2 Actions, coactions and their (exotic) crossed products

For terminology and notation concerning (co)actions, their (exotic) crossed products, and duality – particularly Landstad duality for coactions in terms of generalized fixed-point algebras – we refer the reader to our previous paper [Reference Buss and Echterhoff7].

Let us just recall some basic notation and terminology. Throughout the paper, G will usually denote a locally compact group, with a fixed Haar measure. Continuous actions of G on a $C^*$ -algebra B will usually be written as $\beta \colon G\curvearrowright B$ .

In what follows, by an (exotic) crossed product $B\rtimes _{\beta ,\mu }G$ for an action $\beta :G\curvearrowright B$ we understand a $C^*$ -completion of $C_c(G,B)$ by a norm that satisfies $\|f\|_r\leq \|f\|_\mu \leq \|f\|_{\mathrm \max }$ for all $f\in C_c(G,B)$ , where the $\|\cdot \|_r$ (respectively, $\|\cdot \|_{\mathrm \max }$ ) denote the reduced (respectively, maximal) crossed-product norms on $C_c(G,B)$ . A crossed product $B\rtimes _{\beta ,\mu }G$ is called a duality crossed product if the dual coaction $\widehat {\beta }$ on $B\rtimes _{\beta ,\mathrm \max }G$ factors through a coaction $\widehat {\beta }_\mu $ on $B\rtimes _{\beta ,\mu }G$ . A crossed-product functor is a functor $(B,\beta )\mapsto B\rtimes _{\beta ,\mu }G$ from the category of G-algebras to the category of $C^*$ -algebras that sends actions $\beta :G\curvearrowright B$ to crossed products $B\rtimes _{\beta ,\mu }G$ such that for any G-equivariant $^*$ -homomorphism $\Phi :(B,\beta )\to (B',\beta ')$ the associated $^*$ -homomorphism $\Phi \rtimes _\mu G:B\rtimes _{\beta ,\mu }G\to B'\rtimes _{\beta ',\mu }G$ extends $\Phi \rtimes _{\mathrm {alg}}G: C_c(G,B)\to C_c(G,B'); f\mapsto \Phi \circ f$ . If all $\rtimes _\mu $ -crossed products are duality crossed products, then $\rtimes _\mu $ is called a duality crossed-product functor, which are the crossed-product functors we consider in this work. However, in Section 8 below, we need to restrict our attention to crossed-product functors that are also functorial with respect to (G-equivariant) correspondences. These are called correspondence crossed-product functors and include the maximal and reduced crossed products. It is shown in [Reference Buss, Echterhoff and Willett8, Theorem 5.6] that all correspondence crossed-product functors are duality functors.

A coaction of G on a $C^*$ -algebra A will usually be denoted by the symbol $\delta \colon A\to \mathcal M(A\otimes C^*(G))$ and its crossed product will be denoted by $A\rtimes _\delta \widehat {G}$ . Recall that $A\rtimes _\delta \widehat {G}$ can be realized as

$$ \begin{align*}\overline{\operatorname{\mathrm{{span}}}}((\operatorname{{\mathrm{id}}}\otimes\lambda)\circ\delta(A)(1\otimes \mathcal M(C_0(G)))) \subseteq \mathcal M(A\otimes \mathbb K(L^2(G))),\end{align*} $$

where $M:C_0(G)\to \mathbb {B}(L^2(G))$ is the representation by multiplication operators. We often write

$$ \begin{align*}j_A:=(\operatorname{{\mathrm{id}}}\otimes\lambda)\circ \delta:A\to \mathcal M(A\rtimes_\delta\widehat{G})\quad\text{and}\quad j_{C_0(G)}:=1\otimes M:C_0(G)\to \mathcal M(A\rtimes_\delta\widehat{G})\end{align*} $$

for the canonical morphisms from A and $C_0(G)$ into $\mathcal M(A\rtimes _\delta \widehat {G})$ . The dual action $\widehat {\delta }:G\curvearrowright A\rtimes _{\delta }\widehat {G}$ is then determined by the equation

$$ \begin{align*}\widehat\delta_g(j_A(a)j_{C_0(G)}(f))=j_A(a)j_{C_0(G)}(\mathrm{rt}_g(f)),\end{align*} $$

where $\mathrm {rt}:G\curvearrowright C_0(G)$ denotes the action by right translations.

It has been observed by Nilsen in [Reference Nilsen28, Corollary 2.6] that for every coaction $\delta :A\to \mathcal M(A\otimes C^*(G))$ there exists a canonical surjective $^*$ -homomorphism

$$ \begin{align*}\Psi_A: A\rtimes_\delta\widehat{G}\rtimes_{\widehat\delta,\mathrm\max}G\twoheadrightarrow A\otimes \mathbb K(L^2(G))\end{align*} $$

given as the integrated form of the covariant representation $(j_A\rtimes j_{C_0(G)}, 1\otimes \rho )$ . The coaction $\delta $ is called maximal if $\Psi _A$ is an isomorphism, and it is called normal if it factors through an isomorphism $ A\rtimes _\delta \widehat {G}\rtimes _{\widehat \delta ,r}G\xrightarrow \sim A\otimes \mathbb K(L^2(G))$ . In general, it factors through an isomorphism

(2-1) $$ \begin{align} A\rtimes_\delta\widehat{G}\rtimes_{\widehat\delta,\mu}G\xrightarrow\sim A\otimes \mathbb K(L^2(G)) \end{align} $$

for some (possibly exotic) duality crossed product $\rtimes _\mu $ . We then say that $(A,\delta )$ is a $\mu $ -coaction to indicate that it satisfies Katayama duality for the $\mu $ -crossed product.

The triple $(A\rtimes _\delta \widehat {G},\widehat {\delta }, j_{C_0(G)})$ is what we call a weak $G\rtimes G$ -algebra. More generally, a weak $G\rtimes G$ -algebra is a triple $(B, \beta , \phi )$ where:

• • B is a $C^*$ -algebra,

• • $\beta :G\curvearrowright B$ is an action of G on B,

• • $\phi :C_0(G)\to \mathcal M(B)$ is a nondegenerate, $\mathrm {rt}-\beta $ -equivariant $*$ -homomorphism.

As a variant of the classical Landstad duality for reduced coactions [Reference Quigg30], it is shown in [Reference Buss and Echterhoff5] that for any given duality crossed product $B\rtimes _{\beta ,\mu }G$ , there exists a unique (up to isomorphism) $\mu $ -coaction $(A_\mu , \delta _\mu )$ of G such that

$$ \begin{align*} (A_\mu\rtimes_{\delta_\mu}\widehat{G}, \widehat{\delta}_\mu, j_{C_0(G)})\cong (B, \beta, \phi). \end{align*} $$

This provides the main tool for deformation by coactions, as introduced in [Reference Buss and Echterhoff7], which serves as the foundation for most of the results presented in this paper. The construction of $(A_\mu , \delta _\mu )$ is carried out using the theory of generalized fixed-point algebras and depends on the choice of the crossed product $B\rtimes _{\beta ,\mu }G$ . Moreover, if we start with a duality crossed-product functor $\rtimes _\mu $ on the category of G- $C^*$ -algebras, it is shown in [Reference Buss and Echterhoff7, Proposition 2.9] (see also [Reference Buss and Echterhoff5, Lemma 7.1]) that the assignment

$$ \begin{align*} (B,\beta,\phi) \mapsto (A_\mu,\delta_\mu) \end{align*} $$

is functorial. More precisely, given weak $G\rtimes G$ -algebras $(B,\beta ,\phi )$ and $(B',\beta ',\phi ')$ , if $\Psi :B\to B'$ is a $\beta -\beta '$ -equivariant $*$ -homomorphism satisfying $\Psi \circ \phi =\phi '$ , then $\Psi $ induces a canonical $\delta _\mu -\delta ^{\prime }_\mu $ -equivariant $*$ -homomorphism

$$ \begin{align*} \psi:A_\mu\to A^{\prime}_\mu \end{align*} $$

between the corresponding $\mu $ -fixed-point algebras. In particular, we obtain the following result.

The above proposition will serve as our main tool for comparing different deformation procedures in this work.

3 Fell bundles

Recall from [Reference Doran and Fell12, Reference Doran and Fell13] that a Fell bundle $\mathcal A$ over the locally compact group G is a collection of Banach spaces $\{A_s: s\in G\}$ together with a set of pairings (called multiplications) $A_s\times A_t\to A_{st}: (a_s, a_t)\mapsto a_sa:t$ and involutions $A_s\to A_{s^{-1}}; a_s\mapsto a_s^*$ that are compatible with the linear structures in the usual sense known from $C^*$ -algebras, including the condition $\|a_sa:s^*\|=\|a_s\|^2$ for all $a_s\in A_s$ . In particular, it follows that the fibre $A_e$ over the unit $e\in G$ is a $C^*$ -algebra. If G is not discrete, the topological structure of $\mathcal A$ is determined by the set $C_c(\mathcal A)$ of compactly supported continuous sections $a: G\to \mathcal A$ , $s\mapsto a_s\in A_s$ . We refer to [Reference Doran and Fell12, Reference Doran and Fell13, Reference Exel15] for more details on this structure. The space $C_c(\mathcal A)$ becomes a $^*$ -algebra when equipped with convolution and involution given by

$$ \begin{align*} (a*b)_t=\int_G a_sb:{s^{-1}t}\,\,d s\quad\text{and}\quad a^*_t=\Delta(t^{-1})a_{t^{-1}}^*, \quad a,b\in C_c(\mathcal A).\end{align*} $$

A representation of $\mathcal A$ into $\mathcal M(D)$ for some $C^*$ -algebra D is a mapping ${\pi :\mathcal A\to \mathcal M(D)}$ that preserves multiplication and involution, and such that for every $d\in D$ and every $a\in C_c(\mathcal A)$ the map $G\to D; s\mapsto \pi (a_s)d$ is continuous. Such representation is nondegenerate if $\pi (A_e)D=D$ , that is, the restriction of $\pi $ to the unit fibre $A_e$ of $\mathcal A$ is a nondegenerate representation of the $C^*$ -algebra $A_e$ . Every representation ${\pi :\mathcal A\to \mathcal M(D)}$ integrates to give a $^*$ -representation

$$ \begin{align*}\pi:C_c(\mathcal A)\to \mathcal M(D);\quad \pi(a)d:=\int_G \pi(a_s)d\, \,d s \quad\text{for all } a\in C_c(\mathcal A), d\in D,\end{align*} $$

which is nondegenerate (in the sense that $\overline {\operatorname {span}}\, \pi (C_c(\mathcal A))D=D$ ) if and only if $\pi $ is nondegenerate. The maximal (or universal) cross-sectional $C^*$ -algebra $C^*(\mathcal A)$ is then defined as the completion of $C_c(\mathcal A)$ with respect to the $C^*$ -norm

$$ \begin{align*}\|a\|_{\mathrm\max}:=\sup_{\pi}\|\pi(a)\|\end{align*} $$

where $\pi $ runs through all possible (nondegenerate) representations of $\mathcal A$ . Passing to the integrated form and extension to the completion $C^*(\mathcal A)$ of $C_c(\mathcal A)$ then gives a one-to-one correspondence between nondegenerate representations of $\mathcal A$ and nondegenerate $^*$ -representations of $C^*(\mathcal A)$ . We refer to [Reference Buss and Echterhoff6, Proposition 2.1] for a list of alternative characterizations of the representations of $\mathcal A$ .

At the other extreme we have the reduced cross-sectional $C^*$ -algebra $C_r^*(\mathcal A)$ , which can be described as the image of $C^*(\mathcal A)$ under the (left) regular representation $\Lambda _{\mathcal A}:C^{*}(\mathcal A)\to {\mathbb B}_{A_e}(L^2(\mathcal A))$ , with $\Lambda _{\mathcal A}(a)\xi =a^{*}\xi $ for $a\in {C_c}(\mathcal A)\subseteq C^{*}(\mathcal A)$ and $\xi \in {C_c}(\mathcal A)\subseteq L^2(\mathcal A)$ , where $L^2(\mathcal A)$ denotes the Hilbert $A_e$ -module obtained as a completion of $C_c(\mathcal A)$ by the $A_e$ -valued inner product

$$ \begin{align*}\langle \xi, \eta\rangle_{A_e}=\int_G \xi(s)^{*} \eta(s)\,\,d s.\end{align*} $$

There exists a dual coaction

$$ \begin{align*}\delta_{\mathcal A}: C^*(\mathcal A)\to \mathcal M(C^*(\mathcal A)\otimes C^*(G))\end{align*} $$

given by the integrated form of the representation $s\mapsto a_s\otimes u_s$ , where, as above, $u:G\to U\mathcal M(C^*(G))$ denotes the universal representation of G and each $a_s\in A_s$ acts as a multiplier of $C^*(\mathcal A)$ by $(a_s\cdot f)(t):=a_sf(s^{-1}t)$ . It has been shown in [Reference Buss and Echterhoff6, Theorem 3.1] that $\delta _{\mathcal A}$ is a maximal coaction, that is, it satisfies Katayama duality (2-1) with respect to the maximal crossed product $\rtimes _{\mathrm \max }$ .

Now, similar to [Reference Buss and Echterhoff6, Definition 4.1], given any duality crossed-product functor $\rtimes _\mu $ , there is a unique quotient $C_\mu ^*(\mathcal A)$ of $C^*(\mathcal A)$ such that $\delta _{\mathcal A}$ factors through a coaction

$$ \begin{align*}\delta_\mu:C_\mu^*(\mathcal A)\to \mathcal M(C_\mu^*(\mathcal A)\otimes C^*(G))\end{align*} $$

and such that $(C_\mu ^*(\mathcal A),\delta _\mu )$ satisfies Katayama duality for the $\mu $ -crossed product as in (2-1). In particular, the dual coaction $(C_r^*(\mathcal A), \delta _r)$ on the reduced cross-sectional algebra $C_r^*(\mathcal A)$ corresponds to the reduced crossed-product functor $\rtimes _r$ in this way. It is the normalization of $(C^*(\mathcal A), \delta _{\mathcal A})$ in the sense of Quigg (see, for example, [Reference Buss and Echterhoff6]). We also have $(C^*_{\mathrm \max }(\mathcal A),\delta _{\mathrm \max })=(C^*(\mathcal A),\delta _{\mathcal A})$ . In particular, $C_\mu ^*(\mathcal A)$ is an ‘exotic’ completion of the cross-sectional $^*$ -algebra $C_c(\mathcal A)$ that sits between $C^*(\mathcal A)$ and $C^*_r(\mathcal A)$ in the sense that the identity map on $C_c(\mathcal A)$ extends to surjective $^*$ -homomorphisms

$$ \begin{align*}C^*(\mathcal A)\twoheadrightarrow C^*_\mu(\mathcal A)\twoheadrightarrow C^*_r(\mathcal A).\end{align*} $$

Combining this with [Reference Buss and Echterhoff5, Theorem 4.3] we obtain the following proposition.

3.1 The C*-algebra of kernels of a Fell bundle

We need to recall the realization of $C_r^*(\mathcal A)\rtimes _{\delta _r} \widehat {G}$ (and hence also of $C_\mu ^*(\mathcal A)\rtimes _{\delta _\mu }\widehat {G}$ ) as a completion of a certain $^*$ -algebra of kernel functions $k:G\times G\to \mathcal A$ due to Abadie (see [Reference Abadie3, Section 5]). For this let

$$ \begin{align*} \Bbbk_c({\mathcal A}):=\{k:G\times G\to \mathcal A:k \text{ is cont. with compact supports and } k(s,t)\in A_{st^{-1}}\}. \end{align*} $$

In other words, $\Bbbk _c({\mathcal A})=C_c(\nu ^*(\mathcal A))$ , the space of compactly supported continuous sections of the pullback $\nu ^*(\mathcal A)$ of $\mathcal A$ along $\nu \colon G\times G\to G,\, (s,t)\mapsto st^{-1}$ . This is a $^*$ -algebra with convolution and involution given by

$$ \begin{align*} k*l(s,t)=\int_G k(s,r)l(r,t)\,d r\quad\text{and}\quad k^*(s,t)=k(t,s)^* \end{align*} $$

for all $k,l\in \Bbbk _c({\mathcal A})$ and $s,t\in G$ . By a continuous $^*$ -representation of $\Bbbk _c({\mathcal A})$ we understand a $^*$ -homomorphism $\pi : \Bbbk _c({\mathcal A})\to D$ for some $C^*$ -algebra D such that

$$ \begin{align*}\|\pi(k)\|\leq \|k\|_2:=\bigg(\int_{G\times G} \|k(s,t)\|^2\,\,d(s,t)\bigg)^{1/2}\end{align*} $$

for all $k\in \Bbbk _c({\mathcal A})$ . Let $\|k\|_u:=\sup _{\pi }\|\pi (k)\|$ , where $\pi $ runs through all continuous $^*$ -representations of $\Bbbk _c({\mathcal A})$ . Then the completion $\Bbbk ({\mathcal A})$ of $\Bbbk _c({\mathcal A})$ by this norm is a $C^*$ -algebra; this was introduced by Abadie in [Reference Abadie3]. As noticed there, $\Bbbk ({\mathcal A})$ can also be viewed as the enveloping $C^*$ -algebra of the Banach $^*$ -algebra obtained as the completion of $\Bbbk _c({\mathcal A})$ with respect to $\|\cdot \|_2$ . We write $\Bbbk _2 ({\mathcal A})$ for this $L^2$ -completion and note that it coincides with the $L^2$ -completion of the space $\Bbbk _{c,b} ({\mathcal A})$ of all bounded compactly supported measurable functions $k\colon G\times G\to \mathcal A$ that satisfy the condition $k(s,t)\in A_{st^{-1}}$ . The following result is due to Abadie [Reference Abadie3].

