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Morita Enveloping Fell Bundles

  • Fernando Abadie
  • Alcides BussEmail author
  • Damián Ferraro
Article
  • 115 Downloads

Abstract

We introduce notions of weak and strong equivalence for non-saturated Fell bundles over locally compact groups and show that every Fell bundle is strongly (resp. weakly) equivalent to a semidirect product Fell bundle for a partial (resp. global) action. Equivalences preserve cross-sectional \({\mathrm {C}}^*\)-algebras and amenability. We use this to show that previous results on crossed products and amenability of group actions carry over to Fell bundles.

Keywords

Fell bundle Equivalence Partial action Crossed product Group Amenabilty 

Mathematics Subject Classification

Primary 46L08 Secondary 46L55 

1 Introduction

A Fell bundle over a locally compact group G is a continuous bundle \(\mathcal {A}\rightarrow G\) of Banach spaces \((A_t)_{t\in G}\) together with continuous multiplications \(A_s\times A_t\rightarrow A_{st}\), \((a,b)\mapsto a\cdot b\), and involutions \(A_s\rightarrow A_{s^{-1}}\), \(a\mapsto a^*\), satisfying properties similar to those valid for a \({\mathrm {C}}^*\)-algebra, like the positivity \(a^*\cdot a\ge 0\) and the \({\mathrm {C}}^*\)-axiom \(\Vert a^*\cdot a\Vert =\Vert a\Vert ^2\). Fell bundles generalise partial actions of groups. Indeed, in Exel (1997) Exel defines twisted partial actions of groups on \({\mathrm {C}}^*\)-algebras and to each such action, a Fell bundle is constructed, the so-called semidirect product of the twisted partial action. This is, of course, a generalisation of the semidirect product construction for ordinary (global) actions already introduced by Fell in his first papers on the subject, see Fell (1969a, b, 1977), Doran and Fell (1988a, b). In Exel (1997) only continuous twisted partial actions are considered, but even the measurable twisted actions of Busby and Smith (1970) can be turned into (continuous) Fell bundles, see Exel and Laca (1997).

The main result of Exel (1997) already indicates that Fell bundles are very close to partial actions: it asserts that every regular Fell bundle is isomorphic to a semidirect product by a twisted partial action. It is moreover shown that every separable Fell bundle can be “regularised” via stabilisation. The stabilisation procedure should be viewed as a form of producing an “equivalent” Fell bundle in the spirit of Morita equivalence of \({\mathrm {C}}^*\)-algebras. Hence we may say that the results in Exel (1997) show that every separable Fell bundle is equivalent to one associated to a twisted partial action. A very basic question appears: can the twist be “removed”, that is, is every Fell bundle equivalent to a partial action semidirect product Fell bundle? For saturated Fell bundles this, indeed, follows from the famous Packer–Raeburn Stabilisation Trick which asserts that every twisted (global) action is stably isomorphic to an untwisted action. As a result, every saturated separable Fell bundle is equivalent to one coming from an ordinary action. This version is also known for (separable) saturated Fell bundles over groupoids as proved in Buss et al. (2013), Ionescu et al. (2015), where precise notions of equivalence of saturated Fell bundles are introduced. A version of the stabilisation trick for non-saturated Fell bundles is only known for discrete groups: it is proved in the master thesis of Sehnem (2014) and reproduced in Exel’s book Exel (2017) that, after stabilisation, every separable Fell bundle becomes isomorphic to one coming from an (untwisted) partial action.

In Buss et al. (2013) a new point of view is introduced from which saturated Fell bundles are interpreted as actions of the underlying group(oid) in the bicategory of \({\mathrm {C}}^*\)-correspondences. Indeed, the algebraic structure of the Fell bundle can be used to turn each fibre \(A_t\) into a Hilbert bimodule over the unit fiber \({\mathrm {C}}^*\)-algebra \(A:=A_e\), and these bimodules are imprimitivity (or equivalence) bimodules if (and only if) the Fell bundle is saturated. Hence we may view a saturated Fell bundle as an action of G on A by equivalences. A non-saturated Fell bundle should be viewed as a partial action of G on A by (partial) equivalences.

Although the notion of equivalence between saturated Fell bundles over groups is already well established nowadays, little is known for non-saturated Fell bundles. Only recently a more general notion of equivalence has been introduced in Abadie and Ferraro (2017), which in the present work we call weak equivalence. This notion originates in Abadie (2003): the relationship between the Fell bundles of a partial action and its enveloping action is precisely that of weak equivalence. We introduce yet another notion of equivalence, the strong equivalence. As the name suggests, strong equivalence is stronger than weak equivalence. Strong equivalence is the natural extension of the notion of (Morita) equivalence for partial actions as introduced by the first named author in Abadie (2003). Indeed, we are going to extend one of the main results in Abadie (2003) and prove that every (not necessarily saturated or separable) Fell bundle over G is strongly equivalent to a semidirect product Fell bundle by a partial action of G (Theorem 3.5). On the other hand we will prove that, as long as saturated Fell bundles are concerned, there is no difference between weak equivalence and strong equivalence of Fell bundles (Corollary 4.10); they extend the usual notion of equivalence for global actions.

