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\begin{document}

\title{Noncommutative numerable principal bundles\\ from group actions on C*-algebras}
\author{
  Mariusz Tobolski
}

\date{}

\newcommand{\Addresses}{
  \bigskip
  \footnotesize

  \textsc{Instytut Matematyczny, Uniwersytet Wroc\l{}awski, pl. Grunwaldzki 2/4, 50-384 Poland}\par\nopagebreak \textit{E-mail address}:
  \texttt{mariusz.tobolski@math.uni.wroc.pl}
}}

\maketitle

\begin{abstract}
We introduce the notion of a noncommutative (or quantum) numerable principal bundle in the setting of actions of locally compact Hausdorff groups on C*-algebras. More precisely, we give a definition of the locally trivial \mbox{$G$-C*-algebra}, which is a noncommutative counterpart of the total space of a~locally compact Hausdorff numerable principal \mbox{$G$-bundle}. To obtain this generalization, we have to go beyond the Gelfand--Naimark duality and use the multipliers of the Pedersen ideal. In the case of an action of a compact Hausdorff group on a unital \mbox{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.
\end{abstract}

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

\vspace{.5cm}

\noindent{MSC: 37A55, 46L09, 46L55, 46L85}

\section{Introduction}

Noncommutative mathematics (see e.g.~\cite{a-c94,fs-16}), 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~\Cref{nctop}). 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 above 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~\cite{wz-10}, 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.~\cite{gl-n11}). 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:

\begin{enumerate}
\item[(I)]{\bf 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~\cite{hj-s90} in terms of Hopf--Galois extensions~\cite{kt-81}. Brzezi\'nski and Majid~\cite{bm-93} 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.~\cite{m-d96,pm-h96,c-k04,lvs-05,alp-20,hm-21}). In fact, the notion that Schneider generalized was the one of a~principal $G$-bundle due to H.~Cartan~\cite{h-c49}, where the group action is free and proper but the local triviality (see e.g.~\cite{n-s51}) of the bundle is not assumed. The work of Aschieri, Fioresi, and Latini~\cite{afl-21} 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.

\item[(II)]{\bf 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~\cite{pr-84} (see also~\Cref{classlt}). The work of Connes and Rieffel~\cite{cr-87} was the first attempt to consider gauge theory, specifically the Yang--Mills theory, in the C*-algebraic framework~\cite{a-c94}.  Furthermore, in~\cite{ma-r90}, 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~\cite{da-e00} proposed a different generalization of a Cartan principal $G$-space in the context of actions of locally compact quantum groups on C*-algebras.
In~\cite{dcy-13}, 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~\cite{enoo-09}. 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~\cite{ghtw-18}. Let us mention that the previous work of Budzy\'nski and Kondracki~\cite{bk-96}, while successful in many aspects, gave a generalization of a different concept, i.e. the piecewise triviality of the bundle (see e.g.~\cite{bhms-07}). This was observed by Hajac, Kr\"{a}hmer, Matthes, and Zieli\'nski in~\cite{hkmz-11} at the algebraic level.
\end{enumerate}

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~\cite{bdch-17} that under additional assumptions the Hopf--Galois condition of Schneider is equivalent to the Ellwood's condition. This equivalence is possible due to the existence of the Peter--Weyl functor.

In the paper, we introduce a definition of the {\em noncommutative numerable principal bundle} in the setting of actions of locally compact Hausdorff groups on C*-algebras.   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 there is no satisfactory analog of an open subset of a noncommutative space. Recall, however, that only numerable principal bundles can be classified using the Milnor universal space~\cite{j-m56}. Unless otherwise stated, $G$ will denote a~locally compact Hausdorff group and $A$ will denote a C*-algebra. 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~\cite[p.~332]{t-td08}, 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~\cite{gg-k88}. We say that $A$ is a~{\em (numerable) classically locally trivial $G$-C*-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~\Cref{classlt}). This definition covers all classical principal $G$-bundles with a locally compact Hausdorff total space and actions of dual groups on \mbox{C*-crossed} products coming from locally unitary actions considered in~\cite{pr-84}. However, classical local triviality does not apply to simple \mbox{C*-al}\-ge\-bras. Furthermore, as discussed in~\Cref{classlt}, classically locally trivial $G$-C*-algebras are only partially related to the well-established notion of a trivial noncommutative principal bundle.
It seems that the notion of a~classically locally trivial $G$-C*-algebra is more of a~classical flavour and a truly noncommutative generalization is needed. The aim of the paper is to find a such a generalization.

Let $\cC G$ denote the unreduced cone over $G$ (see~\Cref{secone}), let $\Gamma(K_A)$ be the Pedersen multiplier algebra of $A$ (see~\Cref{sec2.2}), and let ${\rm t}$ denote the height function in $C(\cC G)$ (see~\Cref{height}). Equip $C(\cC G)$ with the compact-open topology and $\Gamma(K_A)$ with the $\kappa$-topology (see~\Cref{kappatop}).

\vspace*{4mm}
{\bf The total space of a noncommutative numerable principal bundle}
\vspace*{2mm}

{\it We say that $A$ is a {\em locally trivial $G$-C*-algebra} if there exist unital $G$-equivariant continuous \mbox{$*$-ho}\-momorphisms $\rho_i:C(\cC G)\to\Gamma(K_A)$, $i\in\bN$ such that $(\sum_{i=0}^N\rho_i({\rm t}))_{N\in\bN}$ converges to $1$ in the \mbox{$\kappa$-topol}\-ogy.}
\vspace*{4mm}

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~\cite{ghtw-18}, which relies on the notion of a partition of unity. Therefore, we have to use unreduced cones or topological joins (see e.g.~\cite{j-m56}) one way or another. Observe that the unreduced cone $\cC G$ of a locally compact Hausdorff group $G$ is no longer locally compact. This means that $\cC G$ cannot be described using the Gelfand--Naimark duality. However, $\cC G$ is still a compactly generated completely regular Hausdorff space. Hence, we can exploit the compact-open topology defined on the $*$-algebra $C(\cC G)$ of all continuous complex-valued functions on $\cC G$. In turn, this requires us to use the Pedersen multiplier algebra $\Gamma(K_A)$, which for $A=C_0(X)$, where $X$ is a locally compact Hausdorff space, is $*$-isomorphic to $C(X)$.

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, all numerable classically locally trivial $G$-C*-algebras are locally trivial. Finally, we prove a generalization of a result known from topology, namely if $A$ is a locally trivial $G$-C*-algebra then the $G$-action on $A$ is {\em $\kappa$-free}. The $\kappa$-freeness is a generalization of freeness of the action of a group on a space (see~\Cref{setthemfree}). If either $A$ is commutative or $G$ is compact and $A$ a PCS-algebra, 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 between completely regular Hausdorff spaces and continuous $*$-homomorphisms between their unital $*$-algebras of all continuous complex-valued functions equipped with the compact-open topology. Results of 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. First, we discuss noncommutative trivial principal bundles and provide an example of a trivial $G$-C*-algebra. Next, we consider classically locally trivial $G$-C*-algebras and show that for every numerable principal bundle $Y\to Y/G$ there is a classically locally trivial $G$-C*-algebra which is in fact trivial. Finally, we introduce three incarnations of local triviality of $G$-C*-algebras and prove some basic facts.
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.

\section{Preliminaries}\label{prelim}

We refer the reader to~\cite{gj-m90} for basic material on C*-algebras.

\subsection{The equivariant Gelfand--Naimark duality}\label{nctop}

Let $A$ be a C*-algebra and let $\widehat{A}$ be its {\em 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~{\em 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 {\em (jointly) continuous} if the assignment
$(a,g)\mapsto \alpha_g(a):=\alpha(g)(a)$ 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 {\em $G$-C*-algebra}. Let $A$ and $B$ be $G$-C*-algebras with actions $\alpha$ and $\beta$ respectively. \mbox{A~$*$-homo}\-morphism $\Phi:A\to B$ is called {\em $G$-equivariant} or \mbox{a~{\em $G$-$*$-ho}\-{\em mo}}\-{\em mor}\-{\em phism} if $\Phi\circ\alpha_g=\beta_g\circ\Phi$ for all $g\in G$. A {\em 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
\begin{equation}\label{specact}
\widehat{A}\times G\longrightarrow\widehat{A}:\qquad ([\pi],g)\longmapsto [\pi\circ\alpha_g].
\end{equation}
The action $\alpha$ can also be extended to an action $M(\alpha)$ on $M(A)$ in the following way
\begin{equation}\label{multiaction}
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.
\end{equation}
The action $M(\alpha)$ is continuous with respect to the {\em strict topology} $\beta$, i.e. the topology generated by the seminorms
\begin{equation}\label{semistrict}
\phi_a(x):=\|xa\|,\quad \psi_a(x):=\|ax\|,\qquad x\in M(A),\quad a\in A.
\end{equation}
However, $M(\alpha)$ may not be continuous with respect to the norm topology.
A {\em $G$-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
\begin{equation}\label{calgact}
(\alpha_g(f))(x):=f(xg),\qquad g\in G,\quad f\in C_0(X),\quad x\in X.
\end{equation}

C*-algebras are central objects studied in noncommutative topology which arose from the Gelfand--Naimark theorem~\cite{gn-43}. The equivariant version of the theorem can be stated as the following anti-equivalence of categories:
\begin{equation}\label{Ggelfnaim}
\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\}.
\end{equation}
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~\cite{sl-w80}:
\begin{equation}\label{GWor}
\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\}.
\end{equation}
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~\Cref{funind} and~\Cref{spind} below.

