal and maximal. Then
the dual action $\hat{\delta}$ is amenable in the sense that the regular
representation

$A\times_{\delta}G\times_{\hat{\delta}}G\to
A\times_{\delta}G\times_{\hat{\delta},r}G$

is an isomorphism.

###### Proof.

This is an immediate consequence of [maximal]*Proposition 2.2, which the
authors illustrate pictorally with the commutative diagram

${A\times_{\delta}G\times_{\hat{\delta}}G}$${A\otimes\mathbb{K}(L^{2}(G))}$${A\times_{\delta}G\times_{\hat{\delta},r}G}$${A^{n}\otimes\mathbb{K}(L^{2}(G)).}$$\scriptstyle{\Phi}$$\scriptstyle{\Lambda}$$\scriptstyle{\psi\otimes\text{id}}$$\scriptstyle{\Psi}$$\scriptstyle{\Upsilon}$

In the diagram, $\Lambda$ is the regular representation for the crossed
product, $\Phi$ is the canonical surjection, and $\psi$ is the quotient map
$\psi:A\to A/\ker{j_{A}}$ associated to the universal map $j_{A}:A\to
A\times_{\delta}G$. Their proof furnishes the isomorphism $\Upsilon$ and the
surjection $\Psi$.

Since $\delta$ is a maximal coaction, the map $\Phi$ is injective, and since
$\delta$ is normal, $\ker{j_{A}}=\\{0\\}$ and $\psi$ is the identity. It now
follows that

$\Lambda=\Psi\circ\Phi=\Upsilon^{-1}\circ(\psi\otimes\text{id})\circ\Phi,$

a composition of isomorphisms. ∎

Finally, we will make use of the notion of coactions for
$C^{*}$-correspondences. The relevant definition is supplied below.

######  Definition 2.21.

Let $X$ be an $A-B$ correspondence, $G$ a locally compact group, and
$\delta,\epsilon$ coactions of $G$ on $A$ and $B$ respectively. A coaction of
$G$ on $X$ is a nondegenerate correspondence homomorphism

$\sigma:X\to M(X\otimes C^{*}(G))$

which has $\delta$ and $\epsilon$ as coefficient maps, and such that

  1. (i)

$\overline{\text{span}}((1_{M(A)}\otimes C^{*}(G))\sigma(X))=X\otimes
C^{*}(G)$

  2. (ii)

$(\sigma\otimes\text{id}_{G})\circ\sigma=(\text{id}_{X}\otimes\delta_{G})\circ\sigma$.

##  3\. Skew Product Quivers

###  3.1. Definition of the Skew Product Quiver

We wish to define a skew product quiver in analogy with the skew product
graphs and skew product topological graphs studied to date [kqr:graph,
tgcoact]. So let $G$ be a locally compact group, and let $\kappa:E^{1}\to G$
be a continuous map, called a cocycle of $G$ on $\mathcal{Q}$. Then if
$F^{i}=E^{i}\times G$ with continuous range map
$r_{\kappa}(e,g)=(r(e),\kappa(e)g)$ and continuous, source map
$s_{\kappa}(e,g)=(s(e),g)$, we have most of the data required for a
topological quiver. All that remains is to find a family of measures on
$F^{1}$ satisfying our conditions above. Since
$s^{-1}(v,g)=s^{-1}(v)\times\\{g\\}$, we define
$\lambda_{\kappa}=\\{\lambda_{v,g}\\}_{(v,g)\in F^{0}}$ by
$\lambda_{v,g}(E)=\lambda_{v}(\\{e\in E^{1}|(e,g)\in E\\})$. First, we handle
a technical lemma.

######  Lemma 3.1.

Let $\xi\in C_{c}(E^{1}\times G)$, and $(g_{i})_{i\in I}$ a convergent net in
$G$ with limit $g$. Then the net of functions $(\xi(\cdot,g_{i}))_{i\in I}$
converges uniformly to $\xi(\cdot,g)$.

###### Proof.

Certainly, $\xi(\cdot,g_{i})$ converges pointwise to $\xi(\cdot,g)$ by the
continuity of $\xi$. The claim is that this convergence is uniform.

Consider the space $K\times L\subset E^{1}\times G$ defined by
$K=\text{proj}_{E^{1}}(\text{supp}(\xi))$ and
$L=\text{proj}_{G}(\text{supp}(\xi))$, which is a compact subset of the
product. Choosing some precompact open set $W$ in $E^{1}\times G$ containing
the slice $K\times\\{g\\}$, we apply the Tube Lemma [munkres]*Lemma 26.8 to
$W\cap K\times L$ find a product $K\times V$ of open sets with
$K\times\\{g\\}\subset K\times V\subset W$. Writing
$U=\text{proj}_{E^{1}}(W)$, the product $U\times V$ is a precompact open set
containing $\text{supp}(\xi(\cdot,g))$.

We now claim that the family of functions
$\\{\xi(\cdot,g)\\}_{g\in\overline{V}}$ satisfies the hypotheses of Ascoli’s
Theorem. That the family is pointwise bounded is clear, and equicontinuity
follows from a standard $\epsilon/3$ argument. So the family of functions is
precompact in $C_{0}(E^{1})$. Since the net $(\xi(\cdot,g_{i}))$ is eventually
in the family, there is a convergent subnet for the uniform topology. Since
the net converges pointwise to $\xi(\cdot,g)$, the net must converge uniformly
to the same limit. ∎

######  Proposition 3.2.

The ensemble
$\mathcal{Q}\times_{\kappa}G=\\{F^{0},F^{1},r_{\kappa},s_{\kappa},\lambda_{\kappa}\\}$
defines a topological quiver.

###### Proof.

Clearly $r_{\kappa}$ and $s_{\kappa}$ are continuous, with $s_{\kappa}$ an
open map. Since

$s_{\kappa}^{-1}(v,g)=\\{(e,g)|\,s(e)=v\\}=s^{-1}(v)\times\\{g\\},$

we have $\text{supp}(\lambda_{v,g})=s^{-1}(v,g)$. So all that remains is to
show that the family $\lambda_{\kappa}$ is a continuous $s_{\kappa}$-system.

Let $\xi\in C_{c}(F^{1})$, and take a net $(v,g)_{i}=(v_{i},g_{i})$ converging
to $(v,g)$ in $F^{0}$. Then compute

$\displaystyle\left|\int\xi\,d\lambda_{v,g}-\int\xi\,d\lambda_{v_{i},g_{i}}\right|=\left|\int\xi(\cdot,g)\,d\lambda_{v}-\int\xi(\cdot,g_{i})\,d\lambda_{v_{i}}\right|$
$\displaystyle\leq$
$\displaystyle\left|\int\xi(\cdot,g)\,d\lambda_{v}-\int\xi(\cdot,g)\,d\lambda_{v_{i}}\right|+\left|\int\xi(\cdot,g)\,d\lambda_{v_{i}}-\int\xi(\cdot,g_{i})\,d\lambda_{v_{i}}\right|$
$\displaystyle=$
$\displaystyle\left|\int\xi(\cdot,g)\,d\lambda_{v}-\int\xi(\cdot,g)\,d\lambda_{v_{i}}\right|+\left|\int\xi(\cdot,g)-\xi(\cdot,g_{i})\,d\lambda_{v_{i}}\right|$
$\displaystyle\leq$
$\displaystyle\left|\int\xi(\cdot,g)\,d\lambda_{v}-\int\xi(\cdot,g)\,d\lambda_{v_{i}}\right|+\int\left|\xi(\cdot,g)-\xi(\cdot,g_{i})\right|\,d\lambda_{v_{i}}$