In the recent paper by Kwaśniewski and Meyer (2018) the notion of Morita globalization of a Fell bundle over a discrete group is introduced, and it is shown that every Fell bundle (over a discrete group) has a Morita globalization. It can be shown that the Fell bundle associated to the action involved in the definition of a Morita globalization of a Fell bundle is weakly equivalent, in our sense, to the original Fell bundle. As a result a Morita globalization of a Fell bundle is an instance of what we call here a Morita enveloping Fell bundle (see Definition 2.11).

The notion of weak equivalence will allow us to show that every partial action of G, once viewed as a Fell bundle, is weakly equivalent to a global action. As a conclusion, every Fell bundle is weakly equivalent to one associated to a global action. We shall prove this in one step, showing directly that the Fell bundle is weakly equivalent to the semidirect product Fell bundle of a global action. This global action is directly constructed from the Fell bundle. Indeed, it is the same action \(\alpha \) appearing in Abadie (2003) which takes place on the \({\mathrm {C}}^*\)-algebra of kernels \({\mathbb {k}(\mathcal {A})}\) of the Fell bundle \(\mathcal {A}\). As already shown in Abadie (2003), \({\mathbb {k}(\mathcal {A})}\) can be canonically identified with the crossed product \(C^*(\mathcal {A})\rtimes _{\delta _\mathcal {A}}G\) by the dual coaction \(\delta _\mathcal {A}\) of G on the full cross-sectional \({\mathrm {C}}^*\)-algebra \(C^*(\mathcal {A})\) of \(\mathcal {A}\); one can also use the reduced \({\mathrm {C}}^*\)-algebra \(C^*_\mathrm r(\mathcal {A})\) together with its dual coaction \(\delta _\mathcal {A}^\mathrm r\) of G, which is a normalisation of \(\delta _\mathcal {A}\). Since the dual coaction \(\delta _\mathcal {A}\) is maximal, \({\mathbb {k}(\mathcal {A})}\rtimes _\alpha G\cong C^*(\mathcal {A})\otimes \mathbb {K}(L^2(G))\) and similarly \({\mathbb {k}(\mathcal {A})}\rtimes _{\alpha ,\mathrm r}G\cong C^*_\mathrm r(\mathcal {A})\otimes \mathbb {K}(L^2(G))\). As a consequence of this, the notion of weak (hence strong) equivalence preserves full and reduced cross-sectional \({\mathrm {C}}^*\)-algebras, that is, weakly equivalent Fell bundles have (strongly Morita) equivalent full and reduced cross-sectional \({\mathrm {C}}^*\)-algebras. Using the same idea, we also derive a version of this result for certain exotic completions \(C^*_\mu (\mathcal {A})\) introduced in Buss and Echterhoff (2015) (some of the latter results were also obtained in Abadie and Ferraro (2017), though with different methods). Moreover, we show that two Fell bundles \(\mathcal {A}\) and \(\mathcal {B}\) are weakly equivalent if and only if the corresponding actions on their \({\mathrm {C}}^*\)-algebras of kernels \({\mathbb {k}(\mathcal {A})}\) and \({\mathbb {k}(\mathcal {B})}\) are equivariantly Morita equivalent, if and only if their dual coactions on \(C^*(\mathcal {A})\) and \(C^*(\mathcal {B})\) are equivariantly Morita equivalent. Strong equivalence of Fell bundles can also be characterised in a similar fashion by the restriction of the global actions on \({\mathbb {k}(\mathcal {A})}\) and \({\mathbb {k}(\mathcal {B})}\) to the partial actions on the \({\mathrm {C}}^*\)-algebras of compact operators \(\mathbb {K}(L^2(\mathcal {A}))\) and \(\mathbb {K}(L^2(\mathcal {B}))\).

Section 5 of the paper can be viewed as a sample of the potential applications of our main results from the previous sections. We study the partial action associated to Fell bundles on spectra level: a Fell bundle \(\mathcal {A}\) over G induces a partial action of G on the spectrum (both primitive and irreducible representations) of the unit fibre \(A_e\). This partial actions have already been introduced in Abadie and Abadie (2017) for discrete groups. We extend the construction to all locally compact groups, proving that the partial action is always continuous. We then show that the enveloping action of the spectral partial action associated to a Fell bundle \(\mathcal {A}\) is precisely the global action on the spectrum of \({\mathbb {k}(\mathcal {A})}\) induced by its canonical action \(\alpha \). In particular, many results of Abadie and Abadie (2017) can be obtained from the already existing results for crossed products by ordinary actions. In the same spirit we extend some results about amenability and nuclearity of crossed products to the realm of Fell bundles.