\subsection{Beyond local compactness}\label{beyondGN}
In this paper, we have to go beyond the anti-equivalences~\Cref{Ggelfnaim} and~\Cref{GWor}. 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
\begin{equation}\label{cosemi}
p_K(f):=\sup_{y\in K}|f(y)|,
\end{equation}
where $K\subset Y$ is compact and $f\in C(Y)$. Let ${\rm Aut}_\kappa(C(Y))$ denote the group of $\kappa$-continuous  \mbox{$*$-auto}\-morphisms of $C(Y)$. Much in the same way as in~\Cref{calgact}, 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
\begin{equation}\label{kappact}
(\alpha_g(f))(y):=f(yg),\qquad g\in G,\quad f\in C(Y), \quad y\in Y.
\end{equation}
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 {\em 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 \mbox{$G$-action} $\alpha:G\to{\rm Aut}_\kappa(C(Y))$ given by \Cref{kappact} induces a $G$-action on $\Omega(C(Y))$ as follows
\begin{equation}\label{charact}
G\longrightarrow {\rm Homeo}(\Omega(C(Y))):\quad \chi\longmapsto \chi\circ\alpha_g\,.
\end{equation}
One can prove that the map
\begin{equation}\label{comregchar}
Y\longrightarrow\Omega(C(Y)),\qquad y\longmapsto m_y\,,\quad m_y(f):=f(y),
\end{equation}
is a continuous bijection, where complete regularity of $Y$ is only needed to prove injectivity~(see e.g.~\cite[Corollary~2.3]{mw-67}). Since the topology on a completely regular Hausdorff space $Y$ is the initial topology induced by $C(Y)$, we conclude that~\Cref{comregchar} 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~\Cref{comregchar} is $G$-equivariant (this also proves that~\Cref{charact} 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
\begin{equation}\label{funind}
f^*:C(Z)\longrightarrow C(Y):\qquad \omega\longmapsto \omega\circ f.
\end{equation}
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
\begin{equation}\label{spind}
f:\Omega(C(Y))\longrightarrow \Omega(C(Z)):\qquad \chi\longmapsto \chi\circ\Phi.
\end{equation}
Hence, using the $G$-homeomorphism~\Cref{comregchar}, we obtain a one-to-one correspondence between
\begin{itemize}
\item continuous $G$-maps $Y\to Z$ between completely regular Hausdorff $G$-spaces
\item and unital $G$-equivariant $\kappa$-continuous $*$-homomorphisms $C(Z)\to C(Y)$ between their $*$-algebras of continuous functions.
\end{itemize}
Note that here we did not distinguished any particular category of commutative topological algebras that is anti-equivalent to the category of completely regular Hausdorff spaces, we only obtained a~certain one-to-one correspondence. However, this is enough for our purposes.

\subsection{Multipliers of the Pedersen ideal}\label{sec2.2}

Let us recall the definition of the Pedersen ideal of a C*-algebra $A$~\cite{gk-p69}. We first introduce the subsets
\[
K_0^+:=\{a\in A_+~:~ab=a\text{ for some $b\in A_+$}\},
\]
\[
K^+:=\left\{a\in A_+~:~\text{there exists $a_0\,,\ldots, a_n\in K_0^+$ such that }a\leq \sum_{i=0}^na_n\right\}.
\]
The {\em Pedersen ideal} is defined to be $K_A:={\rm span}\,K^+$. It is the minimal dense ideal of $A$.
A~{\em 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 {\em 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 $\Gamma(K_A)$~\cite[\S~3.6]{lt-76},
i.e. the locally convex topology given by the seminorms
\begin{equation}\label{kappatop}
\widetilde{\phi}_k((S,T)):=\|S(k)\|,\qquad \widetilde{\psi}_k((S,T)):=\|T(k)\|,\qquad k\in K_A\,.
\end{equation}
In this topology, $\Gamma(K_A)$ is a locally convex complete topological $*$-algebra~\cite[Theorem~3.8]{lt-76}. If $A=C_0(X)$ then the $\kappa$-topology on $\Gamma(K_A)\cong C(X)$ agrees with the compact-open topology~\cite[\S~4.1]{lt-76}.
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, we have the following result:

\begin{lemma}\label{diftop}
Let $A$ be a C*-algebra. Then the strict topology $\beta$ coincides with the $\kappa$-topology on norm-bounded subsets of $M(A)$.
\end{lemma}
\begin{proof}
Recall that the seminorms defining the strict topology are defined as follows
\[
\phi_a((S,T)):=\|S(a)\|,\qquad \psi_a((S,T)):=\|T(a)\|,
\]
where $a\in A$ and $(S,T)\in M(A)$. Since $K_A\subset A$, it is evident that on $M(A)$ the strict topology is always stronger then the restriction of the $\kappa$-topology. For the other direction, observe that every norm-bounded subset of $M(A)$ is of the form
\[
B=\{(S,T)\in M(A)~:~\|S\|=\|T\|\leq m\}
\]
for some $m> 0$. Let $(S_0,T_0)$ be in the $\kappa$-closure of $B$. Pick $\varepsilon>0$. Then there exists $(S,T)\in B$ such that $\widetilde{\phi}_k((S,T)-(S_0,T_0))<\varepsilon$
for all $k\in K_A$.
Next, take $a\in A$. Since $K_A$ is norm-dense in $A$, there is $k_{a,\varepsilon}\in K_A$ such that $\|a-k_{a,\varepsilon}\|<\varepsilon$. Observe that
\begin{align*}
\phi_a((S,T)-(S_0,T_0))&=\|S(a)-S_0(a)+(S-S_0)(k_{a,\varepsilon})-(S-S_0)(k_{a,\varepsilon})\|\\
&\leq \|S(k_{a,\varepsilon})-S_0(k_{a,\varepsilon})\|+(\|S\|+\|S_0\|)\|a-k_{a,\varepsilon}\|\\
&\leq (1+m+\|S_0\|)\varepsilon.
\end{align*}
Since analogous inequalities hold for the seminorms $\psi_a$ for all $a\in A$, we obtain that $(S_0,T_0)$ lies in the closure of $B$ in the strict topology and we are done.
\end{proof}
\begin{remark}
The above proposition generalizes~\cite[Theorem~1(iii)]{rc-b58}.
\end{remark}

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.~\cite[Theorem~1.4]{gk-p69}), we can define an action $\Gamma(\alpha)$ of $G$ on $\Gamma(K_A)$ by $\kappa$-continuous $*$-automorphisms as follows (see e.g.~\cite[\S~2.13]{lt-76})
\begin{equation}\label{gammact}
 \Gamma(\alpha)_g(S)(\alpha_g(k)):=\alpha_g(S(k)),\qquad\Gamma(\alpha)_g(T)(\alpha_g(k)):=\alpha_g(T(k)),
\end{equation}
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 \mbox{$\kappa$-con}\-tin\-u\-ous $*$-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 \Cref{multiaction} and~\Cref{gammact} 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~\Cref{kappact} and~\Cref{gammact} respectively.

Let us now turn to the relation between $G$-morphisms of C*-algebras and unital $\kappa$-continuous $G$-$*$-homomorphisms between their Pedersen multiplier algebras. This discussion is important for~\Cref{trivialpb}, where we consider triviality of noncommutative principal bundles.
\begin{proposition}\label{onewayfunct}
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 \mbox{$G$-equi}\-variant $\kappa$-continuous $*$-homo\-morphism. Then the restriction of $\Phi$ to $A$ is a \mbox{$G$-mor}\-phism.
\end{proposition}
\begin{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~\Cref{diftop} (note that $\phi$ is bounded as a $*$-homomorphism between C*-algebras).
\end{proof}

It is not clear whether all $G$-morphisms can be extended to unital $G$-equivariant \mbox{$\kappa$-con}\-tin\-u\-ous \mbox{$*$-homo}\-morphisms. However, there is a class of morphisms for which this extension exist. A~morphism between C*-algebras $\phi:A\to M(B)$ is {\em strictly non-degenerate} if $K_B\subseteq \phi(K_A)B$~(see \cite[Definition~3.1]{m-a04}). Since any morphism $\phi$ preserves the spectral theory, we have that $\phi(K_A)\subseteq K_B$. Hence, for any strictly non-degenerate $*$-homomorphism $\phi:A\to M(B)$, we obtain that $K_B=\phi(K_A)B$. Furthermore, since $K_B$ is self adjoint, we also have that $K_B=B\phi(K_A)$.