We begin by analysing the second term above. Let $\epsilon>0$. By Lemma 3.1,
there is an index $i$ such that $j\geq i$ implies
$\|\xi(\cdot,g)-\xi(\cdot,g_{j})\|\leq\epsilon$. Meanwhile, by Urysohn’s Lemma
we may find a compactly supported function $\eta$ with $\eta\equiv 1$ on
$\text{proj}_{E^{1}}(\text{supp}(\xi))$. It now follows for the second term
that sufficiently far along the sequence

$\displaystyle\int\left|\xi(\cdot,g)-\xi(\cdot,g_{i})\right|\,d\lambda_{v_{i}}$
$\displaystyle\leq\int\epsilon\eta\,d\lambda_{v_{i}}=\epsilon\int\eta\,d\lambda_{v_{i}}$

which converges by continuity to $\epsilon\int\eta\,d\lambda_{v}.$ Now,

$\displaystyle\left|\int\xi(\cdot,g)\,d\lambda_{v}-\int\xi(\cdot,g)\,d\lambda_{v_{i}}\right|+\int\left|\xi(\cdot,g)-\xi(\cdot,g_{i})\right|\,d\lambda_{v_{i}}$
$\displaystyle\leq$
$\displaystyle\left|\int\xi(\cdot,g)\,d\lambda_{v}-\int\xi(\cdot,g)\,d\lambda_{v_{i}}\right|+\epsilon\int\eta\,d\lambda_{v_{i}}$
$\displaystyle=$
$\displaystyle\left|\int\xi(\cdot,g)\,d\lambda_{v}-\int\xi(\cdot,g)\,d\lambda_{v_{i}}\right|+\epsilon\left(\int\eta\,d\lambda_{v_{i}}-\int\eta\,d\lambda_{v}\right)+\epsilon\left(\int\eta\,d\lambda_{v}\right).$

Now, each of the three terms vanish along the sequence by either continuity of
the $s$-system or Lemma 3.1, establishing continuity for $\lambda_{\kappa}$.

∎

######  Remark 3.3.

Now that we have seen that $\mathcal{Q}\times_{\kappa}G$ is a topological
quiver, observe how $G$ interacts with the skew product. The edge and vertex
sets $E^{i}\times G$ are trivial bundles over $G$, and we can associate to
each individual space $E^{i}\times G$ an action of $G$ on the right by
$(e,h)\cdot g=(e,hg)$. Since $G$ acts on itself and only on the second
coordinate, this defines a principal $G$-bundle $q^{i}:F^{i}\to E^{i}$. In
practice, we will freely write $q$ to refer to either of these quotient maps.
Moreover, the range and source maps each define $G$-bundle morphisms from the
edge spaces to the vertex spaces.

Since we have these quotient maps between topological spaces, we also have an
induced nondegenerate $*$-homomorphism $C_{0}(E^{0})\to M(C_{0}(F^{0}))$ by
$f\mapsto f\circ q$.

###  3.2. A Classification of Skew Products

In this section we extend the notions of [dkq, section 3] with the aim of
showing that the only obstructions to a topological quiver being a skew
product are the existence of a free and proper action by $G$ and a global
section for the quotient. We remark that [dkq] goes considerably further in
their analysis of group actions on topological graphs, which we leave to
pursue in future work.

Given a topological quiver $\mathcal{Q}$, a locally compact group $G$, and a
cocycle $\kappa:E^{1}\to G$, we have shown how to construct a skew product
quiver. One lifts the associated structures (being the range map, source map,
and the $s$-system $\lambda$) to the appropriate trivial bundles $E^{i}\times
G$, allowing some interference by the cocycle. Investigating this construction
further, we have seen that each of the trivial bundles is equipped with a free
and proper action of $G$ by right translation in the second variable, and that
the quotient of each space by this action recovers the original edge and
vertex spaces. Our first motions demonstrate the connection shared between
these nominally distinct bundles.

######  Definition 3.4.

Given topological quivers
$\mathcal{Q}=(E^{0},E^{1},r,s,\lambda),\mathcal{R}=(F^{0},F^{1},r,s,\mu)$, a
_quiver morphism_ $\phi:\mathcal{Q}\to\mathcal{R}$ is a pair of continuous
maps $\phi=(\phi^{0},\phi^{1})$, where $\phi^{0}:E^{0}\to F^{0}$ and
$\phi^{1}:E^{1}\to F^{1}$ such that $r\circ\phi^{1}=\phi^{0}\circ r$ and
$s\circ\phi^{1}=\phi^{1}\circ s$.

An _isomorphism of topological quivers_ is a quiver morphism with each map
$(\phi^{0},\phi^{1})$ a homeomorphism, and which preserves the measurements of
the $s$-system. By this, we mean that for every vertex $v$ and every
measurable set $E\subseteq s^{-1}(v)$,
$\lambda_{v}(E)=\mu_{\phi^{0}(v)}(\phi^{1}(E))$. It follows that the inverse
pair $\phi^{-1}=((\phi^{0})^{-1},(\phi^{1})^{-1})$ is also a quiver morphism.
The group of all automorphisms of a topological quiver will be denoted by
$\text{Aut}(\mathcal{Q})$.

A locally compact group acts on a topological quiver $\mathcal{Q}$ provided
there is a continuous group homomorphism $\alpha:G\to\text{Aut}\mathcal{Q}$,
with $g\mapsto\alpha_{g}=(\alpha_{g}^{0},\alpha_{g}^{1})$. The vertex action
$\alpha^{0}_{g}(v)$ will often be abbreviated by $g\cdot v$, and likewise for
the edge action. The action is _free_ provided it is free on the vertices; it
follows that the action is free on the edges. The action is _proper_ provided
the translation map is proper on the vertex space. This is also enough to
deduce properness of the action on the edge space by [amenable]*Proposition
2.1.14.

######  Example 3.5.

Given a topological quiver $\mathcal{Q}$, group $G$ and a cocycle $\kappa$,
the skew product quiver $\mathcal{Q}\times_{\kappa}G$ comes equipped with an
action of $G$ by right translation on the second factor, $(v,h)\cdot
g=(v,hg)$. Indeed,

$\displaystyle s(e,h)\cdot g$ $\displaystyle=(s(e),h)\cdot
g=(s(e),hg)=s(e,hg)=s((e,h)\cdot g),$ $\displaystyle r(e,h)\cdot g$
$\displaystyle=(r(e),\kappa(e)h)\cdot
g=(r(e),\kappa(e)hg)=r(e,hg)=r((e,h)\cdot g),$

demonstrating covariance of the action with the range and source maps.
Continuity is clear, and since right translation on a group is a free and
proper action, it follows that the this action of $G$ on $\mathcal{Q}$ is both
free and proper.

Given a free and proper action of a group on a topological quiver, it is easy
to identify the quotient by this action.

######  Proposition 3.6.

If a group $G$ acts freely and properly on a topological quiver
$\mathcal{Q}=(E^{0},E^{1},r,s,\lambda)$, then the ensemble
$q(\mathcal{Q})=(q(E^{0}),q(E^{1}),\dot{r},\dot{s},\dot{\lambda})$ is a
topological quiver.

###### Proof.

Since $G$ acts freely and properly on the vertex and edge spaces the orbit
spaces are locally compact Hausdorff, and the orbit maps $q:E^{i}\to
q(E^{i})$, assigning the points to their orbits are quotient maps. The
covariance of the range and source maps for the actions on the vertex and edge
spaces guarantees unique continuous maps $\dot{r}$ and $\dot{s}$ given by
composition with $q$, and $\dot{s}$ remains open since compositions of open
maps are open. Since the $s$-system is equivariant by assumption, there is
also an associated $\dot{s}$-system $\dot{\lambda}$ by Theorem 2.8. ∎

######  Corollary 3.7.

Given a skew product $\mathcal{Q}\times_{\kappa}G$, the quotient quiver
$q(\mathcal{Q}\times_{\kappa}G)$ is isomorphic to $\mathcal{Q}$.