We also add an appendix where we use the tensor product construction of equivalence bundles from Abadie and Ferraro (2017) to prove that strong equivalence of Fell bundles is an equivalence relation.

2 Fell Bundles and Globalization of Weak Group Partial Actions

Let \(\mathcal {B}\) be a Fell bundle over a locally compact group G, both fixed for the rest of this article. We denote by \(\,\mathrm dt\) the integration with respect to a fixed left invariant Haar measure on G and write \(\Delta \) for the modular function of G.

Notation 2.1

Given two sets X and Y for which a product xy between elements \(x\in X\) and \(y\in Y\) is defined and is contained in a normed vector space, we write XY to mean the closed linear space of all such products, that is,
$$\begin{aligned} XY:=\overline{{\text {span}}}\, \{xy:x\in X,\ y\in Y\}. \end{aligned}$$
This applies, for instance, if X and Y are subsets of two fibers \(B_r\) and \(B_s\) of a Fell bundle \(\mathcal {B}\), in which case XY is a closed linear subspace of \(B_{rs}\).

Following Buss et al. (2013), we view a saturated Fell bundle \(\mathcal {B}\) as an action by equivalences of G on the \({\mathrm {C}}^*\)-algebra \(B_e\) (the unit fiber). A non-saturated Fell bundle is viewed as a partial action by equivalences of G on \(B_e\).

To explain this idea explicitly take \(t\in G\) and define \(D^\mathcal {B}_t:=B_tB_t^*=B_t B_{t^{-1}}\) and note that \(B_t\) is a \(D^\mathcal {B}_t-D^\mathcal {B}_{t^{-1}}\)-equivalence bimodule with the natural structure inherited from \(\mathcal {B}\). The key is to think of \(B_t\) as an arrow from \(D^\mathcal {B}_{t^{-1}}\) to \(D^\mathcal {B}_t:\)where the arrows go from right to left to be consistent with the example below.
Given \(r,s\in G\), the \(B_e\)-tensor product \(B_r\otimes _{B_e}B_s\) is the composition of arrows and implements a Morita equivalence between \(J_{r,s}:=B_rB_s(B_rB_s)^*\) and \(J_{{s^{-1}},{r^{-1}}}\). Moreover, \(B_r\otimes _{B_e}B_s\) is isomorphic to \(J_{r,s}B_{rs}=B_{rs}J_{{s^{-1}},{r^{-1}}}\) through the unique unitary U such that
$$\begin{aligned} U:B_r\otimes _{B_e}B_s\rightarrow J_{r,s}B_{rs},\ a\otimes b\mapsto ab. \end{aligned}$$
Then the composition of \(B_r\) with \(B_s\), namely \( B_r\otimes _{B_e}B_s\), is (isomorphic to) a restriction of \(B_{rs}\) to an ideal.

In this way every Fell bundle becomes a \({\mathrm {C}}^*\)-partial action by equivalence bimodules and the bundle is saturated if and only if the action is global (meaning that \(D_t^\mathcal {B}=D_e^\mathcal {B}=\mathcal {B}_e\) for all \(t\in G\)).

Example 2.2

Let \(\alpha =\left( \{A_t\}_{t\in G},\{\alpha _t\}_{t\in G}\right) \) be a partial action of G on the \({\mathrm {C}}^*\)-algebra A and let \(\mathcal {B}_\alpha \) be its semidirect product bundle, see Exel (1997). The fiber \(B_t\) is \(A_t\times \{t\}=A_t\delta _t\) and \(D^{\mathcal {B}_\alpha }_t = A_t\times \{e\}\cong A_t\). The operations on the equivalence \(A_t-A_{t^{-1}}\)-bimodule \(B_t\) are given by
$$\begin{aligned} {}_{A_t}\langle x\delta _t ,y\delta _t\rangle&= xy^*&a\cdot x\delta _t&= ax\delta _t\\ \langle x\delta _t,y\delta _t\rangle _{A_{t^{-1}}}&= \alpha _{t^{-1}}(x^*y)&x\delta _t\cdot b&= \alpha _t(\alpha _{t^{-1}}(x)b)\delta _t. \end{aligned}$$

Next we extend the notion of equivalence between partial actions to the context of Fell bundles.

Definition 2.3

Let \(\mathcal {B}\) be a Fell bundle over G. A right Hilbert\(\mathcal {B}\)-bundle is a Banach bundle \(\mathcal {X}\) over G with continuous functions
$$\begin{aligned} \langle \ ,\ {\rangle _\mathcal {B}}:\mathcal {X}\times \mathcal {X}\rightarrow \mathcal {B},\ (x,y)\mapsto \langle x,y{\rangle _\mathcal {B}},\qquad \mathcal {X}\times \mathcal {B}\rightarrow \mathcal {X},\ (x,b)\mapsto xb, \end{aligned}$$
(2.1)
such that:
  1. (1R)

    For all \(r,s\in G\), \(X_rB_s\subseteq X_{rs}\) and \(\langle X_r,X_s{\rangle _\mathcal {B}}\subseteq B_{{r^{-1}}s}\).