First, let us identify a class of morphisms which are automatically strictly non-degenerate.
\begin{proposition}\label{pedcom}
Let $\phi:A\to M(B)$ be a morphism. If $\phi(A)\subseteq Z(M(B))$, then $\phi$ is strictly non-degenerate.
\end{proposition}
\begin{proof}
The result follows a part of~\cite[Lemma~3.6]{m-a04}, which states that for every morphism $\phi$ we have that $B\phi(K_A)B=K_B$.
\end{proof}
Next, we show that strictly non-degenerate $G$-morphisms extend to unital $G$-equivariant $\kappa$-continuous $*$-homo\-morphisms.
\begin{proposition}\label{stricnondeg}
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 $\Phi:\Gamma(K_A)\to\Gamma(K_B)$.
\end{proposition}
\begin{proof}
We follow~\cite[p.~21]{m-a04} 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 $K_B=\phi(K_A)B=B\phi(K_A)$. 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 $k_A, k'_A\in K_A$ and $b,b'\in B$ such that $k_B=\phi(k_A)b=b'\phi(k'_A)$, we obtain that
\[
\lim_\lambda U_\lambda(k_B)=\lim_\lambda U_\lambda(\phi(k_A)b)=\lim_\lambda \phi(S_\lambda(k_A))b= \phi(S(k_A))b=U(\phi(k_A)b)=U(k_B),
\]
\[
\lim_\lambda V_\lambda(k_B)=\lim_\lambda V_\lambda(b'\phi(k'_A))=\lim_\lambda b'\phi(T_\lambda(k'_A))= b'\phi(T(k'_A))=V(b'\phi(k'_A))=V(k_B).
\]
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
\begin{align*}
\Gamma(\beta)_g(\Phi(S))(\phi(\alpha_g(k))\beta_g(b))&=\Gamma(\beta)_g(\Phi(S))(\beta_g(\phi(k)b))=\beta_g(\Phi(S)(\phi(k)b))
\\&=\beta_g(\phi(S(k))b)=\phi(\alpha_g(S(k)))\beta_g(b)
\\&=\phi(\Gamma(\alpha)_g(S)(\alpha_g(k)))\beta_g(b)=\Phi(\Gamma(\alpha)_g(S))(\phi(\alpha_g(k))\beta_g(b))
\end{align*}
and the computation is the same for $\Gamma(\beta)_g(\Phi(T))$. Therefore, the $*$-homomorphism $\Phi$ is $G$-equi\-variant.
\end{proof}

We conclude the subsection by considering a class of C*-algebras for which $G$-mor\-phisms extend to  unital $G$-equivariant $\kappa$-continuous $*$-homomorphisms despite the fact that they might not be strictly non-degenerate. A C*-algebra $A$ for which $\Gamma(K_A)=M(A)$ is called {\em a PCS-algebra}~\cite[Definition~10.2]{lt-76}. If $A$ is a PCS-algebra, then the $\kappa$-topology and the strict topology $\beta$ coincide on $\Gamma(K_A)=M(A)$. By the result of Mack~\cite[Theorem~3.5]{mack} (see also~\cite[Theorem~2.3]{kw-03}), we know that $A$ is a PCS-algebra if and only if its spectrum $\widehat{A}$ is feebly (or weakly) compact. We immediately arrive at
\begin{proposition}\label{pcs}
Let $\phi:A\to M(B)$ be a $G$-morphism between PCS-algebras. Then $\phi$ extends uniquely to a unital $G$-equivariant $\kappa$-continuous $*$-homomorphism $\Phi:\Gamma(K_A)\to\Gamma(K_B)$.\hfill$\blacksquare$
\end{proposition}

\section{Topological aspects}

\subsection{Unreduced cones of locally compact Hausdorff groups}\label{secone}

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 {\em unreduced cone} $\cC G$ of $G$ is the quotient topological space
\[
\cC G:=([0,1]\times G)/(\{0\}\times G).
\]
We denote the quotient topology on $\cC G$ by $\tau_q$, the quotient map by $\pi:[0,1]\times G\to\cC G$, and call $[(0,g)]\in\cC G$ the {\em tip} of the cone. There is a continuous (right) $G$-action on $\cC G$ defined as follows
\[
\cC G\times G\longrightarrow \cC G:\qquad ([(t,g)],h)\longmapsto [(t,gh)].
\]
Indeed, since $G$ is locally compact Hausdorff, this action is continuous by an old lemma of Whitehead~\cite[Lemma~4]{jhc-w48} (see~\cite{e-m68} for a modern formulation of the result).

Recall that every locally compact Hausdorff group is paracompact Hausdorff~(see e.g.~\cite[Exercise~9, p.~261]{jr-m00} or~\cite[Corollary~3.1.4]{at-08} 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 $\cC G$ is completely regular Hausdorff. However, it may no longer be locally compact (the tip in $\cC G$ does not have any compact neighborhood). These observations are crucial from the point of view of noncommutative topology, since it implies the anti-equivalences~\Cref{Ggelfnaim} and~\Cref{GWor} do not apply in this case.

Consider the maps
\begin{equation}\label{height}
{\rm t}:\cC G\to [0,1],\qquad [(t,g)]\mapsto t,
\end{equation}
\vspace*{-3mm}
\begin{equation}\label{gfunct}
{\rm g}:{\rm t}^{-1}((0,1])\to G,\qquad [(t,g)]\mapsto g.
\end{equation}
We call ${\rm t}$ the {\em 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 $\cC G$, i.e. the initial topology with respect to the maps~\Cref{height} and~\Cref{gfunct} 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~\cite{j-m56} (observe that $\cC G$ 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 $\pi:[0,1]\times G\to\cC G$ is continuous with respect to the topology $\tau_M$ on $\cC G$. Our aim is to show that these topologies are in fact the same for every locally compact Hausdorff group $G$.
\begin{lemma}\label{fcg}
Let $G$ be a locally compact Hausdorff group. Then the family of compact subsets
\[
\mathcal{F}_{\cC G}:=\{\pi(K)~:~K\text{ is a compact subset of $[0,1]\times G$}\}
\]
generates the topology on $(\cC G,\tau_M)$.
\end{lemma}
\begin{proof}
We need to prove that
\[
A\subseteq \cC G\text{ is closed}\quad\iff\quad A\cap B\text{ is closed in }B\text{ for all }B\in\mathcal{F}_{\cC G}.
\]
($\Rightarrow$) Assume that $A$ is closed and take $B\in\mathcal{F}_{\cC G}$. 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.

\noindent($\Leftarrow$) It suffices to show that any limit point of $A$ in $\cC G$ 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 $[(t,g)]\in\cC G\setminus\{[(0,g)]\}$. 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 $(\cC G,\tau_M)$ (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
\[
U:=\cC G\setminus {\rm g}^{-1}(G\setminus V)
\]
of $[(0,g)]$ in $(\cC G,\tau_M)$. 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 $A\cap\pi(K)$.
\end{proof}

\begin{proposition}\label{topequiv}
Let $G$ be a locally compact Hausdorff group. Then the topologies $\tau_q$ and $\tau_M$ on $\cC G$ are equivalent.
\end{proposition}
\begin{proof}
By~\Cref{fcg}, the topology on $(\cC G,\tau_M)$ 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 $(\cC G,\tau_q)$.
First, note that $(\cC G,\tau_q)$ is compactly generated as a Hausdorff quotient of a compactly generated space (see e.g.~\cite[\S~2.6]{ne-s67}).
Next, since $G$ is paracompact Hausdorff, from~\cite[Theorem~9]{r-g03} we infer that $\pi$ is semiproper (or compact-covering), i.e. for every compact subset $B$ of $(\cC G,\tau_q)$, there exists a~compact subset $K$ of $[0,1]\times G$ such that $B\subseteq\pi(K)$. This concludes the proof.
\end{proof}
\begin{remark}
Note that by~\cite[Theorems~5~and~6]{r-g03} the above result might not be true for arbitrary locally compact Hausdorff spaces.
\end{remark}

In conclusion, we have proved that for a locally compact Hausdorff group $G$, the topology on $\cC G$ 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 $G$.

\subsection{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 {\em partition of unity} on $X$ if
\begin{enumerate}
\item it is {\em 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 $\lambda\in\Lambda$,
\item and the finite sum $\sum_{\lambda\in\Lambda} f_\lambda(x)$ equals $1$ for all $x\in X$.
\end{enumerate}
A covering $\{U_\lambda\}$ of $X$ is called {\em numerable} if there is a partition of unity $\{f_\lambda\}$ on $X$ {\em 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~\cite[Corollary~13.1.9]{t-td08} we can always reduce arbitrary partitions of unity to countable ones. Hence, in what follows we assume that $\Lambda=\bN$.

Following the terminology of~\cite[p.~332]{t-td08}, we call a topological $G$-space $X$ {\em trivial} if there is a~$G$-map $X\to G$ and {\em locally trivial} if it has an open covering by trivial $G$-spaces (such a cover is called {\em trivializing}). If $X$ is a locally trivial $G$-space, we say that $X\to X/G$ is a {\em principal $G$-bundle}.
A~principal bundle $X\to X/G$ is called {\em numerable} if there is a numerable trivializing cover of $X$. In particular, if $X$ is paracompact Hausdorff, then any principal $G$-bundle $X\to X/G$ is numerable.

Let us now reformulate the definition of a numerable locally compact Hausdorff principal bundle using the construction of the unreduced cone. We start with trivial
principal $G$-bundles, which are described by the following elementary proposition.
\begin{proposition}\label{trivcone}
Let $G$ be a locally compact Hausdorff group and let $X$ be a locally compact Hausdorff $G$-space. Then the following conditions are equivalent
\begin{enumerate}
\item[{\rm (T1)}] $X$ is $G$-equivariantly homeomorphic to $X/G\times G$ with the natural $G$-action.
\item[{\rm (T2)}] There exists a $G$-equivariant continuous map $\varphi:X\to G$.
\item[{\rm (T3)}] There exists a $G$-equivariant continuous map $\rho:X\to \cC G$ such that
${\rm t}\circ\rho=1$, where ${\rm t}$ is the height function~\Cref{height}.
\end{enumerate}
\end{proposition}
\begin{proof}
Since the equivalence of (T1) and (T2) can be found in most textbooks dealing with principal bundles, we only show that (T2) is equivalent to (T3).

