# Noncommutative numerable principal bundles
from group actions on C*-algebras

Mariusz Tobolski

###### Abstract

We introduce a definition of the locally trivial $G$-C*-algebra, which is a
noncommutative counterpart of the total space of a locally compact Hausdorff
numerable principal $G$-bundle. To obtain this generalization, we have to go
beyond the Gelfand–Naimark duality and use the multipliers of the Pedersen
ideal. Our new concept enables us to investigate local triviality of
noncommutative principal bundles coming from group actions on non-unital
C*-algebras, which we illustrate through examples coming from
$C_{0}(Y)$-algebras and graph C*-algebras. In the case of an action of a
compact Hausdorff group on a unital C*-algebra, local triviality in our sense
is implied by the finiteness of the local-triviality dimension of the action.
Furthermore, we prove that if $A$ is a locally trivial $G$-C*-algebra, then
the $G$-action on $A$ is free in a certain sense, which in many cases
coincides with the known notions of freeness due to Rieffel and Ellwood.

Key words: noncommutative topology, C*-algebra, multipliers of the Pedersen
ideal, C*-dynamical system, quantum principal bundle

MSC: 46L85, 37A55

##  1 Introduction

Noncommutative mathematics (see e.g. [15, 22]), motivated by the Heisenberg
approach to quantum mechanics, is a way of generalizing certain classical
mathematical theories which can be described in terms of commutative algebras,
e.g. topology, measure theory, differential geometry, and group theory. In
particular, the theory of locally compact Hausdorff spaces is
(anti-)equivalent to the theory of commutative C*-algebras via the
Gelfand–Naimark duality (see § 2.1). This leads to the philosophy of viewing
noncommutative C*-algebras as algebras of functions on noncommutative spaces.
The main idea of noncommutative topology is to transfer the important notions
and results about locally compact Hausdorff spaces via the aforementioned
equivalence and then check whether the assumption of commutativity can be
dropped. In most cases, this process is far from obvious and it profits both
sides of the equivalence. For instance, the recent breakthrough in the Elliott
classification program of simple separable nuclear unital C*-algebras was made
using the notion of a nuclear dimension [54], which generalizes the covering
dimension of a locally compact Hausdorff space.

The importance of principal bundles in topology and differential geometry can
hardly be overestimated. They also form the mathematical language of the
classical gauge theory in physics (see e.g. [42]). The aim of the paper is to
find and study the most general noncommutative analog of a principal bundle.
This goal has already been achieved in the case when the spaces involved are
compact Hausdorff. There are two main lines of research which focus on
generalizing principal bundles to noncommutative mathematics:

  1. (I)

Compact principal bundles in noncommutative algebraic geometry. In this
setting, the role of a topological group and a topological space are played by
a Hopf algebra and its unital comodule algebra respectively. The first
definition of a noncommutative principal bundle was given by Schneider [48] in
terms of Hopf–Galois extensions [33]. Brzeziński and Majid [10] used
Schneider‘s definition and considered a noncommutative gauge theory. The
definition of a compact quantum principal bundle is now well established in
the literature (see e.g. [5, 18, 28, 30, 32, 34]). In fact, the notion that
Schneider generalized was the one of a principal $G$-bundle due to H. Cartan
[13], where the group action is free and proper but the local triviality (see
e.g. [50]) of the bundle is not assumed. The work of Aschieri, Fioresi, and
Latini [4] introduces a notion of local triviality which is suitable for
noncommutative algebraic geomtery, but it cannot be used in the C*-algebraic
context. Furthermore, the setting of Hopf–Galois extensions cannot be used to
obtain a generalization of principal bundles beyond the compact case.

  2. (II)

Principal bundles in the theory of C*-algebras. The interplay between
principal bundles and C*-crossed products was studied by J. Phillips and
Raeburn in [44] (see also § 4.2.2). The work of Connes and Rieffel [16] was
the first attempt to consider gauge theory, specifically the Yang–Mills
theory, in the C*-algebraic framework [15]. Furthermore, in [47], Rieffel
defined free and proper actions of locally compact Hausdorff groups on
C*-algebras generalizing the aforementioned notion of a principal bundle due
to H. Cartan. Some years later, Ellwood [21] proposed a different
generalization of a Cartan principal $G$-space in the context of actions of
locally compact quantum groups on C*-algebras. In [17], de Commer and
Yamashita proved that, for compact (quantum) group actions on a unital
C*-algebras, the definitions of Ellwood and Rieffel are equivalent. It is also
worth mentioning that Echterhoff, Nest, and Oyono-Oyono introduced
noncommutative principal torus bundles based on the Green theorem [19].
However, the notion of local triviality was still missing. In the case of a
compact (quantum) group actions on a unital C*-algebras, a solution to the
problem was proposed by the author, Gardella, Hajac, and Wu [24]. Let us
mention that the previous work of Budzyński and Kondracki [12], while
successful in many aspects, gave a generalization of a different concept, i.e.
the piecewise triviality of the bundle (see e.g. [9]). This was observed by
Hajac, Krähmer, Matthes, and Zieliński in [29] at the algebraic level.

There are various interplays between the two approches. In the case of compact
quantum group actions on unital C*-algebras, Baum, de Commer, and Hajac proved
in [8] that under additional assumptions the Hopf–Galois condition of
Schneider is equivalent to the Ellwood condition. This equivalence is possible
due to the existence of the Peter–Weyl functor.

In the paper, we introduce a definition of the noncommutative numerable
principal bundle in the setting of actions of locally compact Hausdorff groups
on C*-algebras. Unless otherwise stated, $G$ will denote a locally compact
Hausdorff group and $A$ will denote a C*-algebra. Recall that a principal
bundle is called numerable if it admits a trivializing cover with a partition
of unity subordinated to it. The assumption of numerability is a crucial one.
We have to rely on partitions of unity, since it is difficult to talk about
open subsets of a noncommutative space. Recall, however, that only numerable
principal bundles can be classified using the Milnor universal space [38]. It
is clear that already in topology it is important to go beyond the compact
case. Indeed, although many principal $G$-bundles can be reduced to compact
subgroups of $G$, there exist genuinely locally compact (i.e. non-reducible)
ones. We use the terminology of [52, p. 332], where total spaces of
(numerable) principal $G$-bundles are called locally trivial $G$-spaces.

One way of obtaining a generalization of a principal bundle would be to use
the Kasparov notion of a $C_{0}(Y)$-algebra [31]. We say that $A$ is a
(numerable) principal $G$-$C_{0}(Y)$-algebra if there is a $G$-equivariant
non-degenerate $*$-homomorphism $\chi:C_{0}(Y)\to ZM(A)$, where $Y$ is a
locally compact Hausdorff $G$-space such that $Y\to Y/G$ is a (numerable)
principal $G$-bundle (see § 4.2.2) and $ZM(A)$ denotes the center of the
multiplier algebra of $A$. This definition covers all classical principal
$G$-bundles with a locally compact Hausdorff total space and actions of dual
groups on C*-crossed products coming from locally unitary actions considered
in [44]. However, this concept does not apply to simple C*-algebras. Since
there are examples of simple C*-algebras giving rise to noncommutative
principal bundles (see [14, Subsection 5.1]), it seems that the notion of a
principal $G$-$C_{0}(Y)$-algebra is more of a classical flavour and a truly
noncommutative object is needed.

The key difficulty in obtaining the right concept is that one has to go beyond
the C*-algebra theory. This is prompted by several facts. First of all, the
only known generalization of the local triviality condition for a principal
$G$-bundle is the finiteness of the local-triviality dimension introduced in
[24], which relies on the notion of a partition of unity. Therefore, we have
to use unreduced cones or topological joins (see e.g. [38]) one way or
another. Let ${\mathcal{C}}G$ denote the unreduced cone over $G$ (see § 3.1).
Observe that if $G$ is not compact then ${\mathcal{C}}G$ is no longer locally
compact. This means that ${\mathcal{C}}G$ cannot be described using the
Gelfand–Naimark duality. However, ${\mathcal{C}}G$ is still a compactly
generated completely regular Hausdorff space. Hence, we can exploit the
compact-open topology defined on the $*$-algebra $C({\mathcal{C}}G)$ of all
continuous complex-valued functions on ${\mathcal{C}}G$. In turn, this
requires us to use the Pedersen multiplier algebra $\Gamma(K_{A})$ (see §
2.3), which for $A=C_{0}(X)$, where $X$ is a locally compact Hausdorff space,
is $*$-isomorphic to $C(X)$. Let us equip $C({\mathcal{C}}G)$ with the
compact-open topology and $\Gamma(K_{A})$ with the $\kappa$-topology (see
(13)). We introduce the following definition (see Definition 4.2).

The total space of a noncommutative numerable principal bundle

We say that $A$ is a locally trivial $G$-C*-algebra if there exist unital
$G$-equivariant continuous $*$-homomorphisms
$\rho_{i}:C({\mathcal{C}}G)\to\Gamma(K_{A})$, $i\in{\mathbb{N}}$ such that
$(\sum_{i=0}^{N}\rho_{i}({\rm t}))_{N\in{\mathbb{N}}}$ converges to $1$ in the
$\kappa$-topology.

If $G$ is compact and $A$ is unital, then the finiteness of the local
triviality dimension immediately implies local triviality of the
$G$-C*-algebra in question. We also have the following desired results. If
$A=C_{0}(X)$ for some locally compact Hausdorff space $X$, then $A$ is a
locally trivial $G$-C*-algebra if and only if $X\to X/G$ is a numerable
principal $G$-bundle. Furthermore, numerable principal $C_{0}(Y)$-algebras are
locally trivial as $G$-C*-algebras. Finally, if $A$ is a locally trivial
$G$-C*-algebra then the $G$-action on $A$ is $\kappa$-free. The
$\kappa$-freeness is a generalization of freeness of the action of a group on
a space (see § 5). If either $A$ is commutative or $G$ is compact and $A$ a
unital, then $\kappa$-freeness is equivalent to the Ellwood freeness
condition.

The paper is organized as follows. Section 2 contains some preliminaries on
the Gelfand–Naimark duality and the multipliers of the Pedersen ideal, that
will be needed in the sequel. We also describe the one-to-one correspondence
between continuous maps of completely regular Hausdorff spaces and continuous
$*$-homomorphisms of their unital $*$-algebras of all continuous complex-
valued functions equipped with the compact-open topology. Results of a purely
topological nature, including the reformulation of the notion of a numerable
principal bundle, are in Section 3. Section 4 is the main part of the paper.
We introduce three incarnations of local triviality of $G$-C*-algebras. We
also consider classes of locally trivial $G$-C*-algebras coming from classical
numerable principal bundles and principal numerable $G$-$C_{0}(Y)$-algebras.
Finally, using the theory of graph C*-algebras, we provide examples of a non-
unital locally trivial $\mathbb{Z}/2\mathbb{Z}$-C*-algebras that are not
principal $C_{0}(X)$-algebras. Section 5 deals with freeness of actions. We
introduce $\kappa$-freeness which is closely related to the Ellwood condition
and prove that it is implied by local triviality.

##  2 Preliminaries

We refer the reader to [41] for basic material on C*-algebras.

###  2.1 The equivariant Gelfand–Naimark duality

Let $A$ be a C*-algebra and let $\widehat{A}$ be its spectrum, i.e. the set of
unitary equivalence classes of irreducible $*$-representations of $A$. Let
${\rm Prim}(A)$ be the space of primitive ideals taken with the Jacobson
topology. Recall that ${\rm Prim}(A)$ is always locally compact and that there
is a surjective map $\widehat{A}\to{\rm Prim}(A)$, which enables us to turn
$\widehat{A}$ into a locally compact space. If $\widehat{A}$ is a
$T_{0}$-space, i.e. a space in which points are closed, then the
aforementioned map is a homeomorphism. Furthermore, if $A$ is commutative,
then $\widehat{A}$ is automatically Hausdorff.