Proposition 3.2. For every duality crossed-product functor $\rtimes _\mu $ , there is a canonical isomorphism $C_\mu ^*(\mathcal A)\rtimes _{\delta _\mu }\widehat {G}\cong \Bbbk ({\mathcal A})$ that sends a typical element of the form $j_{C^*_\mu (\mathcal A)}(a) j_{C_0(G)}(f)$ with $a\in C_c(\mathcal A)$ and $f\in C_c(G)$ to the kernel function

(3-1) $$ \begin{align} k_{a,f}(s,t) = a(st^{-1})f(t)\Delta(t^{-1}). \end{align} $$

Viewing $\Bbbk _c({\mathcal A})$ as a dense subalgebra of $C_\mu ^*(\mathcal A)\rtimes _{\delta _\mu }\widehat {G}$ using this isomorphism, the dual action $\widehat {\delta }_\mu $ of G is given on $\Bbbk _c({\mathcal A})$ by the formula

$$ \begin{align*} \widehat{\delta}_{\mu,r}(k)(s,t)=\Delta(r)k(sr,tr). \end{align*} $$

Moreover, we have

(3-2) $$ \begin{align} (j_{C_0(G)}(f)k)(s,t)=f(s)k(s,t)\quad\text{and}\quad (kj_{C_0(G)}(f))(s,t)=k(s,t)f(t) \end{align} $$

for all $k\in \Bbbk _c({\mathcal A})$ and $f\in C_0(G)$ .

Proof. Abadie only considers reduced coactions, that is, injective coactions of the reduced group $C^*$ -algebra $C^*_r(G)$ . It is well known that such coactions correspond bijectively (in a naturally functorial way) to normal coactions of $C^*(G)$ in our sense (see, for example, [Reference Echterhoff, Kaliszewski, Quigg and Raeburn14, Appendix A.9]). Moreover, this bijective correspondence preserves crossed products and their representation theory. That said, what Abadie proves in [Reference Abadie3] is that there is an isomorphism $C^*_r(\mathcal A)\rtimes _{\delta _r}\widehat {G}\cong \Bbbk ({\mathcal A})$ that is given as in the statement (see the proof of Proposition 8.1 in [Reference Abadie3]). But this implies the general version as in the statement for every exotic $C^*$ -norm associated to a duality crossed-product functor $\rtimes _\mu $ by [Reference Buss and Echterhoff7, Theorem 2.4].

The following result describes the representations of $\Bbbk ({\mathcal A})$ .

Proposition 3.3. Let D be a $C^*$ -algebra. A pair $(\pi ,\kappa )\colon (C^*(\mathcal A),C_0(G))\to \mathcal M(D)$ of nondegenerate representations is covariant for a dual coaction $\delta _{\mathcal A}$ of G on the cross-sectional $C^*$ -algebra of a Fell bundle $\mathcal A$ , and hence extends to a nondegenerate representation $\pi \rtimes \kappa \colon C^*(\mathcal A)\rtimes _{\delta _{\mathcal A}}\widehat {G}\cong \Bbbk ({\mathcal A})\to \mathcal M(D)$ if and only if

(3-3) $$ \begin{align} \pi(a)\kappa(_sf)=\kappa(f)\pi(a)\quad\text{for all } a\in A_s, f\in C_0(G), \end{align} $$

where $_s f(t):=f(st)$ denotes left translation, and we use the same notation $\pi \colon \mathcal A\to \mathcal M(D)$ for the disintegrated form of $\pi \colon C^*(\mathcal A)\to \mathcal M(D)$ .

Proof. By definition, $(\pi ,\kappa )$ is covariant if and only if

$$ \begin{align*} (\pi\otimes\operatorname{{\mathrm{id}}})(\delta_{\mathcal A}(a))=w_\kappa(\pi(a)\otimes 1)w_\kappa^*, \end{align*} $$

for all $a\in C^*(\mathcal A)$ , where $w_\kappa :=(\kappa \otimes \operatorname {{\mathrm {id}}})(w_G)$ and $w_G\in \mathcal M(C_0(G)\otimes C^*(G))$ is given by the universal representation $s\mapsto u_s$ . Since $\pi $ is nondegenerate, it extends to the multiplier algebra $\mathcal M(C^*(\mathcal A))$ . Using the inclusion $A_s\hookrightarrow \mathcal M(C^*(\mathcal A))$ given by $(a_s\cdot \xi )(t)=a_s\xi (s^{-1}t)$ , for $a_s\in A_s$ , $\xi \in C_c(\mathcal A)$ , and the formula $\delta _{\mathcal A}(a_s)=a_s\otimes u_s$ , the covariance condition for $(\pi ,\kappa )$ is equivalent to

$$ \begin{align*}(\pi(a_s)\otimes u_s)w_\kappa=w_\kappa(\pi(a_s)\otimes 1)\quad \text{for all } a_s\in A_s,\, s\in G.\end{align*} $$

Now we remark that if we view the Fourier algebra $A(G)\subseteq C_0(G)$ as a subalgebra of $B(G)\cong C^*(G)'$ , the set of continuous linear functionals on $C^*(G)$ , then $\kappa (f)=(\operatorname {{\mathrm {id}}}\otimes f)(w_\kappa )$ for all $f\in A(G)$ . Taking slices with $f\in A(G)$ , we see that the covariance condition is equivalent to

$$ \begin{align*}\pi(a_s)(\operatorname{{\mathrm{id}}}\otimes f)((1\otimes u_s)w_\kappa)=\kappa(f)\pi(a_s)\quad\text{for all } a_s\in A_s, s\in G, f\in A(G).\end{align*} $$

But

$$ \begin{align*} (\operatorname{{\mathrm{id}}}\otimes f)((1\otimes u_s)w_\kappa)&=(\kappa\otimes f)((1\otimes u_s)w_G)\\ &=(\kappa\otimes \operatorname{{\mathrm{id}}})(\operatorname{{\mathrm{id}}}\otimes f)((1\otimes u_s)w_G)=\kappa(f\cdot u_s), \end{align*} $$

where $(f\cdot u_s)(x):=f(u_sx)$ for all $x\in C^*(G)$ . Viewed as a function on G, this gives $f\cdot u_s={_s}f$ . Therefore, the covariance condition is equivalent to

$$ \begin{align*}\pi(a_s)\kappa(_s f)=\kappa(f)\pi(a_s)\quad\text{for all } a_s\in A_s, s\in G, f\in A(G).\end{align*} $$

Since $A(G)$ is dense in $C_0(G)$ , this is equivalent to (3-3).

It was shown by Abadie [Reference Abadie3] that there exists a well-defined $^*$ -representation $\Lambda _{\Bbbk ({\mathcal A})}:\Bbbk ({\mathcal A})\to {\mathbb B}_{A_e}(L^2(\mathcal A))$ , called the regular representation of $\Bbbk ({\mathcal A})$ , given by

$$ \begin{align*} (\Lambda_{\Bbbk ({\mathcal A})}(k)\xi)(t)=\int_G k(s,t)\xi(t)\,\,d t \quad\text{for }k\in \Bbbk_c({\mathcal A}),\xi\in C_c(\mathcal A).\end{align*} $$

It is stated in [Reference Abadie3, Theorem 5.1(2)] that $\Lambda _{\Bbbk ({\mathcal A})}$ is always faithful. But, unfortunately, this is not true in general (although it is true in several important cases, for example, if $\mathcal A$ is saturated). One can find more information about this in the arXiv version of the paper [Reference Abadie3] available at https://arxiv.org/pdf/math/0007109.pdf, where the incorrect statements have been fixed. As a counterexample let $\mathcal A$ be the Fell bundle over $\mathbb Z_2$ with fibres ${\mathcal A}_0=\mathbb C$ and ${\mathcal A}_1=0$ . Then $\Bbbk ({\mathcal A})=\mathbb C\oplus \mathbb C$ , while $L^2(\mathcal A)=\mathbb C$ .

We now provide a different representation of $\Bbbk ({\mathcal A})$ that will be faithful in general. For this we use the fact that $\Bbbk ({\mathcal A})$ is isomorphic to the crossed product $C^*(\mathcal A)\rtimes _{\delta _{\mathcal A}}\widehat {G}$ . Let

$$ \begin{align*}\delta_{\mathcal A}^\lambda:=(\Lambda_{\mathcal A}\otimes\lambda)\circ\delta_{\mathcal A}\colon C^*(\mathcal A)\to \mathcal M(C_r^*(\mathcal A)\otimes C^*_r(G))\end{align*} $$

denote the reduction of the dual coaction $\delta _{\mathcal A}$ of G on $C^*(\mathcal A)$ (which factors through a genuine reduced coaction of $C_r^*(G)$ on $C_r^*(\mathcal A)$ ). The algebra $\mathcal M(C_r^*(\mathcal A)\otimes C^*_r(G))$ is clearly represented faithfully on $L^2(\mathcal A)\otimes L^2(G)$ via $\Lambda _{\mathcal A}\otimes \lambda $ . Now $L^2(\mathcal A)\otimes L^2(G)$ can be identified with $L^2(\mathcal A\times G)$ if $\mathcal A\times G$ denotes the Fell bundle over $G\times G$ given by the pullback of $\mathcal A$ via the projection $G\times G\to G;(g,h)\mapsto g$ , so in what follows we regard $\Lambda _{\mathcal A}\otimes \lambda $ as a representation of $\mathcal M(C_r^*(\mathcal A)\otimes C_r^*(G))$ into $\mathbb B_{A_e}(L^2(\mathcal A\times G))$ . Then a short computation on the fibres $A_s\subseteq \mathcal A$ shows (see [Reference Exel and Ng16, Lemma 2.9 and Proposition 2.10]) that

$$ \begin{align*}\delta_{\mathcal A}^\lambda(a)=W_{\mathcal A}(\Lambda(a)\otimes 1)W_{\mathcal A}^*\quad\text{for all }a\in C^*(\mathcal A),\end{align*} $$

where $W_{\mathcal A}\in \mathbb B_{A_e}(L^2(\mathcal A\times G))$ is the unitary operator defined by

$$ \begin{align*}W_{\mathcal A}\zeta(s,t):=\zeta(s,s^{-1}t).\end{align*} $$

Now recall that the crossed product $C^*(\mathcal A)\rtimes _{\delta _{\mathcal A}}\widehat {G}\cong C_r^*(\mathcal A)\rtimes _{\delta _r}\widehat {G}$ can be realized as

$$ \begin{align*}C^*(\mathcal A)\rtimes_{\delta_{\mathcal A}}\widehat{G}=\operatorname{\mathrm{\overline{span}}}\{\delta_{\mathcal A}^\lambda(a)(1\otimes M_f): a\in C_c(\mathcal A), f\in C_c(G)\}\subseteq\mathbb B_{A_e}(L^2(\mathcal A\times G)).\end{align*} $$

A simple computation shows that

$$ \begin{align*}W_{\mathcal A}^*(1\otimes M_f)W_{\mathcal A}=\tilde M_{f}\quad \text{where }\tilde M_f\zeta(s,t):=f(st)\zeta(s,t),\end{align*} $$

for all $f\in C_0(G)$ and $\zeta \in C_c(\mathcal A\times G)\subseteq L^2(\mathcal A\times G)$ . It follows that

$$ \begin{align*}\delta_{\mathcal A}^\lambda(a)(1\otimes M_f)=W_{\mathcal A}(\Lambda(a)\otimes 1)\tilde M_f W_{\mathcal A}^*.\end{align*} $$

Hence, conjugation by the unitary $W_{\mathcal A}$ yields an isomorphism

$$ \begin{align*}C^*(\mathcal A)\rtimes_{\delta_{\mathcal A}}\widehat{G}\cong \operatorname{\mathrm{\overline{span}}}\{(\Lambda(a)\otimes 1)\tilde M_f: a\in C_c(\mathcal A), f\in C_0(G)\}\subseteq \mathbb B_{A_e}(L^2(\mathcal A\times G)).\end{align*} $$

On the other hand, the operators $(\Lambda (a)\otimes 1)\tilde M_f$ can be computed as

$$ \begin{align*} ((\Lambda(a)\otimes 1)\tilde M_f\zeta)(s,t)&=\int_G a(r)(\tilde M_f\zeta)(r^{-1}s,t)\,d r=\int_G a(r)f(r^{-1}st)\zeta(r^{-1}s,t)\,d r\\ &=\int_G a(sr^{-1})f(rt)\Delta(r)^{-1}\zeta(r,t)\,d r=\int_G \beta_t(k_{a,f})(s,r)\zeta(r,t)\,d r, \end{align*} $$

where $k_{a,f}(s,r):=a(sr^{-1})f(r)\Delta (r^{-1})$ as in (3-1), and $\beta $ denotes the dual action of G on $\Bbbk ({\mathcal A})$ as in Proposition 3.2. Using the isomorphism $\Bbbk ({\mathcal A})\cong C^*(\mathcal A)\rtimes _{\delta _{\mathcal A}}\widehat {G}$ provided by this proposition, we obtain the following proposition.

Proposition 3.4. Given a Fell bundle $\mathcal A$ over G, there exists a faithful representation

$$ \begin{align*}T\colon \Bbbk ({\mathcal A})\to {\mathbb B}_{A_e}(L^2(\mathcal A\times G))\end{align*} $$

given on $\Bbbk _c({\mathcal A})$ (respectively, $\Bbbk _{c,b} ({\mathcal A})$ ) by the formula

$$ \begin{align*}T_k\zeta(s,t):=\int_G \beta_t(k)(s,r)\zeta(r,t)\,d r=\int_G k(st,rt)\Delta(t)\zeta(r,t)\,d r.\end{align*} $$

Corollary 3.5. The structural homomorphism $\phi \colon C_0(G)\to \mathcal M(\Bbbk ({\mathcal A}))$ extends to a G-equivariant faithful unital $^*$ -homomorphism

$$ \begin{align*}\bar\phi\colon L^\infty(G)\to \mathcal M(\Bbbk ({\mathcal A}))\end{align*} $$

given by the formulas $(\bar \phi (f)k)(s,t){\kern-1pt}={\kern-1pt}f(s)k(s,t)$ , $(k\bar \phi (f))(s,t){\kern-1pt}={\kern-1pt}k(s,t)f(t)$ for $f{\kern-1pt}\in{\kern-1pt} L^\infty (G), k\in \Bbbk _{c,b} ({\mathcal A})$ .

Proof. Consider the $*$ -representation $\Phi :L^\infty (G)\to {\mathbb B}_{A_e}(L^2(\mathcal A\times G))$ given by $(\Phi (f)\xi )(s,t)=f(s)\xi (s,t)$ for $f\in L^\infty (G)$ and $\xi \in L^2(\mathcal A\times G)$ . We then compute

$$ \begin{align*}T_{\bar\phi(f)k}=\Phi(f)T_k\quad\text{and}\quad T_{k\bar\phi(f)}=T_k\Phi(f)\end{align*} $$

for all $f\in L^\infty (G)$ and $k\in \Bbbk _{c,b} ({\mathcal A})$ . This gives the result.

4 Deformation

In [Reference Buss and Echterhoff7] we described a general process to deform $C^*$ -algebras via coactions as follows: if $\delta \colon A\to \mathcal M(A\otimes C^*(G))$ is a coaction of the locally compact group G on the $C^*$ -algebra A, we consider the associated weak $G\rtimes G$ -algebra

$$ \begin{align*}(B,\beta,\phi)=(A\rtimes_\delta\widehat{G},\widehat{\delta}, j_{C_0(G)}).\end{align*} $$

We assume that $(A,\delta )$ satisfies Katayama duality for a given duality crossed-product functor $\rtimes _\mu $ for actions of G, as in (2-1). Given a suitable deformation parameter – such as a (Borel) $2$ -cocycle on G or an action of G on A that commutes with $\delta $ – we deform the weak $G\rtimes G$ -algebra $(B,\beta ,\phi )$ into another weak $G\rtimes G$ -algebra $(B',\beta ',\phi ')$ accordingly. Applying Landstad duality to $(B',\beta ',\phi ')$ , as developed in [Reference Buss and Echterhoff5], with respect to the crossed-product functor $\rtimes _\mu $ , we obtain a deformed coaction $(A',\delta ')$ such that

$$ \begin{align*} (B',\beta', \phi') = (A'\rtimes_{\delta'}\widehat{G}, \widehat{\delta'}, j^{\prime}_{C_0(G)}), \end{align*} $$

where $(A',\delta ')$ also satisfies Katayama duality with respect to $\rtimes _\mu $ .