     
  2. (2R)

    For all \(r,s\in G\) and \(x\in X_r\) the function \(X_r\times B_s\rightarrow X_{rs}\), \((x,b)\mapsto xb\), is bilinear and \(X_s\rightarrow B_{{r^{-1}}s}\), \(y\mapsto \langle x,y{\rangle _\mathcal {B}}\), is linear.

     
  3. (3R)

    For all \(x,y\in X\) and \(b\in B\), \(\langle x,y{\rangle _\mathcal {B}}^* =\langle y,x{\rangle _\mathcal {B}}\), \(\langle x,yb{\rangle _\mathcal {B}}=\langle x,y{\rangle _\mathcal {B}}b\), \(\langle x,x{\rangle _\mathcal {B}}\ge 0\) (in \(B_e\)) and \(\Vert x\Vert ^2 = \Vert \langle x,x{\rangle _\mathcal {B}}\Vert \).

     
We say that \(\mathcal {X}\) is full if
$$\begin{aligned} \overline{{\text {span}}}\, \{\langle X_r,X_r{\rangle _\mathcal {B}}:r\in G\}=B_e. \end{aligned}$$
(2.2)
We say that \(\mathcal {X}\) is strongly full if
$$\begin{aligned} \overline{{\text {span}}}\, \langle X_r,X_r{\rangle _\mathcal {B}}=B_r^*B_r\quad \text{ for } \text{ all } r\in G. \end{aligned}$$
(2.3)

Remark 2.4

  1. (1)

    By a Banach bundle we mean a continuous Banach bundle in the sense of Doran–Fell, see Doran and Fell (1988a). In particular, a Fell bundle is a continuous Banach bundle, by definition. However, the main axiom concerning the continuity of the bundle, namely, the continuity of the norm function \(\mathcal {B}\rightarrow [0,\infty )\), \(b\mapsto \Vert b\Vert \), is somehow automatic, see (Buss et al. 2013, Lemma 3.16). A similar observation holds for every Hilbert \(\mathcal {B}\)-bundle: the continuity of the norm function \(x\mapsto \Vert x\Vert \) on \(\mathcal {X}\) follows from the continuity of the norm function on \(\mathcal {B}\) because \(\Vert x\Vert =\Vert \langle x,x{\rangle _\mathcal {B}}\Vert ^{1/2}\).

     
  2. (2)
    The fullness condition (2.2) is equivalent to the condition that
    $$\begin{aligned} B_r=\overline{{\text {span}}}\, \{\langle X_s,X_{sr}{\rangle _\mathcal {B}}:s\in G\}=\overline{{\text {span}}}\, \{\langle X_s,X_t{\rangle _\mathcal {B}}: s^{-1}t=r\} \end{aligned}$$
     
for all \(r\in G\) because if (2.2) holds, then
$$\begin{aligned} B_r = B_eB_r=\overline{{\text {span}}}\, \{\langle X_s,X_s{\rangle _\mathcal {B}}B_r:s\in G\}\subseteq \overline{{\text {span}}}\, \{\langle X_s,X_{sr}{\rangle _\mathcal {B}}:s\in G\}\subseteq B_r. \end{aligned}$$
In general, \(\langle X_r,X_r{\rangle _\mathcal {B}}\) is only contained in \(B_e\), not necessarily in the ideal \(B_r^*B_r\subseteq B_e\). Hence the strong fullness condition (2.3) requires that \(\langle X_r,X_r{\rangle _\mathcal {B}}\) is contained and is linearly dense in the ideal \(B_r^*B_r\). Moreover, if \(\mathcal {X}\) is strongly full, then \(\overline{{\text {span}}}\, \langle X_r,X_s{\rangle _\mathcal {B}}= B_r^*B_s\) for all \(r,s\in G\) because:
$$\begin{aligned} \overline{{\text {span}}}\, \langle X_r,X_s{\rangle _\mathcal {B}}&=\overline{{\text {span}}}\, \langle X_r\langle X_r,X_r{\rangle _\mathcal {B}},X_s\langle X_{s},X_s{\rangle _\mathcal {B}}{\rangle _\mathcal {B}}\\&\subseteq \overline{{\text {span}}}\, \langle X_r,X_r{\rangle _\mathcal {B}}\langle X_r,X_s{\rangle _\mathcal {B}}\langle X_s,X_s{\rangle _\mathcal {B}}\\&\subseteq \overline{{\text {span}}}\, B_r^*(B_r\langle X_r,X_s{\rangle _\mathcal {B}})B_s^*B_s \subseteq \overline{{\text {span}}}\, B_r^*B_sB_s^*B_s = B_r^*B_s \end{aligned}$$
and
$$\begin{aligned} B_r^*B_s&= B_r^* (B_{r^{-1}})^* B_{r^{-1}}B_sB_{s^{-1}}B_s = \overline{{\text {span}}}\, B_r^* \langle X_{r^{-1}},X_{r^{-1}}{\rangle _\mathcal {B}}B_s\langle X_s,X_s{\rangle _\mathcal {B}}\\&= \overline{{\text {span}}}\, \langle X_{r^{-1}}B_r,X_{r^{-1}}{\rangle _\mathcal {B}}\langle X_s B_s^*,X_s{\rangle _\mathcal {B}}\subseteq \overline{{\text {span}}}\, \langle X_e,X_{r^{-1}}{\rangle _\mathcal {B}}\langle X_e,X_s{\rangle _\mathcal {B}}\\&\subseteq \overline{{\text {span}}}\, \langle X_e\langle X_{r^{-1}},X_e{\rangle _\mathcal {B}},X_s{\rangle _\mathcal {B}}\subseteq \overline{{\text {span}}}\, \langle X_r,X_s{\rangle _\mathcal {B}}. \end{aligned}$$