An action of a locally compact Hausdorff group $G$ on $A$ is given by a group
homomorphism $\alpha:G\to{\rm Aut}(A)$. We say that $\alpha$ is (jointly)
continuous if the map $(a,g)\mapsto\alpha_{g}(a):=\alpha(g)(a)$ is continuous,
where we consider the product topology on the domain. A C*-algebra equipped
with a continuous action of a locally compact Hausdorff group $G$ is called a
$G$-C*-algebra. Let $A$ and $B$ be $G$-C*-algebras with actions $\alpha$ and
$\beta$ respectively. A $*$-homomorphism $\Phi:A\to B$ is called
$G$-equivariant or a $G$-$*$-homomorphism if
$\Phi\circ\alpha_{g}=\beta_{g}\circ\Phi$ for all $g\in G$. A morphism between
two C*-algebras $A$ and $B$ is a non-degenerate $*$-homomorphism $\phi:A\to
M(B)$, i.e. $\phi(A)B$ is norm-dense in $B$. Here $M(B)$ is the C*-algebra of
multipliers of $B$. Every action $\alpha$ of a locally compact Hausdorff group
$G$ on a C*-algebra $A$ induces a continuous (right) action on $\widehat{A}$
by the formula

$\widehat{A}\times
G\longrightarrow\widehat{A}:\qquad([\pi],g)\longmapsto[\pi\circ\alpha_{g}].$
(1)

The action $\alpha$ can also be extended to an action $M(\alpha)$ on $M(A)$ in
the following way

$M(\alpha)_{g}(x)a:=\alpha_{g}(xa),\qquad
aM(\alpha)_{g}(x):=\alpha_{g}(ax),\qquad g\in G,\;x\in M(A),\;a\in A.$ (2)

The action $M(\alpha)$ is continuous with respect to the strict topology
$\beta$, i.e. the topology generated by the seminorms

$\phi_{a}(x):=\|xa\|,\quad\psi_{a}(x):=\|ax\|,\qquad x\in M(A),\quad a\in A.$
(3)

However, $M(\alpha)$ may not be continuous with respect to the norm topology.
A $G$-morphism is a $G$-equivariant morphism, where we consider the extension
of the action to the multiplier algebra. In what follows, we will use the
terms $G$-morphism and $G$-equivariant non-degenerate $*$-homomorphism
interchangeably.

Finally, let $X$ be a locally compact Hausdorff space. Then the space
$C_{0}(X)$ of all continuous complex-valued functions vanishing at infinity
forms a commutative C*-algebra. If $G$ is a locally compact Hausdorff group
acting on $X$ (on the right), then there is an induced continuous action on
$C_{0}(X)$ given by

$(\alpha_{g}(f))(x):=f(xg),\qquad g\in G,\quad f\in C_{0}(X),\quad x\in X.$
(4)

C*-algebras are central objects studied in noncommutative topology which arose
from the Gelfand–Naimark theorem [25]. The equivariant version of the theorem
can be stated as the following anti-equivalence of categories:

$\left\\{\begin{matrix}\text{compact Hausdorff $G$-spaces}\\\ \text{and
continuous
$G$-maps}\end{matrix}\right\\}\quad{\longleftrightarrow}\quad\left\\{\begin{matrix}\text{unital
commutative $G$-C*-algebras}\\\ \text{and unital
$G$-$*$-homomorphisms}\end{matrix}\right\\}.$ (5)

Here by a compact Hausdorff $G$-space we mean that both the space and the
group $G$ are compact Hausdorff. The above result can be extended to locally
compact Hausdorff spaces in at least two different ways. Let us present the
equivariant version of a generalization due to Woronowicz [55]:

$\left\\{\begin{matrix}\text{locally compact Hausdorff $G$-spaces}\\\
\text{and continuous
$G$-maps}\end{matrix}\right\\}\quad{\longleftrightarrow}\quad\left\\{\begin{matrix}\text{commutative
$G$-$C^{*}$-algebras}\\\ \text{and $G$-morphisms}\end{matrix}\right\\}.$ (6)

In the above anti-equivalences, the functor from left to right to each locally
compact Hausdorff space $X$ assigns the commutative C*-algebra $C_{0}(X)$ and
the inverse functor to each commutative C*-algebra $A$ assigns its spectrum
$\widehat{A}$. We omit the assignments of the morphisms for now, since they
are analogous to (11) and (12) below.

###  2.2 Beyond local compactness

In this paper, we have to go beyond the anti-equivalences (5) and (6). Let $Y$
be a completely regular Hausdorff space and let $C(Y)$ denote the unital
$*$-algebra of all continuous complex-valued functions on $Y$. We equip $C(Y)$
with the compact-open topology $\kappa$, i.e. the topology generated by the
seminorms

$p_{K}(f):=\sup_{y\in K}|f(y)|,$ (7)

where $K\subset Y$ is compact and $f\in C(Y)$. This topology turns $C(Y)$ into
a locally convex $*$-algebra. Let ${\rm Aut}_{\kappa}(C(Y))$ denote the group
of $\kappa$-continuous $*$-automorphisms of $C(Y)$. Much in the same way as in
(4), if $G$ is a locally compact Hausdorff group and $Y$ is a (right)
$G$-space, we define a $G$-action $\alpha:G\to{\rm Aut}_{\kappa}(C(Y))$ by the
formula

$(\alpha_{g}(f))(y):=f(yg),\qquad g\in G,\quad f\in C(Y),\quad y\in Y.$ (8)

One can show that this action is continuous, i.e. that the natural map
$G\times C(Y)\to C(Y)$ it induces is continuous, where we consider the product
topology on the domain. Let us define the set of characters

$\Omega(C(Y)):=\\{\chi:C(Y)\to\mathbb{C}~{}:~{}\chi\text{ is a non-zero
$\kappa$-continuous $*$-homomorphism}\\},$

which we can equip with the weak $*$-topology. The above $G$-action
$\alpha:G\to{\rm Aut}_{\kappa}(C(Y))$ given by (8) induces a $G$-action on
$\Omega(C(Y))$ as follows

$G\longrightarrow{\rm
Homeo}(\Omega(C(Y))):\quad\chi\longmapsto\chi\circ\alpha_{g}\,.$ (9)

One can prove that the map

$Y\longrightarrow\Omega(C(Y)),\qquad y\longmapsto m_{y}\,,\quad
m_{y}(f):=f(y),$ (10)

is a continuous bijection, where complete regularity of $Y$ is only needed to
prove injectivity (see e.g. [39, Corollary 2.3]). Since the topology on a
completely regular Hausdorff space $Y$ is the initial topology induced by
$C(Y)$, we conclude that (10) is a homeomorphism. Finally, since

$m_{yg}(f)=f(yg)=(\alpha_{g}(f))(y)=m_{y}(\alpha_{g}(f)),\qquad g\in G,\quad
y\in Y,\quad f\in C(Y),$

we infer that (10) is $G$-equivariant (this also proves that (9) defines a
continuous $G$-action).

If $F:Y\to Z$ is a continuous $G$-map between completely regular Hausdorff
$G$-spaces, then we can define a unital $G$-equivariant $\kappa$-continuous
$*$-homomorphism

$F^{*}:C(Z)\longrightarrow C(Y):\qquad f\longmapsto f\circ F.$ (11)

Next, if we start with a unital $G$-equivariant $\kappa$-continuous
$*$-homomorphism $F^{*}:C(Z)\to C(Y)$, then we can obtain a continuous $G$-map

$F:\Omega(C(Y))\longrightarrow\Omega(C(Z)):\qquad\chi\longmapsto\chi\circ\Phi.$
(12)

Hence, using the $G$-homeomorphism (10), we obtain a one-to-one correspondence
between

  * •

continuous $G$-maps $Y\to Z$ between completely regular Hausdorff $G$-spaces

  * •

and unital $G$-equivariant $\kappa$-continuous $*$-homomorphisms $C(Z)\to
C(Y)$ between their locally convex $*$-algebras of all continuous functions.

###  2.3 Multipliers of the Pedersen ideal

Let us recall the definition of the Pedersen ideal of a C*-algebra $A$ [43].
We first introduce the subsets

$K_{0}^{+}:=\\{a\in A_{+}~{}:~{}ab=a\text{ for some $b\in A_{+}$}\\},$
$K^{+}:=\left\\{a\in A_{+}~{}:~{}\text{there exists $a_{0}\,,\ldots,a_{n}\in
K_{0}^{+}$ such that }a\leq\sum_{i=0}^{n}a_{n}\right\\}.$

The Pedersen ideal is defined to be $K_{A}:={\rm span}\,K^{+}$. It is the
minimal dense ideal of $A$. A multipier of $K_{A}$ is a pair of linear maps
$(S,T)$ from $K_{A}$ to $K_{A}$ such that

$xS(y)=T(x)y\qquad\text{for all $x,y\in K_{A}$}\,.$

The set of all multipiers of $K_{A}$, denoted by $\Gamma(K_{A})$, is a unital
$*$-algebra with natural operations induced from $K_{A}$. Throughout the paper
we call $\Gamma(K_{A})$ the Pedersen multiplier algebra (or the algebra of
multipliers of the Pedersen ideal) of $A$. The subset of bounded multipliers,
i.e. when $S$ and $T$ are bounded maps, equals to the multiplier algebra
$M(A)$. If $A=C_{0}(X)$ for some locally compact Hausdorff space $X$, then
$K_{A}\cong C_{c}(X)$, the $*$-algebra of continuous compactly supported
complex-valued functions on $X$, and $\Gamma(K_{A})\cong C(X)$, the
$*$-algebra of all continuous complex-valued functions on $X$. Observe that
$\Gamma(K_{A})$ is commutative if and only if $A$ is commutative.

For the purposes of the sequel, let us discuss the $\kappa$-topology on
$\Gamma(K_{A})$ [36, § 3.6], i.e. the locally convex topology given by the
seminorms

$\widetilde{\phi}_{k}((S,T)):=\|S(k)\|,\qquad\widetilde{\psi}_{k}((S,T)):=\|T(k)\|,\qquad
k\in K_{A}\,.$ (13)

In this topology, $\Gamma(K_{A})$ is a locally convex complete topological
$*$-algebra [36, Theorem 3.8]. If $A=C_{0}(X)$ then the $\kappa$-topology on
$\Gamma(K_{A})\cong C(X)$ agrees with the compact-open topology [36, § 4.1].
The restriction of the $\kappa$-topology to $M(A)$ can be compared with the
norm and the strict topology $\beta$ on it. In general, all these topologies
are different, e.g. the norm and the strict topologies are complete, while
$M(A)$ is only $\kappa$-dense in $\Gamma(K_{A})$. However, [11, Theorem
1(iii)] shows that in the commutative case the two topologies coincide on
norm-bounded subsets of $M(A)$. Closely following the proof of this result,
one obtains the following generalization:

######  Lemma 2.1.

Let $A$ be a C*-algebra. Then the strict topology $\beta$ coincides with the
$\kappa$-topology on norm-bounded subsets of $M(A)$. $\blacksquare$

Let $A$ be a C*-algebra equipped with a continuous action $\alpha:G\to{\rm
Aut}(A)$ of a locally compact Hausdorff group $G$. Since
$\alpha_{g}(K_{A})=K_{A}$ for every $g\in G$ (see e.g. [43, Theorem 1.4]), we
can define an action $\Gamma(\alpha)$ of $G$ on $\Gamma(K_{A})$ by
$\kappa$-continuous $*$-automorphisms as follows (see e.g. [36, § 2.13])

$\Gamma(\alpha)_{g}(S)(\alpha_{g}(k)):=\alpha_{g}(S(k)),\qquad\Gamma(\alpha)_{g}(T)(\alpha_{g}(k)):=\alpha_{g}(T(k)),$
(14)

where $g\in G$, $(S,T)\in\Gamma(K_{A})$, and $k\in K_{A}$. For $G$-C*-algebras
$A$ and $B$ with actions $\alpha$ and $\beta$ respectively, when we say that a
unital $\kappa$-continuous $*$-homomorphism $\Gamma(K_{A})\to\Gamma(K_{B})$ is
$G$-equivariant, we always mean the equivariance with respect to the extended
actions $\Gamma(\alpha)$ and $\Gamma(\beta)$. Note that if $A$ is a
$G$-C*-algebra with an action $\alpha$, then the inclusion
$M(A)\hookrightarrow\Gamma(K_{A})$ is $G$-equivariant with respect to the
extended actions $M(\alpha)$ and $\Gamma(\alpha)$ (see (2) and (14)
respectively). In the commutative case, the $*$-isomorphism
$C(X)\cong\Gamma(K_{C_{0}(X)})$ of topological $*$-algebras is $G$-equivariant
if we consider the $G$-actions given by (8) and (14) respectively.

Let us now turn to the relation between $G$-morphisms of C*-algebras and
unital $\kappa$-continuous $G$-$*$-homomorphisms of their Pedersen multiplier
algebras.

######  Proposition 2.2.

Let $A$ and $B$ be $G$-C*-algebras, where $G$ is a locally compact Hausdorff
group, and let $\Phi:\Gamma(K_{A})\to\Gamma(K_{B})$ be a unital
$G$-equivariant $\kappa$-continuous $*$-homomorphism. Then the restriction of
$\Phi$ to $A$ is a $G$-morphism.