(T2)$\Rightarrow$(T3). The following $G$-map
\[
\iota:G\longrightarrow \cC G:\qquad g\longmapsto [(1,g)]
\]
is continuous. Define $\rho:=\iota\circ\varphi$. Then $\rho:X\to \cC G$ is a $G$-equivariant continuous map such that ${\rm t}\circ\rho=1$.

(T3)$\Rightarrow$(T2). Compose $\rho$ with the continuous $G$-map ${\rm g}:{\rm t}^{-1}((0,1])\to G$ given by~\Cref{gfunct}. The condition ${\rm t}\circ\rho=1$ ensures that ${\rm g}\circ\rho$ is well defined.
\end{proof}
Next, we give a new description of numerable principal bundles, which admits a noncommutative generalization (see~\Cref{mainsubsection}).
\begin{theorem}\label{numprin}
Let $G$ be a locally compact Hausdorff group and let $X$ be a locally compact Hausdorff \mbox{$G$-space}. Then the following conditions are equivalent:
\begin{enumerate}
\item[{\rm (1)}] $X\to X/G$ is a numerable principal $G$-bundle.
\item[{\rm (2)}] There exist $G$-equivariant continuous maps $\rho_i:X\to\cC G$, $i\in\bN$, such that
the sequence of continuous functions $(\sum_{i=0}^N({\rm t}\circ\rho_i))_{N\in\bN}$ converges to $1$ in the compact-open topology.
\end{enumerate}
\end{theorem}
\begin{proof}
First, assume that $X\to X/G$ is a numerable principal $G$-bundle. Let $\{U_i\}$, $i\in\bN$, 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. Let $\widetilde{\varphi}_i$ be any (not necessarily continuous) extension of $\varphi_i$ to whole $X$.
Next, we define the maps
\[
\rho_i:X\overset{f_i\times\widetilde{\varphi}_i}{\longrightarrow}[0,1]\times G\overset{\pi}{\longrightarrow} \cC G:\quad x\longmapsto  [(f_i(x),\widetilde{\varphi}_i(x))] .
\]
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~\Cref{topequiv}, 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
\[
\sum_{i=0}^N({\rm t}\circ\rho_i)=\sum_{i=0}^Nf_i\qquad\text{for all $N\in\bN$}.
\]
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 $\rho_i:X\to \cC G$ 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~\cite[p.~320]{t-td08}. By~\cite[Lemma~13.1.7]{t-td08} the covering $\{U_i:=({\rm t}\circ\rho_i)^{-1}((0,1])\}$ is numerable. It is left to check that 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~\Cref{gfunct}. Each $\varphi_i$ is continuous and $G$-equivariant as a~composition of two continuous $G$-equivariant maps, so we are done.
\end{proof}
\begin{remark}
We formulate~\Cref{numprin} 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.
\end{remark}

\section{Noncommutative numerable principal bundles}\label{nnpb}

\subsection{Trivial $G$-C*-algebras}\label{trivialpb}

In this subsection, we generalize the notion of a trivial $G$-space to the realm of noncommutative topology via the Gelfand--Naimark duality. Our first step is to rewrite the conditions (T1)-(T3) from~\Cref{trivcone} in terms of commutative C*-algebras of continuous functions. However, the condition (T1) does not admit such a~generalization since $X/G$ might fail to be Hausdorff. Even when we ensure that it is Hausdorff by insisting that the $G$-action is proper, it is not clear how to define a C*-algebraic analog of $X/G$ for a locally compact $X$ (see e.g. \cite{ma-r90,a-b13}). Let us briefly discuss the compact case. For a~compact Hausdorff group $G$ and a unital C*-algebra $A$ equipped with a $G$-action $\alpha$, we define the {\em fixed-point subalgebra} by
\[
A^G:=\{a\in A~:~\alpha_g(a)=a\}.
\]
If $A=C(X)$ for a compact Hausdorff space $X$, then $A^G\cong C(X/G)$. Taking this into account, in the unital case, we could generalize the condition (T1) by insisting that $A$ is $G$-equivariantly isomorphic to $A^G\otimes C(G)$ taken with the natural $G$-action.
However, one can show that this condition is equivalent to the existence of a unital $G$-equivariant $*$-homo\-morphism $\varphi:C(G)\to A$ such that the image of $\varphi$ commutes with $A^G$. For general noncommutative \mbox{C*-al}\-gebras this is a~strong requirement and it excludes many interesting examples, e.g. dual group actions on crossed products. Hence, the condition $A\cong A^G\otimes C(G)$ is not a good candidate for a generalization of triviality of $G$-spaces even in the compact case.

Based on the anti-equivalence of categories~\Cref{GWor} and the discussions in~\Cref{beyondGN} and~\Cref{sec2.2}, we propose the following generalizations of  the condition (T2) from~\Cref{trivcone}.
\begin{definition}\label{triviality}
Let $G$ be a locally compact Hausdorff group acting on a \mbox{C*-al}\-ge\-bra $A$. We say that $A$ is a {\em trivial $G$-C*-algebra} if there exists a $G$-morphism $\phi:C_0(G)\to M(A)$. We call $A$ a {\em $\kappa$-trivial $G$-C*-algebra} if there exists a unital $G$-equivariant $\kappa$-continuous $*$-homomorphism $\Phi:C(G)\to\Gamma(K_A)$
\end{definition}
Few comments regarding~\Cref{triviality} are in order:
\begin{itemize}
\item From~\Cref{onewayfunct}, we infer that $\kappa$-triviality implies triviality.
\item If $A=C_0(X)$ is commutative, then from~\Cref{pedcom} and~\Cref{stricnondeg} it follows that triviality is equivalent to $\kappa$-triviality. Furthermore, by the anti-equivalence~\Cref{GWor}, both triviality and $\kappa$-triviality are also equivalent to the conditions~(T1)-(T3) from~\Cref{trivcone} for $X$.
\item Finally, \Cref{pcs} implies that triviality is also equivalent to $\kappa$-triviality for PCS-algebras.
\end{itemize}
\begin{example}\label{excomp}
For a locally compact Hausdorff group $G$,
let $\cK:=\cK(L^2(G))$ be the C*-algebra of compact operators on $L^2(G)$ (where we take the right Haar measure on $G$), and let $r:G\to U(B(L^2(G))$ be the right-regular representation. We equip $\cK$ with the following $G$-action:
\[
\alpha_g(T):=r(g)Tr(g)^*,\qquad g\in G,\quad T\in\cK.
\]
In~\cite[Example~2.1]{ma-r90}, Rieffel proved that this action is saturated and proper in a suitable sense (we use the right-regular representation instead of the left one, but the result is unchaged). Let us now prove that it turns $\cK$ into a $\kappa$-trivial $G$-C*-algebra. Since simplicity of $\cK$ implies that is a~PCS-algebra, $\kappa$-triviality is equivalent to triviality in this case by the above discussion. Recall from~\cite[Example~4.2]{lt-76} that
\[
\Gamma\left(K_{\cK}\right)\cong B(L^2(G))
\]
and that the $\kappa$-topology on $\Gamma(K_\cK)$ coincides with the strong operator topology on $B(L^2(G))$. Consider the following map
\[
\Phi:C_c(G)\longrightarrow B(L^2(G)):\quad f\longmapsto M_f\,,\qquad (M_f\psi)(g):=f(g)\psi(g),
\]
where we identified $f\in C_c(G)$ with its class in $L^2(G)$. Let $\{q_\psi\}_{\psi\in L^2(G)}$ denote the family of seminorms defining the strong operator topology on $B(L^2(G))$. Note that
\[
q_\psi(\Phi(f))^2=\|M_f\psi\|^2_2=\int_{G} |f(g)\psi(g)|^2d\mu(g)\leq \left(\sup_{h\in K}|f(h)|\right)^2\int_{G}|\psi(g)|^2d\mu(g)=\|\psi\|^2_2\;p_{K}(f)^2\,,
\]
where $K$ is the support of $f$ and $p_K$ is defined by~\Cref{cosemi}. Hence, $\Phi$ is a~well-defined $\kappa$-continuous map. It is also a $G$-equivariant $*$-homomorphism, where we consider the continuous action $\Delta$ of $G$ on $C_c(G)$ given by $(\Delta_h(f))(g):=f(gh)$ and the extended action $\Gamma(\alpha)$ on $B(L^2(G))$.
By $\kappa$-density of $C_c(G)$ in $C(G)$, $\Phi$ extends to a~unital $\kappa$-continuous linear map $C(G)\to B(L^2(G))$ (see~e.g.~\cite[Theorem~5.1]{f-t67}). Since the multiplication is separately $\kappa$-continuous, the $*$-operation is $\kappa$-continuous, and $G$ acts on $B(L^2(G))$ by $\kappa$-continuous $*$-automorphisms, we conclude that this extension is a~$G$-equivariant $*$-homomorphism. Therefore, $\mathcal{K}$ is a $\kappa$-trivial $G$-C*-algebra.
\end{example}