4.1 Abadie–Exel deformation

Following ideas of Abadie and Exel from [Reference Abadie and Exel1], we introduced in [Reference Buss and Echterhoff7, Section 3.2] a deformation process for a given coaction $\delta \colon A\to \mathcal M(A\otimes C^*(G))$ that uses a (strongly) continuous action $\alpha :G\curvearrowright A$ commuting with $\delta $ in the sense that

$$ \begin{align*}\delta(\alpha_s(a))=(\alpha_s\otimes \operatorname{{\mathrm{id}}}_G)(\delta(a))\quad\text{ for all }a\in A\text{ and }s\in G.\end{align*} $$

Given such an action $\alpha $ , the equation

$$ \begin{align*} \tilde\alpha_s(j_A(a)j_{C_0(G)}(f)):=j_A(\alpha_s(a))j_{C_0(G)}(f),\quad a\in A, f\in C_0(G), \end{align*} $$

determines an action $\tilde \alpha :G\curvearrowright B=A\rtimes _{\delta }\widehat {G}$ that commutes with the dual action $\widehat {\delta }$ (meaning $\tilde \alpha _s\circ \widehat {\delta }_t=\widehat {\delta }_t\circ \tilde \alpha _s$ for all $s,t\in G$ ) and fixes $\phi =j_{C_0(G)}$ . Starting then from the weak $G\rtimes G$ -algebra $(B,\beta ,\phi )=(A\rtimes _\delta \widehat {G}, \widehat {\delta }, j_{C_0(G)})$ , we obtain a new weak $G\rtimes G$ -algebra $(B, \tilde \alpha \cdot \beta ,\phi )$ with $(\tilde \alpha \cdot \beta )_s:=\tilde \alpha _s\circ \beta _s$ for all $s\in G$ . Conversely, [Reference Buss and Echterhoff7, Lemma 3.3] shows that every action $\eta :G\curvearrowright B$ that commutes with $\beta =\widehat {\delta }$ and fixes $\phi $ is equal to $\tilde \alpha $ for some action $\alpha :G\curvearrowright A$ commuting with $\delta $ .

Definition 4.1. Let $(A,\delta )$ be a $\mu $ -coaction with respect to a duality crossed-product functor $\rtimes _\mu $ and let $\alpha :G\curvearrowright A$ be an action that commutes with $\delta $ as above. Let $(B,\beta , \phi )=(A\rtimes _\delta \widehat {G}, \widehat {\delta }, j_{C_0(G)})$ and let $(A^\alpha , \delta ^\alpha )$ be the coaction obtained from the weak $G\rtimes G$ -algebra $(B, \tilde \alpha \cdot \beta ,\phi )$ via Landstad duality with respect to $\rtimes _\mu $ . Then we call $(A^\alpha , \delta ^\alpha )$ the Abadie–Exel deformation of $(A,\delta )$ with respect to $\alpha $ .

We call this the Abadie–Exel deformation because it covers the deformation of cross-sectional algebras of Fell bundles as studied by Abadie and Exel in [Reference Abadie and Exel1], as we show in the next subsection.

4.2 Abadie–Exel deformation via Fell bundles

Let $\mathcal A=(A_s)_{s\in G}$ be a Fell bundle over the locally compact group G. Abadie and Exel considered in [Reference Abadie and Exel1] continuous actions $\alpha :G\curvearrowright \mathcal A$ of G by automorphisms of the Fell bundle $\mathcal A$ . This means that for each $s,t\in G$ we get a Banach-space isomorphism

$$ \begin{align*}\alpha_s^t:A_t\to A_t; a_t\mapsto \alpha^t_s(a_t)\end{align*} $$

such that:

1. (1) $\alpha ^t_s(a_t)\alpha ^r_s(a_r)=\alpha ^{tr}_s(a_ta_r)$ and $\alpha ^{t^{-1}}_s(a_t^*)=\alpha _s^t(a_t)^*$ for all $a_t\in A_t, a_r\in A_r$ ;

2. (2) for each fixed section $a\in C_c(\mathcal A)$ the section $\alpha _s(a)$ defined by

   $$ \begin{align*}(\alpha_s(a))_t:=\alpha_s^t(a_t)\end{align*} $$
   is continuous; and

3. (3) the function $s\mapsto \alpha _s(a)$ is continuous with respect to $\|\cdot \|_{\mathrm \max }$ , the universal norm on $C_c(\mathcal A)$ (for this it suffices that the function $s\mapsto \alpha _s(a)$ is continuous in the $L^1$ -norm on $C_c(\mathcal A)$ ).

These properties imply that $\alpha $ induces an action (also called $\alpha $ ) of G on $C^*(\mathcal A)$ , which commutes with $\delta _{\mathcal A}$ since $\delta _{\mathcal A}\circ \alpha _s$ is the integrated form of the representation $a_t\mapsto \alpha ^t_s(a_t)\otimes u_t$ that coincides with the representation $(\alpha _s\otimes \operatorname {{\mathrm {id}}}_G)\circ \delta _{\mathcal A}$ .

Now let $\rtimes _\mu $ be a duality crossed-product functor for G. Since

$$ \begin{align*}(C_\mu^*(\mathcal A)\rtimes_{\delta_\mu}\widehat{G}, \widehat{\delta_\mu}, j_{C_0(G)})=(C^*(\mathcal A)\rtimes_{\delta_{\mathcal A}}\widehat{G}, \widehat{\delta_{\mathcal A}}, j_{C_0(G)})=:(B,\beta,\phi)\end{align*} $$

it follows from [Reference Buss and Echterhoff7, Lemma 3.3] (using both directions of that lemma) that the action $\alpha $ factors through an action (still called $\alpha $ ) on $C_\mu ^*(\mathcal A)$ , which commutes with $\delta _\mu $ . We then get the deformed cosystem $(A^\alpha , \delta ^\alpha )$ as in Definition 4.1 starting from $(A,\delta ):=(C_\mu ^*(\mathcal A), \delta _\mu )$ .

On the other hand, using the action $\alpha $ on $\mathcal A$ , we can now define a new multiplication and involution on the Fell bundle $\mathcal A$ by

$$ \begin{align*}a_s*_\alpha a_t=a_s\alpha_s(a_t)\quad\text{and}\quad a_s^{*_\alpha}=\alpha_{s^{-1}}(a_s^*),\end{align*} $$

where from now on we simply write $\alpha _s(a_t)$ instead of $\alpha _s^t(a_t)$ . We write ${\mathcal A}_\alpha $ for the new Fell bundle obtained in this way. We write $\delta ^\alpha _\mu $ for the dual coaction on $C_\mu ^*({\mathcal A}_\alpha )$ . Similar to our notation for the Fell bundle $\mathcal A$ , we write $(C^*({\mathcal A}_\alpha ), \delta _{{\mathcal A}_\alpha })$ for the maximal coaction $(C_{\mathrm \max }^*({\mathcal A}_\alpha ), \delta ^\alpha _{\mathrm \max })$ .

Theorem 4.2. There is a canonical isomorphism between the cosystems $(C_\mu ^*({\mathcal A}_\alpha ),\delta _\mu ^\alpha )$ and $(A^\alpha ,\delta ^\alpha )$ as constructed above from the action $\alpha :G\curvearrowright \mathcal A$ .

Remark 4.3. In [Reference Abadie and Exel1] Abadie and Exel define the deformed Fell bundles ${\mathcal A}_\alpha $ as above for arbitrary locally compact groups, but only discuss their cross-sectional $C^*$ -algebras for discrete abelian groups. They then define the deformation of $(A,\delta )$ via $\alpha :G\curvearrowright \mathcal A$ as the cosystem $(C^*({\mathcal A}_\alpha ), \delta _{{\mathcal A}_\alpha })$ . The above construction extends this to arbitrary locally compact groups and other possible completions of $C_c(\mathcal A)$ and $C_c({\mathcal A}_\alpha )$ , respectively.

Proof of Theorem 4.2.

Recall that we constructed $(A^\alpha , \delta ^\alpha )$ via Landstad duality for the weak $G\rtimes G$ -algebra $(B, \tilde \alpha \cdot \beta , \phi )$ with respect to $\rtimes _\mu $ , with

$$ \begin{align*}(B,\beta ,\phi ):=(C^*(\mathcal A)\rtimes _{\delta _{\mathcal A}}\widehat {G}, \widehat {\delta _{\mathcal A}}, j_{C_0(G)}).\end{align*} $$

On the other hand, by Proposition 3.1, the cosystem $(C_\mu ^*({\mathcal A}_\alpha ),\delta _\mu ^\alpha )$ can be constructed similarly via the weak $G\rtimes G$ algebra

$$ \begin{align*}(B^\alpha, \beta^\alpha, \phi^\alpha):=(C^*({\mathcal A}_\alpha)\rtimes_{\delta_{{\mathcal A}_\alpha}}\widehat{G},\widehat{\delta_{{\mathcal A}_\alpha}}, j^\alpha_{C_0(G)}).\end{align*} $$

So the result will follow immediately from Proposition 2.1 if we can show that

$$ \begin{align*} (C^*(\mathcal A)\rtimes_{\delta_{\mathcal A}}\widehat{G}, \tilde\alpha\cdot \widehat{\delta_{\mathcal A}}, j_{C_0(G)})\cong (C^*({\mathcal A}_\alpha)\rtimes_{\delta_{{\mathcal A}_\alpha}}\widehat{G},\widehat{\delta_{{\mathcal A}_\alpha}}, j^\alpha_{C_0(G)}) \end{align*} $$

as weak $G\rtimes G$ -algebras. Using Proposition 3.2, for this it suffices to show that there is an isomorphism between $\Bbbk _c({\mathcal A})$ and $\Bbbk _c({{\mathcal A}_\alpha })$ that is isometric for $\|\cdot \|_2$ and that intertwines the relevant actions of G and $C_0(G)$ , respectively. For this we define

$$ \begin{align*}\Phi_\alpha:\Bbbk_c({{\mathcal A}_\alpha})\to \Bbbk_c({\mathcal A}),\quad \Phi_\alpha(k)(s,t)=\alpha_{s^{-1}}(k(s,t))\end{align*} $$

for all $k\in \Bbbk _c({{\mathcal A}_\alpha })$ . This is clearly a linear bijection that is isometric for the norm $\|\cdot \|_2$ . But it is also a $^*$ -homomorphism: for $k,l\in \Bbbk _c({{\mathcal A}_\alpha })$ we compute

$$ \begin{align*} \Phi_\alpha(k*l)(s,t)&=\alpha_{s^{-1}}(k*l(s,t))=\alpha_{s^{-1}}\bigg(\int_G k(s,r)\alpha_{sr^{-1}}(l(r,t))\,d r\bigg)\\ &=\int_G\alpha_{s^{-1}}(k(s,r))\alpha_{r^{-1}}(l(r,t))\,d r\\ &=\Phi_\alpha(k)*\Phi_\alpha(l)(s,t), \end{align*} $$

for all $s,t\in G$ . We also have

$$ \begin{align*} \Phi_\alpha(k^*)(s,t)&=\alpha_{s^{-1}}(k^*(s,t))=\alpha_{s^{-1}}(k(t,s)^{*_\alpha})\\ &=\alpha_{t^{-1}}(k(t,s))^*=\Phi_\alpha(k(t,s))^*=\Phi_\alpha(k)^*(s,t). \end{align*} $$

This proves the existence of the isomorphism $\Phi _\alpha $ as stated in the theorem. It is trivial to check that $\Phi _\alpha $ intertwines the $C_0(G)$ -action $j_{C_0(G)}^\alpha $ with $j_{C_0(G)}$ and that it is ${\widehat {\delta _{{\mathcal A}_\alpha }}-\tilde \alpha \cdot \widehat {\delta }_{\mathcal A}}$ -equivariant. Hence, it preserves the weak $G\rtimes G$ -structures on both algebras. Since Landstad duality with respect to $\rtimes _\mu $ is functorial by [Reference Buss and Echterhoff5], this finishes the proof.

Example 4.4. Let A be a $C^*$ -algebra and consider the trivial Fell bundle $\mathcal A=A\times G$ over G with fibres $A_t=A\times \{t\}\cong A$ for all $t\in G$ and with multiplication and involution given by $(a,s)(b,t)=(ab,st)$ and $(a,s)^*=(a^*, s^{-1})$ . Any G-action $\alpha \colon G\curvearrowright A$ extends to a G-action on $\mathcal A$ by $\alpha ^t_s(a, t):=(\alpha _s(a),t)$ . The induced G-action on $C^*(\mathcal A)\cong A\otimes _{\mathrm \max } C^*(G)$ is just $\alpha \otimes \operatorname {{\mathrm {id}}}$ , and the dual G-coaction on $C^*(\mathcal A)$ corresponds to $\operatorname {{\mathrm {id}}}\otimes \delta _G$ . The $\alpha $ -deformed Fell bundle ${\mathcal A}_\alpha $ is easily seen to be isomorphic to the semidirect product Fell bundle ${\mathcal A}_\alpha =A\times _\alpha G$ associated to the action $\alpha \colon G\curvearrowright A$ . From our general Theorem 4.2 we get that the $\alpha $ -deformation of the coaction $(C^*(\mathcal A),\delta _{\mathcal A})\cong (A\otimes _{\mathrm \max } C^*(G),\operatorname {{\mathrm {id}}}\otimes \delta _G)$ is isomorphic to $(C^*({\mathcal A}_\alpha ),\delta _{{\mathcal A}_\alpha })\cong (A\rtimes _{\alpha ,\mathrm \max }G,\widehat {\alpha })$ . Similarly, taking reduced norms, we get that $(C^*_r(\mathcal A),\delta _{\mathcal A}^r)\cong (A\otimes C^*_r(G),\operatorname {{\mathrm {id}}}\otimes \delta _G^r)$ is deformed into $(A\rtimes _{\alpha ,r}G,\widehat {\alpha }_r)$ , and a similar result holds for exotic crossed products (where, in general, $A\rtimes _{\operatorname {{\mathrm {id}}},\mu }G$ has no obvious description as a ‘standard’ tensor product).

More generally, we can start with an arbitrary action $\beta \colon G\curvearrowright A$ on a fixed $C^*$ -algebra A, and then any other G-action $\alpha \colon G\curvearrowright A$ commuting with $\beta $ gives an action on the Fell bundle ${\mathcal A}_\beta =A\times _\beta G$ with $\alpha $ -deformed Fell bundle $({\mathcal A}_\beta )_\alpha ={\mathcal A}_{\beta \cdot \alpha }=A\times _{\beta \cdot \alpha }G$ , where $\beta \cdot \alpha \colon G\curvearrowright A$ is defined by $(\beta \cdot \alpha )_t:=\beta _t\circ \alpha _t=\alpha _t\circ \beta _t$ . In this situation, given a duality crossed-product functor $\rtimes _\mu $ for G, we deform the dual G-coaction on $C^*_\mu ({\mathcal A}_\beta )=A\rtimes _{\beta ,\mu }G$ into the dual G-coaction on $C^*_\mu ({\mathcal A}_{\beta \cdot \alpha })=A\rtimes _{\beta \cdot \alpha ,\mu }G$ .

Example 4.5. As a particular example, consider the trivial Fell bundle $C(\mathbb T)\times \mathbb Z$ and the action $\alpha _\theta $ of $\mathbb Z$ on $C(\mathbb T)$ by rotations for a fixed angle $\theta \in \mathbb R$ . In this case we deform $C(\mathbb T)\otimes C^*(\mathbb Z)\cong C(\mathbb T^2)$ with the dual $\mathbb T$ -action (that is, a $\mathbb Z$ -coaction) into the rotation algebra $C(\mathbb T^2_\theta ):=A\rtimes _{\alpha _\theta }\mathbb Z$ with its dual $\mathbb T$ -action. It is interesting to notice that $\mathbb T$ -actions (that is, $\mathbb Z$ -coactions) cannot be deformed using $2$ -cocycles (that is, twists) as in the next section because the group $\mathbb Z$ carries no nontrivial $2$ -cocycles. But we do get $C(\mathbb T^2_\theta )$ as a cocycle deformation of $C(\mathbb T^2)\cong C^*(\mathbb Z^2)$ , considering it endowed with its canonical (dual) $\mathbb T^2$ -action and deforming it with respect to the $2$ -cocycle on $\mathbb Z^2$ given by $\omega _\theta ((n,m),(k,l))=e^{2\pi i\theta mk}$ .

We see from the above family of examples that deformation of Fell bundles in the Abadie–Exel sense is quite general. In particular, we should not expect as many permanence results to hold in this setting, as we have for deformation by $2$ -cocycles as studied in [Reference Buss and Echterhoff7] or in Section 5 below.

Of course, one can also combine Abadie–Exel deformations with cocycle deformations to obtain more examples. The numerous examples as discussed by Abadie and Exel in [Reference Abadie and Exel1] show that this theory has many interesting applications.

5 Deformation by twists

Here we recall the deformation of coactions via twists, following the approach introduced in [Reference Buss and Echterhoff7]. Let G be a locally compact group. We first recall that a twist over G is a central extension

(5-1) $$ \begin{align} \sigma=(\mathbb T\stackrel{\iota}{\hookrightarrow} G_\sigma\stackrel{q}{\twoheadrightarrow} G) \end{align} $$

of G by the circle group $\mathbb T$ . In what follows, we often omit the embedding $\iota $ in our notations by simply identifying $\mathbb T$ as a (central) normal subgroup of $G_\sigma $ and q with the quotient map $G_\sigma \twoheadrightarrow G_\sigma /\mathbb T\cong G$ .