######  Proof.

Let $\phi:=\Phi|_{A}$. Then $\phi$ is a $G$-equivariant $*$-homomorphism and
$\phi(A)\subseteq M(B)$. We only need to prove that it is non-degenerate,
which is equivalent to showing that $\phi(e_{\lambda})$ strictly converges to
$1$ for some approximate unit $(e_{\lambda})$ of $A$. Observe that if
$(e_{\lambda})$ is an approximate unit of $A$, then $e_{\lambda}\to 1$ in the
$\kappa$-topology in $\Gamma(K_{A})$. Without loss of generality we may assume
that $(e_{\lambda})$ is bounded. We infer that

$\beta\lim_{\lambda}\phi(e_{\lambda})=\kappa\lim_{\lambda}\phi(e_{\lambda})=\kappa\lim_{\lambda}\Phi(e_{\lambda})=\Phi(\kappa\lim_{\lambda}e_{\lambda})=\Phi(1)=1,$

where in the first equality we used Lemma 2.1 (note that $\phi$ is bounded as
a $*$-homomorphism between C*-algebras). $\blacksquare$

It is not clear whether all $G$-morphisms can be extended to unital
$G$-equivariant $\kappa$-continuous $*$-homomorphisms. However, there is a
class of morphisms for which this extension exist. A morphism between
C*-algebras $\phi:A\to M(B)$ is strictly non-degenerate if
$K_{B}\subseteq\phi(K_{A})B$ (see [2, Definition 3.1]). First, let us identify
a class of morphisms which are automatically strictly non-degenerate. In what
follows, $ZA$ denotes the center of a C*-algebra $A$.

######  Proposition 2.3.

Let $\phi:A\to M(B)$ be a morphism. If $\phi(A)\subseteq ZM(B)$, then $\phi$
is strictly non-degenerate.

######  Proof.

First, note that $B\phi(A)B$ is a two-sided ideal of $B$ which is dense by
non-degeneracy of $\phi$. By minimality of the Pedersen ideal, we obtain that
$K_{B}\subseteq B\phi(A)B$. Since $\phi(A)\subseteq ZM(B)$ and $B^{2}=B$, we
conclude that $\phi$ is strictly non-degenerate. $\blacksquare$

Next, we show that strictly non-degenerate $G$-morphisms extend to unital
$G$-equivariant $\kappa$-continuous $*$-homomorphisms.

######  Proposition 2.4.

Let $\phi:A\to M(B)$ be a strictly non-degenerate $G$-morphism. Then $\phi$
extends uniquely to a unital $G$-equivariant $\kappa$-continuous
$*$-homomorphism $\Phi:\Gamma(K_{A})\to\Gamma(K_{B})$.

######  Proof.

We follow [2, p. 21] and define the map $\Phi:\Gamma(K_{A})\to\Gamma(K_{B})$
by

$\Phi(S)(\phi(k)b):=\phi(S(k))b,\qquad\Phi(T)(b\phi(k)):=b\phi(T(k)),\qquad
k\in K_{A},\quad b\in B.$

Here we use that $K_{B}\subseteq\phi(K_{A})B=B\phi(K_{A})$. It is plain to
show that $(\Phi(S),\Phi(T))\in\Gamma(K_{B})$. Then it is equally
straightforward to show that $\Phi$ is a unital $*$-homomorphism. Its
uniqueness follows for example from the uniqueness of the extension to the
multiplier algebras $M(A)\to M(B)$. We need to show the $\kappa$-continuity of
$\Phi$. Let $(S_{\lambda}\,,T_{\lambda})\to(S,T)$ in the $\kappa$-topology and
let

$(U_{\lambda}\,,V_{\lambda}):=\Phi((S_{\lambda}\,,T_{\lambda})),\qquad(U,V):=\Phi((S,T)).$

Since for every $k_{B}\in K_{B}$ there exist $k_{A},k^{\prime}_{A}\in K_{A}$
and $b,b^{\prime}\in B$ such that
$k_{B}=\phi(k_{A})b=b^{\prime}\phi(k^{\prime}_{A})$, we obtain that

$\lim_{\lambda}U_{\lambda}(k_{B})=\lim_{\lambda}U_{\lambda}(\phi(k_{A})b)=\lim_{\lambda}\phi(S_{\lambda}(k_{A}))b=\phi(S(k_{A}))b=U(\phi(k_{A})b)=U(k_{B}),$
$\lim_{\lambda}V_{\lambda}(k_{B})=\lim_{\lambda}V_{\lambda}(b^{\prime}\phi(k^{\prime}_{A}))=\lim_{\lambda}b^{\prime}\phi(T_{\lambda}(k^{\prime}_{A}))=b^{\prime}\phi(T(k^{\prime}_{A}))=V(b^{\prime}\phi(k^{\prime}_{A}))=V(k_{B}).$

Assume that the $G$-actions on $A$ and $B$ are denoted by $\alpha$ and $\beta$
respectively. For all $g\in G$, $(S,T)\in\Gamma(K_{A})$, $k\in K_{A}$, and
$b\in B$, we have that

$\displaystyle\Gamma(\beta)_{g}(\Phi(S))(\phi(\alpha_{g}(k))\beta_{g}(b))$
$\displaystyle=\Gamma(\beta)_{g}(\Phi(S))(\beta_{g}(\phi(k)b))=\beta_{g}(\Phi(S)(\phi(k)b))$
$\displaystyle=\beta_{g}(\phi(S(k))b)=\phi(\alpha_{g}(S(k)))\beta_{g}(b)$
$\displaystyle=\phi(\Gamma(\alpha)_{g}(S)(\alpha_{g}(k)))\beta_{g}(b)=\Phi(\Gamma(\alpha)_{g}(S))(\phi(\alpha_{g}(k))\beta_{g}(b))$

and the computation is the same for $\Gamma(\beta)_{g}(\Phi(T))$. Therefore,
the $*$-homomorphism $\Phi$ is $G$-equivariant. $\blacksquare$

##  3 Topological aspects

In this section, we discuss various topological subtleties that arise when one
tries to find a right analog of a principal bundle in the theory of group
actions on C*-algebras.

###  3.1 Unreduced cones of locally compact Hausdorff groups

Although some of the concepts and results of this subsection apply to general
topological spaces, we focus on the case of a locally compact Hausdorff group
$G$. The unreduced cone ${\mathcal{C}}G$ of $G$ is the quotient topological
space

${\mathcal{C}}G:=([0,1]\times G)/(\\{0\\}\times G).$

We denote the quotient topology on ${\mathcal{C}}G$ by $\tau_{q}$, the
quotient map by $\pi:[0,1]\times G\to{\mathcal{C}}G$, and call
$[(0,g)]\in{\mathcal{C}}G$ the tip of the cone. There is a continuous (right)
$G$-action on ${\mathcal{C}}G$ defined as follows

${\mathcal{C}}G\times
G\longrightarrow{\mathcal{C}}G:\qquad([(t,g)],h)\longmapsto[(t,gh)].$

Indeed, since $G$ is locally compact Hausdorff, this action is continuous by
an old lemma of Whitehead [53, Lemma 4] (see [37] for a modern formulation of
the result).

Recall that every locally compact Hausdorff group is paracompact Hausdorff
(see e.g. [40, Exercise 9, p. 261] or [3, Corollary 3.1.4] for a stronger
statement). Then $[0,1]\times G$ is paracompact Hausdorff as a product of a
compact Hausdorff space and a paracompact Hausdorff space. Taking this into
account, it is straightforward to show that the space ${\mathcal{C}}G$ is
completely regular Hausdorff. However, it may no longer be locally compact
(the tip in ${\mathcal{C}}G$ does not have any compact neighborhood). These
observations are crucial from the point of view of noncommutative topology,
since it implies the anti-equivalences (5) and (6) do not apply in this case.

Consider the maps

${\rm t}:{\mathcal{C}}G\to[0,1],\qquad[(t,g)]\mapsto t,$ (15) ${\rm g}:{\rm
t}^{-1}((0,1])\to G,\qquad[(t,g)]\mapsto g.$ (16)

We call ${\rm t}$ the height function. The above maps are continuous with
respect to $\tau_{q}$. Indeed, observe that ${\rm t}\circ\pi:[0,1]\times
G\to[0,1]$ is the projection onto the first factor. Similarly, since ${\rm
t}^{-1}((0,1])\cong(0,1]\times G$, the map ${\rm g}\circ\pi|_{(0,1]\times G}$
is the projection onto the second factor. For the purposes of the next
subsection, we consider a different topology $\tau_{M}$ on ${\mathcal{C}}G$,
i.e. the initial topology with respect to the maps (15) and (16) with a
subbase given by the collection

$\\{{\rm t}^{-1}(U),\;{\rm g}^{-1}(V)~{}:~{}U\text{ is open in
}[0,1],\;V\text{ is open in }G\\}.$

The above topology was introduced by Milnor for the topological join in [38]
(observe that ${\mathcal{C}}G$ as a set is equal to the topological join of
$G$ with the singleton set). Note that $\tau_{q}$ is finer than $\tau_{M}$.
Furthermore, the map $\pi:[0,1]\times G\to{\mathcal{C}}G$ is continuous with
respect to the topology $\tau_{M}$ on ${\mathcal{C}}G$. Our aim is to show
that these topologies are in fact the same for every locally compact Hausdorff
group $G$.

######  Lemma 3.1.

Let $G$ be a locally compact Hausdorff group. Then the family of compact
subsets

$\mathcal{F}_{{\mathcal{C}}G}:=\\{\pi(K)~{}:~{}K\text{ is a compact subset of
$[0,1]\times G$}\\}$

generates the topology on $({\mathcal{C}}G,\tau_{M})$.

######  Proof.

We need to prove that

$A\subseteq{\mathcal{C}}G\text{ is closed}\quad\iff\quad A\cap B\text{ is
closed in }B\text{ for all }B\in\mathcal{F}_{{\mathcal{C}}G}.$

($\Rightarrow$) Assume that $A$ is closed and take
$B\in\mathcal{F}_{{\mathcal{C}}G}$. The set $B$ is of the form $\pi(K)$, where
$K$ is a compact subset of $[0,1]\times G$. Since $\pi$ is continous with
respect to $\tau_{M}$, $B$ is closed as a compact subset of a Hausdorff space.
Hence, $A\cap B$ is closed.

($\Leftarrow$) It suffices to show that any limit point of $A$ in
${\mathcal{C}}G$ is a limit point of $A\cap\pi(K)$ for some compact
$K\subseteq[0,1]\times G$. First, assume that $A$ has a limit point
$[(t,g)]\in{\mathcal{C}}G\setminus\\{[(0,g)]\\}$. Since the continuous map
$\pi$ is a homeomorphism when restricted to $(0,1]\times G$, the result
follows by taking $K$ to be any compact neighbourhood of $(t,g)$ in the
locally compact product space $(0,1]\times G$. Next, assume that the tip
$[(0,g)]$ is a limit point of $A$, i.e. that there is a net
$(x_{\lambda})_{\lambda\in\Lambda}$ in $A$ that converges to $[(0,g)]$ in
$({\mathcal{C}}G,\tau_{M})$ (without loss of generality we assume that this
net is not eventually constant). By local compactness, pick any $g\in G$ and
its open neighbourhood $V$ such that its closure $\overline{V}$ is compact.
Consider the following open neighbourhood

$U:={\mathcal{C}}G\setminus{\rm g}^{-1}(G\setminus V)$

of $[(0,g)]$ in $({\mathcal{C}}G,\tau_{M})$. Then there exists
$\lambda_{0}\in\Lambda$ such that for all $\mu\geq\lambda_{0}$ we have that
$x_{\mu}\in U$. The non-empty set

$\Lambda_{U}:=\\{\mu\in\Lambda~{}:~{}\mu\geq\lambda_{0}\\}$

is cofinal, so we can consider a subnet $(x_{\mu})_{\mu\in\Lambda_{U}}$ that
has to converge to $[(0,g)]$ as well. Take $K:=[0,1]\times\overline{V}$. Since
$x_{\mu}\in\pi(K)$ for all $\mu\in\Lambda_{U}$, we obtain that $[(0,g)]$ is a
limit point of $A\cap\pi(K)$. $\blacksquare$

######  Remark 3.2.

Note that the above lemma is true for any locally compact Hausdorff space.
$\blacklozenge$

######  Proposition 3.3.

Let $G$ be a locally compact Hausdorff group. Then the topologies $\tau_{q}$
and $\tau_{M}$ on ${\mathcal{C}}G$ are equivalent.

######  Proof.

By Lemma 3.1, the topology on $({\mathcal{C}}G,\tau_{M})$ is generated by
compact subsets of the form $\pi(K)$, where $K$ is a compact subset of
$[0,1]\times G$. Let us observe that the same is true for
$({\mathcal{C}}G,\tau_{q})$. First, note that $({\mathcal{C}}G,\tau_{q})$ is
compactly generated as a Hausdorff quotient of a compactly generated space
(see e.g. [49, § 2.6]). Next, since $G$ is paracompact Hausdorff, from [26,
Theorem 9] we infer that $\pi$ is semiproper (or compact-covering), i.e. for
every compact subset $B$ of $({\mathcal{C}}G,\tau_{q})$, there exists a
compact subset $K$ of $[0,1]\times G$ such that $B\subseteq\pi(K)$. This
concludes the proof. $\blacksquare$

######  Remark 3.4.

Note that by [26, Theorems 5 and 6] the above result might not be true for
arbitrary locally compact Hausdorff spaces. However, it will be true for any
locally compact Hausdorff space that is paracompact. $\blacklozenge$

In conclusion, we have proved that for a locally compact Hausdorff group $G$,
the topology on ${\mathcal{C}}G$ is both an initial topology with respect to
the maps ${\rm t}$ and ${\rm g}$ and a final topology with respect to the map
$\pi$. This means that we are well placed for deciding continuity of functions
going both from and into the unreduced cone of $G$.