Left Hilbert bundles are similarly defined. We spell out the complete definition for convenience.

Definition 2.5

Let \(\mathcal {A}\) be a Fell bundle over G. A left Hilbert \(\mathcal {A}\)-bundle is a Banach bundle \(\mathcal {X}\) over G with continuous functions
$$\begin{aligned} {}_\mathcal {A}\langle \ ,\ {\rangle }:\mathcal {X}\times \mathcal {X}\rightarrow \mathcal {A},\ (x,y)\mapsto {}_\mathcal {A}\langle x,y{\rangle },\qquad \mathcal {A}\times \mathcal {X}\rightarrow \mathcal {X},\ (a,x)\mapsto ax, \end{aligned}$$
(2.4)
such that:
  1. (1L)

    For all \(r,s\in G\), \(A_rX_s\subseteq X_{rs}\) and \({}_\mathcal {A}\langle X_r,X_s{\rangle }\subseteq A_{r{s^{-1}}}\).

     
  2. (2L)

    For all \(r,s\in G\) and \(x\in X_r\) the function \(A_r\times X_s\rightarrow X_{rs}\), \((a,x)\mapsto ax\), is bilinear and \(X_s\rightarrow A_{s{r^{-1}}}\), \(y\mapsto {}_\mathcal {A}\langle y,x{\rangle }\), is linear.

     
  3. (3L)

    For all \(x,y\in X\) and \(a\in A\), \({}_\mathcal {A}\langle x,y{\rangle }^* ={}_\mathcal {A}\langle y,x{\rangle }\), \({}_\mathcal {A}\langle ax,y{\rangle }=a{}_\mathcal {A}\langle x,y{\rangle }\), \({}_\mathcal {A}\langle x,x{\rangle }\ge 0\) (in \(A_e\)) and \(\Vert x\Vert ^2 = \Vert {}_\mathcal {A}\langle x,x{\rangle }\Vert \).

     
If
$$\begin{aligned} A_e=\overline{{\text {span}}}\, \{{}_\mathcal {A}\langle X_r,X_r{\rangle }:r\in G\}, \end{aligned}$$
(2.5)
\(\mathcal {X}\) is called full, and if
$$\begin{aligned} A_rA_r^* =\overline{{\text {span}}}\, {}_\mathcal {A}\langle X_r,X_r{\rangle }\quad \text{ for } \text{ all } r\in G, \end{aligned}$$
(2.6)
\(\mathcal {X}\) is called strongly full.

Definition 2.6

Let \(\mathcal {A}\) and \(\mathcal {B}\) be Fell bundles over G. A weak\(\mathcal {A}-\mathcal {B}\)-equivalence bundle is a Banach bundle \(\mathcal {X}\) which is a full left Hilbert \(\mathcal {A}\)-bundle, a full right Hilbert \(\mathcal {B}\)-bundle and \({}_\mathcal {A}\langle x,y{\rangle }z=x\langle y,z{\rangle _\mathcal {B}}\) for all \(x,y,z\in \mathcal {X}\). In this case we say that \(\mathcal {A}\) and \(\mathcal {B}\) are weakly equivalent. If, in addition, \(\mathcal {X}\) is strongly full, both as a left and right bundle, we say that \(\mathcal {X}\) is a strong \(\mathcal {A}-\mathcal {B}\)-equivalence and that \(\mathcal {A}\) and \(\mathcal {B}\) are strongly equivalent.

We have included an appendix where we show several properties regarding tensor products of equivalence bundles. For example we show, in Theorem A.1, that strong equivalence is an equivalence relation. Weak equivalence was shown to be an equivalence relation in Abadie and Ferraro (2017).