Given a Borel section $\mathfrak {s}\colon G\to G_\sigma $ for q which satisfies $\mathfrak {s}(e)=1$ , we obtain a (normalized) Borel cocycle $\omega =\omega _\sigma \in Z^2(G,\mathbb T)$ by $\omega (g,h):=\mathfrak {s}(g)\mathfrak {s}(h)\mathfrak {s}(gh)^{-1}$ . The cohomology class $[\omega _\sigma ]\in H^2(G,\mathbb T)$ does not depend on the choice of the Borel section $\mathfrak {s}$ and we obtain an isomorphism $[\sigma ]\leftrightarrow [\omega _\sigma ]$ between the group $\operatorname {{\mathrm {Twist}}}(G)$ of isomorphism classes $[\sigma ]$ of twists, equipped with the Baer multiplication of extensions, and the group $H^2(G,\mathbb T)$ . We refer to [Reference Buss and Echterhoff7, Section 4] for a detailed discussion. There it is also shown that $\operatorname {{\mathrm {Twist}}}(G)\cong H^2(G,\mathbb T)$ is isomorphic to the Brauer group $\operatorname {{\mathrm {Br}}}(G)$ of Morita equivalence classes of actions $\alpha :G\curvearrowright \mathbb K({\mathcal H})$ on algebras of compact operators on some Hilbert space ${\mathcal H}$ . It sends a class $[\omega ]\in H^2(G,\mathbb T)$ to the Morita equivalence class $[\alpha ]$ of the action $\alpha =\text {Ad}\rho ^{\bar \omega }:G\curvearrowright \mathbb K(L^2(G))$ , where ${\rho ^{\bar \omega }:G\curvearrowright \mathcal U(L^2(G))}$ is the $\bar \omega $ -twisted right regular representation of G given by the formula $(\rho ^{\bar \omega }_s\xi )(t)=\Delta (s)^{1/2}\bar \omega (t,s)\xi (ts)$ , and $\bar \omega $ is the complex conjugate (that is, inverse) of $\omega $ . In this paper we mostly use the picture of twists given by central extensions (5-1), and sometimes also use $2$ -cocycles.

Given a twist $G_\sigma $ over G, we can consider the Green twisted action of $(G_\sigma ,\mathbb T)$ on $\mathbb C$ given by the pair $(\operatorname {{\mathrm {id}}}_{\mathbb C}, \iota ^\sigma )$ in which $\iota ^\sigma :\mathbb T\to \mathbb T=U(\mathbb C)$ denotes the identity map. Then, if $\beta :G\curvearrowright B$ is any action of G, we can twist $\beta $ with $(\operatorname {{\mathrm {id}}}_{\mathbb C},\iota ^\sigma )$ by taking the diagonal twisted action

(5-2) $$ \begin{align} (\beta,\iota^\sigma):=\beta\otimes (\operatorname{{\mathrm{id}}}_{\mathbb C},\iota^\sigma):(G_\sigma, \mathbb T)\curvearrowright B\otimes \mathbb C\cong B, \end{align} $$

where we identify $\beta $ with its inflation $\beta \circ q\colon G_\sigma \curvearrowright B$ . Note that in this setting we have $\iota ^\sigma _z b=zb$ for all $z\in \mathbb T, b\in B$ .

We now fix a duality crossed-product functor $\rtimes _\mu $ for G and a $\mu $ -coaction $(A,\delta )$ . Consider, as before, the corresponding weak $G\rtimes G$ -algebra

$$ \begin{align*}(B,\beta,\phi)=(A\rtimes_{\delta}\widehat{G}, \widehat{\delta}, j_{C_0(G)}).\end{align*} $$

We want to use this twisted action to construct a deformed weak $G\rtimes G$ -algebra $(B_\sigma ,\beta _\sigma , \phi _\sigma )$ as follows. We first observe that the twist $\sigma =(\mathbb T\hookrightarrow G_\sigma \stackrel q\twoheadrightarrow G)$ determines a complex line bundle ${\mathcal L}_\sigma $ over G given by the quotient space

(5-3) $$ \begin{align} {\mathcal L}_\sigma=G_\sigma\times_{\mathbb T}\mathbb C:=(G_\sigma\times \mathbb C)/\mathbb T, \end{align} $$

with respect to the action $\mathbb T\curvearrowright G_\sigma \times \mathbb C; \;z\cdot (x, u)=(\bar zx, zu)$ . The $C_0$ -sections of this bundle then naturally identify with the space of functions

$$ \begin{align*} C_0(G_\sigma, \iota):=\{f\in C_0(G_\sigma): f(zx)=\bar{z} f(x)\text{ for all } x\in G_\sigma, z\in \mathbb T\}. \end{align*} $$

We observe in [Reference Buss and Echterhoff7, Remark 5.6] that $C_0(G_\sigma , \iota )$ becomes a $C_0(G)-C_0(G)$ imprimitivity Hilbert bimodule and the right translation action $\mathrm {rt}^\sigma :G_\sigma \curvearrowright C_0(G_\sigma , \iota )$ implements a (Morita) equivalence between the right translation action $\mathrm {rt}: G\curvearrowright C_0(G)$ on the left and the twisted right translation action $(\mathrm {rt},\iota ^\sigma ):(G_\sigma ,\mathbb T)\curvearrowright C_0(G)$ on the right. We then observe in [Reference Buss and Echterhoff7, Remark 5.11] that for the weak $G\rtimes G$ -algebra $(B,\beta , \phi )$ , the internal tensor product

$$ \begin{align*}{\mathcal L}(G_\sigma,B):= C_0(G_\sigma,\iota)\otimes_{C_0(G)}B\end{align*} $$

is a full Hilbert B-module and the diagonal action $\mathrm {rt}^\sigma \otimes \beta $ of $G_\sigma $ on ${\mathcal L}(G_\sigma ,B)$ is compatible with the twisted action $(\beta ,\iota ^\sigma ):(G_\sigma ,\mathbb T)\curvearrowright B$ as introduced in (5-2) above. The adjoint action $\beta _\sigma :=\text {Ad}(\mathrm {rt}^\sigma \otimes \beta )$ then factors through an ordinary action of G on $B_\sigma :=\mathbb K({\mathcal L}(G_\sigma ,B))$ and $({\mathcal L}(G_\sigma ,B), \mathrm {rt}^\sigma \otimes \beta )$ becomes a $\beta _\sigma -(\beta , \iota ^\sigma )$ -equivariant Morita equivalence.

Together with the left action of $\phi _\sigma : C_0(G)\to \mathbb B({\mathcal L}(G_\sigma ,B))\cong \mathcal M(B_\sigma )$ , which is induced from the left action of $C_0(G)$ on $C_0(G_\sigma ,\iota )$ , the triple $(B_\sigma , \beta _\sigma , \phi _\sigma )$ becomes a weak $G\rtimes G$ -algebra. The following definition is [Reference Buss and Echterhoff7, Definition 5.8].

In other words, $(A^\sigma _\mu ,\delta ^\sigma _\mu )$ is the unique $\mu $ -coaction for which there exists an isomorphism of weak $G\rtimes G$ -algebras

$$ \begin{align*}(A^\sigma_\mu\rtimes_{\delta^\sigma_\mu}\widehat{G},\widehat{\delta^\sigma_\mu},\phi_{A^\sigma_\mu})\cong (B_\sigma,\beta_\sigma,\phi_\sigma).\end{align*} $$

If the twist $\sigma =(\mathbb T\hookrightarrow G_\sigma \stackrel {q}\twoheadrightarrow G)$ splits via a continuous section $\mathfrak {s}\colon G\to G_\sigma $ , then $\sigma $ is represented in $H^2(G,\mathbb T)$ by the class of the continuous $2$ -cocycle $\omega (g,h)=\partial \mathfrak {s}(g,h)=\mathfrak {s}(g)\mathfrak {s}(h)\mathfrak {s}(gh)^{-1}$ . In this case the deformed weak $G\rtimes G$ -algebra $(B_\sigma ,\beta _\sigma ,\phi _\sigma )$ is isomorphic to the triple $(B, \beta ^\omega ,\phi )$ in which $\beta ^\omega :G\curvearrowright B$ is given by the formula

(5-4) $$ \begin{align} \beta^\omega:G\to \operatorname{\mathrm{Aut}}(B); \quad\beta^\omega(s)=\text{Ad} U_\omega(s)\circ \beta(s), \end{align} $$

where $U_\omega :G\to U\mathcal M(B)$ is defined by $U_\omega (s)= \phi (u_\omega (s))$ , with $u_\omega (s)\in C_b(G,\mathbb T)=U\mathcal M(C_0(G))$ given by

$$ \begin{align*} u_\omega(s)(r)=\overline{\omega(r,s)}. \end{align*} $$

We refer to [Reference Buss and Echterhoff7, Section 3.3] for further details.

Indeed, for a general twist $\sigma =(\mathbb T\hookrightarrow G_\sigma \stackrel {q}\to G)$ , by [Reference Feldman and Greenleaf17, Theorem 1], we can always choose a Borel section $\mathfrak {s}\colon G\to G_\sigma $ for q with corresponding Borel cocycle $\omega =\partial \mathfrak {s}$ and it may happen that the action (5-4) makes sense even if $\omega $ is not continuous. This is the case, for example, if the structural homomorphism $\phi \colon C_0(G)\to \mathcal M(B)$ extends to a G-equivariant unital homomorphism $\bar \phi \colon L^\infty (G)\to \mathcal M(B)$ ; this is the content of [Reference Buss and Echterhoff7, Proposition 5.13]. We will apply this in the next section to the kernel algebras $\Bbbk ({\mathcal A})$ using Corollary 3.5 in order to deform dual coactions on cross-sectional $C^*$ -algebras of Fell bundles $\mathcal A$ over G.

6 Deformation of Fell bundles by cocycles

Let $\rtimes _\mu $ be a duality crossed-product functor for a locally compact group G. Given a Fell bundle $\mathcal A$ over G, we consider its $C^*$ -cross-sectional algebra $C_\mu ^*(\mathcal A)$ , which comes equipped with the dual $\mu $ -coaction $\delta :C_\mu ^*(\mathcal A)\to \mathcal M(C_\mu ^*(\mathcal A)\otimes C^*(G))$ given by the integrated form of the map $a_s\mapsto a_s\otimes u_s$ for $a_s\in A_s$ .

In this section we want to show that the deformation of $C_\mu ^*(\mathcal A)$ via this coaction and a twist $\sigma =(\mathbb T\hookrightarrow G_\sigma \twoheadrightarrow G)$ can be described directly on the level of the Fell bundle $\mathcal A$ itself – similarly to what happened in Section 4.1 for the case of Abadie–Exel deformation. This result will generalize [Reference Buss and Echterhoff7, Proposition 5.16], which in turn generalizes [Reference Bhowmick, Neshveyev and Sangha4, Proposition 4.3], where a similar result has been shown for dual coactions of reduced crossed products, and also results from [Reference Yamashita36] for the case of reduced cross-sectional $C^*$ -algebras of Fell bundles over discrete groups.

So in what follows let $\mathcal A=(A_g)_{g\in G}$ be a Fell bundle over G and $A=C_\mu ^*(\mathcal A)$ with respect to some duality crossed-product functor $\rtimes _\mu $ for G. We construct the deformed Fell bundle ${\mathcal A}_\sigma $ via the twist $\sigma =(\mathbb T\stackrel {\iota }{\hookrightarrow } G_\sigma \stackrel {q}{\twoheadrightarrow } G)$ as follows. Consider the pullback Fell bundle over $G_\sigma $ that can be realized as

$$ \begin{align*}q^*(\mathcal A)=\{(a,\tilde g)\in \mathcal A\times G_\sigma: p(a)=q(\tilde g)\},\end{align*} $$

where $p\colon \mathcal A\to G$ denotes the bundle projection. The circle group $\mathbb T$ acts ‘diagonally’ on $q^*(\mathcal A)$ via

$$ \begin{align*}z\cdot (a,\tilde g):=(\bar z a,z\tilde g),\quad z\in\mathbb T,\, a\in \mathcal A,\, \tilde g\in G_\sigma.\end{align*} $$

Definition 6.1. We define the twisted Fell bundle ${\mathcal A}_\sigma $ as the quotient Fell bundle

$$ \begin{align*}{\mathcal A}_\sigma:=q^*(\mathcal A)/\mathbb T\end{align*} $$

with respect to the diagonal $\mathbb T$ -action defined above. More concretely, this consists of equivalence classes $[a,\tilde g]$ , with the bundle projection

$$ \begin{align*}p^\sigma\colon {\mathcal A}_\sigma\to G; \; p^\sigma[a,\tilde g]:=p(a)=q(\tilde g).\end{align*} $$

The algebraic operations on ${\mathcal A}_\sigma $ are also induced from those in $q^*(\mathcal A)$ , that is,

$$ \begin{align*}\lambda\cdot [a,\tilde g]+[a',\tilde g]:=[\lambda a+a',\tilde g],\end{align*} $$

$$ \begin{align*}[a,\tilde g]\cdot [b,\tilde h]:=[ab,\tilde g\tilde h], \quad [a,\tilde g]^*:=[a^*,\tilde g^{-1}],\end{align*} $$

for all $\lambda \in \mathbb C$ , $a,a'\in A_g$ , $b\in A_h, \tilde g,\tilde h\in G_\sigma $ , $g=q(\tilde g)$ and $h=q(\tilde h)$ .

It is a well-known result that ${\mathcal A}_\sigma $ is a Fell bundle over G with respect to the above operations; see, for instance, [Reference Kaliszewski, Muhly, Quigg and Williams21, Corollary A.12].

We will now describe the space of sections of the twisted Fell bundle ${\mathcal A}_\sigma $ . We start with the following lemma.

Lemma 6.3. There is a bijection between the sections $\xi :G\to {\mathcal A}_\sigma $ and the set of sections

$$ \begin{align*} S(G_\sigma, \mathcal A,\iota):=\{\xi:G_\sigma\to \mathcal A: \xi({\tilde{g}})\in A_{q({\tilde{g}})}\text{ and }\xi({\tilde{g}} z)=\bar{z}\xi({\tilde{g}})\; \text{ for all } {\tilde{g}}\in G_\sigma, z\in \mathbb T\} \end{align*} $$

for $q^*\mathcal A$ that assigns to each section $\xi \in S(G_\sigma , \mathcal A,\iota )$ the section

(6-1) $$ \begin{align} \tilde\xi:G\to {\mathcal A}_{\sigma}; \; \tilde\xi(g)=[\xi({\tilde{g}}), {\tilde{g}}]\quad \text{with }{\tilde{g}}\in q^{-1}(g). \end{align} $$

Moreover, $\xi $ is continuous (respectively, measurable, compactly supported) if and only if $\tilde \xi $ is continuous (respectively, measurable, compactly supported).

Proof. We first check that the section $\tilde \xi :G\to {\mathcal A}_\sigma $ as in (6-1) is well defined. Indeed, if ${\tilde {g}}, g'\in q^{-1}(g)\in G_{\sigma }$ , then there exists a unique $z\in \mathbb T$ such that $g'={\tilde {g}} z$ , so that

$$ \begin{align*}[\xi(g'), g']=[\tilde\xi({\tilde{g}} z), {\tilde{g}} z]=[\bar{z}\xi({\tilde{g}}), z{\tilde{g}}]=[\xi({\tilde{g}}), {\tilde{g}}].\end{align*} $$

Thus $\tilde \xi $ sends the element $g\in G$ to a well-defined element in the fibre $A_{\sigma ,g}$ of $A_\sigma $ . Conversely, if $\tilde \xi :G\to {\mathcal A}_\sigma $ is any section, then for each $g\in G$ there exist $a_g\in A_g$ and $g'\in q^{-1}(g)$ such that $\tilde \xi (g)=[a_g, g']\in A_{\sigma ,g}$ . But then, for any ${\tilde {g}}\in q^{-1}(g)$ , there is a unique $z\in \mathbb T$ such that $g'={\tilde {g}} z$ and then we get

$$ \begin{align*}[a_g, g']=[a_g, {\tilde{g}} z]=[za_g, {\tilde{g}}].\end{align*} $$

It follows that $\xi ({\tilde {g}}):=za_g$ is the unique element in $A_g$ such that $\tilde \xi (g)=[\xi ({\tilde {g}}), {\tilde {g}}]$ . We therefore obtain a well-defined element $\xi \in S(G_\sigma ,\mathcal A,\iota )$ that satisfies (6-1).

We now show that continuity of $\tilde \xi $ implies continuity of $\xi $ . For this let $({\tilde {g}}_i)$ be a net in $G_\sigma $ that converges to ${\tilde {g}}\in G_\sigma $ , and let us write $g_i=q({\tilde {g}}_i)$ and $g={\tilde {g}}$ . Then $g_i\to g$ and continuity of $\tilde \xi $ implies that $\tilde \xi (g_i)=[\xi ({\tilde {g}}_i), {\tilde {g}}_i]\to [\xi ({\tilde {g}}), {\tilde {g}}]=\tilde \xi (g)$ . Since the quotient map $q^*(\mathcal A)\twoheadrightarrow q^*(\mathcal A)/\mathbb T={\mathcal A}_\sigma $ is open we may assume, after passing to a subnet if necessary, that there is a net $(z_i)$ in $\mathbb T$ such that $z_i(\xi ({\tilde {g}}_i),{\tilde {g}}_i)= (\xi (z_i{\tilde {g}}_i), z_i{\tilde {g}}_i) \to (\xi ({\tilde {g}}),{\tilde {g}})$ in $q^*(\mathcal A)$ . But since $\mathbb T$ is compact we may assume without loss of generality that $z_i\to z$ for some $z\in \mathbb T$ . But then it follows that ${\tilde {g}}_i\to \bar {z}{\tilde {g}}$ , and since ${\tilde {g}}_i\to {\tilde {g}}$ by assumption, we get $z=1$ and hence $\lim _i \xi ({\tilde {g}}_i)=\lim _i\xi (z_i{\tilde {g}}_i)=\xi ({\tilde {g}})$ , which proves continuity of $\xi $ .