###### Proof.

Since any skew product quiver $\mathcal{Q}\times_{\kappa}G$ is naturally
equipped with a free and proper action, it follows from Proposition 3.6 that
the quotient is another topological quiver. In this context the orbit maps $q$
take the particularly simple form of projection onto the first factor of the
products $E^{i}\times G$, the quotient maps $\dot{r},\dot{s}$ coincide with
the range and source maps of the original quiver, and the equivariant system
of measures agrees with the system from the original quiver, so that
$q(\mathcal{Q}\times_{\kappa}G)\cong\mathcal{Q}$. ∎

So we see that given a quiver $\mathcal{Q}$, a group $G$ and a cocycle
$\kappa:E^{1}\to G$, we can generate a new topological quiver
$\mathcal{Q}\times_{\kappa}G$, and this quiver comes equipped with a free and
proper action by $G$. For the case of directed graphs and discrete groups,
[GrossTucker]*Theorem 2.2.2 shows that free actions by $G$ are in one to one
correspondence with the skew products by some cocycle (see also
[dkq]*Corollary 3.7). We have shown that for topological quivers, every skew
product construction naturally associates a free and proper action by $G$, so
we can ask under what criteria this free and proper action recovers the
underlying quiver and cocycle. It turns out that there is only one.

######  Theorem 3.8.

Given a topological quiver $\mathcal{Q}$, a locally compact group $G$, and a
cocycle $\kappa:E^{1}\to G$, the skew product quiver
$\mathcal{Q}\times_{\kappa}G$ comes equipped with a free and proper action of
$G$ by right translation in the second coordinate, and the quiver of orbits is
isomorphic to the initial quiver $\mathcal{Q}$.

Conversely, suppose we are given a topological quiver $\mathcal{R}$ together
with a free and proper action by a group $G$. If the vertex orbit map $q$ is
trivial (so that $F^{0}\cong q(F^{0})\times G$), then $\mathcal{R}$ is
isomorphic to a skew product quiver.

###### Proof.

The first part is Corollary 3.7. We proceed with the other direction.

Denote the components of $\mathcal{R}$ by $(F^{0},F^{1},r,s,\lambda)$, and the
quiver of orbits by
$q(\mathcal{R})=(E^{0},E^{1},\dot{r},\dot{s},\dot{\lambda})$ which is
guaranteed by Proposition 3.6. By hypothesis, we have an isomorphism

$\phi:F^{0}\to E^{0}\times G,$

and we freely identify a vertex from $F^{0}$ with its image under $\phi$.
Since the source map $s$ is $G$-equivariant, Theorem 2.6 produces an
isomorphism $l$ from $F^{1}$ to the pullback $E^{1}*_{\dot{s}}F^{0}$.
Considering that the assignment $(e,(v,g))\mapsto(e,g)$ with inverse
$(e,g)\mapsto(e,s(e),g)$ constructs a homeomorphism $E^{1}*_{\dot{s}}G\to
E^{1}\times G$, post-composition of $l$ by this map constructs a homeomorphism

$\sigma:F^{1}\to E^{1}\times G.$

We claim that the following diagram commutes

${F^{1}}$${F^{0}}$${E^{1}\times G}$${E^{0}\times
G}$${E^{1}}$${E^{0}.}$$\scriptstyle{q}$$\scriptstyle{s}$$\scriptstyle{\sigma}$$\scriptstyle{q}$$\scriptstyle{\phi}$$\scriptstyle{\text{pr}_{1}}$$\scriptstyle{\dot{s}\times\text{id}}$$\scriptstyle{\text{pr}_{1}}$$\scriptstyle{\dot{s}}$

It is clear that the bottom square commutes according to the definition of the
central map $\dot{s}\times\text{id}$. Meanwhile, the right triangle commutes
essentially by definition, and the left triangle commutes as a consequence of
Theorem 2.6. For the top square, we compute that

$\phi\circ s\circ\sigma^{-1}(e,g)=\phi\circ
s(f)=(s(e),g)=\dot{s}\times\text{id}(e,g),$

where $f$ is the unique element of $F^{1}$ associated to the data
$(e,(s(e),g))$ in $E^{1}*_{s}F^{0}$. Precomposition by $\sigma$ in the
expression above concludes the commutativity of the diagram.

A similar argument shows that the analogous diagram for range maps also
commutes, so that we have a commutative diagram

${F^{0}}$${F^{1}}$${F^{0}}$${E^{0}\times G}$${E^{1}\times G}$${E^{1}\times
G}$${E^{0}\times
G}$${E^{0}}$${E^{1}}$${E^{0}.}$$\scriptstyle{q}$$\scriptstyle{\phi}$$\scriptstyle{q}$$\scriptstyle{r}$$\scriptstyle{s}$$\scriptstyle{\sigma}$$\scriptstyle{\rho}$$\scriptstyle{q}$$\scriptstyle{\phi}$pr$\scriptstyle{\dot{r}\times\text{id}}$pr$\scriptstyle{\dot{s}\times\text{id}}$prpr$\scriptstyle{\dot{r}}$$\scriptstyle{\dot{s}}$

This diagram now suggests our candidate for a quiver isomorphism with a skew
product. We define
$\tilde{r}=(\dot{r}\times\text{id})\circ\rho\circ\sigma^{-1}$, define
$\mathcal{Q}=(E^{0}\times G,E^{1}\times
G,\tilde{r},\dot{s}\times\text{id},\\{\lambda_{\phi(v)}\\}_{v\in F^{0}})$, and
then construct the quiver isomorphism

$(\sigma,\phi):\mathcal{R}\to\mathcal{Q}.$

Indeed, the maps $\sigma$ and $\phi$ are homeomorphisms by construction, and
it is clear that at least the source maps factor through the isomorphism. As
for the range map, we compute

$\tilde{r}\circ\sigma=(\dot{r}\times\text{id})\circ\rho=\phi\circ r,$

verifying the range maps also factor through the isomorphism. The source map
$\dot{s}\times\text{id}=\phi\circ s\circ\sigma^{-1}$ is open as a composition
of open maps, and the $\dot{s}\times\text{id}$-system agrees with the original
$s$-system up to identification of the vertices under $\phi$. Thus,
$(\sigma,\phi)$ is a quiver isomorphism.

It remains to show that the image of this isomorphism is in fact a skew
product. We claim that $\mathcal{Q}\cong q(\mathcal{R})\times_{\kappa}G$ for
some cocycle $\kappa$. To find this cocycle, we investigate the edge space
automorphism $\rho\circ\sigma^{-1}$.