Example 2.7

Every equivalence of partial actions (see Abadie 2003, Section 4.2 and Exel 2017, Definition 15.7) can be turned into a strong equivalence between the associated Fell bundles. Suppose \(\alpha =\{I_{t^{-1}}{\mathop {\rightarrow }\limits ^{\alpha _t}}I_t\}_{t\in G}\) and \(\beta =\{J_{t^{-1}}{\mathop {\rightarrow }\limits ^{\beta _t}}J_t\}_{t\in G}\) are partial actions on the \({\mathrm {C}}^*\)-algebras A and B respectively, and suppose X is an \(A-B\)-equivalence bimodule such that \(I_tX=XJ_t\) for all \(t\in G\). For \(t\in G\), define \(X_t:=I_tX=XJ_t\) and suppose \(\gamma =\{X_{t^{-1}}{\mathop {\rightarrow }\limits ^{\gamma _t}}X_t\}_{t\in G}\) is a partial action of G on X such that
$$\begin{aligned} \alpha _t(\langle x,y\rangle _A)\gamma _t(z) =\gamma _t(\langle x,y\rangle _Az) =\gamma _t(x\langle y,z\rangle _B) =\gamma _t(x)\beta _t(\langle y,z\rangle _B) \end{aligned}$$
for all \(t\in G\) and \(x,y,z\in X_t\). Then \(\alpha \) and \(\beta \) are said to be Morita equivalent, and the following notations are used: \(X^l:=A\), \(X^r:=B\), \(\gamma ^l=\alpha \), and \(\gamma ^r=\beta \) (in fact X determines A and B up to isomorphism, and then \(\gamma ^l\) and \(\gamma ^r\) are determined by \(\gamma \); see Abadie (2003) for details). Let \(\mathbb {L}(\gamma )\) be the linking partial action of \(\gamma \) (see the proof of Proposition 4.5 of Abadie (2010)) and let \(\mathcal {B}_{\mathbb {L}(\gamma )}\) be the Fell bundle associated with \(\mathbb {L}(\gamma )\). Define \(\mathcal {X}_\gamma \) as the Banach subbundle of \(\mathcal {B}_{\mathbb {L}(\gamma )}\)
$$\begin{aligned} X_\gamma :=\left\{ \left( \begin{array}{ll} 0 &{}\quad x\\ 0 &{}\quad 0 \end{array} \right) \delta _t:x\in X_t,\ t\in G \right\} . \end{aligned}$$
With the structure inherited from the identity \(\mathcal {B}_{\mathbb {L}(\gamma )}-\mathcal {B}_{\mathbb {L}(\gamma )}\)-bundle structure of \(\mathcal {B}_{\mathbb {L}(\gamma )}\), \(\mathcal {X}_\gamma \) is a strong \(\mathcal {B}_\alpha -\mathcal {B}_\beta \)-equivalence bundle, where \(\mathcal {B}_\alpha \) and \(\mathcal {B}_\beta \) denote the Fell bundles associated with \(\alpha \) and \(\beta \), respectively.

The notion of weak equivalence allows us to “identify” partial actions with the corresponding enveloping actions, in case these exist. This is explained in the following example. In particular this shows that a non-saturated Fell bundle may be weakly equivalent to a saturated one.

Example 2.8

Let \(\beta \) be a global action of G on the \({\mathrm {C}}^*\)-algebra B and assume that A is a \({\mathrm {C}}^*\)-ideal of B such that \(B=\overline{{\text {span}}}\, \{\beta _t(A):t\in G\}\). This means that \(\beta \) is the enveloping (global) action of the partial action \(\alpha \) given as the restriction of \(\beta \) to A (see Abadie (2003)). In this situation, \(\mathcal {B}_\alpha \) is weakly equivalent to \(\mathcal {B}_\beta \). The equivalence is implemented by the bundle \(\mathcal {X}=A\times G\), considered as a Banach subbundle of \(\mathcal {B}_\beta \) and viewing \(\mathcal {B}_\alpha \) as a Fell subbundle of \(\mathcal {B}_\beta \). The operations are the ones inherited from the identity \(\mathcal {B}_\beta -\mathcal {B}_\beta \)-bundle. Notice that \(\mathcal {B}_\alpha \) is, in general, not strongly equivalent to \(\mathcal {B}_\beta \) because a strong equivalence between Fell bundles implies in a (strong) Morita equivalence between their unit fibers A and B. And it is easy to produce examples where this is not the case. For instance, one may take a commutative \({\mathrm {C}}^*\)-algebra \(B=\mathrm {C_0}(X)\) and an ideal \(A\subseteq B\) which is not isomorphic to B, like \(B=\mathrm {C_0}(\mathbb {R})\) and \(A=\mathrm {C_0}((0,1)\cup (1,2))\) with \(G=\mathbb {R}\) acting by translation.