Conversely, if $\xi $ is continuous and $g_i\to g$ in G, then using openness of $q:G_\sigma \twoheadrightarrow G$ , we may pass to a subnet to find a net $({\tilde {g}}_i)$ in $G_\sigma $ and ${\tilde {g}}\in G_\sigma $ such that ${\tilde {g}}_i\to {\tilde {g}}$ and $q({\tilde {g}}_i)=g_i, q({\tilde {g}})=g$ . It then follows that $\tilde \xi (g_i)=[\xi ({\tilde {g}}_i), {\tilde {g}}_i]\to [\xi ({\tilde {g}}),{\tilde {g}}]=\tilde \xi (g)$ and therefore $\tilde \xi $ is continuous as well.

To see that $\tilde \xi $ is measurable if and only if $\xi $ is measurable we recall from [Reference Doran and Fell12, Proposition 15.4] that $\xi $ is measurable if for every compact subset K of $G_\sigma $ there exists a sequence $(\xi _n)$ of continuous sections such that $\xi _n({\tilde {g}})\to \xi ({\tilde {g}})$ for almost all ${\tilde {g}} \in K$ (and similarly for $\tilde \xi $ ). But $\xi _n({\tilde {g}})\to \xi ({\tilde {g}})$ for almost all ${\tilde {g}}\in K$ if and only if $\tilde \xi _n(g)\to \tilde \xi (g)$ for almost all $g\in q(K)$ , and the result follows.

Finally, it follows from (6-1) that $\operatorname {{\mathrm {supp}}}\xi =q^{-1}(\operatorname {{\mathrm {supp}}}\tilde \xi )$ . Thus $\xi $ has compact support if and only if $\tilde \xi $ has compact support.

Lemma 6.4. The convolution algebra $C_c({\mathcal A}_\sigma )$ is isomorphic to the algebra

$$ \begin{align*}C_c(G_\sigma, \mathcal A, \iota):=\{a\in S(G_\sigma,\mathcal A,\iota): a\text{ is continuous with compact supports}\}\end{align*} $$

with convolution and involution given by

$$ \begin{align*}(a*b)_{{\tilde{g}}}=\int_{G} a_{\tilde h} b_{\tilde{h}^{-1}{\tilde{g}}}\, \,d h\quad\text{and}\quad a^*_{{\tilde{g}}}=\Delta({\tilde{g}}) (a_{{\tilde{g}}^{-1}})^*,\end{align*} $$

where $\tilde {h}\in G_\sigma $ with $h=q(\tilde h)$ . Moreover, every section $a\in C_c(G_\sigma ,\mathcal A,\iota )$ can be written as a finite linear combination of functions of the form ${\tilde {g}}\mapsto f({\tilde {g}})\tilde {a}_g$ , $g=q({\tilde {g}})$ , with $f\in C_c(G_\sigma , \iota )$ and $\tilde {a}\in C_c(\mathcal A)$ .

Proof. It follows from Lemma 6.3 that

$$ \begin{align*}C_c(G_\sigma,\mathcal A,\iota)\to C_c({\mathcal A}_\sigma); a\mapsto (g\mapsto [a_{{\tilde{g}}},{\tilde{g}}])\;\text{if }g=q({\tilde{g}})\end{align*} $$

is a linear bijection. We need to check that it preserves convolution and involution. We first observe that by the conditions on the elements in $S(G_\sigma , \mathcal A, \iota )$ the integrand in the convolution integral in the lemma is constant on $\mathbb T$ -cosets in $G_\sigma $ . Therefore, the integral makes sense. Now, if $\tilde {a}:G\to {\mathcal A}_\sigma $ is given by $\tilde {a}_g=[a_{{\tilde {g}}}, {\tilde {g}}]$ for ${\tilde {g}}\in G_\sigma $ with $q({\tilde {g}})=g$ , and similarly for b, we compute

$$ \begin{align*} (\tilde{a}*\tilde{b})_g&=\int_G \tilde{a}_h\tilde{b}_{h^{-1}g}\,\,d h = \int_G [a_{\tilde{h}}, \tilde{h}][b_{\tilde{h}^{-1}{\tilde{g}}}, \tilde{h}^{-1}{\tilde{g}}]\,\,d h\\ &=\int_G [a_{\tilde{h}}b_{\tilde{h}^{-1}{\tilde{g}}},{\tilde{g}}]\,\,d h =[(a*b)_{{\tilde{g}}},{\tilde{g}}], \end{align*} $$

which settles the claim on the convolution. We leave it as an exercise to check that the involution is preserved as well.

For the final assertion we use the fact that by Gleason’s theorem [Reference Gleason18, Theorem 4.1] there always exists a local continuous section for the quotient map $q:G_\sigma \to G$ . Thus, given an element $a\in C_c(G_\sigma ,\mathcal A,\iota )$ , we can find a finite open cover $U_1,\ldots , U_l$ of $K:=q(\operatorname {{\mathrm {supp}}} a)\subseteq G$ together with continuous maps $\mathfrak {s}_i:U_i\to q^{-1}(U_i)$ such that $q\circ \mathfrak {s}_i=\operatorname {{\mathrm {id}}}_{U_i}$ for all $1\leq i\leq l$ . Let $\{\chi _i: 1\leq i\leq l\}\subseteq C_c^+(G)$ be a partition of the unit of K subordinate to $\{U_i: 1\leq i\leq l\}$ . For each ${\tilde {g}}\in q^{-1}(U_i)$ there exists a unique element $z_{{\tilde {g}},i}\in \mathbb T$ such that ${\tilde {g}}=z_{{\tilde {g}},i}\mathfrak {s}_i(g)$ , with $g=q({\tilde {g}})$ . We use this to define $f_i:G_\sigma \to \mathbb C$ by $f_i({\tilde {g}})=\bar {z}_{{\tilde {g}},i}\sqrt {\chi _i(g)}$ for ${\tilde {g}}=z_{{\tilde {g}},i}\mathfrak {s}_i(g)\in q^{-1}(U_i)$ and $0$ otherwise. Moreover, we define $\tilde {a}_i\in C_c(\mathcal A)$ by $\tilde {a}_i(g)=\sqrt {\chi _i(g)}a_{\mathfrak {s}_i(g)}$ for all $g\in U_i$ and $0$ otherwise. Then for all ${\tilde {g}}\in G_\sigma $ we obtain, with $g=q({\tilde {g}})$ ,

$$ \begin{align*} a({{\tilde{g}}})&=\sum_{i=1}^l\chi_i(g)a({{\tilde{g}}})=\sum_{g\in U_i}\chi_i(g)a({z_{{\tilde{g}},i}\mathfrak{s}_i(g)})\\ &=\sum_{g\in U_i} \bar{z}_{{\tilde{g}},i}\sqrt{\chi_i(g)}\sqrt{\chi_i(g)}a({\mathfrak{s}_i(g)}) =\sum_{i=1}^l f_i({\tilde{g}})\tilde{a}_i(g) \end{align*} $$

and the result follows.

In the special case where the extension $\sigma =(\mathbb T\hookrightarrow G_\sigma \stackrel {q}{\twoheadrightarrow } G)$ admits a continuous section $\mathfrak {s}:G\to G_\sigma $ for the quotient map q with $\mathfrak {s}(e)=e$ , the twisted Fell bundle ${\mathcal A}_\sigma $ has a much more direct description as follows. Let $\omega =\partial \mathfrak {s}\in Z^2(G,\mathbb T)$ denote the cocycle corresponding to $\sigma $ and $\mathfrak {s}$ . Since $\mathfrak {s}$ is continuous, $\omega $ is continuous as well. We may then define an $\omega $ -twisted multiplication on the Banach bundle $\mathcal A$ by the formulas

$$ \begin{align*} a_g\cdot_\omega a_h:=\omega(g,h)a_ga_h,\quad a_g^{*,\omega}:=\overline\omega(g^{-1},g)a_g^*,\quad a_g\in A_g,a_h\in A_h. \end{align*} $$

This gives a new Fell bundle structure on the Banach bundle $\mathcal A$ , which we denote by ${\mathcal A}_\omega $ . For the following proposition recall that for any fixed element ${\tilde {g}}\in q^{-1}(G)\subseteq G_\sigma $ the elements in the fibre $A_{\sigma ,g}$ have a unique representative of the form $(a_g, {\tilde {g}})\in q^*\mathcal A$ . In particular, if we apply this to ${\tilde {g}}=\mathfrak {s}(g)$ we can identify $A_g$ with $A_{\sigma ,g}$ via $a_g\mapsto [a_g, \mathfrak {s}(g)]$ .

Proposition 6.5. Let $\mathcal A$ , $\sigma $ and $\omega $ be as above. Then the map

$$ \begin{align*}\Theta: {\mathcal A}_\omega\to {\mathcal A}_\sigma: a_g\mapsto [a_g, \mathfrak{s}(g)]\end{align*} $$

is an isomorphism of Fell bundles.

Proof. As observed above, the map $\Theta $ is a bijection on each fibre, so we only need to see that it is a homeomorphism and preserves multiplication and involution. Indeed, arguments similar to those given in the proof of Lemma 6.3 show that $\Theta $ is a homeomorphism.

To check that $\Theta $ is multiplicative let $[a_g, \mathfrak {s}(g)]\in A_{\sigma , g}$ and $[b_h, \mathfrak {s}(h)]\in A_{\sigma ,h}$ . Then

$$ \begin{align*} \Theta(a_g\cdot_{\omega}b_h)&=\Theta(\overline{\omega(g,h)}a_gb_h)\\ &=[\overline{\omega(g,h)}a_gb_h, \mathfrak{s}(gh)]=[a_gb_h, \omega(g,h)\mathfrak{s}(gh)]\\ &=[a_gb_h, \mathfrak{s}(g)\mathfrak{s}(h)]=[a_g, \mathfrak{s}(g)][b_h, \mathfrak{s}(h)]. \end{align*} $$

A similar computation shows that $\Theta $ also preserves involution.

In what follows, if $\mathcal A$ is a Fell bundle over G, and $\sigma =(\mathbb T\hookrightarrow G_\sigma \stackrel {q}\twoheadrightarrow G)$ is a twist for G, then we want to prove that the algebras of kernels $\Bbbk ({\mathcal A})$ and $\Bbbk ({{\mathcal A}_\sigma })$ are isomorphic. Recall that $\Bbbk ({\mathcal A})\cong C^*(\mathcal A)\rtimes _{\delta _{\mathcal A}}\widehat {G}$ and that under this isomorphism the structural homomorphism $j_{C_0(G)}$ and the dual action $\widehat {\delta _{\mathcal A}}$ are given by

$$ \begin{align*}(j_{C_0(G)}(f)k)(g,h)=f(g)k(g,h) \quad \text{and}\quad (\widehat{\delta_{\mathcal A}}(g)k)(s,t)=\Delta(g)k(sg,tg).\end{align*} $$

Thus the dual weak $G\rtimes G$ -algebra $(C^*(\mathcal A)\rtimes _{\delta _{\mathcal A}}\widehat {G}, \widehat {\delta _{\mathcal A}}, j_{C_0(G)})$ is given by

$$ \begin{align*} (B,\beta,\phi)=(\Bbbk ({\mathcal A}), \widehat{\delta_{\mathcal A}}, j_{C_0(G)}). \end{align*} $$

Similarly, the dual weak $G\rtimes G$ -algebra for $(C^*({\mathcal A}_\sigma ), \delta _{{\mathcal A}_\sigma })$ is given by the triple

$$ \begin{align*}(C,\gamma,\psi)=(\Bbbk ({{\mathcal A}_\sigma}), \widehat{\delta_{{\mathcal A}_\sigma}}, j^\sigma_{C_0(G)}).\end{align*} $$

We aim to show that the $\sigma $ -twisted weak $G\rtimes G$ -algebra $(B_\sigma ,\beta _\sigma ,\phi _\sigma )$ of Definition 5.1 is isomorphic to $(C,\gamma ,\psi )$ . This will then show that for $(A,\delta )=(C_\mu ^*(\mathcal A), \delta _\mu )$ and any duality crossed-product functor $\rtimes _\mu $ the deformed cosystem $(A^\sigma , \delta ^\sigma )$ is isomorphic to $(C_\mu ^*({\mathcal A}_\sigma ), \delta ^\sigma _\mu )$ .

We start with a description of $\Bbbk _c({{\mathcal A}_\sigma })$ and $\Bbbk _{c,b} ({{\mathcal A}_\sigma })$ as follows.

Lemma 6.6. The algebra $\Bbbk _c({{\mathcal A}_\sigma })$ can be identified with the space of continuous compactly supported functions $k: G_\sigma \times G_\sigma \to \mathcal A$ satisfying the relations

(6-2) $$ \begin{align} k(z\tilde g, u\tilde h)=\bar{z}u k(\tilde g, \tilde h)\in A_{gh^{-1}}\quad \text{for all } z,u\in \mathbb T, {\tilde{g}} \in q^{-1}(g), {\tilde{h}}\in q^{-1}(h). \end{align} $$

Under this identification, convolution and involution are given by the formulas

(6-3) $$ \begin{align} \begin{aligned} k*l({\tilde{g}}, {\tilde{h}})&=\int_{G} k({\tilde{g}},{\tilde{r}})\cdot l({\tilde{r}},{\tilde{h}})\,d r\quad\text{and}\\ k^*({\tilde{g}}, {\tilde{h}})&=k({\tilde{h}}, {\tilde{g}})^*, \end{aligned} \end{align} $$

where $r=q({\tilde {r}})$ and the product $\cdot $ and involution $^*$ are taken from $\mathcal A$ . Similarly, the algebra $\Bbbk _{c,b} ({{\mathcal A}_\sigma })$ can be identified with the bounded measurable compactly supported functions satisfying (6-2).

Note that it follows from (6-2) that the integrand in the convolution integral only depends on $r=q({\tilde {r}})$ , so the formula makes sense.

Remark 6.7. Using Lemma 6.3, it follows from (3-1) that the kernel $k_{a,f}$ for $a\in C_c({\mathcal A}_\sigma )$ and $f\in C_c(G)$ can be identified with the function $k_{a,f}:G_\sigma \times G_\sigma \to \mathcal A$ given by

$$ \begin{align*} k_{a,f}(\tilde g,\tilde h)= a(\tilde g \tilde h^{-1})f(g)\Delta(g)^{-1}. \end{align*} $$

Since $(z\tilde g)(u\tilde h)^{-1}=z\overline u \tilde g\tilde h^{-1}$ and $a(z\tilde g)=\overline z a(\tilde g)\in A_g$ , these kernels satisfy (6-2).

From now on, we always identify $\Bbbk _c({{\mathcal A}_\sigma })$ with the set of functions $k:G_\sigma \times G_\sigma \to \mathcal A$ as in Lemma 6.6 above. In the following proposition let $\mathfrak {s}:G\to G_\sigma $ be any Borel cross-section for the quotient map $q:G_\sigma \to G$ , and let $\omega =\partial \mathfrak {s}\in Z^2(G,\mathbb T)$ be the associated Borel cocycle.

Proposition 6.8. Given a twist $\sigma =(\mathbb T\hookrightarrow G_\sigma \twoheadrightarrow G)$ and a Fell bundle $\mathcal A$ over G, with associated twisted Fell bundle ${\mathcal A}_\sigma $ , we have an isomorphism of kernel $C^*$ -algebras

(6-4) $$ \begin{align} \Phi\colon \Bbbk ({{\mathcal A}_\sigma})\xrightarrow\sim\Bbbk ({\mathcal A}),\quad k\mapsto \Phi(k)(g,h):=k(\mathfrak{s}(g),\mathfrak{s}(h)). \end{align} $$

The isomorphism $\Phi $ commutes with the canonical structural homomorphisms from $C_0(G)$ and sends the dual G-action $\widehat {\delta _{{\mathcal A}_\sigma }}$ on $\Bbbk ({{\mathcal A}_\sigma })$ to the G-action $\widehat {\delta }^\omega $ on $\Bbbk ({\mathcal A})$ given by

(6-5) $$ \begin{align} \widehat{\delta}^\omega_t(k)(g,h):=\Delta(t)\overline{\omega(g,t)}\omega(h,t)k(gt,ht) \end{align} $$

for all $k\in \Bbbk _{c,b} ({\mathcal A})$ , $t,g,h\in G$ .

Proof. The discussion preceding Proposition 3.2 shows that we can regard $\Bbbk ({\mathcal A})$ also as the enveloping $C^*$ -algebra of the Banach $^*$ -algebra $\Bbbk _2 ({\mathcal A})$ , the completion of the space $\Bbbk _{c,b} ({\mathcal A})$ with respect to the $L^2$ -norm $\|k\|_2=(\int _{G\times G} \| k(g,h)\|^2\,\,d g\,d h)^{1/2}$ . Now given the Borel section $\mathfrak {s}\colon G\to G_\sigma $ , we define

$$ \begin{align*}\Phi: \Bbbk_{c,b} ({{\mathcal A}_\sigma})\to\Bbbk_{c,b} ({\mathcal A});\quad \Phi(k)(g,h)=k(\mathfrak{s}(g), \mathfrak{s}(h)).\end{align*} $$

Straightforward computations show that this map preserves convolution and involution, and is isometric for $\|\cdot \|_2$ . To see that it is surjective, observe that for every $l\in \Bbbk _{c,b} ({\mathcal A})$ we can define $k\in \Bbbk _{c,b} ({{\mathcal A}_\sigma })$ by $k(\mathfrak {s}(g)z, \mathfrak {s}(h)u):=\bar {z}ul(g,h)$ , and then $l=\Phi (k)$ .