Since the inner diamond of the above diagram commutes, we know that
$\text{pr}\circ\rho\circ\sigma^{-1}=\text{pr}$, so $\rho\circ\sigma^{-1}$
fixes the first factor. Since $\rho$ and $\sigma$ are $G$-equivariant, this
map depends only on the choice of $e\in E^{1}$, and we define $\kappa$
according to

$\kappa(e)=\text{pr}_{2}\circ\rho\circ\sigma^{-1}(e,1),$

where we treat $1$ as the neutral element of the group. Now, let us identify
$q(\mathcal{R})\times_{\kappa}G$. As spaces, the edge and vertex spaces are
$E^{1}\times G$ and $E^{0}\times G$, and the source map for the skew product
is $(e,g)\mapsto(\dot{s}(e),g)=\dot{s}\times\text{id}(e,g)$. For the range
map, we write

$(e,g)\mapsto(\dot{r}(e),\kappa(e)g),$

and compute

$\displaystyle(\dot{r}(e),\kappa(e)g)$ $\displaystyle=g(\dot{r}(e),\kappa(e))$
$\displaystyle=g[\dot{r}\times\text{id}(e,\kappa(e))]$
$\displaystyle=g[\dot{r}\times\text{id}(\rho\circ\sigma^{-1}(e,1))]$
$\displaystyle=g[\tilde{r}(e,1)]$ $\displaystyle=\tilde{r}(e,g),$

and verify that the range map for the skew product agrees with the range map
defined for $\mathcal{Q}$. Finally, the equivariance of the $s$-system
$\lambda$ under the group action guarantees that the orbit quiver has the
$\dot{s}$-system $\dot{\lambda}$, and the skew product quiver by construction
reproduces an equivariant $\dot{s}\times\text{id}$-system $\tilde{\lambda}$
which agrees with the original $s$-system by the commutativity of the top-
right square in the diagram. We thus conclude that

$\displaystyle\mathcal{R}$ $\displaystyle=(F^{0},F^{1},s,r,\lambda)$
$\displaystyle\cong(E^{0}\times G,E^{1}\times
G,\dot{s}\times\text{id},\tilde{r},\tilde{\lambda})$
$\displaystyle=\mathcal{Q}$ $\displaystyle=q(\mathcal{R})\times_{\kappa}G,$

and $\mathcal{R}$ is isomorphic to a skew-product quiver.

∎

##  4\. Coactions and Skew Products

Throughout this section we fix a topological quiver $\mathcal{Q}$ together
with a cocycle $\kappa:E^{1}\to G$, and write $(X,A)$ for the correspondence
associated to $\mathcal{Q}$. Furthermore, we write $(Y,B)$ for the
correspondence associated to $\mathcal{Q}\times_{\kappa}G$.

Dual to the skew product construction for topological quivers, a cocycle also
determines an action by pointwise multiplication of $C_{0}(G)$ on $(X,A)$
according to the induced homomorphism $\kappa^{*}:C_{0}(G)\to
M(C_{0}(E^{1}))$. It is clear that this action commutes with the action coming
from the range map, since their image is in the representation of
$C_{0}(E^{1})$ in $\mathcal{L}(X)$.

Since $\kappa^{*}$ is a homomorphism of $C_{0}(G)$, we also have a
homomorphism

$\overline{\kappa^{*}\otimes\text{id}}:M(C_{0}(G)\otimes C^{*}(G))\to
M(A^{\prime}\otimes C^{*}(G))\subset\mathcal{L}(X\otimes C^{*}(G)),$

since $\kappa^{*}$ is nondegenerate (that is,
$\overline{\text{span}}(\kappa^{*}(C_{0}(G))\cdot A^{\prime})=A^{\prime}$).
Define $v\in M(A^{\prime}\otimes C^{*}(G))$ a unitary by
$v=\overline{\kappa^{*}\otimes\text{id}}(w_{G})$.

######  Remark 4.1.

Because of the identification $M(C_{0}(X)\otimes C^{*}(G))\cong
C_{b}(X,M^{\beta}(C^{*}(G)))$, we recognize $v$ as the function $v:E^{1}\to
M^{\beta}(C^{*}(G))$ with $v(e)=\iota(\kappa(e))$. So $v$ provides a
$C^{*}$-algebraic expression of the cocycle on the associated correspondence.

######  Proposition 4.2.

Given a cocycle $\kappa$ of $G$ on $\mathcal{Q}$, there is an $(X,A)$
correspondence coaction $\sigma$ by $G$ on $X$ defined by
$\sigma(\xi)=v\cdot(\xi\otimes 1)$.

Moreover, $\sigma$ induces a coaction $\zeta$ by $G$ on $C^{*}(\mathcal{Q})$
which is determined on generators by

$\displaystyle\zeta\circ k_{X}(\xi)$
$\displaystyle=\overline{k_{X}\otimes\text{id}}\circ\sigma(\xi)$
$\displaystyle\zeta\circ k_{A}(\xi)$ $\displaystyle=k_{A}\otimes\text{id}$

where $(k_{X},k_{A})$ is the universal representation of $(X,A)$ in
$C^{*}(\mathcal{Q})$.

###### Proof.

Because $X$ is nondegenerate as an $A-A$ correspondence, the results of
[kqrcoact] apply. In particular, since $\mu=M\circ\kappa^{*}$ commutes with
$\phi_{A}$, we apply [kqrcoact, Corollary 3.5] to generate $\zeta$. ∎

With this coaction in hand, we may generate the coaction crossed product
$C^{*}(\mathcal{Q})\times_{\zeta}G$ following the theory introduced in the
Preliminaries. Our main purpose is to show that this crossed product agrees
with the $C^{*}$-algebra associated to the skew product quiver.

######  Theorem 4.3.

$C^{*}(\mathcal{Q})\times_{\zeta}G\cong C^{*}(\mathcal{Q}\times_{\kappa}G)$.

######  Remark 4.4.

Here is our strategy to prove this theorem. We will first produce a covariant
representation $(\Pi,\psi):(C^{*}(\mathcal{Q}),\zeta)\to
M(C^{*}(\mathcal{Q}\times_{\kappa}G)$, thereby generating a unique
homomorphism

$\Pi\times\psi:C^{*}(Q)\times_{\zeta}G\to C^{*}(\mathcal{Q}\times_{\kappa}G).$

Then, we will invoke the Dual Invariant Uniqueness Theorem [boiler, Theorem
5.4] to show that the integrated form is injective. To achieve this end, our
approach will be through a series of lemmas and propositions. As a guide, it
may be constructive to consult the following diagram:

(4.1)
${C^{*}(\mathcal{Q})}$${M(C^{*}(\mathcal{Q})\times_{\zeta}G)}$${C_{0}(G)}$${M(C^{*}(\mathcal{Q}\times_{\kappa}G)).}$$\scriptstyle{\Pi}$$\scriptstyle{j_{C^{*}(\mathcal{Q})}}$$\scriptstyle{\Pi\times\psi}$$\scriptstyle{j_{G}}$$\scriptstyle{\psi}$

We begin with the construction of the homomorphism $\Pi$, which will require
the most significant resources.

######  Proposition 4.5.

There is a Cuntz-Pimsner covariant homomorphism
$(\mu,\nu):(X,A)\to(M(Y),M(B))$. By the functoriality of Cuntz-Pimsner
covarience [kqrfunctor], there is a unique associated homomorphism
$\Pi:C^{*}(\mathcal{Q})\to M(C^{*}(\mathcal{Q}\times_{\kappa}G))$.

We define $\nu:A\to M(B)$ according to $\nu:f\mapsto f\circ q$, and turn our
attention to the map $\mu$. Throughout this discussion, fix $\xi\in
C_{c}(E^{1})$ and $f\in C_{c}(F^{0})$.

######  Lemma 4.6.

The function $(\xi\circ q)\cdot(f\circ s):F^{1}\to\mathbb{C}$ has compact
support.

###### Proof.

Let $K=\text{supp}(\xi)\subseteq E^{1}$ and $L=\text{supp}(f)\subseteq F^{0}$.
We may identify $F^{1}$ with $E^{1}*_{s}F^{0}$ by the map
$(e,g)\mapsto(e,s(e),g)$ by Theorem 2.6. Then

$(K\times L)\cap(E^{1}*_{s}F^{0})=\\{(e,v,g):e\in K,(v,g)\in L,\text{ and
}s(e)=q(v,g)=v\\}$

is compact in $E^{1}*_{s}F^{0}\cong F^{1}$. This is exactly the support of
$\xi\circ q\cdot f\circ s$. ∎

######  Lemma 4.7.

The map $\mu(\xi):C_{c}(F^{0})\to C_{c}(F^{1})$ defined by $\mu(\xi)\cdot
f=\xi\circ q\cdot f\circ s$ is uniform to Hilbert-norm bounded, and hence
extends to a bimodule map $\mu(\xi):B\to Y$.