###  3.2 Numerable principal bundles

Let $X$ be a topological space and let $\\{f_{\lambda}\\}$ be a family of
continuous functions from $X$ to $[0,1]$ indexed by a set $\Lambda$. Recall
that $\\{f_{\lambda}\\}$ is a partition of unity on $X$ if

  1. 1.

it is locally finite, i.e. for every $x\in X$ there is a neighbourhood $U$ of
$x$ such that $f_{\lambda}|_{U}\neq 0$ for only finitely many
$\lambda\in\Lambda$,

  2. 2.

and the finite sum $\sum_{\lambda\in\Lambda}f_{\lambda}(x)$ equals $1$ for all
$x\in X$.

A covering $\\{U_{\lambda}\\}$ of $X$ is called numerable if there is a
partition of unity $\\{f_{\lambda}\\}$ on $X$ subordinated to it, i.e. ${\rm
supp}f_{\lambda}\subseteq U_{\lambda}$ for all $\lambda\in\Lambda$, where
${\rm supp}f_{\lambda}$ denotes the closed support of $f_{\lambda}$. By [52,
Corollary 13.1.9] we can always reduce arbitrary partitions of unity to
countable ones. Hence, in what follows we assume that $\Lambda={\mathbb{N}}$.

Following the terminology of [52, p. 332], we call a topological $G$-space $X$
trivial if there is a $G$-map $X\to G$ and locally trivial if it has an open
covering by trivial $G$-spaces (such a cover is called trivializing). If $X$
is a locally trivial $G$-space, we say that $X\to X/G$ is a principal
$G$-bundle. A principal bundle $X\to X/G$ is called numerable if there is a
numerable trivializing cover of $X$. In particular, when $X$ is paracompact
Hausdorff, then any principal $G$-bundle $X\to X/G$ is numerable.

Let us now give a new description of numerable principal bundles, which admits
a noncommutative generalization (see § 4.1).

######  Theorem 3.5.

Let $G$ be a locally compact Hausdorff group and let $X$ be a locally compact
Hausdorff $G$-space. Then the following conditions are equivalent:

  1. (1)

$X\to X/G$ is a numerable principal $G$-bundle.

  2. (2)

There exist $G$-equivariant continuous maps $\rho_{i}:X\to{\mathcal{C}}G$,
$i\in{\mathbb{N}}$, such that the sequence of continuous functions
$(\sum_{i=0}^{N}({\rm t}\circ\rho_{i}))_{N\in{\mathbb{N}}}$ converges to $1$
in the compact-open topology.

######  Proof.

First, assume that $X\to X/G$ is a numerable principal $G$-bundle. Let
$\\{U_{i}\\}_{i\in{\mathbb{N}}}$ be a numerable trivializing cover of $X$ with
trivializing maps $\varphi_{i}:U_{i}\to G$ and let $\\{f_{i}\\}$ be the
$G$-invariant partition of unity subordinated to that cover. One can always
find such a partition of unity since every trivializing cover is
$G$-invariant. Let $\widetilde{\varphi}_{i}$ be any (not necessarily
continuous) extension of $\varphi_{i}$ to whole $X$. Next, we define the maps

$\rho_{i}:X\overset{f_{i}\times\widetilde{\varphi}_{i}}{\longrightarrow}[0,1]\times
G\overset{\pi}{\longrightarrow}{\mathcal{C}}G:\quad
x\longmapsto[(f_{i}(x),\widetilde{\varphi}_{i}(x))].$

Since $({\rm t}\circ\rho_{i})(x)=f_{i}(x)$ and $({\rm
g}\circ\rho_{i})(x)=\varphi_{i}(x)$ (whenever it is well defined), from the
definition of the Milnor topology $\tau_{M}$ and Proposition 3.3, we infer
that the maps $\rho_{i}$ are continuous. They are also $G$-equivariant as each
$\varphi_{i}$ is $G$-equivariant and the tip $[(0,g)]$ is a fixed point of the
$G$-action on $\mathcal{C}G$. Observe that

$\sum_{i=0}^{N}({\rm t}\circ\rho_{i})=\sum_{i=0}^{N}f_{i}\qquad\text{for all
$N\in{\mathbb{N}}$}.$

Since $\\{f_{i}\\}$ is a partition of unity, the sequence
$(\sum_{i=0}^{N}({\rm t}\circ\rho_{i}))_{N}$ converges locally uniformly to
$1$. Hence, by local compactness of $X$, it also converges to $1$ in the
compact-open topology.

Next, assume that there are $G$-equivariant maps $\rho_{i}:X\to{\mathcal{C}}G$
such that $(\sum_{i=0}^{N}({\rm t}\circ\rho_{i}))_{N}$ converges to $1$ in the
compact-open topology. Compact convergence implies pointwise convergence, so
that the family $\\{{\rm t}\circ\rho_{i}\\}$ is a $G$-invariant generalized
partition of unity in the sense of [52, p. 320]. By [52, Lemma 13.1.7] the
covering $\\{U_{i}:=({\rm t}\circ\rho_{i})^{-1}((0,1])\\}$ is numerable.
Furthermore, the equivariance of the maps $\rho_{i}$ implies that this cover
is $G$-invariant. It is left to check whether it is trivializing. To this end,
we define the maps

$\varphi_{i}:U_{i}\longrightarrow G:\qquad x\longmapsto{\rm g}(\rho_{i}(x)).$

Here ${\rm g}:{\rm t}^{-1}((0,1])\to G$ is the map given by (16). Each
$\varphi_{i}$ is continuous and $G$-equivariant as a composition of two
continuous $G$-equivariant maps, so we are done. $\blacksquare$

######  Remark 3.6.

We formulate Theorem 3.5 using the compact-open topology, since this topology
can be generalized to the noncommutative setting. However, the statement of
the theorem would remain true if we would use locally uniform or even point-
wise convergence instead. $\blacklozenge$

##  4 Noncommutative numerable principal bundles

In this section, we present a definition which generalizes the notion of a
locally compact Hausdorff numerable principal bundle to the noncommutative
setting. At the same time, this definition will generalize the local-
triviality dimensions to the context of actions of locally compact Hausdorff
groups on (not necessarily unital) C*-algebras.

###  4.1 Locally trivial $G$-C*-algebras

We start with some basics on the local-triviality dimensions. Recall from §
3.1 that ${\rm t}$ denotes the height function in $C({\mathcal{C}}G)$. The
following definition should be compared with [14, 24].

######  Definition 4.1.

Let $G$ be a compact Hausdorff group and let $A$ be a unital $G$-C*-algebra.

  1. 1.

The weak local-triviality dimension $\dim_{\rm WLT}^{G}(A)$ is the smallest
$n$ for which there exist unital $G$-equivariant $*$-homomorphisms
$\rho_{0}\,,\ldots,\rho_{n}:C({\mathcal{C}}G)\to A$ such that
$\sum_{i=0}^{n}\rho_{i}({\rm t})$ is invertible.

  2. 2.

The (plain) local-triviality dimension $\dim_{\rm LT}^{G}(A)$ is the smallest
$n$ for which there exist unital $G$-equivariant $*$-homomorphisms
$\rho_{0}\,,\ldots,\rho_{n}:C({\mathcal{C}}G)\to A$ such that
$\sum_{i=0}^{n}\rho_{i}({\rm t})=1$.

  3. 3.

The strong local-triviality dimension $\dim_{\rm SLT}^{G}(A)$ is the smallest
$n$ for which there exist unital $G$-equivariant $*$-homomorphisms
$\rho_{0}\,,\ldots,\rho_{n}:C({\mathcal{C}}G)\to A$ with commuting images and
such that $\sum_{i=0}^{n}\rho_{i}({\rm t})$=1.

In any of the above cases, if there is no such finite $n$, then the associated
dimension equals $\infty$. $\blacklozenge$

We list some known results about the local-triviality dimensions (see [14,
24]):

  1. (i)

We always have that

$\dim_{\rm SLT}^{G}(A)\geq\dim_{\rm LT}^{G}(A)\geq\dim_{\rm WLT}^{G}(A).$

When $A$ is commutative, then the above inequalities become equalities. There
are also examples showing that these inequalities can be strict.

  2. (ii)

If a compact Hausdorff group $G$ acts on a unital C*-algebra of the form
$A=C(X)$ for some compact Hausdorff space $X$, then finiteness of any of the
local-triviality dimensions is equivalent to $X\to X/G$ being a principal
$G$-bundle.

  3. (iii)

If $\dim_{\rm WLT}^{G}(A)<\infty$, then the $G$-action on $A$ is free in the
sense of Ellwood [21]. Hence by the results of Baum, de Commer, and Hajac [8],
we have that at the algebraic level the triple $(A,A^{G},G)$ is a compact
quantum principal bundle.

  4. (iv)

There are many examples of actions with finite local-triviality dimensions
including antipodal actions on quantum spheres and gauge actions on various
graph C*-algebras.

The following definition is inspired by Theorem 3.5 and Definition 4.1.

######  Definition 4.2.

Let $G$ be a locally compact Hausdorff group acting on a C*-algebra $A$.

  1. 1.

We say that $A$ is a weakly locally trivial $G$-C*-algebra if there exist
unital $G$-equivariant $\kappa$-continuous $*$-homomorphisms
$\rho_{i}:C({\mathcal{C}}G)\to\Gamma(K_{A})$, $i\in{\mathbb{N}}$, such that
the sequence $(\sum_{i=0}^{N}\rho_{i}({\rm t}))_{N}$ converges to an
invertible element $x\in\Gamma(K_{A})$ in the $\kappa$-topology.

  2. 2.

We say that $A$ is a locally trivial $G$-C*-algebra if there exist unital
$G$-equivariant $\kappa$-continuous $*$-homomorphisms
$\rho_{i}:C({\mathcal{C}}G)\to\Gamma(K_{A})$, $i\in{\mathbb{N}}$, such that
the sequence ($\sum_{i=0}^{N}\rho_{i}({\rm t}))_{N}$ converges to $1$ in the
$\kappa$-topology.

  3. 3.

We say that $A$ is a strongly locally trivial $G$-C*-algebra if there exist
unital $G$-equivariant $\kappa$-continuous $*$-homomorphisms
$\rho_{i}:C({\mathcal{C}}G)\to\Gamma(K_{A})$, $i\in{\mathbb{N}}$, with
commuting images and such that ($\sum_{i=0}^{N}\rho_{i}({\rm t}))_{N}$
converges to $1$ in the $\kappa$-topology.

$\blacklozenge$

Here are some observations and remarks:

  1. (i)

If $G$ is compact Hausdorff, then ${\mathcal{C}}G$ is also compact Hausdorff.
This turns $C({\mathcal{C}}G)$ into a unital C*-algebra for which the
$\kappa$-topology coincides with the norm topology. Furthermore, if $A$ is
unital, then $\Gamma(K_{A})=A$ and again the $\kappa$-topology coincides with
the norm topology. Therefore, the unital $G$-equivariant $*$-homomorphisms
$\rho_{i}$ in Definition 4.2 are automatically continuous. We obtain that

$\displaystyle\dim_{\rm WLT}^{G}(A)<\infty\quad$
$\displaystyle\iff\quad\text{$A$ is a unital weakly locally trivial
$G$-C*-algebra},$ $\displaystyle\dim_{\rm LT}^{G}(A)<\infty\quad$
$\displaystyle~{}~{}~{}\Rightarrow~{}\quad\text{$A$ is a unital locally
trivial $G$-C*-algebra},$ $\displaystyle\dim_{\rm SLT}^{G}(A)<\infty\quad$
$\displaystyle\iff\quad\text{$A$ is a unital strongly locally trivial
$G$-C*-algebra}.$

The implication from left to right for $\dim^{G}_{\rm LT}$ is not clear at the
moment, but we conjecture that it holds.

  2. (ii)

It might happen even beyond the compact/unital case that the number of
$*$-homomorphisms $\rho_{i}$ from Definition 4.2 is finite and then we could
talk about the local-triviality dimension of a given action. However, this
time finiteness of this dimension in the commutative case will not recover the
notion of a principal bundle. This was possible only for unital C*-algebras
generalizing compact spaces, since then every trivilizing cover has a finite
subcover.

  3. (iii)

Note that we have the following implications for $G$-C*-algebras:

$\text{strong local triviality}\quad\Rightarrow\quad\text{local
triviality}\quad\Rightarrow\quad\text{weak local triviality}.$

  4. (iv)

Let $G$ be a locally compact Hausdorff group, let $A$ be a weakly locally
trivial $G$-C*-algebra, and let $B$ be a $G$-C*-algebra. If there is a
strictly non-degenerate $G$-morphism $A\to M(B)$, then it follows from
Proposition 2.4 that $B$ is a weakly locally trivial $G$-C*-algebra. Hence, if
$I$ is a $G$-invariant ideal of $A$, then $A/I$ is a weakly locally trivial
$G$-C*-algebra. The same results hold if everywhere above we replace weakly
locally trivial with either locally trivial or strongly locally trivial.