Let us present a different characterization of $\kappa$-triviality, which resembles the condition (T3) of~\Cref{trivcone} and will be needed for our discussion of local triviality in~\Cref{mainsubsection}. First, for every $w\in C_0((0,1])$ and $f\in C(G)$, we define a function $wf$ on $\cC G$ by the formula
\begin{equation}\label{wformul}
(wf)([(t,g)]):=\begin{cases} 0~~& t=0,\\ w(t)f(g)~~& t\neq 0.\end{cases}
\end{equation}
Note that $wf\in C(\cC G)$. In particular, let
\begin{equation}\label{tformul}
({\rm t}f)([(t,g)]):=\begin{cases} 0~~& t=0,\\ tf(g)~~& t\neq 0.\end{cases}
\end{equation}
Observe that ${\rm t}1$ equals to the height function~\Cref{height}. Next, let $C_0((0,1],C_c(G))$ be a $*$-algebra, where operations are defined pointwise and we consider the $\kappa$-topology on $C_c(G)$. Observe that there is an injective $*$-homo\-morphism
\begin{equation}\label{obvi0}
\iota_0:C_0((0,1],C_c(G))\longrightarrow C(\cC G),\qquad \iota_0(u)([(t,g)]):=\begin{cases} 0 ~~& t=0,\\ (u(t))(g)~~& t\neq 0.\end{cases}
\end{equation}
Hence, we can view $C_0((0,1],C_c(G))$ as a $*$-subalgebra of $C(\cC G)$ and restrict both the $\kappa$-topology and the $G$-action to it. Finally, for our next lemma, we need the following generalization of the Stone--Weierstrass theorem, which follows from a result of Timofte~\cite[Corollary~14]{v-t05}.
\begin{theorem}\label{sw1}
Let $Y$ be a Hausdorff topological space and let $W$ be a~\mbox{$*$-sub}\-algebra of $C(Y)$. If $W$ separates points and $W(y):=\{f(y)~:~f\in W\}\neq \{0\}$ for every $y\in Y$, then $W$ is dense in $C(Y)$ in the compact-open topology.\hfill $\blacksquare$
\end{theorem}
\begin{lemma}\label{sublemma}
Let $G$ be a locally compact Hausdorff group and consider $C(\cC G)$ with the compact-open topology. Then the $*$-sub\-al\-ge\-bra $W:=\bC1+\iota_0(C_0((0,1],C_c(G)))\subset C(\cC G)$ is $\kappa$-dense, where $\iota_0$ is defined by~\Cref{obvi0}.
\end{lemma}
\begin{proof}
Every element of $W$ is of the form $\lambda1+\iota_0(u)$, where $\lambda\in\bC$ and $u\in C_0((0,1],C_c(G))$. Let $w\in C_0((0,1])$ and $f\in C_c(G)$. Note that the function $wf$ defined by~\Cref{wformul} belongs to $W$. Indeed, if $u\in C_0((0,1],C_c(G))$ is a function defined by $u(t):=w(t)f$ then $\iota_0(u)=wf$. By~\Cref{sw1}, it suffices to show that $W$ separates points and that
\[
W([(t,g)])=\{u([(t,g)])~:~u\in W\}\neq \{0\}
\]
for all points $[(t,g)]\in \cC G$. The latter condition is immediate since $1\in W$. Next, since $G$ is locally compact Hausdorff, for any pair of distinct points $g_1, g_2\in G$ we can find a function $f_{12}\in C_c(G)$ such that $f_{12}(g_1)\neq f_{12}(g_2)$ (see e.g.~\cite[\S 2.12, p.~39]{w-r87}). Pick any pair $[(t_1,g_2)]\neq [(t_2,g_2)]$ of distinct points in $\cC G$. Then, $({\rm t}f_{12})([(t_1,g_1)])\neq ({\rm t}f_{12})([(t_2,g_2)])$, where ${\rm t}f_{12}\in W$ is the function defined by~\Cref{tformul}. This concludes the proof.
\end{proof}
\begin{theorem}\label{thm4.5}
Let $G$ be a locally compact Hausdorff group acting on a C*-algebra $A$. Then $A$ is a $\kappa$-trivial $G$-C*-algebra if and only if there is a unital $G$-equivariant continuous $*$-homomorphism $\rho:C(\cC G)\to\Gamma(K_A)$ such that $\rho({\rm t})=1$, where we consider the compact-open topology on $C(\cC G)$ and the $\kappa$-topology on $\Gamma(K_A)$. \end{theorem}
\begin{proof} ($\Rightarrow$) Since $A$ is $\kappa$-trivial, there exists a unital $G$-equivariant $\kappa$-continuous $*$-ho\-mo\-mor\-phism $\Phi:C(G)\to \Gamma(K_A)$. Write $\phi:=\Phi|_{C_c(G)}$ and consider the $*$-subalgebra $W$ of $C(\cC G)$ as in~\Cref{sublemma}. We define the map
\[
\widetilde{\rho}:W\longrightarrow\Gamma(K_A):\; \lambda1+\iota_0(u)\longmapsto \lambda1+\phi (u(1)), \qquad \lambda\in\bC,\quad u\in C_0((0,1],C_c(G)).
\]
It is clear that $\widetilde{\rho}$ is a unital $G$-equivariant $\kappa$-continuous $*$-homomorphism. Due to the fact that $W$ is dense in $C(\cC G)$, $\widetilde{\rho}$ extends to a unital $\kappa$-continuous linear map \mbox{$\rho:C(\cC G)\to\Gamma(K_A)$} (see~e.g.~\cite[Theorem~5.1]{f-t67}). Since the multiplication is separately $\kappa$-continuous, the $*$-operation is $\kappa$-continuous, and $G$ acts by $\kappa$-continuous $*$-automorphisms, we conclude that $\rho$ is a $G$-equivariant $*$-ho\-mo\-mor\-phism. To prove that $\rho(\rm t)=1$, we take an approximate unit $(e_\lambda)\subseteq C_c(G)$. Then $({\rm t}e_\lambda)$ converges to ${\rm t}$ in $C(\cC G)$ in the compact-open topology, where ${\rm t}e_\lambda$ is defined by~\Cref{tformul}. Observe that
\[
\rho({\rm t})=\kappa\lim_\lambda\rho({\rm t}e_\lambda)=\kappa\lim_\lambda\phi(e_\lambda)=\kappa\lim_\lambda\Phi(e_\lambda)=\Phi(1)=1.
\]
($\Leftarrow$) Consider the linear map
\[
\iota_{\rm t}:C(G)\longrightarrow C(\cC G):\quad f\longmapsto {\rm t}f.
\]
Here ${\rm t}f$ is defined by~\Cref{tformul}. It is immediate that $\iota_{\rm t}$ is $*$-preserving and $G$-equivariant. We check that it is also continuous with respect to the compact-open topologies on $C(G)$ and $C(\cC G)$.
Recall from~\Cref{fcg} that the topology of $\cC G$ is generated by the family $\mathcal{F}_{\cC G}$, i.e. the compact subsets $L$ for which there exists a compact subset $K\subset [0,1]\times G$ such that $L\subseteq \pi(K)$. Furthermore, by compactness of $K$, there exist compact subsets $D\subseteq [0,1]$ and $H\subset G$ such that $K\subseteq D\times H$ as a~closed subset. It follows that for all $f\in C(G)$ and $L\in\mathcal{F}_{\cC G}$ we have that
\[
\sup_{p\in L}|(\iota_{\rm t}(f))(p)|=\sup_{[(t,g)]\in L\setminus\{[(0,g)]\}}|(\iota_{\rm t}(f))([(t,g)])|\leq\sup_{(t,g)\in D\times H}|tf(g)|\leq \sup_{g\in H}|f(g)|,
\]
which implies the $\kappa$-continuity of $\iota_{\rm t}$\,. Next, define $\Phi:=\rho\circ\iota_{\rm t}$\,. It is automatically $*$-preserving, \mbox{$G$-equivariant}, and $\kappa$-continuous as a composition of maps satisfying these properties. Since $\rho({\rm t})=1$, we obtain that $\varphi(1)=1$. Finally, the following computation
\[
\Phi(f_1f_2)=\rho({\rm t}f_1f_2)=\rho(\sqrt{\rm t}f_1)\rho({\rm t})\rho(\sqrt{\rm t}f_2)=\Phi(f_1)\Phi(f_2),\qquad f_1\,,f_2\in C(G),
\]
shows that $\Phi$ is also a homomorphism. Here $\sqrt{\rm t}\in C_0((0,1])$ is the function $\sqrt{\rm t}(t):=\sqrt{t}$.
\end{proof}

\subsection{Classically locally trivial $G$-C*-algebras}\label{classlt}

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.
For a C*-algebra $A$, let $ZM(A)$ denotes the center of its multiplier algebra $M(A)$.

\begin{definition}\label{centerprin}
Let $G$ be a locally compact Hausdorff group acting on a C*-algebra $A$. We say that $A$ is a {\em classically locally trivial $G$-C*-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 {\em numerable classically locally trivial $G$-C*-algebra}.
\end{definition}
In the above definition, if $Y$ is second-countable, then $A$ is numerable. If $G$ is additionally second-countable and torsion-free, then~\Cref{centerprin} is equivalent to the definition of a proper $G$-C*-algebra of Guentner, Higson, and Trout~\cite[Definition~8.2]{ght-00}, which generalizes the notion of a~proper action in the sense of Baum, Connes, and Higson~\cite[Definition~1.3]{bch-94}.