###### Proof.

Lemma 4.6 shows that $\mu(\xi)$ is well defined, and linearity is clear. So
see that

$\displaystyle\langle\mu(\xi)\cdot f,\mu(\xi)\cdot f\rangle(v,g)$
$\displaystyle=\int\overline{(\xi\circ q)\cdot(f\circ s)}\cdot(\xi\circ
q)\cdot(f\circ s)\,d\lambda_{v,g}$ $\displaystyle=\int|\xi\circ q|^{2}|f\circ
s|^{2}\,d\lambda_{v,g}$
$\displaystyle=\langle\xi,\xi\rangle(q(v,g))|f(v,g)|^{2}\leq\|\xi\|^{2}\|f\|_{u}^{2}.$

Since this holds for all vertices $(v,g)$, we see that

(4.2) $\|\mu(\xi)\cdot f\|\leq\|\xi\|\|f\|_{u}.$

So $\mu(\xi)$ is bounded, and thus extends uniquely to a bounded linear map
$B\to Y$. ∎

######  Proposition 4.8.

There is a Hilbert to uniform-norm bounded map $\mu(\xi)^{*}:C_{c}(F^{1})\to
C_{c}(F^{0})$. Moreover, the unique continuous linear extension is an adjoint
for $\mu(\xi)$.

###### Proof.

Fix $\eta\in C_{c}(F^{1})$ and define $\mu(\xi)^{*}$ by

$\mu(\xi)^{*}\cdot\eta(v,g)=\int\overline{\xi\circ
q(\alpha,g)}\eta(\alpha,g)\,d\lambda_{v,g}(\alpha,g).$

Notice that the restriction of $\xi\circ q$ to $s^{-1}(v,g)$ (and we will
denote all such restrictions by, e.g. $\xi\circ q|$) agrees with the
restriction of $\xi$ to $s^{-1}(v)$ so that $\xi\circ q|\in
C_{c}(s^{-1}(v,g))$, and $\mu(\xi)^{*}\cdot\eta(v,g)=\langle\xi\circ
q|,\eta|\rangle$, the inner product for $(L^{2}(F^{1}),\lambda_{v,g})$. Then

$\displaystyle|\mu(\xi)^{*}\cdot\eta(v,g)|^{2}$
$\displaystyle=|\langle\xi\circ q|,\eta|\rangle|^{2}\leq\left[\int|\xi\circ
q|^{2}\,d\lambda_{v,g}\right]\cdot\left[\int|\eta|^{2}\,d\lambda_{v,g}\right]$
$\displaystyle=\langle\xi\circ q,\xi\circ
q\rangle(v,g)\cdot\langle\eta,\eta\rangle(v,g)\leq\|\xi\|^{2}\cdot\|\eta\|^{2},$

so $\mu(\xi)^{*}\eta$ is bounded from $C_{c}(F^{1})$ with the Hilbert norm to
$C_{c}(F^{0})$ with the uniform norm. So $\mu(\xi)^{*}$ extends uniquely to
$\mu(\xi)^{*}:Y\to B$.

Lastly, to check adjoints, for $f\in C_{0}(F^{0})$,

$\displaystyle\langle\mu(\xi)^{*}\cdot\eta,f\rangle(v,g)=\overline{\int\overline{\xi\circ
q}\cdot\eta\,d\lambda_{v,g}}\cdot f(v,g)=\int(\xi\circ
q)\cdot\overline{\eta}\,d\lambda_{v,g}\cdot f(v,g)$
$\displaystyle=\int(\xi\circ q)\cdot\overline{\eta}\cdot f\circ
s\,d\lambda_{v,g}=\langle\eta,(\xi\circ q)\cdot(f\circ
s)\rangle(v,g)=\langle\eta,\mu(\xi)\cdot f\rangle(v,g).$

It follows that $\mu(\xi)^{*}$ is an adjoint for $\mu(\xi)$. ∎

######  Remark 4.9.

It now follows that $\mu(\xi)\in\mathcal{L}(B,Y)=M(Y)$ for any $\xi\in
C_{c}(E^{1})$. Since the map $\mu:C_{c}(E^{1})\to M(Y)$ is bounded by (4.2),
$\mu$ extends to a linear map $\mu:X\to M(Y)$. Now, for any $\xi\in X$, we
already have a natural right action of $B$ on $\mu(\xi)$ given by evaluation,
but we also have a left action of $B$ on $\mu(\xi)$, defined by
$f\cdot\mu(\xi)=(f\circ r)\cdot(\xi\circ q)$. Arguments analogous to the
proofs of Lemmas 4.6 and 4.7 show that $f\cdot\mu(\xi)\in Y$, and thus $\mu$
maps $X$ to the relative multiplier algebra $M_{B}(Y)$.

######  Proposition 4.10.

The pair $(\mu,\nu):(X,A)\to(M(Y),M(B))$ is a correspondence homomorphism.

###### Proof.

It suffices to confirm that
$\langle\mu(\upsilon),\mu(\xi)\rangle=\nu(\langle\upsilon,\xi\rangle)$, and
that $\mu(f\cdot\xi)=\nu(f)\cdot\mu(\xi)$. We first check covariance of the
left action, by computing

$\displaystyle\mu(f\cdot\xi)$ $\displaystyle=(f\circ r\cdot\xi)\circ q=(f\circ
r)\circ q)\cdot(\xi\circ q)$ $\displaystyle=(f\circ q)\circ r\cdot(\xi\circ
q)=\nu(f)\cdot\mu(\xi).$

To confirm the relationships between the inner products, we first choose $f\in
B$ and compute that for any vertex $(v,g)$,

$\displaystyle\langle\mu(\eta),\mu(\xi)\rangle\cdot f(v,g)$
$\displaystyle=\mu(\eta)^{*}\mu(\xi)\cdot f(v,g)=\mu(\eta)^{*}[\xi\circ q\cdot
f\circ s](v,g)$ $\displaystyle=\int\overline{\eta\circ q}\cdot(\xi\circ
q)\cdot(f\circ s)\,d\lambda_{v,g}=\int\overline{\eta\circ q}\cdot(\xi\circ
q)\,d\lambda_{v,g}f(v,g)$ $\displaystyle=\int(\overline{\eta}\cdot\xi)\circ
q\,d\lambda_{v,g}f(v,g)=\left[\int\overline{\eta}\cdot\xi\,d\lambda_{v}\right]\circ
q\cdot f(v,g)$ $\displaystyle=\langle\eta,\xi\rangle\circ q\cdot
f(v,g)=\nu(\langle\eta,\xi\rangle)\cdot f(v,g).$

Since $f$ and $(v,g)$ were arbitrary, the result follows. ∎

With this correspondence homomorphism now in hand, we now turn our attention
to Cuntz-Pimsner covariance for this homomorphism.

######  Lemma 4.11.

We have $\nu(J_{X})\subseteq M(B;J_{Y})$.