  5. (v)

The idea of describing the action of a group on a C*-algebra by equivariant
maps into a different object that can be obtained from that C*-algebra is not
new. The Rokhlin dimension of an action of a second-countable compact group
$G$ on a $\sigma$-unital C*-algebra $A$ is defined using $G$-equivariant
completely positive contractive order zero maps from $C(G)$ to the central
sequence algebra $A_{\infty}\cap A^{\prime}$ (see e.g. [23]).

###  4.2 Classes of examples of locally trivial $G$-C*-algebras

Definition 4.2 enables us to talk about local triviality of non-unital
$G$-C*-algebras. Recall that up to this point even actions of finite groups on
non-unital C*-algebras were beyond the scope of applicability of the local-
triviality dimensions. Therefore, we obtain a plethora of new examples.

####  4.2.1 Classical numerable principal bundles

First, observe that Definition 4.2 generalizes the notion of a locally compact
Hausdorff numerable principal $G$-bundle.

######  Proposition 4.3.

Let $G$ be a locally compact Hausdorff group acting on a locally compact
Hausdorff space $X$. Then the following are equivalent

  1. (1)

$X\to X/G$ is a numerable principal $G$-bundle,

  2. (2)

$C_{0}(X)$ is a strongly locally trivial $G$-C*-algebra,

  3. (3)

$C_{0}(X)$ is a locally trivial $G$-C*-algebra,

  4. (4)

$C_{0}(X)$ is a weakly locally trivial $G$-C*-algebra.

######  Proof.

(1) $\Rightarrow$ (2). Assume that $X\to X/G$ is a numerable principal bundle.
By Theorem 3.5, there exist $G$-equivariant continuous maps

$\rho_{i}:X\longrightarrow{\mathcal{C}}G,\qquad i\in{\mathbb{N}},$

such that $(\sum_{i=0}^{N}({\rm t}\circ\rho_{i}))_{N\in{\mathbb{N}}}\subset
C(X)$ converges to $1$ in the $\kappa$-topology. Since both $X$ and
${\mathcal{C}}G$ are completely regular Hausdorff spaces, we use (11) to
obtain unital $G$-equivariant $\kappa$-continuous $*$-homomorphisms

$\rho^{*}_{i}:C({\mathcal{C}}G)\longrightarrow
C(X)\cong\Gamma(K_{C_{0}(X)}),\qquad i\in{\mathbb{N}},$

Since $C(X)$ is commutative and $\rho^{*}_{i}({\rm t})={\rm t}\circ\rho_{i}$ ,
the result follows.

The implications (2) $\Rightarrow$ (3) and (3) $\Rightarrow$ (4) are
immediate.

(4) $\Rightarrow$ (1). Assume that $C_{0}(X)$ is a weakly locally trivial
$G$-C*-algebra. This means that there are unital $G$-equivariant
$\kappa$-continuous $*$-homomorphisms

$\rho^{*}_{i}:C({\mathcal{C}}G)\longrightarrow\Gamma(K_{C_{0}(X)})\cong
C(X),\qquad i\in{\mathbb{N}},$

such that $(\sum_{i=0}^{N}\rho^{*}_{i}({\rm t}))_{N}$ converges to an
invertible element $\sigma\in C(X)$ in the $\kappa$-topology. Since both $X$
and ${\mathcal{C}}G$ are completely regular Hausdorff spaces, we use (12) to
obtain $G$-equivariant continuous maps

$\rho_{i}:X\longrightarrow{\mathcal{C}}G,\qquad i\in{\mathbb{N}},$

such that $(\sum_{i=0}^{N}({\rm t}\circ\rho_{i}))_{N\in{\mathbb{N}}}$
converges to an invertible element $\sigma\in C(X)$ in the $\kappa$-topology.
Since $\sum_{i=0}^{N}({\rm t}\circ\rho_{i})$ is positive for every
$N\in{\mathbb{N}}$, then $\sigma$ is also positive (see e.g. [36, Proposition
8.5] for a more general statement about Pedersen multiplier algebras).
Furthermore, $\sigma(xg)=\sigma(x)$ for all $x\in X$ and $g\in G$. Without
loss of generality, we assume that $0<\sigma(x)^{-1}\leq 1$ for all $x\in X$.
Consider the following continuous $G$-equivariant maps

$l_{x}:{\mathcal{C}}G\longrightarrow{\mathcal{C}}G:\qquad[(t,g)]\mapsto[(\sigma(x)^{-1}t,g)],\qquad
x\in X,$

which can be used to define

$\widetilde{\rho}_{i}:X\longrightarrow{\mathcal{C}}G,\qquad\widetilde{\rho}_{i}(x):=l_{x}(\rho_{i}(x)),\qquad
i\in{\mathbb{N}}.$

Each $\widetilde{\rho}_{i}$ is clearly $G$-equivariant. Since $({\rm
t}\circ\widetilde{\rho}_{i})(x)=\sigma(x)^{-1}({\rm t}\circ\rho_{i})(x)$ and
${\rm g}\circ\widetilde{\rho}_{i}={\rm g}\circ\rho_{i}$ (whenever it is well-
defined), we conclude from Proposition 3.3 that each $\widetilde{\rho}_{i}$ is
continuous. Observe that $(\sum_{i=0}^{N}({\rm t}\circ\widetilde{\rho}_{i}))$
converges to $1$ in the $\kappa$-topology. From Theorem 3.5 we conclude that
$X\to X/G$ is a numerable principal $G$-bundle. $\blacksquare$

Since partitions of unity enable us to formulate the notion of a locally
trivial $G$-C*-algebra, we do not see how to get rid of the numerability
assumption. Recall, however, that only numerable principal bundles can be
classified using the Milnor universal space.

####  4.2.2 Principal $G$-$C_{0}(Y)$-algebras

In this subsection, we consider a semiclassical generalization of principal
bundles to the world of noncommutative topology. The next definition is a
natural one and it was probably considered by other authors, but we include it
for completeness.

######  Definition 4.4.

Let $G$ be a locally compact Hausdorff group acting on a C*-algebra $A$. We
say that $A$ is a principal $G$-$C_{0}(Y)$-algebra if there is a
$G$-equivariant $*$-homomorphism

$\chi:C_{0}(Y)\longrightarrow ZM(A)$

for some locally compact Hausdorff principal $G$-bundle $Y\to Y/G$, such that
$\chi(C_{0}(Y))A$ is norm-dense in $A$. If the bundle $Y\to Y/G$ is numerable,
we say that $A$ is a numerable principal $G$-$C_{0}(Y)$-algebra.
$\blacklozenge$

In the above definition, if $Y$ is second-countable, then $A$ is numerable. If
$G$ is additionally second-countable and torsion-free, then Definition 4.4 is
equivalent to the definition of a proper $G$-C*-algebra of Guentner, Higson,
and Trout [27, Definition 8.2], which generalizes the notion of a proper
action in the sense of Baum, Connes, and Higson [7, Definition 1.3].

Let $\alpha:G\to{\rm Aut}(A)$ be a continuous action of a locally compact
Hausdorff group on $A$. In what follows, we assume that $\widehat{A}$ is
Hausdorff. The continuous action of $G$ on $\widehat{A}$ given by (1) induces
(via (4)) an action $\beta:G\to{\rm Aut}(C_{0}(\widehat{A}\,))$ such that

$\alpha_{g}(f\cdot a)=\beta_{g}(f)\cdot\alpha_{g}(a),\qquad g\in G,\quad f\in
C_{0}(\widehat{A}\,),\quad a\in A,$ (17)

(see e.g. [46, Lemma 7.1]). Here $f\cdot a$ is the element given by the
Dauns–Hofmann theorem (see e.g. [20, Theorem 3]). Define the following map

$\chi:C_{0}(\widehat{A}\,)\longrightarrow ZM(A),\qquad\chi(f)a:=f\cdot a.$

It is evident that $\chi$ is a non-degenerate $*$-homomorphism. Furthermore,
the equation (17) implies that it is $G$-equivariant. Therefore, if
$\widehat{A}\to\widehat{A}/G$ is a (numerable) principal $G$-bundle, then $A$
is a (numerable) principal $G$-$C_{0}(\widehat{A}\,)$-algebra. Next, assume
that $A$ is a (numerable) principal $G$-$C_{0}(Y)$-algebra with Hausdorff
spectrum $\widehat{A}$, so there is a $G$-equivariant non-degenerate
$*$-homomorphism $\chi:C_{0}(Y)\to ZM(A)\cong C_{b}(\widehat{A}\,)$. From the
anti-equivalence (6), we obtain a continuous $G$-map $\widehat{A}\to Y$. Since
$Y$ is a (numerable) principal $G$-bundle, then so is $\widehat{A}$ (it
suffices to pull back the trivializing cover on $Y$ to the spectrum
$\widehat{A}$ ). Therefore, we arrived at the following proposition.

######  Proposition 4.5.

Let $G$ be a locally compact Hausdorff group acting on a C*-algebra $A$ with
Hausdorff spectrum $\widehat{A}$. Then $\widehat{A}\to\widehat{A}/G$ is a
(numerable) principal $G$-bundle if and only if $A$ is a (numerable) principal
$G$-$C_{0}(Y)$-algebra for some locally compact Hausdorff $Y$. $\blacksquare$

As a corollary, we obtain that Definition 4.4 indeed generalizes the notion of
a locally compact Hausdorff principal bundle.

######  Corollary 4.6.

Let $G$ be a locally compact Hausdorff group acting on a locally compact
Hausdorff space $X$. Then the following are equivalent

  1. (1)

$C_{0}(X)$ is a (numerable) principal $G$-$C_{0}(Y)$-algebra for some locally
compact Hausdorff $Y$.

  2. (2)

$X\to X/G$ is a (numerable) principal $G$-bundle. $\blacksquare$

Next, we prove that every principal numerable $G$-$C_{0}(Y)$-algebra is
locally trivial in the sense of Definition 4.2.

######  Proposition 4.7.

Let $G$ be a locally compact Hausdorff group and let $Y\to Y/G$ be a locally
compact Hausdorff numerable principal $G$-bundle. Then every principal
numerable $G$-$C_{0}(Y)$-algebra is a locally trivial $G$-C*-algebra.

######  Proof.

Let $A$ be a principal $G$-$C_{0}(Y)$-algebra. There is a $G$-equivariant
$*$-homomorphism

$\chi:C_{0}(Y)\longrightarrow ZM(A)$

such that $\chi(C_{0}(Y))A$ is norm-dense in $A$. We infer from Proposition
2.3 and Proposition 2.4 that $\chi$ extends to a unital $G$-equivariant
$\kappa$-continuous $*$-homomorphism $\widetilde{\chi}:C(Y)\to\Gamma(K_{A})$.
Furthermore, since $Y\to Y/G$ is a numerable principal $G$-bundle, we know
from Proposition 4.3 that there exist unital $G$-equivariant $*$-homomorphisms
$\rho_{i}:C({\mathcal{C}}G)\to\Gamma(K_{C_{0}(Y)})\cong C(Y)$,
$i\in{\mathbb{N}}$, such that $(\sum_{i=0}^{N}\rho_{i}({\rm t}))_{N}$
converges to $1$ in the $\kappa$-topology. Define
$\widetilde{\rho}_{i}:=\widetilde{\chi}\circ\rho_{i}$. Then

$\widetilde{\rho}_{i}:C({\mathcal{C}}G)\longrightarrow\Gamma(K_{A}),\qquad
i\in{\mathbb{N}},$

are unital $G$-equivariant $\kappa$-continous $*$-homomorphisms. Finally,
observe that

$\kappa\lim_{N\to\infty}\sum_{i=0}^{N}\widetilde{\rho}_{i}({\rm
t})=\kappa\lim_{N\to\infty}\sum_{i=0}^{N}\widetilde{\chi}(\rho_{i}({\rm
t}))=\widetilde{\chi}\left(\kappa\lim_{N\to\infty}\sum_{i=0}^{N}\rho_{i}({\rm
t})\right)=\widetilde{\chi}(1)=1.$

$\blacksquare$

Let us end this subsection by discussing a certain class of locally trivial
$G$-C*-algebras that was already present in the literature. In [44] and [35] a
class of actions (and coactions) for which the induced action on the spectrum
gives rise to a principal $G$-bundle was studied. Let $\alpha:G\to{\rm
Aut}(A)$ be a continuous action of an abelian locally compact Hausdorff group
on a C*-algebra with Hausdorff spectrum. The action $\alpha$ is called
pointwise unitary if for each $[\pi]\in\widehat{A}$ there is a (strongly
continuous) unitary representation $U$ of $G$ on the Hilbert space $H_{\pi}$
associated with the $*$-representation $\pi$ such that

$\pi(\alpha_{g}(a))=U_{g}\pi(a)U_{g}^{*}\qquad g\in G,\qquad a\in A.$

If the above holds, we say that $U$ implements $\alpha$ in the
$*$-representation $\pi$. The action $\alpha$ is locally unitary if for each
$[\pi]\in\widehat{A}$ there is a neighborhood $N$ of $[\pi]$ and a strictly
continuous map $u:G\to M(A)$ such that, for each $[\rho]\in N$,
$\overline{\rho}\circ u$ implements $\alpha$ in the representation $\rho$.
Here $\overline{\rho}$ denotes the extension of $\rho$ to $M(A)$.