It follows that $\Phi $ extends to an isometric isomorphism between $\Bbbk _2 ({{\mathcal A}_\sigma })$ and $\Bbbk _2 ({\mathcal A})$ , and hence to their enveloping $C^*$ -algebras $\Bbbk ({{\mathcal A}_\sigma })$ and $\Bbbk ({\mathcal A})$ . It follows from (3-2) that this isomorphism preserves the structure maps $j_{C_0(G)}$ and $j_{C_0(G)}^\sigma $ . Finally, for the dual actions we compute

$$ \begin{align*} \Phi(\widehat{\delta^\sigma}_t(k))(g,h)&=\widehat{\delta^\sigma}_t(\mathfrak{s}(g),\mathfrak{s}(h))=\Delta(t)k(\mathfrak{s}(g)\mathfrak{s}(t),\mathfrak{s}(h)\mathfrak{s}(t))\\ &=\Delta(t)k(\omega(g,t)\mathfrak{s}(gt),\omega(h,t)\mathfrak{s}(h,t))=\Delta(t)\overline{\omega(g,t)}\omega(h,t)k(gt,ht). \end{align*} $$

This yields the final assertion involving the dual actions.

The above proposition provides the main step of the proof of the following theorem.

For use in the next section, we now want to give an alternative description of the isomorphism $(B_\sigma ,\beta _\sigma ,\phi _\sigma )\cong (\Bbbk ({{\mathcal A}_\sigma }), \widehat {\delta _{{\mathcal A}_\sigma }}, j_{C_0(G)}^\sigma )$ of the above theorem, which does not depend on a choice of a Borel section $\mathfrak {s}:G\to G_\sigma $ . The need for this comes from the fact that in general, if

$$ \begin{align*}\Sigma=(X\times \mathbb T\hookrightarrow \mathcal G_\Sigma\twoheadrightarrow X\times G)\end{align*} $$

is a continuous family of twists $\sigma _x=(\mathbb T\hookrightarrow G_{\sigma _x}\twoheadrightarrow G)$ , $x\in X$ , as introduced in [Reference Buss and Echterhoff7] to study continuity properties of our deformation process, we do not know whether there exists a global Borel cross-section $\mathfrak {S}:X\times G\to \mathcal G_\Sigma $ that induces Borel cross-sections $\mathfrak {s}_x:G\to G_{\sigma _x}$ in the fibres.

To overcome this problem, if $(B,\beta ,\phi )=(\Bbbk ({\mathcal A}), \widehat {\delta _{\mathcal A}}, j_{C_0(G)})$ as above, recall the Hilbert B-module

$$ \begin{align*}{\mathcal L}(G_\sigma,B)={\mathcal L}(G_\sigma,\Bbbk ({\mathcal A}))=C_0(G_\sigma,\iota)\otimes_{C_0(G)}\Bbbk ({\mathcal A}).\end{align*} $$

Given a section $\mathfrak {s}:G\to G_\sigma $ , it follows from Corollary 3.5, [Reference Buss and Echterhoff7, Proposition 5.15] and (3-2) that ${\mathcal L}(G_\sigma ,B)\cong B$ as Hilbert B-modules with isomorphism given by

(6-6) $$ \begin{align} \Theta\colon {\mathcal L}(G_\sigma,\Bbbk ({\mathcal A}))\xrightarrow\sim \Bbbk ({\mathcal A}), \quad f\otimes k\mapsto \Theta(f\otimes k)(g,h):=f(\mathfrak{s}(g))k(g,h). \end{align} $$

In order to provide an alternative description of ${\mathcal L}(G_\sigma ,B)$ , we let $\mathcal X_c(\mathcal A)$ denote the set of compactly supported continuous functions

$$ \begin{align*}\xi: G_\sigma\times G\to \mathcal A\quad\text{with} \quad\xi(\tilde g, h)\in A_{gh^{-1}}\quad\text{and}\quad \xi(z\tilde g, h)=\bar{z}\xi(\tilde g, h),\end{align*} $$

for all $z\in \mathbb T$ , $\tilde g\in G_\sigma , h\in G$ with $g=q(\tilde g)$ . Define a $\Bbbk _c({\mathcal A})$ -valued inner product on $\mathcal X_c(\mathcal A)$ by

$$ \begin{align*} \langle\xi\!\mid\!\eta\rangle_{\Bbbk_c({\mathcal A})}(g,h)= \int_{G} \xi({\tilde{r}}, g)^*\eta({\tilde{r}}, h)\,d r. \end{align*} $$

with $r=q({\tilde {r}})$ . As before, one checks that the integrand only depends on $r=q({\tilde {r}})\in G$ , so the integral makes sense. On the other hand, we have a $\Bbbk _c({{\mathcal A}_\sigma })$ -valued left inner product on $\mathcal X_c(\mathcal A)$ given by the formula

$$ \begin{align*}{_{\Bbbk_c({{\mathcal A}_\sigma})}\langle\xi\!\mid\!\eta\rangle}({\tilde{g}}, {\tilde{h}})=\int_G \xi({\tilde{g}}, r)\eta({\tilde{h}}, r)^*\,d r.\end{align*} $$

One easily checks that ${_{\Bbbk _c({{\mathcal A}_\sigma })}\langle \xi , \eta \rangle }$ satisfies (6-2). Moreover, we have left and right actions of $\Bbbk _c({{\mathcal A}_\sigma })$ and $\Bbbk _c({\mathcal A})$ on $\mathcal X_c(\mathcal A)$ given on kernels by

$$ \begin{align*} k\cdot\xi({\tilde{g}}, t)=\int_{G}k({\tilde{g}}, {\tilde{h}})\xi({\tilde{h}}, t)\,d h \end{align*} $$

and

$$ \begin{align*} \xi\cdot l({\tilde{g}}, t)=\int_{G}\xi({\tilde{g}}, r) l(r,t)\,d r. \end{align*} $$

One checks that these formulas satisfy the usual algebraic compatibility conditions for a pre-equivalence bimodule, as for instance the relation $_{\Bbbk _c({{\mathcal A}_\sigma })}\langle \xi \!\mid \!\eta \rangle \cdot \zeta =\xi \cdot \langle \eta \!\mid \!\zeta \rangle _{\Bbbk _c({\mathcal A})}$ for all $\xi ,\eta ,\zeta \in \mathcal X_c(\mathcal A)$ . Finally, we define an action $\gamma :G_\sigma \curvearrowright \chi _c(\mathcal A)$ by

$$ \begin{align*} ( \gamma_{{\tilde{g}}}\xi)({\tilde{h}}, r)={\Delta(g)}\xi({\tilde{h}}{\tilde{g}}, rg)\quad \text{with }g=q({\tilde{g}}). \end{align*} $$

An easy computation shows that it is compatible with (the inflations to $G_\sigma $ of) the dual actions $\widehat {\delta _{{\mathcal A}_\sigma }}$ on $\Bbbk _c({{\mathcal A}_\sigma })$ on the left and the Green twisted action $(\widehat {\delta _{\mathcal A}},\iota ^\sigma )$ on $\Bbbk ({\mathcal A})$ on the right.

Proposition 6.10. The $\Bbbk _c({{\mathcal A}_\sigma })-\Bbbk _c({\mathcal A})$ -pre-equivalence bimodule $\mathcal X_c(\mathcal A)$ completes to give a $\widehat {\delta _{{\mathcal A}_\sigma }}-(\widehat {\delta _{\mathcal A}}, \iota ^\sigma )$ -equivariant $\Bbbk ({{\mathcal A}_\sigma })-\Bbbk ({\mathcal A})$ -equivalence bimodule $(\mathcal X(\mathcal A), \gamma )$ . Moreover, the map

$$ \begin{align*} \Psi: C_0(G_\sigma,\iota)\odot \Bbbk_c({\mathcal A})\to \mathcal X_c(\mathcal A);\quad \Psi(f\otimes k)({\tilde{g}}, t):= f({\tilde{g}})k(q({\tilde{g}}),t) \end{align*} $$

extends to an $(\mathrm {rt}^{\sigma }\otimes \widehat {\delta _{\mathcal A}})-\gamma $ -equivariant isomorphism of right Hilbert $\Bbbk ({\mathcal A})$ -modules ${\mathcal L}(G_\sigma ,\Bbbk ({\mathcal A}))\xrightarrow \sim \mathcal X(\mathcal A)$ , which then induces an isomorphism of weak $G\rtimes G$ -algebras $(B_\sigma ,\beta _\sigma ,\phi _\sigma )\cong (\Bbbk ({{\mathcal A}_\sigma }), \widehat {\delta _{{\mathcal A}_\sigma }}, j_{C_0(G)}^\sigma )$ for $(B,\beta ,\phi ):=(\Bbbk ({\mathcal A}), \widehat {\delta _{\mathcal A}}, j_{C_0(G)})$ .

Proof. We use the second assertion for the proof of the first. Indeed, it is straightforward to check that $\Psi $ preserves the right inner product and actions and an argument as in the proof of Lemma 6.4, using continuous local sections and partitions of the unit, shows that it is also surjective. This implies that $\mathcal X_c(\mathcal A)$ is a right pre-Hilbert $\Bbbk ({\mathcal A})$ -module and that $\Psi $ extends to an isomorphism $\Psi : {\mathcal L}(G_\sigma ,\Bbbk ({\mathcal A}))\xrightarrow \sim \mathcal X(\mathcal A)$ of right Hilbert $\Bbbk ({\mathcal A})$ -modules. By the compatibility conditions of the pairings for $\mathcal X_c(\mathcal A)$ we further see that $B_\sigma :=\Bbbk ({\mathcal A})_\sigma =\mathbb K({\mathcal L}(G_\sigma ,\Bbbk ({\mathcal A})))\cong \mathbb K(\mathcal X(\mathcal A))$ is a $C^*$ -completion of $\Bbbk _c({{\mathcal A}_\sigma })$ . We need to show that it coincides with $\Bbbk ({{\mathcal A}_\sigma })$ . For this it suffices to show that the left action of $\Bbbk _c({{\mathcal A}_\sigma })$ on $\mathcal X_c(\mathcal A)$ extends faithfully to a left action of $\Bbbk ({{\mathcal A}_\sigma })$ on $\mathcal X(\mathcal A)$ . For this we choose a Borel section $\mathfrak {s}:G\to G_\sigma $ and recall the isomorphism $\Theta : {\mathcal L}(G_\sigma , \Bbbk ({\mathcal A}))\xrightarrow \sim \Bbbk ({\mathcal A})$ as right Hilbert $\Bbbk ({\mathcal A})$ -modules as in (6-6). Then the composition of isomorphisms

$$ \begin{align*}{\mathcal X}(\mathcal A)\stackrel{\Psi^{-1}}{\xrightarrow\sim}{\mathcal L}(G_\sigma,\Bbbk{{\mathcal A}})\stackrel{\Theta}{\xrightarrow\sim} \Bbbk{{\mathcal A}}\end{align*} $$

sends a function $\xi \in \mathcal X_c(\mathcal A)$ to the function $\tilde \xi \in \Bbbk _c({\mathcal A})$ given by $\tilde \xi (g,h):=\xi (\mathfrak {s}(g),h)$ . A simple computation then shows that

$$ \begin{align*}k\cdot \xi =\Phi(k)\cdot \tilde\xi\quad \text{for all } k\in \Bbbk_c({{\mathcal A}_\sigma}), \xi\in \mathcal X_c(\mathcal A),\end{align*} $$

where $\Phi :\Bbbk ({{\mathcal A}_\sigma })\xrightarrow \sim \Bbbk ({\mathcal A})$ is the isomorphism of (6-4). This proves that the left action is bounded and extends faithfully to a left action of $\Bbbk ({{\mathcal A}_\sigma })$ on $\mathcal X(\mathcal A)$ . To complete the proof, we need to check that $\Psi $ is $\gamma -\mathrm {rt}^\sigma \otimes \widehat {\delta _{\mathcal A}}$ -equivariant and that it intertwines the left structure maps $j_{C_0(G)}^\sigma :C_0(G)\to {\mathbb B}_{\Bbbk ({\mathcal A})}(\chi (\mathcal A))$ with the left action $\phi _\sigma :C_0(G)\to {\mathbb B}_{\Bbbk ({\mathcal A})}({\mathcal L}(G_\sigma ,\Bbbk ({\mathcal A})))$ induced from the left action of $C_0(G)$ on $C_0(G_\sigma ,\iota )$ by pointwise multiplication. We do the first and leave the second to the reader: for all $f\in C_0(G_\sigma )$ and $k\in \Bbbk _c({\mathcal A})$ we compute

$$ \begin{align*} \Psi((\mathrm{rt}^{\sigma}\otimes\widehat{\delta_{\mathcal A}})_{{\tilde{g}}}(f\otimes k))({\tilde{h}}, r)&= \Psi(\mathrm{rt}^\sigma_{{\tilde{g}}}(f)\otimes \widehat{\delta_{\mathcal A}}_{{\tilde{g}}}(k))({\tilde{h}},r)\\ &=f({\tilde{h}}{\tilde{g}})\Delta(g)k(hg,rg)\\ &=\Delta(g)(f({\tilde{h}}{\tilde{g}})k(hg, rg))\\ &=\gamma_{{\tilde{g}}}(\Psi(f\otimes k))({\tilde{h}},r) \end{align*} $$

for all ${\tilde {g}},{\tilde {h}}\in G_\sigma $ , $r\in G$ such that $g=q({\tilde {g}}), h=q({\tilde {h}})$ . This finishes the proof.

7 Continuous fields

In this section we study continuity properties of deformation of cross-sectional algebras of Fell bundles with respect to a continuous family of twists. Our results extend results of Raeburn in [Reference Raeburn32] who considered deformation of Fell bundles by continuous families of circle-valued $2$ -cocycles of discrete amenable groups G. Recall from [Reference Buss and Echterhoff7] that by a continuous family of twists of G over a locally compact Hausdorff space X, or simply a twist over $X\times G$ , we understand a groupoid central extension

$$ \begin{align*} \Sigma:=(X\times \mathbb T\stackrel{\iota}{\hookrightarrow} \mathcal G\stackrel{q}\twoheadrightarrow X\times G) \end{align*} $$

of the trivial group bundle $X\times G$ by the central trivial group bundle $X\times \mathbb T$ . Then for each $x\in X$ , we obtain a central extension $\sigma _x:=(\mathbb T\hookrightarrow G_{\sigma _x}\twoheadrightarrow G)$ of G by $\mathbb T$ . For ease of notation, we denote the elements in $\mathcal G$ generally by $\tilde g$ , indicate by $(x,\tilde g)$ that $\tilde g$ lies in the fibre $G_{\sigma _x}=q^{-1}(\{x\}\times G)\subseteq \mathcal G$ over $x\in X$ , and write $(x, g)$ for its image in $X\times G$ under the quotient map.

As explained in [Reference Buss and Echterhoff7], continuous families of twists of G over X are closely related to $C_0(X)$ -linear actions on continuous trace $C^*$ -algebras. More precisely, the following construction provides natural examples of continuous families of twists.

Example 7.1. Let $\mathbb K=\mathbb K({\mathcal H})$ for a Hilbert space ${\mathcal H}$ and suppose that $\alpha :G\curvearrowright C_0(X,\mathbb K)$ is a $C_0(X)$ -linear (that is, fibrewise) action of G on $C_0(X,\mathbb K)$ as considered in [Reference Buss and Echterhoff7, Definition 4.23]. For all $x\in X$ let $\alpha ^x:G\curvearrowright \mathbb K$ denote the action on the fibre at x. Then

$$ \begin{align*}\Sigma_{\alpha}:=\{(x,g,v)\in X\times G\times \mathcal U({\mathcal H}): \alpha_g^x=\text{Ad} v\}\end{align*} $$

together with the embedding $X\times \mathbb T\hookrightarrow \mathcal G_\alpha; (x,z)\mapsto (x,e, \bar {z}1_{{\mathcal H}})$ and the quotient map $\mathcal G_\alpha \twoheadrightarrow X\times G; (x,g,v)\mapsto (x,g)$ defines a groupoid central extension

$$ \begin{align*}\Sigma_\alpha=(X\times \mathbb T\hookrightarrow \mathcal G_\alpha\twoheadrightarrow X\times G)\end{align*} $$

as above. We refer to [Reference Buss and Echterhoff7, Lemma 4.28] for further details of this construction.