###### Proof.

For topological quivers, the ideal $J_{X}$ is characterized in the
preliminaries as $J_{X}=C_{0}(E^{0}_{\text{rg}})$, the set of regular vertices
of the topological quiver. It suffices to show that for any $f\in
J_{X}=C_{0}(E^{0}_{\text{rg}})$ and $b\in B$, we have $\nu(f)\cdot b\in
J_{Y}$. So suppose $\nu(f)\cdot b(v,g)\neq 0$, and we will show that $(v,g)$
is a regular vertex of $\mathcal{Q}\times_{\kappa}G$. It follows that $f\circ
q(v,g)=f(v)\neq 0$, so $v\in E^{0}_{\text{rg}}$. Since $v$ is not a sink, we
can easily find an edge emitted by $(v,g)$. Meanwhile we may choose a suitable
compact neighborhood $W\subseteq E^{0}$ of $v$ with $r^{-1}(W)\subseteq E^{1}$
compact and $s$ restricted to this subset a local homeomorphism.

Define a compact subset $U=W\times V$ of $E^{0}\times G$. We claim that $U$ is
a precompact neighborhood of $(v,g)$ which has $r^{-1}(\overline{U})$ compact
and $s|_{r^{-1}(U)}$ a local homeomorphism. U is closed, so
$r^{-1}(\overline{U})=r^{-1}(U)$ is closed in $E^{1}\times G$. Moreover, if
$(e,g)\in r^{-1}(U)$, then $r(e,g)=(r(e),k(e)g)$, and it follows that

$r^{-1}(U)\subseteq r^{-1}(W)\times\kappa(r^{-1}(W))^{-1}\cdot V.$

Note that since the second factor in the above product is compact, and thus
$r^{-1}(U)$ is compact. So it remains to show that the restriction of the
source map to $r^{-1}(U)$ is a local homeomorphism. This is easy, since the
source map acts like the source map for $\mathcal{Q}$ on the first factor and
the identity on the second factor, so the restriction is the coordinatewise
product of a local homeomorphism and a global homeomorphism.

∎

######  Lemma 4.12.

The diagram

${J_{X}}$${M(B;J_{Y})}$${\mathbb{K}(X)}$${M_{B}(\mathbb{K}(Y)),}$$\scriptstyle{\phi_{A}}$$\scriptstyle{\nu}$$\scriptstyle{\bar{\phi}_{B}|}$$\scriptstyle{\mu^{(1)}}$

commutes.

###### Proof.

Choose $f\in J_{X}$, and fix $\epsilon>0$. Then there is a finite-rank
operator $\Theta=\sum\theta_{\eta_{k},\xi_{k}}\in\mathbb{K}(X)$ with
$\|\phi_{A}(f)-\Theta\|<\epsilon$. Now, for any $y\in C_{c}(F^{1})$, we
compute that

$\displaystyle\left\|[\mu^{(1)}\circ\phi_{A}(f)]y-[\bar{\phi}_{B}|\circ\nu]y\right\|$
$\displaystyle\leq\left\|[\mu^{(1)}\circ\phi_{A}(f)]y-[\mu^{(1)}(\Theta)]y\right\|+\left\|\mu^{(1)}(\Theta)y-[\bar{\phi}_{B}|\circ\nu]y\right\|$
$\displaystyle\leq\epsilon\|y\|+\left\|\mu^{(1)}(\Theta)y-[\bar{\phi}_{B}|\circ\nu]y\right\|,$

so that what remains is to bound the second addend. To this end, we see

$\mu^{(1)}(\Theta)y=\mu^{(1)}(\sum\theta_{\eta_{k},\xi_{k}})y=\left[\sum\mu(\eta_{k})\mu(\xi_{k})^{*}\right]y,$

and by evaluating pointwise, that

$\displaystyle\mu^{(1)}(\Theta)y(e,g)$ $\displaystyle=\sum\eta_{k}\circ
q(e,g)\cdot\int\overline{\xi_{k}\circ
q}(\alpha,g)y(\alpha,g)\,d\lambda_{s(e),g}$
$\displaystyle=\sum\eta_{k}(e)\cdot\int\overline{\xi_{k}}(\alpha)y(\alpha,g)\,d\lambda_{s(e)}=\left[\sum\eta_{k}\langle\xi_{k},y(\cdot,g)\rangle\right](e)$
$\displaystyle=\Theta(y(\cdot,g))(e).$

On the other hand,

$\displaystyle[\bar{\phi}_{B}|\circ\nu](f)y(e,g)$
$\displaystyle=[\bar{\phi}_{B}|(f\circ q)y](e,g)=(f\circ q\circ r)(e,g)\cdot
y(e,g)$ $\displaystyle=(f\circ r\circ q)(e,g)\cdot
y(e,g)=[\phi_{A}(f)y(\cdot,g)](e),$

from which it follows that

$\displaystyle\mu^{(1)}(\Theta)y(e,g)-(\bar{\phi}_{B}|\circ\nu)(f)y(e,g)$
$\displaystyle=\Theta(y(\cdot,g))(e)-[\phi_{A}(f)y(\cdot,g)](e)$
$\displaystyle=([\Theta-\phi_{A}(f)]y(\cdot,g))(e).$

So we compute

$\displaystyle\|\mu^{(1)}(\Theta)-(\bar{\phi}_{B}|\circ\nu)(f)y\|^{2}$
$\displaystyle=\sup_{F^{0}}\langle\mu^{(1)}(\Theta)-(\bar{\phi}_{B}|\circ\nu)(f)y,\mu^{(1)}(\Theta)-(\bar{\phi}_{B}|\circ\nu)(f)y\rangle(v,g)$
$\displaystyle=\sup_{E^{0},G}\langle(\Theta-\phi_{A})(f)y(\cdot,g),(\Theta-\phi_{A})(f)y(\cdot,g)\rangle(v)$
$\displaystyle=\sup_{G}\|(\Theta-\phi_{A})(f)y(\cdot,g)\|^{2}$
$\displaystyle\leq\sup_{G}\|(\Theta-\phi_{A})(f)\|^{2}\|y(\cdot,g)\|^{2}$

Taking square roots,

$\displaystyle\left\|[\mu^{(1)}\circ\phi_{A}(f)]y-[\bar{\phi}_{B}|\circ\nu]y\right\|$
$\displaystyle\leq\left\|[\mu^{(1)}\circ\phi_{A}(f)]y-\mu^{(1)}(\Theta)]y\right\|+\left\|\mu^{(1)}(\Theta)y-[\bar{\phi}_{B}|\circ\nu]y\right\|\leq
2\epsilon\|y\|$

∎

We can now show that $(\mu,\nu)$ is Cuntz-Pimsner covariant.

###### Proof of Proposition 4.5.

$\nu$ is clearly nondegenerate, and Remark 4.9 shows that $\mu:X\to M_{B}(Y)$,
satisfying the first two conditions. Lemmas 4.11 and 4.12 supply the next two
conditions of Definition 2.12, so that $(\mu,\nu)$ is a Cuntz-Pimsner
covariant correspondence homomorphism.

The existence of $\Pi$ is now a consequence of [kqrfunctor]*Corollary 3.6. ∎

Having completed the construction of $\Pi$ by Proposition 4.5, we return to
the Diagram 4.1, and introduce the homomorphism $\psi$. Since
$M(C^{*}(\mathcal{Q}\times_{\kappa}G))$ contains a copy of $M(C_{0}(F^{0}))$,
we define $\psi:C_{0}(G)\to M(C_{0}(F^{0}))$ by $\psi(\gamma)=\gamma\circ
q_{2}$, where $q_{2}$ is the projection of $F^{0}$ onto the second factor.
Then the composition $k_{B}\circ\psi:C_{0}(G)\to
M(C^{*}(\mathcal{Q}\times_{\kappa}G))$ induces a coaction, which we denote by
$\delta^{\psi}$.