Next, consider the action $\widehat{\alpha}:\widehat{G}\to{\rm Aut}(A\rtimes
G)$ of the dual group $\widehat{G}$ on the crossed product C*-algebra
$A\rtimes G$ (see e.g. [46, p. 176]). Note that since $G$ is abelian, we do
not have to distinguish between the reduced and maximal crossed product.
Recall that there is an action of $\widehat{G}$ on $(A\rtimes G)^{\wedge}$
(see e.g. (1)). From [35, Theorem 5.9] it follows that if $\alpha$ is locally
unitary then $(A\rtimes G)^{\wedge}$ is Hausdorff and the restriction map
$(A\rtimes G)^{\wedge}\to\widehat{A}$ gives rise to a principal
$\widehat{G}$-bundle. This in turn implies, by Proposition 4.5 and Proposition
4.7, that $A\rtimes G$ is a locally trivial $\widehat{G}$-C*-algebra whenever
$(A\rtimes G)^{\wedge}\to\widehat{A}$ is numerable.

######  Remark 4.8.

Although both cited theorems from [35] are phrased in the language of
coactions, our formulation is equivalent due to [35, Remark 5.7 (2)].
$\blacklozenge$

####  4.2.3 Actions of $U(1)$ and its cyclic subgroups

Let $G$ be a compact Hausdorff abelian group and let $\widehat{G}$ denote its
Pontryagin dual. Then a continuous action $\alpha$ of $G$ on a C*-algebra $A$
induces a $\widehat{G}$-grading as follows

$A_{\chi}:=\\{a\in A~{}:~{}\alpha_{g}(a)=\chi(g)a,\;g\in
G\\},\qquad\chi\in\widehat{G}.$

######  Proposition 4.9.

Let $U(1)$ act on a C*-algebra $A$ and let $\gamma$ denote the generator of
$\mathbb{Z}=\widehat{U(1)}$. If there exist normal elements $a_{i}\in
A_{\gamma}$, $i\in{\mathbb{N}}$, such that
$(\sum_{i=0}^{N}a^{*}_{i}a_{i})_{N\in{\mathbb{N}}}$ is an approximate unit,
then $A$ is a locally trivial $U(1)$-C*-algebra.

######  Proof.

Assume that there are normal elements $a_{i}\in A$, $i\in{\mathbb{N}}$,
satisfying the assumptions of the proposition. Let $u$ be the unitary
generator of the C*-algebra $C(U(1))$ and consider the function $\sqrt{\rm
t}u\in C({\mathcal{C}}U(1))$ defined by $\sqrt{\rm t}u([(t,z)]):=\sqrt{t}z$.
It is straightforward to show that the assignments

$C({\mathcal{C}}U(1))\ni\sqrt{\rm t}u\longmapsto a_{i}\in A,\qquad
i\in{\mathbb{N}},$

give rise to unital $*$-homomorphisms

$\rho_{i}:C({\mathcal{C}}U(1))\longrightarrow
M(A)\subseteq\Gamma(K_{A}),\qquad i\in{\mathbb{N}}.$

The $U(1)$-equivariance of a given $\rho_{i}$ follows from the fact that
$a_{i}\in A_{\gamma}$. Finally, observe that

$\left(\sum_{i=0}^{N}\rho_{i}({\rm
t})\right)_{N\in{\mathbb{N}}}=\left(\sum_{i=0}^{N}\rho_{i}\left(\left(\sqrt{\rm
t}u^{*}\right)\left(\sqrt{\rm
t}u\right)\right)\right)_{N\in{\mathbb{N}}}=\left(\sum_{i=0}^{N}a^{*}_{i}a_{i}\right)_{N\in{\mathbb{N}}}.$

Since $(\sum_{i=0}^{N}a^{*}_{i}a_{i})_{N\in{\mathbb{N}}}$ is an approximate
unit, it converges to 1 in the $\kappa$-topology. $\blacksquare$

Let $n\in{\mathbb{N}}$ and let $\mathbb{Z}/n\mathbb{Z}$ be a cyclic group of
order $n$. Let

${\rm Star}_{n}:=\\{z\in{\mathbb{C}}~{}:~{}z^{n}\in[0,1]\\}$

be the star-convex set with center $0$ and generated by the $n$th roots of
unity. The proof of the following proposition is analogous to the proof of
Proposition 4.9.

######  Proposition 4.10.

Let $\mathbb{Z}/n\mathbb{Z}$ act on a C*-algebra $A$ and let $\gamma$ denote
the generator of $\mathbb{Z}/n\mathbb{Z}=\widehat{\mathbb{Z}/n\mathbb{Z}}$. If
there exist normal elements $a_{i}\in A_{\gamma}$, $i\in{\mathbb{N}}$, such
that the spectrum of each $a_{i}$ is contained in ${\rm Star}_{n}$ and such
that $(\sum_{i=0}^{N}a^{*}_{i}a_{i})_{N\in{\mathbb{N}}}$ is an approximate
unit, then $A$ is a locally trivial $\mathbb{Z}/n\mathbb{Z}$-C*-algebra.

Note that for $n=2$ the above elements $a_{i}$ are odd and self-adjoint and
$(\sum_{i=0}^{N}a_{i}^{2})_{N\in{\mathbb{N}}}$ is an approximate unit.

####  4.2.4 $\mathbb{Z}/2\mathbb{Z}$-actions on non-unital graph C*-algebras

Recall that a simple C*-algebra cannot be a principal $G$-$C_{0}(Y)$-algebra.
However, we know examples of simple unital locally trivial $G$-C*-algebras,
namely one can show that the Cuntz algebra $\mathcal{O}_{n}$ is a locally
trivial ${\mathbb{Z}}/2{\mathbb{Z}}$-C*-algebra (see e.g. [14, Subsection
5.1]). As far as the author knows, there are no examples of this type for non-
unital C*-algebras in the literature. Let us use our new methods to find such
an example. To this end, we need to discuss certain actions on graph
C*-algebras. We refer the reader to [6] for a detailed treatment of the
theory.

A directed graph is a quadruple $(E^{0},E^{1},s,r)$, where $E^{0}$ is the set
of vertices, $E^{1}$ is the set of edges, and $s,r:E^{1}\to E^{0}$ are the
source and range maps respectively. We assume that both $E^{0}$ and $E^{1}$
are countable. A graph is called row-finite if for every $v\in E^{0}$ we have
that $|s^{-1}(v)|<\infty$. By a _path_ $e$ of length $|e|=k\geq 1$ we mean a
directed path, i.e. a sequence of edges $\gamma:=e_{1}\ldots e_{k}$, with
$r(e_{i})=s(e_{i+1})$ for all $i=1,\ldots,k-1$. By convention, vertices are
paths of length $0$. Let us denote the set of finite paths by $FP(E)$. One can
extend the range and source maps to $FP(E)$ by putting $r(\mu):=r(e_{n})$ and
$s(\mu):=s(e_{1})$ for $\mu=e_{1}\ldots e_{n}$ and $r(v)=s(v)=v$ for vertices.

The C*-algebra $C^{*}(E)$ of a row-finite graph $E$ is the universal
C*-algebra generated by mutually orthogonal projections
$\big{\\{}P_{v}\,|\,v\in E_{0}\big{\\}}$ and partial isometries
$\big{\\{}S_{e}\,|\,e\in E_{1}\big{\\}}$ satisfying the _Cuntz–Krieger
relations_ :

$\displaystyle S_{e}^{*}S_{e}$ $\displaystyle=P_{r(e)}$
$\displaystyle\text{for all }e\in E_{1}\text{, and}$ (CK1)
$\displaystyle\sum_{e\in s^{-1}(v)}\\!\\!S_{e}S_{e}^{*}$ $\displaystyle=P_{v}$
$\displaystyle\text{for all }v\in E_{0}\text{ with $s^{-1}(v)\neq\emptyset$.}$
(CK2)

For a path $\mu=e_{1}\ldots e_{n}$, we write $S_{\mu}:=S_{e_{1}}\ldots
S_{e_{n}}$. Let us recall some results on graph C*-algebras that will be
needed in our examples:

  1. (i)

In every graph C*-algebra of a row-finite graph we have that
$P_{s(e)}S_{e}=S_{e}=S_{e}P_{r(e)}$ and $S^{*}_{f}S_{e}=0$ if and only if
$f\neq e$.

  2. (ii)

The graph C*-algebra $C^{*}(E)$ is unital if and only if $|E^{0}|<\infty$.

  3. (iii)

We have that $C^{*}(E)\cong\overline{\rm
span}\\{S_{\mu}S_{\nu}^{*}~{}:~{}\mu,\nu\in FP(E),~{}r(\mu)=r(\nu)\\}$.

  4. (iv)

It is well known that if $X\subseteq E^{0}$, then $\sum_{v\in X}P_{v}$
converges strictly in the multiplier algebra $M(C^{*}(E))$ to a projection
$P_{X}$ [6, Lemma 1.1]. If $X=E^{0}$, then $\sum_{v\in E^{0}}P_{v}$ converges
strictly to the unit in $M(C^{*}(E))$.

  5. (v)

Every graph C*-algebra comes equipped with the gauge $U(1)$-action $\alpha$
defined on the generators as follows

$\alpha_{z}(P_{v}):=P_{v},\qquad\alpha_{z}(S_{e})=zS_{e}\,,\qquad z\in
U(1),\quad v\in E^{0},\quad e\in E^{1}.$

Here we are only interested in the restriction of the gauge action to
$\mathbb{Z}/2\mathbb{Z}$. To find locally trivial
${\mathbb{Z}}/2{\mathbb{Z}}$-C*-algebras among graph C*-algebras, one can use
Proposition 4.10 and the fact that the generalized sum of vertex projections
strictly converges to $1$. Next, we present the aforementioned example of a
simple non-unital locally trivial $\mathbb{Z}/2\mathbb{Z}$-C*-algebra.

######  Example 4.11.

The C*-algebra of compact operators ${\mathcal{K}}(H)$ on a seprarable Hilbert
space $H$ can be realized as the graph C*-algebra (see Figure 1). Recall that
${\mathcal{K}}(H)$ is a simple C*-algebra, so it cannot be a principal
$\mathbb{Z}/2{\mathbb{Z}}$-$C_{0}(Y)$-algebra. However, we will show that it
is a locally trivial ${\mathbb{Z}}/2{\mathbb{Z}}$-C*-algebra. This is a purely
noncommutative phenomenon.

$v_{0}$$v_{1}$$v_{2}$$v_{3}$. . . . . .$e_{0}$$e_{1}$$e_{2}$ Figure 1: Graph
representation of ${\mathcal{K}}$.

Consider the restriction of the gauge action on ${\mathcal{K}}(H)$ to
$\mathbb{Z}/2\mathbb{Z}$. For brevity, we use the following notation
$P_{n}:=P_{v_{n}}$ and $S_{n}:=S_{e_{n}}$ for all $n\in{\mathbb{N}}$. Consider
the following odd self-adjoint elements

$x_{n}:=S_{2n}+S_{2n}^{*}\,,\qquad n\in{\mathbb{N}}.$

Observe that

$\sum_{n=0}^{N}x_{n}^{2}=\sum_{n=0}^{N}(P_{2n}+P_{2n+1})=\sum_{n=0}^{2N+1}P_{n}\,.$

Therefore, $(\sum_{n=0}^{N}x_{n}^{2})_{N\in{\mathbb{N}}}$ is an approximate
unit in ${\mathcal{K}}$. By Proposition 4.10, we obtain that ${\mathcal{K}}$
is a locally trivial ${\mathbb{Z}}/2{\mathbb{Z}}$-C*-algebra. $\blacklozenge$

######  Example 4.12.

We define a graph $E_{\infty}$ as follows (see Figure 2):

$E^{0}_{\infty}:=\\{v_{n}~{}:~{}n\in{\mathbb{N}}\\},\qquad
E^{1}_{\infty}:=\\{e_{n}\,,e_{n,n+1}~{}:~{}n\in{\mathbb{N}}\\},$

where $s(e_{n})=r(e_{n})=v_{n}$, $s(e_{n,n+1})=v_{n}$, and
$r(e_{n,n+1})=v_{n+1}$ for all $n\in{\mathbb{N}}$.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$. .
.$e_{0}$$e_{01}$$e_{1}$$e_{12}$$e_{2}$$e_{23}$$e_{3}$ Figure 2: The graph
$E_{\infty}$.