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~\Cref{specact} induces (via~\Cref{calgact}) an action $\beta:G\to {\rm Aut}(C_0(\widehat{A}\,))$ such that
\begin{equation}\label{equidh}
\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,
\end{equation}
(see e.g.~\cite[Lemma~7.1]{rw-98}). Here $f\cdot a$ is the element given by the Dauns--Hofmann theorem (see e.g.~\cite[Theorem~3]{eo-74}). 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~\Cref{equidh} 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) classically locally trivial $G$-C*-algebra. Next, assume that $A$ is a (numerable) classically locally trivial $G$-C*-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~\Cref{GWor}, 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.

\begin{proposition}\label{claspec}
Let $G$ be a locally compact Hausdorff group acting on a C*-algebra $A$ with the 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) classically locally trivial $G$-C*-algebra.\hfill$\blacksquare$
\end{proposition}
As a corollary, we obtain that~\Cref{centerprin} indeed generalizes the notion of a~locally compact Hausdorff principal bundle.
\begin{corollary}
Let $G$ be a locally compact Hausdorff group acting on a locally compact Hausdorff space $X$. Then the following are equivalent
\begin{enumerate}
\item[{\rm (1)}] $C_0(X)$ is a (numerable) classically locally trivial $G$-C*-algebra.
\item[{\rm (2)}] $X\to X/G$ is a (numerable) principal $G$-bundle.\hfill$\blacksquare$
\end{enumerate}
\end{corollary}

In~\cite{pr-84} and~\cite{lprs-87} a class of actions (and coactions) for which the induced action on the spectrum gives rise to a principal $G$-bundle was studied. We recall some results from these papers, since they provide a class of classically locally trivial $G$-C*-algebras. 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 {\em 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 {\em 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.~\cite[p.~176]{rw-98}). 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.~\Cref{specact}).
From~\cite[Theorem~5.9]{lprs-87} 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 locally trivial principal $\widehat{G}$-bundle. This in turn implies by~\Cref{claspec} that $A\rtimes G$ is a classically locally trivial $\widehat{G}$-C*-algebra. Furthermore, from~\cite[Proposition~5.14]{lprs-87} we obtain that if $A$ is a classically locally trivial $G$-C*-algebra, then the action $\widehat{\alpha}$ of $\widehat{G}$ on $A\rtimes G$ is locally unitary.
\begin{remark}
Although both cited theorems from~\cite{lprs-87} are phrased in the language of coactions, our formulation is equivalent due to~\cite[Remark~5.7 (2)]{lprs-87}.
\end{remark}

We end the subsection with an example which shows that for every locally compact Hausdorff principal $G$-bundle $Y\to Y/G$ there is a trivial $G$-C*-algebra $A$ such that there is a $G$-equivariant non-degenerate $*$-homomorphism $C_0(Y)\to ZM(A)$.

\begin{example}\label{trivloctriv}
Let $Y\to Y/G$ be a locally compact Hausdorff principal $G$-bundle and let $\beta:G\to {\rm Aut}(C_0(Y))$ be the induced $G$-action.  We show that for any such bundle there exists a~noncommutative classically locally trivial $G$-C*-algebra, which is trivial in the sense of~\Cref{triviality}.

Let $A:=C_0(Y,\mathcal{K}(L^2(G)))$, where $\mathcal{K}(L^2(G))$ is the C*-algebra of compact operators on the Hilbert space $L^2(G)$, which is obtained using the right Haar measure on $G$. Define a continuous $G$-action $\alpha$ on $A$ by
\[
(\alpha_g(a))(y):=r(g)a(yg)r(g)^*,\qquad g\in G,\quad a\in A,\quad y\in Y,
\]
where $r:G\to U(B(L^2(G)))$ is the right-regular representation. Observe that
\[
ZM(C_0(Y,\mathcal{K}(L^2(G))))\cong ZC_b(Y,B(L^2(G)))
\]
(see e.g.~\cite[Corollary~3.4]{apt-73}). There is a $*$-homomorphism
\[
\chi:C_0(Y)\longrightarrow ZC_b(Y,B(L^2(G))),\qquad \chi(f)(y):=f(y)1_{B(L^2(G))}.
\]
First, we prove that $\chi(C_0(Y))C_0(Y,\mathcal{K}(L^2(G)))$ is norm-dense in $C_0(Y,\mathcal{K}(L^2(G)))$. The linear span of the elements of the form $(fT)(y):=f(y)T$, where $f\in C_0(Y)$ and $T\in\mathcal{K}(L^2(G))$, is norm-dense in $C_0(Y,\mathcal{K}(L^2(G)))$ (see e.g.~\cite[Lemma~6.4.16]{gj-m90}), so it suffices to approximate each $fT$ by elements in $\chi(C_0(Y))C_0(Y,\mathcal{K}(L^2(G)))$. Let $(e_\lambda)$ be an approximate unit in $C_0(Y)$. Then we obtain that
\[
\lim_\lambda\|\chi(e_\lambda)fT-fT\
=\lim_\lambda\sup_{y\in Y}|e_\lambda(y)f(y)-f(y)|\|T\|=0.
\]
Furthermore, the $*$-ho\-mo\-mor\-phism $\chi$ is $G$-equivariant. Indeed, for all $y\in Y$, we have that
\[
(M(\alpha)_g(\chi(f)))(y)=r(g)f(yg)1_{B(L^2(G))}r(g)^*=f(yg)1_{B(L^2(G))}=(\chi(\beta_g(f)))(y).
\]
Recall that $M(\alpha)$ is the extended action~\Cref{multiaction}.

Finally, we show that the $G$-C*-algebra $A$ is always trivial. Define a $*$-homomorphism
\[
\Phi:C_0(G)\longrightarrow C_b(Y,B(L^2(G))),\qquad (\Phi(\phi))(y):=\rho(\phi),
\]
where $\rho:C_0(G)\to B(L^2(G))$ is the $G$-equivariant non-degenerate $*$-homomorphism defined in~\Cref{excomp}. First, we show that $\Phi$ is non-degenerate, i.e. that $\Phi(C_0(G))C_0(Y,\mathcal{K}(L^2(G)))$ is norm-dense in $C_0(Y,\mathcal{K}(L^2(G)))$. Let $T\in\mathcal{K}(L^2(G))$. Since $\rho(C_0(G))\mathcal{K}(L^2(G))$ is norm-dense in $\mathcal{K}(L^2(G))$, for some $\phi_i\in C_0(G)$ and $T_i\in\mathcal{K}(L^2(G))$ we have that
\[
\left\|\,T-\sum_{i=0}^n\rho(\phi_i)T_i\,\right\|\underset{n\to\infty}{\longrightarrow} 0.
\]
We infer that
\[
\left\|\,\sum_{i=0}^n\Phi(\phi_i)fT_i-fT\right\|=\sup_{y\in Y}|f(y)|\left\|\sum_{i=0}^n\rho(\phi_i)T_i-T\,\right\|\underset{n\to\infty}{\longrightarrow} 0.
\]
The $*$-homomorphism $\Phi$ is also $G$-equvariant. Indeed, observe that
\[
(M(\alpha)_g(\Phi(\phi)))(y)=r(g)\Phi(\phi)(yg)r(g)^*=r(g)\rho(\phi)r(g)^*=\rho(\Delta_g(\phi))=(\Phi(\Delta_g(\phi)))(y).
\]
Here $\Delta$ is the $G$-action on $C_0(G)$ induced by the right multiplication in $G$.
\end{example}

In view of~\Cref{claspec}, \Cref{centerprin} is not applicable to simple C*-algebras. Another drawback is that trivial and $\kappa$-trivial $G$-C*-algebras in the sense of~\Cref{trivialpb} might not be classically locally trivial. \Cref{trivloctriv} is a further illustration that a different generalization of principal bundles to the realm of noncommutative topology is needed. We provide such a generalization in the next subsection.

\subsection{Locally trivial $G$-C*-algebras}\label{mainsubsection}

In this subsection 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. Recall from~\Cref{secone} that ${\rm t}$ denotes the height function in $C(\cC G)$.

We start with some basics on the local-triviality dimensions. The following definition should be compared with~\cite{ghtw-18,cpt-21}.
\begin{definition}\label{ltdim}
Let $G$ be a compact Hausdorff group and let $A$ be a unital $G$-C*-algebra.
\begin{enumerate}
\item The {\em weak local-triviality dimension} $\dim_{\rm WLT}^G(A)$ is the smallest $n$ for which there exist unital $G$-equivariant $*$-homomorphisms $\rho_0\,,\ldots,\rho_n:C(\cC G)\to A$ such that $\sum_{i=0}^n\rho_i({\rm t})$ is invertible.
\item The {\em (plain) local-triviality dimension} $\dim_{\rm LT}^G(A)$ is the smallest $n$ for which there exist unital $G$-equivariant $*$-homomorphisms $\rho_0\,,\ldots,\rho_n:C(\cC G)\to A$ such that $\sum_{i=0}^n\rho_i({\rm t})=1$.
\item The {\em strong local-triviality dimension} $\dim_{\rm SLT}^G(A)$ is the smallest $n$ for which there exist unital $G$-equivariant $*$-homomorphisms $\rho_0\,,\ldots,\rho_n:C(\cC G)\to A$ with commuting images and such that $\sum_{i=0}^n\rho_i({\rm t})$=1.
\end{enumerate}
In any of the above cases, if there is no such finite $n$, then the associated dimension equals $\infty$.
\end{definition}
We list some known results about the local-triviality dimensions (see~\cite{ghtw-18,cpt-21}):
\begin{enumerate}
\item[(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.
\item[(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.
\item[(iii)] If $\dim_{\rm WLT}^G(A)<\infty$, then the $G$-action on $A$ is free in the sense of Ellwood~\cite{da-e00}. Hence by the results of Baum, de Commer, and Hajac~\cite{bdch-17}, we have that at the algebraic level the triple $(A,A^G, G)$ is a compact quantum principal bundle.
\item[(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.
\end{enumerate}