######  Proposition 4.13.

The pair $(\Pi,k_{B}\circ\psi)$ is a covariant representation for the coaction
$(C^{*}(\mathcal{Q}),\zeta)$ in $M(C^{*}(\mathcal{Q}\times_{\kappa}G))$, so we
have a unique integrated form $C^{*}(\mathcal{Q})\rtimes_{\zeta}G\to
M(C^{*}(Q\times_{\kappa}G))$, which we will denote by $\Pi\times\psi.$

###### Proof.

Our objective is to show that the diagram

${C^{*}(\mathcal{Q})}$${M(C^{*}(\mathcal{Q})\otimes
C^{*}(G))}$${M(C^{*}(\mathcal{Q}\times_{\kappa}G))}$${M(C^{*}(\mathcal{Q}\times_{\kappa}G)\otimes
C^{*}(G))}$$\scriptstyle{\zeta}$$\scriptstyle{\Pi}$$\scriptstyle{\overline{\Pi\times\text{id}}}$$\scriptstyle{\delta^{\psi}}$

commutes. Since $C^{*}(\mathcal{Q})$ is generated by the images of the
canonical correspondence maps $(k_{X},k_{A})$, it is enough to confirm
separately that

$\displaystyle\overline{\Pi\times\text{id}}\circ\zeta\circ k_{A}$
$\displaystyle=\delta^{\psi}\circ\Pi\circ k_{A}$
$\displaystyle\overline{\Pi\times\text{id}}\circ\zeta\circ k_{X}$
$\displaystyle=\delta^{\psi}\circ\Pi\circ k_{X}.$

Beginning with the first computation, we witness

$\displaystyle\overline{\Pi\times\text{id}}\circ\zeta\circ k_{A}(a)$
$\displaystyle=\overline{\Pi\times\text{id}}\circ\overline{k_{A}\times\text{id}}(a\otimes
1)=\overline{\Pi\circ k_{A}\otimes\text{id}}(a\otimes 1)$
$\displaystyle=\overline{k_{B}\circ\nu\otimes\text{id}}(a\otimes
1)=\overline{k_{B}\otimes\text{id}}(\nu(a)\otimes 1),$

whose argument we identify as the operator valued function $\nu(a)\otimes 1\in
C_{b}(F^{0},M^{\beta}(C^{*}(G)))$ assigning $(v,g)\mapsto a(v)\iota(1)$.
Attending to this argument $\nu(a)\otimes 1$, we compute pointwise

$\displaystyle\nu(a)\otimes 1(v,g)$
$\displaystyle=a(v)\iota(1)=a(v)\iota(u_{g}u_{g}^{*})=1\otimes u_{g}\cdot
a(v)\otimes 1\cdot 1\otimes u_{g}^{*}$
$\displaystyle=[\overline{\psi\otimes\text{id}}(\omega_{G})\cdot a\otimes
1\cdot\overline{\psi\otimes\text{id}}(\omega_{G})^{*}](v,g).$

Then, since $\overline{k_{B}\otimes\text{id}}$ is a homomorphism, follows that
$\overline{\Pi\times\text{id}}\circ\zeta\circ k_{A}=\delta^{\psi}\circ\Pi\circ
k_{A}$.

We follow up with the second computation, which is a bit more delicate.
Observe that

$\displaystyle\delta^{\psi}\circ\Pi\circ k_{X}(x)$ $\displaystyle=$
$\displaystyle\delta^{\psi}\circ k_{Y}\circ\mu(x)$ $\displaystyle=$
$\displaystyle\overline{k_{B}\circ\psi\otimes\text{id}}(\omega_{G})\big{(}k_{Y}(\mu(x))\otimes
1\big{)}\overline{k_{B}\circ\psi\otimes\text{id}}(\omega_{G})^{*}$
$\displaystyle=$
$\displaystyle\overline{k_{B}\otimes\text{id}}\circ\overline{\psi\otimes\text{id}}(\omega_{G})\big{(}k_{Y}(\mu(x))\otimes
1\big{)}\overline{k_{B}\otimes\text{id}}\circ\overline{\psi\otimes\text{id}}(\omega_{G})^{*}$
$\displaystyle=$
$\displaystyle\overline{k_{B}\otimes\text{id}}\circ\overline{\psi\otimes\text{id}}(\omega_{G})\big{(}\overline{k_{Y}\otimes\text{id}}(\mu(x)\otimes
1)\big{)}\overline{k_{B}\otimes\text{id}}\circ\overline{\psi\otimes\text{id}}(\omega_{G})^{*}$
$\displaystyle=$
$\displaystyle\overline{k_{Y}\otimes\text{id}}\Big{[}\overline{\psi\otimes\text{id}}(\omega_{G})\big{(}\mu(x)\otimes
1\big{)}\overline{\psi\otimes\text{id}}(\omega_{G})^{*}\Big{]},$

where the last line follows since $(k_{Y},k_{B})$ is a Toeplitz
representation. Now, isolating the argument (which is a multiplier of
$Y\otimes C^{*}(G)$), we calculate pointwise for the elementary tensor
$b\otimes\gamma\in B\otimes C^{*}(G)$

$\displaystyle\overline{\psi\otimes\text{id}}(\omega_{G})(\mu(x)\otimes
1)\overline{\psi\otimes\text{id}}(\omega_{G})^{*}(b\otimes\gamma)(e,g)$
$\displaystyle=$
$\displaystyle\overline{\psi\otimes\text{id}}(\omega_{G})(r(e),\kappa(e)g)\cdot
x(e)1\cdot\overline{\psi\otimes\text{id}}(\omega_{G})^{*}(s(e),g)b(s(e),g)\gamma$
$\displaystyle=$ $\displaystyle u_{\kappa(e)g}\cdot x(e)1\cdot u_{g}^{*}\cdot
b(s(e))\gamma$ $\displaystyle=$ $\displaystyle
x(e)u_{\kappa(e)g}u_{g}^{*}\cdot b(s(e),g)\gamma$ $\displaystyle=$
$\displaystyle x(e)u_{\kappa(e)}\cdot b(s(e),g)\gamma$ $\displaystyle=$
$\displaystyle(v\cdot(x\otimes 1))\circ q(e,g)\cdot b\otimes\gamma(e,g)$
$\displaystyle=$ $\displaystyle\overline{\mu\otimes\text{id}}(v\cdot(x\otimes
1))(e,g)\cdot b\otimes\gamma(e,g).$

Thus, the argument agrees with $\overline{\mu\otimes\text{id}}(v\cdot(x\otimes
1))$ on elementary tensors, and consequently for all of $B\otimes C^{*}(G)$.

With this in hand, we observe that

$\displaystyle\delta^{\psi}\circ\Pi\circ k_{X}(x)$ $\displaystyle=$
$\displaystyle\overline{k_{Y}\otimes\text{id}}\left[\overline{\mu\otimes\text{id}}(v\cdot(x\otimes
1))\right]$ $\displaystyle=$
$\displaystyle\overline{k_{Y}\circ\mu\otimes\text{id}}(\sigma(x))$
$\displaystyle=$ $\displaystyle\overline{\Pi\circ
k_{X}\otimes\text{id}}(\sigma(x))$ $\displaystyle=$
$\displaystyle\overline{\Pi\otimes\text{id}}\circ\overline{k_{X}\otimes\text{id}}\circ\sigma(x)$
$\displaystyle=$ $\displaystyle\overline{\Pi\otimes\text{id}}\circ\zeta\circ
k_{X}(x)$

and we thus conclude that the pair $(\Pi,k_{B}\circ\psi)$ is a covariant
representation of the coaction $(C^{*}(\mathcal{Q}),\zeta)$.