Remark 7.2. Conversely, every twist $\Sigma =(X\times \mathbb T\stackrel {\iota }{\hookrightarrow } \mathcal G\stackrel {q}\twoheadrightarrow X\times G)$ gives rise to a fibrewise action of G on a continuous field of compact operators over X. If G and X are second countable, this precisely inverts the construction of the above example. For the construction let $C_c(\mathcal G,\iota )$ denote the space of compactly supported continuous functions $\xi $ on $\mathcal G$ that satisfy the relation

(7-1) $$ \begin{align} \xi(x, {\tilde{g}} z)=\bar{z}\xi(x,{\tilde{g}})\quad\text{for all } (x,{\tilde{g}})\in \mathcal G, z\in \mathbb T. \end{align} $$

We define a $C_0(X)$ -valued inner product on $C_c(\mathcal G,\iota )$ by

$$ \begin{align*} \langle\xi\!\mid\!\eta\rangle_{C_0(X)}(x)=\int_G \overline{\xi(x,{\tilde{g}})}\eta(x,{\tilde{g}})\,d g \end{align*} $$

and we let $L^2(\mathcal G,\iota )$ denote the completion of $C_c(\mathcal G,\iota )$ with respect to this inner product. Then the right translation action $\rho _{\mathcal G}:\mathcal G\curvearrowright L^2(\mathcal G, \iota )$ is given fibrewise by the right translation action $\rho ^{\sigma _x}:G\curvearrowright L^2(G_{\sigma _x}, \iota )$ given by $(\rho ^{\sigma _x}_{\tilde {g}}\xi )({\tilde {h}})=\sqrt {\Delta (g)}\xi ({\tilde {h}}{\tilde {g}})$ for ${\tilde {g}},{\tilde {h}}\in G_{\sigma _x}$ and $g=q_x({\tilde {g}})$ . The adjoint action $\text {Ad}\rho _{\mathcal G}$ then provides a $C_0(X)$ -linear action $\alpha :G\curvearrowright \mathbb K(L^2(\mathcal G,\iota ))$ with fibre actions $\alpha ^x=\text {Ad}\rho ^{\sigma _x}:G\curvearrowright \mathbb K(L^2(G_{\sigma _x}, \iota ))$ . It has been shown in [Reference Buss and Echterhoff7, Lemma 4.33] that if $\Sigma =\Sigma _{\alpha '}$ for some continuous family of actions $\alpha ':G\curvearrowright C_0(X,\mathbb K)$ as in the above example, then $( \mathbb K(L^2(\mathcal G,\iota )), \alpha )$ and $(C_0(X,\mathbb K),\alpha ')$ are $X\times G$ -equivariantly Morita equivalent.

Another source of examples for continuous families of twists is given via continuous families of $2$ -cocycles.

A third class of important examples comes from central group extensions. The following example is [Reference Buss and Echterhoff7, Proposition 4.35].

Example 7.4. Let $Z\stackrel {\iota _Z}{\hookrightarrow } H\stackrel {q_H}{\twoheadrightarrow } G$ be a central extension of G by the abelian group Z. Then Z acts freely and properly on the product space $\widehat {Z}\times H\times \mathbb T$ by

$$ \begin{align*}z(\chi, h, w):= (\chi, zh, \chi(z)w)\quad\text{for all } z\in Z, (\chi, h,w)\in \widehat{Z}\times H\times \mathbb T,\end{align*} $$

and there is a twist

$$ \begin{align*}\Sigma_H:=(\widehat{Z}\stackrel{\iota}{\times} \mathbb T\hookrightarrow \mathcal G_H\stackrel{q}{\twoheadrightarrow} \widehat{Z}\times G)\end{align*} $$

with $\mathcal G_H:=(\widehat {Z}\times H\times \mathbb T)/Z$ and inclusion and quotient maps given by

$$ \begin{align*}\iota:(\chi, w)\mapsto [\chi, e, w]\quad\text{and}\quad q:[\chi, h, w]\mapsto (\chi, q_H(h)).\end{align*} $$

Here e denotes the neutral element of H.

For each $\chi \in \widehat {Z}$ the fibre $\sigma _\chi =(\mathbb T\stackrel {\iota _\chi }\hookrightarrow G_\chi \stackrel {q_\chi }\twoheadrightarrow G)$ at $\chi $ is then given by $G_\chi :={(H\times \mathbb T)/Z}$ with respect to the action $z(h,w)=(zh, \chi (z)w)$ for $z\in Z, (h,w)\in H\times \mathbb T$ . The inclusion and quotient maps are given by $\iota _\chi :w\mapsto [e,w]$ and $q_\chi :[h,w]\mapsto q_H(h)$ , respectively.

This notation was introduced by Calvin Moore in [Reference Moore25] in the case of second countable locally compact groups, where he uses a Borel cross-section $\mathfrak {s}:G\to H$ in order to define a transgression map $\mathrm {tg}:\widehat {Z}\to H^2(G,\mathbb T)$ . In that case, Moore’s transgression map can be obtained from ours by composing with the isomorphism $\operatorname {{\mathrm {Twist}}}(G)\cong H^2(G,\mathbb T)$ . Note that our construction does not need a Borel section, which may not exist if G is not second countable. Notice that many (but by no means all) groups are smooth in the above sense, among them all discrete groups, all semisimple Lie groups, the group $\mathbb R^n$ , and many more. We refer to [Reference Buss and Echterhoff7, Section 4] for a more detailed discussion and for concrete examples.

7.1 Deformation via continuous families of twists

Suppose that $(B, \beta , \phi )$ is a weak $G\rtimes G$ -algebra and that $\rtimes _\mu $ is a duality crossed-product functor for G. Given a twist $\Sigma =(X\times \mathbb T\hookrightarrow \mathcal G\twoheadrightarrow X\times G)$ over $X\times G$ with fibres $\sigma _x=(\mathbb T\hookrightarrow G_x\twoheadrightarrow G)$ , our general deformation process of [Reference Buss and Echterhoff7, Section 6] (see Section 5 above) provides for all $x\in X$ the weak $G\rtimes G$ -algebras $(B_{\sigma _x},\beta _{\sigma _x},\phi _{\sigma _x})$ and then, via Landstad duality, the cosystems $(A^{\sigma _x}_\mu ,\delta ^{\sigma _x}_\mu )$ . In [Reference Buss and Echterhoff7, Theorem 6.16] we prove that the $(B_{\sigma _x},\beta _{\sigma _x},\phi _{\sigma _x})$ are fibres of a continuous field $(\mathcal B_\Sigma ,\beta _\Sigma ,\Phi _\Sigma )$ of weak $G\rtimes G$ -algebras over X, and from this we obtain, depending on certain properties of the given crossed-product functor $\rtimes _\mu $ , continuity properties of the field of coactions $X\ni x\mapsto (A^{\sigma _x}_\mu ,\delta ^{\sigma _x}_\mu )$ over X.

We need to explain in more detail how $(\mathcal B_\Sigma ,\beta _\Sigma ,\Phi _\Sigma )$ is constructed and in what way it can be regarded as a continuous field of weak $G\rtimes G$ -algebras. For this we consider the function space $C_0(\mathcal G,\iota )$ consisting of $C_0$ -functions $\xi \colon \mathcal G\to \mathbb C$ satisfying (7-1). Then $C_0(\mathcal G,\iota )$ becomes an imprimitivity Hilbert bimodule over $C_0(X\times G)$ with respect to the inner products

$$ \begin{align*} {_{C_0(X\times G)}}\langle\xi\!\mid\!\eta\rangle(x, g)=\xi(x,\tilde g)\overline{\eta(x,\tilde g)}\quad \text{and}\quad \langle\xi\!\mid\!\eta\rangle_{C_0(X\times G)}(x, g)=\overline{\xi(x,\tilde g)}\eta(x,\tilde g). \end{align*} $$

We then consider $\mathcal B:=C_0(X,B)$ , viewed as a $C_0(X)$ -algebra with constant fibre B. We define the (balanced) tensor product of Hilbert modules,

$$ \begin{align*} \mathcal E_\Sigma(\mathcal G,\mathcal B):=(C_0(\mathcal G,\iota)\otimes_{C_0(X\times G)}\mathcal B)\otimes_{C_0(X)}L^2(\mathcal G,\iota)^*, \end{align*} $$

where $L^2(\mathcal G,\iota )^*$ denotes the $C_0(X)-\mathbb K(L^2(\mathcal G,\iota ))$ equivalence bimodule dual to the right $C_0(X)$ -Hilbert module $L^2(\mathcal G,\iota )$ as constructed in Remark 7.2. It is shown in [Reference Buss and Echterhoff7] (see the discussion preceding [Reference Buss and Echterhoff7, Theorem 6.16]) that the $\mathcal G$ -action

$$ \begin{align*}\gamma_{\mathcal G}:=\mathrm{rt}_{\mathcal G}\otimes_{C_0(X\times G)}\beta\otimes_{C_0(X\times G)} {\rho_{\mathcal G}}^*:\mathcal G\curvearrowright \mathcal E_\Sigma(C_0(X,B))\end{align*} $$

for the right translation action $\mathrm {rt}_{\mathcal G}:\mathcal G\curvearrowright C_0(\mathcal G,\iota )$ and action $\rho _{\mathcal G}:\mathcal G\curvearrowright L^2(\mathcal G,\iota )$ as in Remark 7.2 is trivial on $X\times \mathbb T$ and therefore factors through a well-defined $C_0(X)$ -linear action $\gamma _\Sigma :G\curvearrowright \mathcal E_\Sigma (C_0(X,B))$ . We denote by $\Phi _\Sigma :C_0(X\times G)\to \mathcal M(\mathbb K(\mathcal E_\Sigma (\mathcal G, B)))$ the nondegenerate $^*$ -homomorphism induced from the left action of $C_0(X\times G)$ on $C_0(\mathcal G,\iota )$ . We then define

$$ \begin{align*}(\mathcal B_\Sigma, \beta_\Sigma,\Phi_\Sigma):=(\mathbb K(\mathcal E_\Sigma(\mathcal G,\mathcal B)) ,\text{Ad} \gamma_\Sigma, \Phi_\Sigma).\end{align*} $$

It is then easy to check that for all $x\in X$ the quotient maps $q_x:\mathcal B_\Sigma \to B_{\sigma _x}$ induce surjective morphisms of weak $G\rtimes G$ -algebras

$$ \begin{align*}q_x: (\mathcal B_\Sigma, \beta_\Sigma,\Phi_\Sigma)\twoheadrightarrow (B_{\sigma_x},\beta_{\sigma_x}, \phi_{\sigma_x}).\end{align*} $$

Alternatively, we can also consider the right Hilbert $\mathcal B$ -module

(7-2) $$ \begin{align} {\mathcal L}(\mathcal G,\mathcal B):=C_0(\mathcal G,\iota)\otimes_{C_0(X\times G)}\mathcal B \end{align} $$

equipped with the diagonal action $\epsilon _\Sigma :=\mathrm {rt}_{\mathcal G}\otimes _{C_0(X\times G)}\beta $ of $\mathcal G$ . Since $L^2(\mathcal G,\iota )$ is an imprimitivity bimodule, we also have

$$ \begin{align*}(\mathcal B_\Sigma, \beta_\Sigma,\Phi_\Sigma)\cong(\mathbb K({\mathcal L}_\Sigma(\mathcal G,\mathcal B)) ,\text{Ad} \epsilon_\Sigma, \Psi_\Sigma),\end{align*} $$

where, similar to $\Phi _\Sigma $ , the $^*$ -homomorphism $\Psi _\Sigma :C_0(X\times G)\to \mathcal M(\mathbb K({\mathcal L}_\Sigma (\mathcal G,B)))$ is also induced from the left action of $C_0(X\times G)$ on $C_0(\mathcal G,\iota )$ . We refer to [Reference Buss and Echterhoff7, Theorem 6.16] for more details.

7.2 Deformation of Fell bundles via continuous families of twists

We want to apply the above deformation procedure to the dual weak $G\rtimes G$ -algebra $(B, \beta ,\phi )=(C^*(\mathcal A)\rtimes _{\delta _{\mathcal A}}\widehat {G}, \widehat {\delta _{\mathcal A}}, j_{C_0(G)})$ of a Fell bundle $\mathcal A$ over G and we want to show that, similar to deformation by a twist $\sigma =(\mathbb T\hookrightarrow G\twoheadrightarrow G)$ as considered in the previous section, the construction of $(\mathcal B_\Sigma , \beta _\Sigma , \Phi _\Sigma )$ can then be done completely on the level of Fell bundles. Given a Fell bundle $p:\mathcal A\to G$ and a twist $\Sigma =(X\times \mathbb T\hookrightarrow \mathcal G\twoheadrightarrow X\times ~G)$ , we construct a Fell bundle ${\mathcal A}_\Sigma $ over $X\times G$ as

$$ \begin{align*} {\mathcal A}_{\Sigma}=(\mathcal A\times_{(X\times G)}\mathcal G)/\sim \end{align*} $$

where $\mathcal A\times _{(X\times G)}\mathcal G=\{(a, (x,{\tilde {g}}))\in \mathcal A\times \mathcal G: p(a)=q_x({\tilde {g}})\}$ and where $\sim $ denotes the equivalence relation

$$ \begin{align*} &(a, (x,{\tilde{g}}))\sim (b, (y,{\tilde{h}}))\Longleftrightarrow x=y\quad\text{and}\nonumber\\ &\quad\text{there exists } z\in \mathbb T\;\text{such that}\; (b, (x,{\tilde{h}}))=(\bar{z}a, (x, z{\tilde{g}})). \end{align*} $$

The projection $p_\Sigma :{\mathcal A}_\Sigma \to X\times G$ is given by $p_\Sigma ([a, (x,{\tilde {g}})])=(x, g)$ if $g=q({\tilde {g}})$ . Observe that the restriction ${\mathcal A}_\Sigma |_{\{x\}\times G_{\sigma _x}}$ coincides with the deformed Fell bundle ${\mathcal A}_{\sigma _x}$ with respect to the fibre $\sigma _x=(\mathbb T\hookrightarrow G_{\sigma _x}\twoheadrightarrow G)$ at $x\in X$ as defined in the previous section. As usual we write $C_c({\mathcal A}_\Sigma )$ for the space of continuous sections $a:X\times G\to {\mathcal A}_\Sigma $ with compact supports. Note that it becomes a $^*$ -algebra with respect to the convolution

$$ \begin{align*}a*b(x,g)=\int_G a(x,h)b(x,h^{-1}g)\, \,d h\quad\text{and}\quad a^*(x,g)=\Delta(g^{-1})a_{(x,g^{-1})}^*\end{align*} $$

for $a,b\in C_c({\mathcal A}_\Sigma )$ . The following lemma is then a complete analogue of Lemma 6.4 above and we omit the proof.

Lemma 7.6. There is a bijection between the elements of $C_c({\mathcal A}_\Sigma )$ and the set of compactly supported continuous functions $a:\mathcal G\to \mathcal A$ satisfying

$$ \begin{align*} a_{(x,z{\tilde{g}})}=\bar{z}a_{(x,{\tilde{g}})}\quad\text{for all } (x,{\tilde{g}})\in \mathcal G, z\in \mathbb T. \end{align*} $$

Under this identification, convolution and involution are given by the formulas

$$ \begin{align*} a*b(x, {\tilde{g}})=\int_G a(x,{\tilde{h}})b(x,{\tilde{h}}^{-1}{\tilde{g}})\, \,d h\quad\text{and}\quad a^*(x,{\tilde{g}})=\Delta(g^{-1})a(x,{\tilde{g}}^{-1})^* \end{align*} $$

where, as usual, we write $h=q_x({\tilde {h}})$ , $g=q_x({\tilde {g}})$ for $q_x:G_{\sigma _x}\to G$ .

There are constructions of full and reduced cross-sectional algebras of Fell bundles over groupoids, but in our situation these constructions can be reduced to the situation of Fell bundles over the group G by associating to ${\mathcal A}_\Sigma $ a Fell bundle, say $\widetilde {\mathcal A}_\Sigma $ , over G as follows: the fibres $\tilde {A}_g$ are given as the $C_0$ -sections $C_0({\mathcal A}_\Sigma |_{X\times \{g\}})$ of the restriction ${\mathcal A}_\Sigma |_{X\times \{g\}}$ of ${\mathcal A}_\Sigma $ to $X\times \{g\}$ and multiplication and involution are defined pointwise over X.

Notation 7.7. Let ${\mathcal A}_\Sigma $ and $\widetilde {\mathcal A}_\Sigma $ be as above and let $\rtimes _\mu $ be any duality crossed-product functor for G. We then write $C^*({\mathcal A}_\Sigma ):=C^*(\widetilde {\mathcal A}_\Sigma )$ , and similarly $C_\mu ^*({\mathcal A}_\Sigma ):=C^*_\mu (\widetilde {\mathcal A}_\Sigma )$ . Then $C^*({\mathcal A}_\Sigma )$ is equipped with a dual coaction $\delta _{{\mathcal A}_\Sigma }: C^*({\mathcal A}_\Sigma )\to \mathcal M(C^*({\mathcal A}_\Sigma )\otimes C^*(G))$ , which factors through a coaction $\delta ^\Sigma _\mu $ on $C_\mu ^*({\mathcal A}_\Sigma )$ .

Note that there is an obvious inclusion

$$ \begin{align*}C_c({\mathcal A}_\Sigma)\hookrightarrow C_c(\widetilde{\mathcal A}_\Sigma); a \mapsto (g\mapsto a|_{X\times\{g\}})\end{align*} $$

and one can show that this map induces an isomorphism of the usually defined full (or reduced) cross-sectional algebra $C^*({\mathcal A}_\Sigma )$ (respectively, $C_r^*({\mathcal A}_\Sigma )$ ) in the general setting of Fell bundles over groupoids and $C^*(\widetilde {\mathcal A}_\Sigma )$ (respectively, $C_r^*(\widetilde {\mathcal A}_\Sigma )$ ), so our definition makes sense; indeed the case of full cross-sectional $C^*$ -algebras is a special case of [Reference Buss and Meyer9, Theorem 6.2] and the reduced case a special case of [Reference LaLonde24, Proposition 5.1]. From now on, we simply identify ${\mathcal A}_\Sigma $ with the Fell bundle $\widetilde {\mathcal A}_\Sigma $ over G whenever it seems convenient.