The following definition, which is the main contribution of the paper, is inspired by~\Cref{ltdim} and~\Cref{numprin}.
\begin{definition}\label{fulllt}
Let $G$ be a locally compact Hausdorff group acting on a C*-algebra $A$.
\begin{enumerate}
\item We say that $A$ is a {\em weakly locally trivial $G$-C*-algebra} if there exist unital $G$-equivariant \mbox{$\kappa$-con}\-tin\-u\-ous $*$-homomorphisms $\rho_i:C(\cC G)\to \Gamma(K_A)$, $i\in\bN$, 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.
\item We say that $A$ is a {\em locally trivial $G$-C*-algebra} if there exist unital \mbox{$G$-equi}\-variant \mbox{$\kappa$-con}\-tin\-u\-ous \mbox{$*$-homo}\-morphisms $\rho_i:C(\cC G)\to \Gamma(K_A)$, $i\in\bN$, such that the sequence ($\sum_{i=0}^N\rho_i({\rm t}))_N$
converges to $1$ in the $\kappa$-topology.
\item We say that $A$ is a {\em strongly locally trivial $G$-C*-algebra} if there exist unital $G$-equivariant \mbox{$\kappa$-con}\-tin\-u\-ous \mbox{$*$-homo}\-morphisms $\rho_i:C(\cC G)\to \Gamma(K_A)$, $i\in\bN$, with commuting images and such that ($\sum_{i=0}^N\rho_i({\rm t}))_N$
converges to $1$ in the $\kappa$-topology.
\end{enumerate}
\end{definition}
Here are some observations and remarks:
\begin{enumerate}
\item[(i)] If $G$ is compact Hausdorff, then $\cC G$ is also compact Hausdorff. This turns $C(\cC G)$ 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~\Cref{fulllt} are automatically continuous. We obtain that
\begin{align*}
\dim_{\rm WLT}^G(A)<\infty\quad&\iff\quad \text{$A$ is a unital weakly locally trivial $G$-C*-algebra},\\
\dim_{\rm LT}^G(A)<\infty\quad&~~~\Rightarrow~\quad\text{$A$ is a unital locally trivial $G$-C*-algebra},\\
\dim_{\rm SLT}^G(A)<\infty\quad&\iff\quad\text{$A$ is a unital strongly locally trivial $G$-C*-algebra}.
\end{align*}
The implication from left to right for $\dim^G_{\rm LT}$ is not clear at the moment, but we conjecture that it holds.

\item[(ii)] It might happen even beyond the compact/unital case that the number of $*$-homomorphisms $\rho_i$ from \Cref{fulllt} 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.
\item[(iii)]
If there is only one $G$-equivariant $\kappa$-continuous $*$-homomorphism $\rho:C(\cC G)\to\Gamma(K_A)$ in~\Cref{fulllt}, then we have that $\rho({\rm t})=1$. By~\Cref{thm4.5} we conclude that $A$ is $\kappa$-trivial (see~\Cref{triviality}), which implies triviality of the noncommutative bundle.
\item[(iv)]
Note that we have the following implications for $G$-C*-algebras:
\[
\text{strong local triviality}\quad\Rightarrow\quad\text{local triviality}\quad\Rightarrow\quad\text{weak local triviality}.
\]
\item[(v)]
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.
\begin{itemize}
\item If there is a strictly non-degenerate $G$-morphism $A\to M(B)$, then it follows from~\Cref{stricnondeg} that $B$ is a weakly locally trivial $G$-C*-algebra.
\item If $A$ and $B$ are both PCS-algebras and there is a non-degenerate $G$-$*$-homomorphism $A\to M(B)$, then it follows from~\Cref{pcs} that $B$ is a weakly locally trivial $G$-C*-algebra.
\end{itemize}
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.
\item[(vi)]
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 $A_\infty\cap A'$~(see e.g.~\cite{e-g19}).
\end{enumerate}

Let us now prove some elementary facts about locally trivial $G$-C*-algebras. First, observe that~\Cref{fulllt} generalizes the notion of a locally compact Hausdorff numerable principal $G$-bundle. 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.

\begin{proposition}\label{commult}
Let $G$ be a locally compact Hausdorff group acting on a locally compact Hausdorff space $X$. Then the following are equivalent
\begin{enumerate}
\item[{\rm (1)}] $X\to X/G$ is a numerable principal $G$-bundle,
\item[{\rm (2)}] $C_0(X)$ is a strongly locally trivial $G$-C*-algebra,
\item[{\rm (3)}] $C_0(X)$ is a locally trivial $G$-C*-algebra,
\item[{\rm (4)}] $C_0(X)$ is a weakly locally trivial $G$-C*-algebra.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) $\Rightarrow$ (2). Assume that $X\to X/G$ is a numerable principal bundle. By~\Cref{numprin}, there exist $G$-equivariant continuous maps
\[
\rho_i:X\longrightarrow \cC G,\qquad i\in\bN,
\]
such that $(\sum_{i=0}^N({\rm t}\circ\rho_i))_{N\in\bN}\subset C(X)$ converges to $1$ in the $\kappa$-topology. Since both $X$ and $\cC G$ are completely regular Hausdorff spaces, we use \Cref{funind} to obtain unital $G$-equivariant $\kappa$-continuous $*$-homomorphisms
\[
\rho^*_i:C(\cC G)\longrightarrow C(X)\cong\Gamma(K_{C_0(X)}),\qquad i\in\bN,
\]
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
\[
\rho^*_i:C(\cC G)\longrightarrow\Gamma(K_{C_0(X)})\cong C(X),\qquad i\in\bN,
\]
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 $\cC G$ are completely regular Hausdorff spaces, we use~\Cref{spind} to obtain $G$-equivariant continuous maps
\[
\rho_i:X\longrightarrow \cC G,\qquad i\in\bN,
\]
such that $(\sum_{i=0}^N({\rm t}\circ\rho_i))_{N\in\bN}$ 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 $N\in\bN$, then $\sigma$ is also positive (see e.g.~\cite[Proposition~8.5]{lt-76} for a~more general statement about Pedersen multiplier algebras). Hence, we have that $0< \sigma(x)^{-1}\leq 1$ for all $x\in X$. Consider the following continuous $G$-equivariant maps
\[
l_x:\cC G\longrightarrow\cC G:\qquad [(t,g)]\mapsto [(\sigma(x)^{-1}t,g)],\qquad x\in X,
\]
which can be used to define
\[
\widetilde{\rho}_i:X\longrightarrow \cC G,\qquad \widetilde{\rho}_i(x):=l_x(\rho_i(x)),\qquad i\in\bN.
\]
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~\Cref{topequiv} 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 \Cref{numprin} we conclude that $X\to X/G$ is a numerable principal $G$-bundle.
\end{proof}

Next, we show that the class of locally trivial $G$-C*-algebras contains the class of numerable classically locally trivial $G$-C*-algebras discussed in~\Cref{classlt}.

\begin{proposition}
Let $G$ be a locally compact Hausdorff group. Then every numerable classically locally trivial $G$-C*-algebra is a locally trivial $G$-C*-algebra.
\end{proposition}
\begin{proof}
Let $A$ be a numerable classically locally trivial $G$-C*-algebra. Then there is a locally compact Hausdorff numerable principal $G$-bundle $Y\to Y/G$ and 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~\Cref{pedcom} and~\Cref{stricnondeg} 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~\Cref{commult} that there exist unital $G$-equivariant $*$-homomorphisms $\rho_i:C(\cC G)\to \Gamma(K_{C_0(Y)})\cong C(Y)$, $i\in\bN$, 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
\[
\widetilde{\rho}_i:C(\cC G)\longrightarrow \Gamma(K_A),\qquad i\in\bN,
\]
are unital $G$-equivariant $\kappa$-continous $*$-homomorphisms. Observe that
\[
\kappa\lim_{N\to\infty}\sum_{i=0}^N\widetilde{\rho}_i({\rm t})=\kappa\lim_{N\to\infty}\sum_{i=0}^N\widetilde{\chi}(\rho_i({\rm t}))=\widetilde{\chi}\left(\kappa\lim_{N\to\infty}\sum_{i=0}^N\rho_i({\rm t})\right)=\widetilde{\chi}(1)=1.
\]
\end{proof}

\subsection{Actions of $U(1)$ and its cyclic subgroups on PCS-algebras}

\Cref{fulllt} 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.

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

\begin{proposition}\label{u1act}
Let $U(1)$ act on a PCS-algebra $A$ and let $\gamma$ denote the generator of $\mathbb{Z}=\widehat{U(1)}$. If there exist normal elements $a_i\in A_\gamma$, $i\in\bN$,
such that $(\sum_{i=0}^Na^*_ia_i)_{N\in\bN}$ is an approximate unit, then $A$ is a~locally trivial $U(1)$-C*-algebra.
\end{proposition}
\begin{proof}
Since $C(U(1))$ is a unital C*-algebra and $A$ is a PCS-algebra, it suffices to find $U(1)$-equivariant unital \mbox{$*$-ho}\-mo\-mor\-phisms $\rho_i:C(\cC U(1))\to M(A)$ such that $(\sum_{i=0}^N\rho_i({\rm t}))_{N\in\bN}$ strictly converges to $1$. Indeed, a unital $*$-homomorphism from a unital C*-algebra $B$ to the multiplier algebra $M(A)$ is automatically continuous with respect to the strict topology (which coincides with the $\kappa$-topology for PCS-algebras).