∎

It remains for us to show that the integrated form $\Pi\times\psi$ is an
isomorphism. We will show that the image of $\Pi\times\psi$ contains all of
the generators of $C^{*}(\mathcal{Q}\times_{\kappa}G)$ to deduce surjectivity,
and employ the Dual Invariant Uniqueness Theorem [boiler, Theorem 5.4] to
prove injectivity.

######  Proposition 4.14.

The map $\Pi\times\psi:C^{*}(\mathcal{Q})\times_{\zeta}G\to
C^{*}(\mathcal{Q}\times_{\kappa}G)$ is surjective.

###### Proof.

As stated above, we need only show that $\Pi\times\psi$ is surjective on the
images of the generators $k_{B}(C_{c}(F^{0})),k_{Y}(C_{c}(F^{1}))$. We check
for $a\in A$ and $f\in C_{0}(G)$ that

$\displaystyle\quad\Pi\times\psi(j_{C^{*}\mathcal{Q}}(k_{A}(a))j_{G}(f))$
$\displaystyle=\Pi(k_{A}(a))k_{B}(\psi(f))$
$\displaystyle=k_{B}(\nu(a))k_{B}(\psi(f))=k_{B}(\nu(a)\psi(f))$
$\displaystyle=k_{B}(a\otimes 1\cdot 1\otimes f)=k_{B}(a\otimes f).$

A similar computation for $x\in C_{c}(E^{1})$ shows

$\displaystyle\quad\Pi\times\psi(j_{\mathcal{Q}}(k_{X}(x))j_{G}(f))$
$\displaystyle=\Pi(k_{X}(x))k_{B}(\psi(f))$
$\displaystyle=k_{Y}(\mu(x))k_{B}(\psi(f))$ $\displaystyle=k_{Y}(x\otimes
1\cdot 1\otimes f)$ $\displaystyle=k_{Y}(x\otimes f),$

Taking these equations together, we see that products such as

$k_{B}(a\otimes f)k_{Y}(x\otimes\phi)$

are contained in the image of $\Pi\times\psi.$ An argument in linearity and
density now shows that the integrated form is surjective. ∎

######  Proposition 4.15.

$\Pi\times\psi$ is injective.

###### Proof.

We apply [boiler, Theorem 5.4]. First, this entails the covariant
representation $(\Pi,\psi)$ in $M(C^{*}(\mathcal{Q}\times_{\kappa}G))$ with
$\text{ker }\Pi=\text{ker }j_{\mathcal{Q}}$. Indeed, the gauge action on the
skew product quiver intertwines the gauge action on $C^{*}(\mathcal{Q})$, so
since $\nu$ is injective, it follows that $\Pi$ is injective. Then, since
$\Pi=(\Pi\times\psi)\circ j_{\mathcal{Q}}$, it also follows that
$j_{\mathcal{Q}}$ is also injective, so that the kernels agree.

Next, we need an action $\beta$ of $G$ on $C^{*}(\mathcal{Q}\times_{\kappa}G)$
whose strict extension fixes $\Pi$ and acts by right translation on $\psi$.
Define $\beta_{h}$ as the action determined on
$C^{*}(\mathcal{Q}\times_{\kappa}G)$ by right translation on $(Y,B)$. Then
$\overline{\beta_{h}}(\Pi)=\Pi$, while
$\overline{\beta_{h}}(\psi(f))=\psi(\text{rt}_{h}f)$, satisfying the
hypotheses of the theorem. It then follows that $\Pi\times\psi$ is injective.
∎

In the course of proving Proposition 4.15, we deduced an auxilary result.

######  Corollary 4.16.

The coaction $\zeta$ is normal.

###### Proof.

In the proof above, we noticed that $j_{Q}$ is injective, which is the
definition of normality of the coaction. ∎

We are now prepared to prove the main theorem.

###### Proof of Theorem 4.3.

Proposition 4.13 populates the diagram 4.1, while Propositions 4.15 and 4.14
demonstrate that the integrated form $\Pi\times\psi$ produces the desired
isomorphism. ∎

##  5\. Application to Duality

Topological quivers are quite abundant, as the following proposition shows.

######  Proposition 5.1.

Suppose that $X,Y$ are two second countable locally compact Hausdorff spaces,
and suppose that $f,g:X\to Y$ are both continuous maps. If $f$ is open and
surjective, then there is a topological quiver with vertex space $Y$ and edge
space $X$.

###### Proof.

This is an immediate consequence of Blanchard’s Theorem [blanchard]. Since $f$
is an open and surjective map, there is an associated $f$-system of
probability measures $\lambda=\\{\lambda_{y}\\}_{y\in Y}$. Then the ensemble
$(Y,X,g,f,\lambda)$ is the desired topological quiver. ∎

In particular, topological quivers generalize directed graphs (where the
spaces above are discrete), as well as topological graphs where the source map
is a local homeomorphism. Theorem 4.3 thus recovers [kqr:graph]*Theorem 2.4 as
well as [tgcoact]*Theorem 3.1. We mention that the approach used in this
present article constructs the isomorphism
$C^{*}\mathcal{Q}\times_{\zeta}G\cong C^{*}(\mathcal{Q}\times_{\kappa}G)$ with
our bare hands, a departure from the technique of [tgcoact] which employs
Landstad duality to recover the coaction, then shows this coaction is
equivalent to the constructed one. Presumably, this technique would work just
as well.

We have shown that the coaction $\zeta$ associated to a cocycle on a
topological quiver is injective, and consequently a normal coaction. As
mentioned in the preliminaries, it is possible for a coaction to be
simultaneously normal and maximal, and Proposition 2.20 shows that this has
strong implications for the dual action. So far, we have been unable to
demonstrate the maximality of $\zeta$, so we offer the claim here as a
conjecture.

######  Conjecture 5.2.

If $\mathcal{Q}$ is a topological quiver equipped with a cocycle
$\kappa:E^{1}\to G$, over a locally compact group, then the natural coaction
$\zeta$ described in Proposition 4.2 is maximal. Consequently, the regular
representation in Proposition 2.20 is an isomorphism.

The conjecture holds trivially for amenable groups, and the analysis in
[kqr:graph] shows that this holds for discrete groups and directed graphs
without restriction. A positive answer this question would generalize the work
in the discrete setting and strengthen the results contained in [tgcoact,
dkq]. As a final application, we deduce consequences for the $C^{*}$-algebra
of a quiver which has a suitable principal action by a locally compact group.

######  Corollary 5.3.

If $\mathcal{Q}$ is a topological quiver equipped with a free and proper
action $\alpha$ by $G$ for which the vertex orbit bundle $(E^{0},q,E^{0}/G)$
is trivial, then

$C^{*}(\mathcal{Q})\times_{\alpha,r}G\cong
C^{*}(q(\mathcal{Q}))\otimes\mathbb{K}(L^{2}G).$

Consequently, the reduced crossed product is Morita equivalent to the
$C^{*}$-algebra of the quotient quiver.

###### Proof.

Write $\mathcal{R}=q(\mathcal{Q})$ the quotient quiver under the action.
According to Theorem 3.8, $\mathcal{Q}$ is a skew product quiver
$\mathcal{R}\times_{\kappa}G$, so
$C^{*}(\mathcal{Q})=C^{*}(\mathcal{R}\times_{\kappa}G)$ is $G$-equivariantly
isomorphic to $C^{*}(\mathcal{R})\times_{\zeta}G$ for the natural coaction
$\zeta$. Since $\zeta$ is a normal coaction (see Corollary 4.16) Katayama
duality applies [enchilada]*Theorem A.69, and

$C^{*}(\mathcal{Q})\times_{\alpha,r}G\cong
C^{*}(\mathcal{R})\times_{\zeta}G\times_{\alpha}G=C^{*}(\mathcal{R})\otimes\mathbb{K}.$

∎

## References

  * quiver