With notation as in Example 2.8, we have \(\mathcal {B}_\alpha \mathcal {X}\subseteq \mathcal {X}\), \(\mathcal {X}\mathcal {B}_\beta =\mathcal {X}\), \(\mathcal {B}_\alpha =\mathcal {X}\mathcal {X}^*\) and \(\mathcal {X}^*\mathcal {X}=\mathcal {B}_\beta \) (where, for example, the equality \(\mathcal {B}_\alpha =\mathcal {X}\mathcal {X}^*\) means that the t-fiber of \(\mathcal {B}_\alpha \) is the closed linear span of all \(\mathcal {X}_s\mathcal {X}_r^*\) with \(sr^{-1}=t\)). This motivates the following.

Definition 2.9

An enveloping bundle of a Fell bundle \(\mathcal {A}\) is a saturated Fell bundle \(\mathcal {B}\) for which there exists a Fell subbundle \(\mathcal {C}\subseteq \mathcal {B}\) and an isomorphism of Fell bundles \(\pi :\mathcal {A}\rightarrow \mathcal {C}\) such that for \(\mathcal {X}:=\mathcal {C}\mathcal {B}\), we have \(\mathcal {X}\mathcal {X}^*=\mathcal {C}\) and \(\mathcal {X}^*\mathcal {X}=\mathcal {B}\).

Remark 2.10

With notation as above, the bundle \(\mathcal {X}\) above is a weak equivalence \(\mathcal {A}-\mathcal {B}\)-bundle with the operations \({}_\mathcal {A}\langle x ,y{\rangle }= \pi ^{-1}(xy^*)\), \((a,x)\mapsto \pi (a)x\), \(\langle x,y{\rangle _\mathcal {B}}= x^*y\) and \( (x,b)\mapsto xb\). Hence every Fell bundle is weakly equivalent to its enveloping bundle (if it admits one). The equivalence is, however, not strong in general (see Example 2.8).

Imitating the notion of Morita enveloping action from Abadie (2003) we state the following.

Definition 2.11

A Morita enveloping bundle of a Fell bundle \(\mathcal {A}\) is a saturated Fell bundle \(\mathcal {B}\) which is the enveloping bundle of a Fell bundle strongly equivalent to \(\mathcal {A}\).

It is shown in Abadie (2003) that every partial action on a \({\mathrm {C}}^*\)-algebra has a Morita enveloping action. In the next section we show that every Fell bundle admits a Morita enveloping Fell bundle. Moreover, we show that this Morita enveloping Fell bundle can be realised as a semidirect product bundle of a global action. This global action is unique up to Morita equivalence of actions on \({\mathrm {C}}^*\)-algebras.

Remark 2.12

Since weak equivalence of Fell bundles is an equivalence relation, the Morita enveloping bundle of a Fell bundle is unique up to weak equivalence. In fact, we will show in Corollary 4.11 that it is unique up to strong equivalence.

The Bundle of Generalized Compact Operators

Given a full right Hilbert \(\mathcal {B}\)-bundle \(\mathcal {X}\) there exists, up to isomorphism, a unique Fell bundle \(\mathbb {K}(\mathcal {X})\) such that \(\mathcal {X}\) is a weak \(\mathbb {K}(\mathcal {X})-\mathcal {B}\) equivalence bundle. We recall next the main lines of the construction of \(\mathbb {K}(\mathcal {X})\), and we refer to Abadie and Ferraro (2017) for complete details.

To describe the fiber over \(t\in G\) of the bundle \(\mathbb {K}(\mathcal {X})\), note first that, given \(x,y\in \mathcal {X}\), say \(x\in X_{ts}\) and \(y\in X_s\) for some \(s,t\in G\), we have a map \([x,y]:\mathcal {X}\rightarrow \mathcal {X}\) such that \([x,y]z:=x\langle y,z\rangle \) for all \(z\in \mathcal {X}\). The map [xy] has the following properties:
  1. (1)

    \([x,y]X_r\subseteq X_{tr}\) for all \(r\in G\)

     
  2. (2)

    [xy] is linear when restricted to each fiber \(X_r\) of \(\mathcal {X}\).

     
  3. (3)

    [xy] is continuous.

     
  4. (4)

    [xy] is bounded: its norm \(\Vert [x,y]\Vert :=\sup _{\{z\in \mathcal {X}:\Vert z\Vert \le 1\}}\Vert [x,y]z\Vert \) is finite with \(\Vert [x,y]\Vert \le \Vert x\Vert \,\Vert y\Vert \).

     
  5. (5)

    [xy] is adjointable: there exists a (necessarily unique) adjoint operator \([x,y]^*:\mathcal {X}\rightarrow \mathcal {X}\) such that \(\langle [x,y]z,z'\rangle =\langle z,[x,y]^*z'\rangle \) for all \(z,z'\in \mathcal {X}\). Moreover, we have \([x,y]^*=[y,x]\).