Consider the restriction of the gauge action $\alpha$ on $C^{*}(E_{\infty})$
to the subgroup $\mathbb{Z}/2\mathbb{Z}$. The graph C*-algebra $C^{*}(E)$ is
non-unital, so we need to use our new methods to establish whether
$C^{*}(E_{\infty})$ is a locally trivial $\mathbb{Z}/2\mathbb{Z}$-C*-algebra.
By Proposition 4.10, we need to find odd self-adjoint elements $x_{i}$,
$i\in{\mathbb{N}}$, such that $(\sum_{i=0}^{N}x^{2}_{i})_{N\in{\mathbb{N}}}$
converges strictly to the unit in $M(C^{*}(E_{\infty}))$. We put

$\chi_{n}:=\frac{1}{3}\left(1+\frac{(-1)^{n}}{2^{n+1}}\right),\qquad
n\in\mathbb{N}.$

Observe that $\chi_{0}=\frac{1}{2}$, $2\chi_{n}+\chi_{n-1}=1$ for
$n\in\mathbb{N}$, and $\lim_{n\to\infty}\chi_{n}=\frac{1}{3}$. Consider the
following elements

$y_{n}:=\sqrt{2\chi_{n}}(S_{n}+S_{n,n+1}),$

where $S_{n}:=S_{e_{n}}$ and $S_{n,n+1}:=S_{e_{n,n+1}}$. We claim that the
real and imaginary parts of the elements $y_{n}$ give us the needed odd self-
adjoint elements, i.e. let

$x_{2n}:=\frac{y_{n}+y_{n}^{*}}{2},\qquad
x_{2n+1}:=\frac{y_{n}-y_{n}^{*}}{2i},\qquad n\in{\mathbb{N}}.$

Note that

$\sum_{n=0}^{2N+1}x^{2}_{n}=\sum_{n=0}^{N}\left(x_{2n}^{2}+x_{2n+1}^{2}\right)=\sum_{n=0}^{N}\chi_{n}(2P_{n}+P_{n+1})=\sum_{n=0}^{N}P_{n}+\chi_{N}P_{N+1},$

where $P_{n}:=P_{v_{n}}$. Note that for any finite path $\mu$ in $E_{\infty}$,
we can find $N\in\mathbb{N}$ such that for every $n>N$ we have that
$P_{n}S_{\mu}=0$. Since every $a\in C^{*}(E_{\infty})$ can be approximated by
a linear combination of the elements of the form $S_{\mu}S_{\nu}^{*}$, where
$\mu,\nu$ are finite paths in $E_{\infty}$ of the same range, we conclude that
$\lim_{N\to\infty}P_{N}a=0$ for all $a\in C^{*}(E_{\infty})$. Therefore, for
any $a\in C^{*}(E_{\infty})$, we have that

$\displaystyle\left\|\sum_{n=0}^{2N+1}x_{n}^{2}a-a\right\|$
$\displaystyle=\left\|\sum_{n=0}^{2N+1}x_{n}^{2}a-\sum_{n=0}^{N}P_{n}a+\sum_{n=0}^{N}P_{n}a-a\right\|$
$\displaystyle\leq\|\chi_{N}P_{N+1}a\|+\left\|\sum_{n=0}^{N}P_{n}a-a\right\|\longrightarrow
0.$

This implies that $(\sum_{n=1}^{2N}x^{2}_{n})_{N\in{\mathbb{N}}}$ converges
strictly to the unit in $M(C^{*}(E_{\infty}))$. A similar calculation shows
that $(\sum_{n=1}^{2N+1}x^{2}_{n})_{N\in{\mathbb{N}}}$ is also an approximate
unit, which implies that $C^{*}(E_{\infty})$ is a locally trivial
$\mathbb{Z}/2\mathbb{Z}$-C*-algebra. $\blacklozenge$

##  5 Relation to free actions

If $X\to X/G$ is a principal $G$-bundle, then the action of $G$ on $X$ is
automatically free. In this subsection we show that in the noncommutative case
local triviality also implies a version of freeness of the action. There are
many generalizations of freeness to the setting of actions of locally compact
Hausdorff groups on C*-algebras (see e.g. [45] for the case of finite group
actions).

###  5.1 $\kappa$-freeness and the Ellwood condition

We begin with a characterization of injective maps between locally compact
Hausdorff spaces in terms of their $*$-algebras of all continuous functions.
The proof we present here is a modification of the proof of [21, Lemma 2.7]
(see also [9]) and heavily relies on the Tietze extension theorem for locally
compact Hausdorff spaces, which states that every continuous function on a
non-empty compact subset of a locally compact Hausdorff space can be extended
uniquely to a compactly supported continuous function on the whole space with
the same sup-norm.

######  Lemma 5.1.

Let $Y$ and $Z$ be locally compact Hausdorff spaces and let $f:Y\to Z$ be a
continuous map. Then $f$ is injective if and only if the induced unital
$\kappa$-continuous $*$-homomorphism $f^{*}:C(Z)\to C(Y)$ (see (11)) has
$\kappa$-dense range.

######  Proof.

($\Rightarrow$) Assume that $f:Y\to Z$ is injective. Since $C_{b}(Y)$ is
$\kappa$-dense in $C(Y)$, it suffices to show that any element in $C_{b}(Y)$
can be approximated by the elements in the range of $f^{*}$. Let $\omega\in
C_{b}(Y)$ and let $\emptyset\neq K\subset Y$ be compact. From the Tietze
extension theorem it follows that there is $c_{K}\in C_{c}(Y)$ such that
$c_{K}(K)\subseteq\\{1\\}$ and $\|c_{K}\|=1$. Take
$\omega_{K}:=\omega\,c_{K}\in C(K)$. By the injectivity of $f$, for any
compact subset $K\subset Y$, there is a $*$-isomorphism of C*-algebras
$\Psi_{K}:C(f(K))\to C(K)$. Again using the Tietze extension theorem, the
function $\Psi_{K}^{-1}(\omega_{K})\in C(f(K))$ can be extended to the
function $\eta_{K}\in C_{b}(Z)$ such that
$\|\Psi_{K}^{-1}(\omega_{K})\|=\|\eta_{K}\|$. Observe that

$\displaystyle\|f^{*}(\eta_{K})-\omega\|$ $\displaystyle=\sup_{y\in
Y}|f^{*}(\eta_{K})(y)-\omega(y)|=\sup_{y\in Y}|\eta_{K}(f(y))-\omega(y)|$
$\displaystyle\leq\sup_{z\in Z}|\eta_{K}(z)|+\sup_{y\in
Y}|\omega(y)|\leq\|\eta_{K}\|+\|\omega\|$
$\displaystyle\leq\|\omega_{K}\|+\|\omega\|\leq\|\omega\|\|c_{K}\|+\|\omega\|=2\|\omega\|.$

Finally, for any $k\in C_{c}(Y)$ with ${\rm supp}\,k=L$, we have that

$\displaystyle p_{L}(f^{*}(\eta_{K})-\omega)$ $\displaystyle=\sup_{y\in
Y}|(\eta_{K}(f(y))-\omega(y))k(y)|=\sup_{y\notin
K}|\eta_{K}(f(y))-\omega(y)||k(y)|$
$\displaystyle\leq\|f^{*}(\eta_{K})-\omega\|\sup_{y\notin K}|k(y)|\leq
2\|\omega\|\sup_{y\notin K}|k(y)|.$

Hence for any $K$ such that $L\subseteq K$, we have that
$p_{L}(f^{*}(\eta_{K})-\omega)=0$. This implies that
$(f^{*}(\eta_{K}))_{\emptyset\neq K\subset Y}$ converges to $\omega$ in the
$\kappa$-topology.

($\Leftarrow$) Assume that $f$ is not injective. Then there are points
$y_{1}\neq y_{2}$ such that $f(y_{1})=f(y_{2})$. Since $\\{y_{1}\,,y_{2}\\}$
is a compact subspace, then by the Tietze extension theorem there is a
function $\omega\in C_{c}(Y)$ such that $\omega(y_{1})=\omega(y_{2})=1$.
Continuous functions on a locally compact Hausdorff space $Y$ separate points,
hence, there is also a function $\psi\in C(Y)$ such that
$\psi(y_{1})\neq\psi(y_{2})$. If the range of $f^{*}$ would be $\kappa$-dense
in $C(Y)$, then there would exist a net $(\eta_{\lambda})_{\lambda\in\Lambda}$
in $f^{*}(C(Z))$ such that

$\sup_{y\in K}|\eta_{\lambda}(y)-\psi(y)\omega(y)|\longrightarrow 0,$

for all compact subsets $K\subset Y$. However, for $K=\\{y_{1},y_{2}\\}$ we
would have that

$\displaystyle\max$
$\displaystyle\\{|\eta_{\lambda}(y_{1})-\psi(y_{1})\omega(y_{1})|,|\eta_{\lambda}(y_{2})-\psi(y_{2})\omega(y_{2})|\\}$
$\displaystyle\geq\frac{1}{2}\left(|\eta_{\lambda}(y_{1})-\psi(y_{1})\omega(y_{1})|+|\eta_{\lambda}(y_{2})-\psi(y_{2})\omega(y_{2})|\right)$
$\displaystyle\geq\frac{|\psi(y_{1})\omega(y_{1})-\psi(y_{2})\omega(y_{2})|}{2}>0,$

where in the last step we used the triangle inequality and the fact that
$\eta_{\lambda}(y_{1})=\eta_{\lambda}(y_{2})$ for all $\lambda\in\Lambda$.
Therefore, we arrive at a contradiction. $\blacksquare$

To proceed, we need some results on the spaces of continuous functions with
values in unital locally convex Hausdorff $*$-algebras, i.e. locally convex
Hausdorff spaces equipped with a compatible unital $*$-algebra structure. Let
$X$ be a locally compact Hausdorff space and let $V$ be a unital locally
convex Hausdorff $*$-algebra with the family of seminorms
$\\{s_{\lambda}\\}_{\lambda\in\Lambda}$ generating its topology. Let $C(X,V)$
denote the space of all continuous functions $X\to V$. The $\kappa$-topology
on $C(X,V)$ is defined using the seminorms

$u\longmapsto\sup_{x\in K}s_{\lambda}(u(x)),$ (18)

where $K\subset X$ is compact. There is also a natural unital $*$-algebra
structure on $C(X,V)$ coming from the unital $*$-algebra structure on $V$. Let
us state a generalization of the Stone–Weierstrass theorem that can be deduced
from [51, Corollary 8].

######  Theorem 5.2.

Let $X$ be a Hausdorff space, let $V$ be a locally convex Hausdorff space, and
let $W$ be a vector subspace of $C(X,V)$. If $W$ separates points, $C(X)\cdot
W\subset W$, and $W(x):=\\{w(x):w\in W\\}$ is dense in $V$ for every $x\in X$,
then $W$ is $\kappa$-dense in $C(X,V)$. $\blacksquare$

Consider the functions

$(fv)(x):=f(x)v,\qquad f\in C(X),\quad v\in V.$ (19)

Then from Theorem 5.2 it follows that

${\rm span}\\{fv~{}:~{}f\in C(X),\;v\in V\\}\quad\text{is $\kappa$-dense
in}\quad C(X,V).$ (20)

The following lemma should be well known, but we present its proof for
completeness.

######  Lemma 5.3.

Let $Y$ and $Z$ be locally compact Hausdorff spaces. There are isomorphisms of
topological $*$-algebras

$C(Y,C(Z))\cong C(Y\times Z)\cong C(Z,C(Y)).$

Here we consider the compact-open topology on $C(Y\times Z)$ and the topology
given by seminorms (18) for the other two $*$-algebras.

######  Proof.

Consider the following assignment

$\Phi:C(Y,C(Z))\longrightarrow C(Y\times Z),\qquad
u\longmapsto(\,(y,z)\longmapsto(u(y))(z)\,).$

Since $Z$ is locally compact Hausdorff, the above map is well defined (see
e.g. [40, Theorem 46.11]). It is straightforward to check that $\Phi$ is a
unital $\kappa$-continuous $*$-homomorphism. We define a map

$\Phi^{-1}:C(Y\times Z)\longrightarrow
C(Y,C(Z)),\qquad\xi\longmapsto(\,y\longmapsto\xi(y,\cdot)\,).$

It is clear that $\Phi^{-1}$ is a unital $\kappa$-continuous $*$-homomorphism
and that $\Phi$ and $\Phi^{-1}$ are mutually inverse. The proof of the other
isomorphism is analogous. $\blacksquare$

######  Theorem 5.4.

Let $G$ be a locally compact Hausdorff group acting on a locally compact
Hausdorff space $X$. The $G$-action on $X$ is free if and only if

${\rm span}\left\\{f_{1}\alpha_{(\cdot)}(f_{2})~{}:~{}f_{1},f_{2}\in
C_{c}(X)\right\\}\quad\text{is $\kappa$-dense in}\quad C(G,C(X)),$

where $\alpha$ is the induced action of $G$ on $C_{0}(X)$. Here
$(f_{1}\alpha_{(\cdot)}(f_{2}))(g):=f_{1}\alpha_{g}(f_{2})$.