Assume that there are normal elements $a_i\in A$, $i\in\bN$, satisfying the assumptions of the proposition. Let $u$ be the unitary generator of the C*-algebra $C(U(1))$ and consider the function $\sqrt{\rm t}u$ (see~\Cref{wformul}), where $\sqrt{\rm t}\in C_0((0,1])$ is defined by $\sqrt{\rm t}(t):=\sqrt{t}$.
It is straightforward to show that the assignments
\[
C(\cC U(1))\ni \sqrt{\rm t}u\longmapsto a_i\in M(A),\qquad i\in\bN,
\]
give rise to unital $*$-homomorphisms
\[
\rho_i:C(U(1))\longrightarrow M(A),\qquad i\in\bN.
\]
The $U(1)$-equivariance of a given $\rho_i$ follows from the fact that $a_i\in A_\gamma$.
Finally, observe that
\[
\left(\sum_{i=0}^N\rho_i({\rm t})\right)_{N\in\bN}=\left(\sum_{i=0}^N\rho_i\left(\left(\sqrt{\rm t}u^*\right)\left(\sqrt{\rm t}u\right)\right)\right)_{N\in\bN}=\left(\sum_{i=0}^Na^*_ia_i\right)_{N\in\bN}.
\]
Since $(\sum_{i=0}^Na^*_ia_i)_{N\in\bN}$ is an approximate unit, the conclusion follows.
\end{proof}

Let $n\in\bN$ and let $\mathbb{Z}/n\mathbb{Z}$ be a cyclic group of order $n$. Let
\[
{\rm Star}_n:=\{z\in\bC~:~z^n\in[0,1]\}
\]
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~\Cref{u1act}.

\begin{proposition}\label{znact}
Let $\mathbb{Z}/n\mathbb{Z}$ act on a PCS-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 $a_i\in A_\gamma$, $i\in\bN$, such that the spectrum of each $a_i$ is contained in ${\rm Star}_n$ and
such that $(\sum_{i=0}^Na^*_ia_i)_{N\in\bN}$ is an approximate unit, then $A$ is a~locally trivial $\mathbb{Z}/n\mathbb{Z}$-C*-algebra.\hfill$\blacksquare$
\end{proposition}
Note that for $n=2$ the above elements $a_i$ are odd and self-adjoint and $(\sum_{i=0}^Na_i^2)_{N\in\bN}$ is an approximate unit.

\section{Relation to free actions}\label{setthemfree}

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.~\cite{nc-p09} for the case of finite group actions).

\subsection{$\kappa$-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~\cite[Lemma~2.7]{da-e00} (see also~\cite{bhms-07}) 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.
\begin{lemma}\label{kapinj}
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 \mbox{$f^*:C(Z)\to C(Y)$} (see~\Cref{funind}) has $\kappa$-dense range.
\end{lemma}
\begin{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
\begin{align*}
\|f^*(\eta_K)-\omega\|&=\sup_{y\in Y}|f^*(\eta_K)(y)-\omega(y)|=\sup_{y\in Y}|\eta_K(f(y))-\omega(y)|\\
&\leq\sup_{z\in Z}|\eta_K(z)|+\sup_{y\in Y}|\omega(y)|\leq \|\eta_K\|+\|\omega\|\\
&\leq \|\omega_K\|+\|\omega\|\leq \|\omega\|\|c_K\|+\|\omega\|=2\|\omega\|.
\end{align*}
Finally, for any $k\in C_c(Y)$ with ${\rm supp}\,k=L$, we have that
\begin{align*}
p_L(f^*(\eta_K)-\omega)&=\sup_{y\in Y}|(\eta_K(f(y))-\omega(y))k(y)|=\sup_{y\notin K}|\eta_K(f(y))-\omega(y)||k(y)|\\
&\leq \|f^*(\eta_K)-\omega\|\sup_{y\notin K}|k(y)|\leq 2\|\omega\|\sup_{y\notin K}|k(y)|.
\end{align*}
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
\begin{align*}
\max&\{|\eta_\lambda(y_1)-\psi(y_1)\omega(y_1)|,|\eta_\lambda(y_2)-\psi(y_2)\omega(y_2)|\}\\
&\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)\\
&\geq \frac{|\psi(y_1)\omega(y_1)-\psi(y_2)\omega(y_2)|}{2}>0,
\end{align*}
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.
\end{proof}

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 \mbox{$*$-al}\-gebra 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
\begin{equation}\label{kapfun}
u\longmapsto \sup_{x\in K}s_\lambda(u(x)),
\end{equation}
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 yet another generalization of the Stone--Weierstrass theorem that can be deduced from~\cite[Corollary~8]{v-t05}.
\begin{theorem}\label{newstone}
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 $C(X,V)$.\hfill$\blacksquare$
\end{theorem}
Consider the functions
\begin{equation}\label{fvnct}
(fv)(x):=f(x)v,\qquad f\in C(X),\quad v\in V.
\end{equation}
Then from~\Cref{newstone} it follows that
\begin{equation}\label{splitdense}
{\rm span}\{fv~:~f\in C(X),\; v\in V\}\quad\text{is $\kappa$-dense in}\quad C(X,V).
\end{equation}

The following lemma should be well known, but we present its proof for completeness.
\begin{lemma}\label{isoprod}
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~\Cref{kapfun} for the other two $*$-algebras.
\end{lemma}
\begin{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.~\cite[Theorem~46.11]{jr-m00}). 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.
\end{proof}

\begin{theorem}\label{frecom}
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 $(f_1\alpha_{(\cdot)}(f_2))(g):=f_1\alpha_{g}(f_2)$.
\end{theorem}
\begin{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~\Cref{kapinj} and~\Cref{isoprod} 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 $C(X\times X)$ by~\Cref{sw1}.
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 $\mathcal{E}=F^*(W)$.

\noindent($\Rightarrow$) 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 $C(G,C(X))$.

\noindent($\Leftarrow$) 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.
\end{proof}

In~\cite{da-e00}, Ellwood introduced a notion of a free action of a locally compact quantum group on \mbox{a~C*-al}\-gebra, 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 {\em 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~\cite[Definition~1.6]{ma-r90}. 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~\cite[Theorem~1.1]{dcy-13}. 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~\cite{bdch-17} 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~\cite[Proposition~2.1]{da-e00} 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~\cite[Theorem~2.9(a)]{da-e00}.

The above discussion and~\Cref{frecom} prompt the following definition.
\begin{definition}
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 $(k_1\alpha_{(\cdot)}(k_2))(g):=k_1\alpha_g(k_2)$.
\end{definition}

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.
\begin{proposition}
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 a PCS-algebra.
\end{proposition}
\begin{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~\cite[Corollary~3.4]{apt-73}, we know that $M(C_0(G,A))=C_b(G,M(A))$ and the \mbox{$\kappa$-topol}\-ogy on $C_b(G,M(A))$ is given by the restriction of seminorms of the form~\Cref{kapfun}. 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, \Cref{frecom} ensures that $\kappa$-freeness of $\alpha$ is equivalent to freeness of the $G$-action on $X$, which is equivalent to the Ellwood condition by~\cite[Theorem~2.9(a)]{da-e00}. Next, if $G$ is compact and $A$ is a PCS-algebra, then $C(G,\Gamma(K_A))=C_b(G,M(A))= M(C(G,A))$ and the strict topology on $M(C(G,A))$ coincides with the $\kappa$-topology on $C(G,\Gamma(K_A))$.
\end{proof}

\subsection{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
\[
\rho:C(\cC G)\longrightarrow \Gamma(K_A).
\]
We define the following unital $G$-equivariant $\kappa$-continuous $*$-homomorphism
\begin{equation}\label{tilderho}
\widetilde{\rho}:C(G,C(\cC G))\longrightarrow C(G,\Gamma(K_A)),\qquad \widetilde{\rho}(u)(g):=\rho(u(g)).
\end{equation}

\begin{theorem}
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.
\end{theorem}
\begin{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~\Cref{splitdense}, we only have to show that the functions $fv$ (see~\Cref{fvnct}), 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 $\rho_i:C(\cC G)\to\Gamma(K_A)$, $i\in\bN$, 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
\[
fx=\kappa\lim_{N\to\infty} \sum_{i=0}^Nf\rho_i({\rm t})=\kappa\lim_{N\to\infty}\sum_{i=0}^N\widetilde{\rho}_i(f{\rm t}),
\]
where each $\widetilde{\rho}_i$ is given by the formula~\Cref{tilderho} and $f{\rm t}\in C(G,C(\cC G))$ is defined by~\Cref{fvnct}.
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 $C(G,C(\cC G))$, where $({\rm t}\gamma_\lambda)(g):={\rm t}\gamma_\lambda(g)$, $g\in G$, and
\[
{\rm t}\gamma_\lambda(g):\cC G\longrightarrow\bC:\qquad {\rm t}\gamma_\lambda(g)([(t,g')]):=\begin{cases}0~~& t=0,\\ t(\gamma_\lambda(g))(g')~~&t\neq 0.\end{cases}
\]
Since $\gamma_\lambda\in\mathcal{G}$ for every $\lambda$, there exist $m_\lambda\in\bN$ 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 $f1_{\Gamma(K_A)}$.
\end{proof}

\section*{Acknowledgments}\noindent

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 {\it Noncommutative universal spaces for groups and quantum groups}.

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