The crossed product $C^*({\mathcal A}_\Sigma )\rtimes _{\delta _{{\mathcal A}_\Sigma }}\widehat {G}$ comes with the dual action $\widehat {\delta _{{\mathcal A}_\Sigma }}$ of G and the inclusion $\Psi _G=j_{C_0(G)}:C_0(G)\to \mathcal M(C^*({\mathcal A}_\Sigma )\rtimes _{\delta _{{\mathcal A}_\Sigma }}\widehat {G})$ . We also have a canonical nondegenerate $^*$ -homomorphism $\Psi _X:C_0(X)\to Z\mathcal M(C^*({\mathcal A}_\Sigma )\rtimes _{\delta _{{\mathcal A}_\Sigma }}\widehat {G})$ , which is induced by pointwise multiplication $(\varphi a)(x,{\tilde {g}})=\varphi (x)a(x,{\tilde {g}})$ of functions $\varphi \in C_0(X)$ with sections $a\in C_c({\mathcal A}_{\Sigma })$ . It is easily checked that it commutes with $j_{C_0(G)}$ . We therefore obtain a well-defined structure map

$$ \begin{align*} \Psi_{X\times G}:=\Psi_X\otimes \Psi_G:C_0(X\times G)\to \mathcal M(C^*({\mathcal A}_\Sigma)\rtimes_{\delta_{{\mathcal A}_\Sigma}}\widehat{G}). \end{align*} $$

Theorem 7.8. Let $(B, \beta ,\phi )=(C^*(\mathcal A)\rtimes _{\delta _{\mathcal A}}\widehat {G}, \widehat {\delta _{\mathcal A}}, j_{C_0(G)})$ be the dual weak $G\rtimes G$ -algebra for a Fell bundle $\mathcal A$ over G. Let $\Sigma =(X\times \mathbb T\hookrightarrow \mathcal G\twoheadrightarrow X\times G)$ be a twist for $X\times G$ and let $(\mathcal B_\Sigma ,\beta _\Sigma , \Phi _\Sigma )$ be the deformation of $(B,\beta ,\phi )$ by $\Sigma $ as in Section 7.1. Then $(B_\Sigma ,\beta _\Sigma , \Phi _\Sigma )$ is isomorphic to the triple

$$ \begin{align*}(C^*({\mathcal A}_\Sigma)\rtimes_{\delta_{{\mathcal A}_\Sigma}}\widehat{G}, \widehat{\delta_{{\mathcal A}_\Sigma}}, \Psi_{X\times G}).\end{align*} $$

Proof. Arguing similarly to the proof of Proposition 6.10, we use the descriptions of $B=C^*(\mathcal A)\rtimes _{\delta _{\mathcal A}}\widehat {G}\cong \Bbbk ({\mathcal A})$ and $C^*({\mathcal A}_\Sigma )\rtimes _{\delta _{{\mathcal A}_\Sigma }}\widehat {G}\cong \Bbbk ({{\mathcal A}_\Sigma })$ in order to construct the desired isomorphism. Let ${\mathcal A}_X=X\times \mathcal A$ be the pullback of $\mathcal A$ to $X\times G$ via the projection $X\times G\to G$ (this can be regarded as a special case of the construction of ${\mathcal A}_\Sigma $ for the trivial twist $X\times \mathbb T\hookrightarrow X\times (G\times \mathbb T)\twoheadrightarrow X\times G$ ). We then have $C^*({\mathcal A}_X)\cong C_0(X,C^*(\mathcal A))$ and

$$ \begin{align*} \mathcal B:=C_0(X,B)= C_0(X,C^*(\mathcal A)\rtimes_{\delta_{\mathcal A}}\widehat{G}) = C^*({\mathcal A}_X)\rtimes_{\delta_{{\mathcal A}_X}}\widehat{G}= \Bbbk ({{\mathcal A}_X}). \end{align*} $$

We write $\Bbbk _c({{\mathcal A}_\Sigma })$ for the compactly supported functions $k:\mathcal G\times _X\mathcal G\to \mathcal A$ satisfying

$$ \begin{align*} k(x,z{\tilde{g}},u{\tilde{h}})=\bar{z}uk(x,{\tilde{g}},{\tilde{h}}),\quad (x,{\tilde{g}},{\tilde{h}})\in \mathcal G\times_X\mathcal G, \end{align*} $$

and observe that $\Bbbk _c({{\mathcal A}_\Sigma })$ can be regarded as a dense subalgebra of $\Bbbk ({{\mathcal A}_\Sigma })$ in a canonical way.

We follow ideas similar to those in the proof of Proposition 6.10 and realize the right Hilbert $\mathcal B$ -module ${\mathcal L}(\mathcal G,\mathcal B)$ of (7-2) as a completion of the space $\mathcal X_c( {\mathcal A}_X)$ of all compactly supported continuous functions $\xi \colon \mathcal G\times G\to \mathcal A$ satisfying

$$ \begin{align*}\xi(x,z\tilde g,h)=\bar z\xi(x,\tilde g,h)\in A_{gh^{-1}}\quad\text{for all }(x,\tilde g)\in\mathcal G, h\in G.\end{align*} $$

We can mimic the formulas of the previous section and define inner products and left and right actions of $\Bbbk _c({{\mathcal A}_\Sigma })$ and $\Bbbk _c({{\mathcal A}_X})$ by the following formulas for $\xi ,\eta \in \mathcal X_c({\mathcal A}_X)$ , $k\in \Bbbk _c({{\mathcal A}_\Sigma })$ and $l\in \Bbbk _c({{\mathcal A}_X})$ :

$$ \begin{align*}\langle\xi\!\mid\!\eta\rangle_{\Bbbk_c({{\mathcal A}_X})}(x,s,t):=\int_{G_x}\xi(g,s)^*\eta(g,t)\,d g,\end{align*} $$

$$ \begin{align*}_{\Bbbk_c({{\mathcal A}_\Sigma})}\langle\xi\!\mid\!\eta\rangle(x,\tilde g,\tilde h):=\int_G\xi(x,\tilde g,t)\eta(x,\tilde h,t)^*\,d t, \end{align*} $$

$$ \begin{align*}(k\cdot\xi)(x,\tilde g,t):=\int_{G_x}k(x,\tilde g,\tilde h)\xi(x,\tilde h,t)\,d \tilde h,\end{align*} $$

$$ \begin{align*}(\xi\cdot l)(x,\tilde g,t):=\int_G\xi(x,\tilde g,h)l(h,t)\,d h.\end{align*} $$

The module ${\mathcal L}(\mathcal G,\mathcal B)$ is $C_0(X)$ -linear with fibres

$$ \begin{align*}{\mathcal L}(G_{\sigma_x},B)=C_0(G_{\sigma_x},\iota_x)\otimes_{C_0(G)}B,\end{align*} $$

where $\sigma _x=(\mathbb T\hookrightarrow G_{\sigma _x}\twoheadrightarrow G)$ is the fibre of $\Sigma $ at $x\in X$ . We know from Proposition 6.10 that the algebra of compact operators of this module is isomorphic to $\Bbbk ({{\mathcal A}_{\sigma _x}})$ , where ${\mathcal A}_{\sigma _x}$ is the Fell bundle deformed from $\mathcal A$ via $\sigma _x$ .

We need to show that $\mathcal B_\Sigma =\mathbb K({\mathcal L}(\mathcal G,\mathcal B))$ is isomorphic to $\Bbbk ({{\mathcal A}_\Sigma })$ , and that this isomorphism intertwines the actions and structure maps. Notice that $C^*({\mathcal A}_\Sigma )$ is a $C_0(X)$ -algebra with fibres $C^*({\mathcal A}_{\sigma _x})$ : this follows from the fact that full cross-sectional $C^*$ -algebras of Fell bundles preserve exact sequences (see, for example, [Reference Exel15, Proposition 21.15], which proves this statement for Fell bundles over discrete groups; a similar proof applies for locally compact groups). It follows then from [Reference Nilsen29, Theorem 4.3] that $\Bbbk ({{\mathcal A}_\Sigma })=C_0({\mathcal A}_\Sigma )\rtimes _{\delta _{{\mathcal A}_\Sigma }}\widehat {G}$ is also a $C_0(X)$ -algebra with fibres

$$ \begin{align*}\Bbbk ({{\mathcal A}_{\sigma_x}})\cong C^*({\mathcal A}_{\sigma_x})\rtimes_{\delta_x}\widehat{G}.\end{align*} $$

Now observe that we have a canonical map

$$ \begin{align*}\Psi_X\colon C_0(\mathcal G,\iota)\odot \Bbbk_c({{\mathcal A}_X})\to \mathcal X_c({\mathcal A}_X);\quad\Psi_X(f\otimes \xi)(x,\tilde g,h):=f(x,\tilde g)\xi(x, g,h).\end{align*} $$

As in the case of a single twist, one checks that this preserves the structures of right modules over $\Bbbk _c({{\mathcal A}_X})$ , so that $\Psi _X$ induces an isomorphism of right Hilbert $\Bbbk ({{\mathcal A}_X})$ -modules

$$ \begin{align*}\Psi_\Sigma\colon {\mathcal L}(\mathcal G,\mathcal B)=C_0(\mathcal G,\iota)\otimes_{C_0(X\times G)}\Bbbk_c({{\mathcal A}_X})\xrightarrow\sim \mathcal X({\mathcal A}_X),\end{align*} $$

where $\mathcal X({\mathcal A}_X)$ denotes the completion of $\mathcal X_c({\mathcal A}_X)$ with respect to the right $\mathcal B$ -module structure (recall that $\mathcal B=\Bbbk ({{\mathcal A}_X})$ ). So again, as in the case of a single twist, to show that $\Bbbk ({{\mathcal A}_\Sigma })=\mathcal B_\Sigma =\mathbb K({\mathcal L}(\mathcal G,\mathcal B))$ it is enough to see that the left action of $\Bbbk _c({{\mathcal A}_\Sigma })$ on $\mathcal X_c({\mathcal A}_X)$ extends to a $^*$ -homomorphism (that is, a left action by adjointable operators)

$$ \begin{align*}\Bbbk ({{\mathcal A}_\Sigma})\to {\mathbb B}_{\mathcal B}({\mathcal L}(\mathcal G,\mathcal B)).\end{align*} $$

But since both the algebra and the module involved carry $C_0(X)$ -linear structures that are preserved by the left action of $\Bbbk _c({{\mathcal A}_{\Sigma }})$ on $\mathcal X({\mathcal A}_X)\cong {\mathcal L}(\mathcal G,\mathcal B)$ , and we already know that the fibrewise left action of $\Bbbk _c({{\mathcal A}_{\sigma _x}})$ on the fibre ${\mathcal L}(G_{\sigma _x},B)$ of ${\mathcal L}(\mathcal G,\mathcal B)$ over x extends to an action by adjointable operators

$$ \begin{align*}\Bbbk ({{\mathcal A}_{\sigma_x}})\to{\mathbb B}_{B}({\mathcal L}(G_x,B)), \end{align*} $$

the result follows. Finally, notice that by Proposition 6.10 we know that the isomorphism $\Bbbk ({{\mathcal A}_{\sigma _x}})\to {\mathbb B}_{B}({\mathcal L}(G_x,B))$ intertwines the actions and structure maps and therefore induces an isomorphism of weak $G\rtimes G$ -algebras

$$ \begin{align*}(B_{\sigma_x}, \beta_{\sigma_x}\phi_{\sigma_x})\cong (\Bbbk ({{\mathcal A}_{\sigma_x}}), \widehat{\delta_{{\mathcal A}_{\sigma_x}}}, j_{C_0(G)}).\end{align*} $$

Since the $C_0(X)$ -linear actions and structure maps for $\mathcal B_\Sigma $ and $\Bbbk ({{\mathcal A}_\Sigma })$ induce these actions and structure maps on the fibres, we can finally conclude the desired isomorphism

$$ \begin{align*}(B_\Sigma,\beta_\Sigma, \Phi_\Sigma)\cong(C^*({\mathcal A}_\Sigma)\rtimes_{\delta_{{\mathcal A}_\Sigma}}\widehat{G}, \widehat{\delta_{{\mathcal A}_\Sigma}}, \Psi_{X\times G}).\\[-35pt]\end{align*} $$

Recall that $C^*({\mathcal A}_\Sigma )$ is a $C_0(X)$ -algebra by extending pointwise multiplication of functions in $C_0(X)$ with sections in $C_c({\mathcal A}_\Sigma )$ . The same holds true for $C^*_\mu ({\mathcal A}_\Sigma )$ for every duality crossed-product functor $\rtimes _\mu $ . Recall that for any $C_0(X)$ -algebra D, the (maximal) fibre $D_x$ of D over x is defined as the quotient $D_x:=D/I_x$ with $I_x=C_0(X\setminus \{x\})D$ . Thus we have the fibres $C_\mu ^*({\mathcal A}_\Sigma )_x$ for each $x\in X$ .

On the other hand, it is clear that evaluation at $x\in X$ induces $\delta ^\Sigma _\mu -\delta ^{\sigma _x}_\mu $ -equivariant quotient maps $Q_x:C^*_\mu ({\mathcal A}_\Sigma )\twoheadrightarrow C^*_\mu ({\mathcal A}_{\sigma _x})$ , and the obvious question arises, under what conditions do the $Q_x$ factor through isomorphisms $C_\mu ^*({\mathcal A}_\Sigma )_x \cong C^*_\mu ({\mathcal A}_{\sigma _x})$ ? By [Reference Williams35, Theorem C.26], this is equivalent to saying that $C_\mu ^*({\mathcal A}_\Sigma )$ is an upper semicontinuous bundle of $C^*$ -algebras with fibres $C_\mu ^*({\mathcal A}_{\sigma ^x})$ . Indeed, as a direct application of the above theorem together with [Reference Buss and Echterhoff7, Theorem 6.16] we now get the following result.

Theorem 7.9. Let $\mathcal A$ be a Fell bundle over G and let $\Sigma =(X\times \mathbb T\hookrightarrow \mathcal G\twoheadrightarrow X\times G)$ be a twist over $X\times G$ with fibres $\sigma _x=(\mathbb T\hookrightarrow G_{\sigma _x}\twoheadrightarrow G)$ . Then the following assertions hold.

1. (1) If $\rtimes _\mu $ is an exact duality crossed-product functor (which always holds for $\rtimes _{\mathrm \max }$ ) then $Q_x:C^*_\mu ({\mathcal A}_\Sigma )\twoheadrightarrow C^*_\mu ({\mathcal A}_{\sigma _x})$ factors through an isomorphism $C_\mu ^*({\mathcal A}_\Sigma )_x \cong C^*_\mu ({\mathcal A}_{\sigma _x})$ .

2. (2) If G is an exact group, then $C_r^*({\mathcal A}_\Sigma )$ is a continuous bundle of $C^*$ -algebras over X with fibres $C_r^*({\mathcal A}_{\sigma _x})$ .

In the special case where G is a discrete amenable group, item (2) of the above theorem can be derived from the results in the paper [Reference Raeburn32, Section 6] by Iain Raeburn. Using the fact that every discrete group admits a representation group in the sense of Moore (see also Notation 7.5 above), he used this result to show that for every discrete amenable group G and Fell bundle $\mathcal A$ over G, there exists a kind of universal continuous bundle of $C^*$ -algebras over $X=H^2(G,\mathbb T)$ (which, in this case, carries a canonical compact Hausdorff topology) with fibres $C^*({\mathcal A}_\omega )$ , the (unique) cross-sectional algebra of the deformed Fell bundle ${\mathcal A}_\omega $ for $\omega \in Z^2(G,\mathbb T)$ as considered in Proposition 6.5 above. This result can now be generalized as follows.

Theorem 7.10. Suppose that G is smooth in the sense of Notation 7.5 and that $Z\hookrightarrow H\twoheadrightarrow G$ is a representation group for G. Let $\Sigma _H$ be the twist for $\widehat {Z} \times G$ constructed in Example 7.4.

Identifying $\operatorname {{\mathrm {Twist}}}(G)$ with $\widehat {Z}$ via the transgression map $\mathrm {tg}:\chi \mapsto [\sigma _\chi ]$ , we obtain that $C^*({\mathcal A}_{\Sigma _H})$ forms an upper semicontinuous bundle of $C^*$ -algebras over $\operatorname {{\mathrm {Twist}}}(G)$ , with fibres isomorphic to $C^*({\mathcal A}_{\sigma })$ for $[\sigma ] \in \operatorname {{\mathrm {Twist}}}(G)$ .

Furthermore, if G is exact, then $C_r^*({\mathcal A}_{\Sigma _H})$ is a continuous bundle of $C^*$ -algebras over $\operatorname {{\mathrm {Twist}}}(G)$ , with fibres isomorphic to $C_r^*({\mathcal A}_{\sigma })$ for $[\sigma ] \in \operatorname {{\mathrm {Twist}}}(G)$ .