######  Proof.

First, we need some preparation. The action of $G$ on $X$ is free if and only
if the following map

$F:X\times G\longrightarrow X\times X:\quad(x,g)\longmapsto(x,xg)$

is injective. By Lemma 5.1 and Lemma 5.3 it is also equivalent to the fact
that the following unital $\kappa$-continuous $*$-homomorphism

$F^{*}:C(X\times X)\longrightarrow C(G,C(X)),\qquad(F^{*}(f)(g))(x):=f(x,xg),$

has $\kappa$-dense range. Observe that

$W:={\rm span}\\{(f_{1}\times f_{2})~{}:~{}f_{1}\,,f_{2}\in C_{c}(X)\\},$

where $(f_{1}\times f_{2})(x_{1}\,,x_{2}):=f_{1}(x_{1})f_{2}(x_{2})$, is
$\kappa$-dense in $C(X\times X)$. We use the notation

$\mathcal{E}:={\rm
span}\left\\{f_{1}\alpha_{(\cdot)}(f_{2})~{}:~{}f_{1},f_{2}\in
C_{c}(X)\right\\}.$

We infer that $\mathcal{E}=F^{*}(W)$.

($\Rightarrow$) Assume that the $G$-action on $X$ is free, which is tantamount
to the range of $F^{*}$ being $\kappa$-dense. Since $\mathcal{E}=F^{*}(W)$, it
is $\kappa$-dense in $F^{*}(C(X\times X))$. In turn, this implies that
$\mathcal{E}$ is $\kappa$-dense in $C(G,C(X))$.

($\Leftarrow$) Since $\mathcal{E}=F^{*}(W)\subset F^{*}(C(X\times X))$, the
$\kappa$-density of $\mathcal{E}$ implies the $\kappa$-density of the range of
$F^{*}$, which is equivalent to the freeness of the action. $\blacksquare$

In [21], Ellwood introduced a notion of a free action of a locally compact
quantum group on a C*-algebra, which we now specify to actions of classical
groups. Let $\alpha:G\to{\rm Aut}(A)$ be a continuous action of a locally
compact Hausdorff group on a C*-algebra. We say that $\alpha$ satisfies the
Ellwood condition if

${\rm span}\\{a\alpha_{(\cdot)}(b)~{}:~{}a,b\in A\\}\quad\text{is strictly
dense in}\quad M(C_{0}(G,A)).$

Here $(a\alpha_{(\cdot)}(b))(g):=a\alpha_{g}(b)$. Under additional assumptions
the Ellwood condition is equivalent to other versions of freeness known from
the literature. If $G$ is compact Hausdorff and $A$ is unital, then the
Ellwood condition is equivalent to the saturation property introduced by
Rieffel [47, Definition 1.6]. The proof of the equivalence of the two
conditions in a more general case of compact quantum group actions on unital
C*-algebras can be found in [17, Theorem 1.1]. The Ellwood condition is also
equivalent to the algebraic Peter–Weyl–Galois condition known from the
Hopf–Galois theory (note that the Hopf $*$-algebra in question have to be
cosemisimple to correspond to a dense Hopf $*$-subalgebra of the C*-algebra of
a compact quantum group). We refer the reader to [8] for a proof of the latter
equivalence.

A canonical example of an action satisfying the Ellwood condition is the
$G$-action $\Delta$ on $C_{0}(G)$ induced by the (right) multiplication (see
[21, Proposition 2.1] for a more general result). It is known that when
$A=C_{0}(X)$, then $\alpha$ satisfies the Ellwood condition if and only if the
induced action of $G$ on $X$ is free [21, Theorem 2.9(a)].

The above discussion and Theorem 5.4 prompt the following definition.

######  Definition 5.5.

Let $\alpha:G\to{\rm Aut}(A)$ be a continuous action of a locally compact
Hausdorff group on a C*-algebra. We say that $\alpha$ is $\kappa$-free if

${\rm span}\\{k_{1}\alpha_{(\cdot)}(k_{2})~{}:~{}k_{1},k_{2}\in
K_{A}\\}\qquad\text{is $\kappa$-dense in}\qquad C(G,\Gamma(K_{A})).$

Here $(k_{1}\alpha_{(\cdot)}(k_{2}))(g):=k_{1}\alpha_{g}(k_{2})$.
$\blacklozenge$

The next result shows that the Ellwood condition always implies
$\kappa$-freeness. In general, it is not clear whether $\kappa$-freeness
implies the Ellwood condition, however, it is true for a big class of
examples.

######  Proposition 5.6.

Let $\alpha:G\to{\rm Aut}(A)$ be a continuous action of a locally compact
Hausdorff group on a C*-algebra. If $\alpha$ satisfies the Ellwood condition,
then it is $\kappa$-free. The converse holds whenever $A$ is commutative or
when $G$ is compact and $A$ is unital.

######  Proof.

Since $\alpha$ satisfies the Ellwood condition and strict density implies
$\kappa$-density, we have that

${\rm span}\\{a\alpha_{(\cdot)}(b)~{}:~{}a,b\in A\\}\quad\text{is
$\kappa$-dense in}\quad M(C_{0}(G,A)).$

Furthermore, from [1, Corollary 3.4], we know that
$M(C_{0}(G,A))=C_{b}(G,M(A))$ and the $\kappa$-topology on $C_{b}(G,M(A))$ is
given by the restriction of seminorms of the form (18). The result follows by
noticing that

${\rm span}\\{k_{1}\alpha_{(\cdot)}(k_{2})~{}:~{}k_{1},k_{2}\in K_{A}\\}$

is $\kappa$-dense in

${\rm span}\\{a\alpha_{(\cdot)}(b)~{}:~{}a,b\in A\\}$

and that $C_{b}(G,M(A))$ is $\kappa$-dense in $C(G,\Gamma(K_{A}))$.

If $A=C_{0}(X)$ is commutative, Theorem 5.4 ensures that $\kappa$-freeness of
$\alpha$ is equivalent to freeness of the corresponding $G$-action on $X$,
which is equivalent to the Ellwood condition by [21, Theorem 2.9(a)]. Next, if
$G$ is compact and $A$ is unital, then $C(G,\Gamma(K_{A}))=C(G,A)$. The result
follows from the fact that on $C(G,A)$ the strict topology and the
$\kappa$-topology coincide. $\blacksquare$

###  5.2 Local triviality implies $\kappa$-freeness

In this subsection, we prove that if $A$ is a weakly locally trivial
$G$-C*-algebra then the considered $G$-action on $A$ is $\kappa$-free. First,
we need the following notation. Assume that there is a unital $G$-equivariant
$\kappa$-continuous $*$-homomorphism

$\rho:C({\mathcal{C}}G)\longrightarrow\Gamma(K_{A}).$

We define the following unital $G$-equivariant $\kappa$-continuous
$*$-homomorphism

$\widetilde{\rho}:C(G,C({\mathcal{C}}G))\longrightarrow
C(G,\Gamma(K_{A})),\qquad\widetilde{\rho}(u)(g):=\rho(u(g)).$ (21)

######  Theorem 5.7.

Let $\alpha:G\to{\rm Aut}(A)$ be a continuous action of a locally compact
Hausdorff group on a C*-algebra $A$. If $A$ is a weakly locally trivial
$G$-C*-algebra, then $\alpha$ is $\kappa$-free.

######  Proof.

Throughout the proof we will use the notation

$\mathcal{E}:={\rm span}\\{k_{1}\alpha_{(\cdot)}(k_{2})~{}:~{}k_{1},k_{2}\in
K_{A}\\}.$

We have to prove that $\mathcal{E}$ is $\kappa$-dense in $C(G,\Gamma(K_{A}))$.
Note that by (20), we only have to show that the functions $fv$ (see (19)),
where $f\in C(G)$ and $v\in\Gamma(K_{A})$, can be approximated in the
$\kappa$-topology by the elements of $\mathcal{E}$. Furthermore, since

$fv=(1_{C(G)}v)(f1_{\Gamma(K_{A})})\qquad\text{and}\qquad(1_{C(G)}v)\mathcal{E}\subset\mathcal{E},$

it suffices to approximate functions of the form $f1_{\Gamma(K_{A})}$.

Since $A$ is a weakly locally trivial $G$-C*-algebra, there exist unital
$G$-equivariant $\kappa$-continuous $*$-homomorphisms
$\rho_{i}:C({\mathcal{C}}G)\to\Gamma(K_{A})$, $i\in{\mathbb{N}}$, such that

$\sum_{i=0}^{N}\rho_{i}({\rm
t})\quad\underset{N\to\infty}{\longrightarrow}\quad x\qquad\text{in the
$\kappa$-topology},$

for some invertible $x\in\Gamma(K_{A})$. The above implies that

$fx=\kappa\lim_{N\to\infty}\sum_{i=0}^{N}f\rho_{i}({\rm
t})=\kappa\lim_{N\to\infty}\sum_{i=0}^{N}\widetilde{\rho}_{i}(f{\rm t}),$

where each $\widetilde{\rho}_{i}$ is given by the formula (21) and $f{\rm
t}\in C(G,C({\mathcal{C}}G))$ is defined by (19). Next, let

$\mathcal{G}:={\rm span}\\{f_{1}\Delta_{(\cdot)}(f_{2})~{}:~{}f_{1},f_{2}\in
C_{c}(G)\\}.$

Since the $G$-action on $C_{0}(G)$ satisfies the Ellwood condition, it is also
$\kappa$-free. Hence, there is a net $(\gamma_{\lambda})\subset\mathcal{G}$
such that

$f1_{C(G)}=\kappa\lim_{\lambda}\gamma_{\lambda}$

in $C(G,C(G))$. This in turn implies that

$f{\rm t}=\kappa\lim_{\lambda}{\rm t}\gamma_{\lambda}$

in $C(G,C({\mathcal{C}}G))$, where $({\rm t}\gamma_{\lambda})(g):={\rm
t}\gamma_{\lambda}(g)$, $g\in G$, and

${\rm
t}\gamma_{\lambda}(g):{\mathcal{C}}G\longrightarrow{\mathbb{C}}:\qquad{\rm
t}\gamma_{\lambda}(g)([(t,g^{\prime})]):=\begin{cases}0~{}~{}&t=0,\\\
t(\gamma_{\lambda}(g))(g^{\prime})~{}~{}&t\neq 0.\end{cases}$

Since $\gamma_{\lambda}\in\mathcal{G}$ for every $\lambda$, there exist
$m_{\lambda}\in{\mathbb{N}}$ and $a^{\lambda}_{j},b^{\lambda}_{j}\in
C_{c}(G)$, with $j=0,1,\ldots,m_{\lambda}$, such that

$\gamma_{\lambda}=\sum_{j=0}^{m_{\lambda}}a^{\lambda}_{j}\Delta_{(\cdot)}(b^{\lambda}_{j}).$

Subsequently, since each $\widetilde{\rho}_{i}$ is a $\kappa$-continuous
$*$-homomorphism and each $\rho_{i}$ is $G$-equivariant, we obtain that

$\sum_{i=0}^{N}\widetilde{\rho}_{i}(f{\rm
t})=\kappa\lim_{\lambda}\sum_{i=0}^{N}\sum_{j=0}^{m_{\lambda}}\rho_{i}\left(\sqrt{\rm
t}a^{\lambda}_{j}\right)\Gamma(\alpha)_{(\cdot)}\left(\rho_{i}\left(\sqrt{\rm
t}b^{\lambda}_{j}\right)\right).$

Since $\rho_{i}\left(\sqrt{\rm t}a^{\lambda}_{j}\right)$ and
$\rho_{i}\left(\sqrt{\rm t}b^{\lambda}_{j}\right)$ belong to $K_{A}$ for all
$\lambda$, this implies that the functions of the form $fx$ can be
approximated in the $\kappa$-topology by the elements of $\mathcal{E}$. By
invertibility of $x\in\Gamma(K_{A})$, the same conclusion follows for the
functions of the form $f1_{\Gamma(K_{A})}$. $\blacksquare$

## Acknowledgments

It is a pleasure to thank Piotr M. Hajac for his continuous support and for
many conversations about (quantum) principal bundles in the non-compact
setting. Discussions with Alexandru Chirvasitu on local triviality,
pro-C*-algebras, and classifying spaces and with Tomasz Maszczyk on unreduced
cones of C*-algebras and the multiplier algebra of the Pedersen ideal, have
been a great help at the beginning of this project. I would also like to thank
Tyrone Cutler for an exhaustive answer to the question about the topology of
the unreduced cones and joins that I posted on Mathematics Stack Exchange.
This work was financed by NCN grant 2020/36/C/ST1/00082 entitled
Noncommutative universal spaces for groups and quantum groups.

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