     
It is not hard to check that the vector space \(\mathbb {B}_t(\mathcal {X})\) of maps \(S:\mathcal {X}\rightarrow \mathcal {X}\) that satisfy properties (1)–(5) above is a Banach space, in fact a \({\mathrm {C}}^*\)-ternary ring with the operation \((S_1,S_2,S_3):=S_1S_2^*S_3\). The elements of \(\mathbb {B}_t(\mathcal {X})\) are called adjointable operators of order t. If \(G_d\) is the group G with the discrete topology, it follows that the family \((\mathbb {B}_t(\mathcal {X}))_{t\in G}\) is a Fell bundle over \(G_d\), where the product is given by composition. Now define \(\mathbb {K}_t(\mathcal {X}):=\overline{\text {span}}\{[x,y]:\, x\in X_{ts}, y\in X_s, s\in G\}\). It is easy to check that \(\mathbb {K}(\mathcal {X}):=(\mathbb {K}_t(\mathcal {X}))_{t\in G}\) is a Fell subbundle of \((\mathbb {B}_t(\mathcal {X}))_{t\in G}\). Finally, there is a suitable topology on \(\mathbb {K}(\mathcal {X})\) making it a Fell bundle over G, and \(\mathcal {X}\) is a weak \(\mathbb {K}(\mathcal {X})-\mathcal {B}\) equivalence bundle with the obvious operations and inner products, see Abadie and Ferraro (2017) for details.

The Fell bundle \(\mathbb {K}(\mathcal {X})\) is unique in the following sense: if \(\mathcal {X}\) is a weak \(\mathcal {A}-\mathcal {B}\) equivalence, then there exists an isomorphism \(\pi :\mathcal {A}\rightarrow \mathbb {K}(\mathcal {X})\) such that \(\pi ({}_{\mathcal {A}}\langle x,y\rangle )=[x,y]\) for all \(x,y\in \mathcal {X}\) (see Abadie and Ferraro 2017, Corollary 3.10).

The Linking Fell Bundle of an Equivalence Bundle

Given a weak \(\mathcal {A}-\mathcal {B}\) equivalence \(\mathcal {X}\), it is possible to define a Fell bundle \(\mathbb {L}(\mathcal {X})=(L_t)_{t\in G}\) which plays a role similar to that of the linking algebra of an imprimitivity bimodule. The fiber \(L_t\) over \(t\in G\) is defined to be \(L_t:=\begin{pmatrix} A_t&{}X_t\\ \tilde{X}_{t^{-1}}&{}B_t\end{pmatrix}\) with entrywise vector space operations (here, given an \(A-B\) Hilbert bimodule X, \(\tilde{X}\) denotes its dual \(B-A\) Hilbert bimodule). The operations and topology on \(\mathbb {L}(\mathcal {X})\) are defined as follows:
  1. (1)
    Product and involution on \(\mathbb {L}(\mathcal {X})\) are given by
    $$\begin{aligned} \begin{pmatrix} a &{}\quad x\\ \widetilde{y} &{}\quad b \end{pmatrix} \begin{pmatrix} c &{}\quad u\\ \widetilde{v} &{}\quad d \end{pmatrix} =\begin{pmatrix} ac + {}_\mathcal {A}\langle x,v{\rangle }&{}\quad au+xd\\ \widetilde{c^*y} + \widetilde{vb^*} &{}\quad \langle y,u{\rangle _\mathcal {B}}+ bd \end{pmatrix} \quad \text{ and } \end{aligned}$$
    $$\begin{aligned} \begin{pmatrix} a &{}\quad x\\ \widetilde{y} &{}\quad b \end{pmatrix}^* =\begin{pmatrix} a^* &{}\quad y\\ \widetilde{x} &{}\quad b^* \end{pmatrix}.\end{aligned}$$
     
  2. (2)
    Given \(\xi \in C_c(\mathcal {A})\), \(\eta \in C_c(\mathcal {B})\) and \(f,g\in C_c(\mathcal {X})\) the function
    $$\begin{aligned}\begin{pmatrix} \xi &{}\quad f\\ g &{}\quad \eta \end{pmatrix}:G\rightarrow \mathbb {L}(\mathcal {X}), \ t\mapsto \begin{pmatrix} \xi (t) &{}\quad f(t)\\ {\widetilde{g({t^{-1}})}} &{}\quad \eta (t) \end{pmatrix} \end{aligned}$$
    is a continuous section (see Doran and Fell 1988a, 13.18).
     
The subbundle \(\mathcal {A}\oplus \mathcal {X}\) of \(\mathbb {L}(\mathcal {X})\) is then a weak \(\mathcal {A}-\mathbb {L}(\mathcal {X})\) equivalence bundle, and the subbundle \(\mathcal {X}\oplus \mathcal {B}\) is a weak \(\mathbb {L}(\mathcal {X})-\mathcal {B}\) equivalence bundle. We refer the reader to the third section of Abadie and Ferraro (2017) for details.

3 Canonical Action on the Kernels and Morita Equivalence

Recall that \(L^2(\mathcal {B})\) is the (full) right Hilbert \(B_e\)-module obtained as the completion of \(\mathrm {C_c}(\mathcal {B})\) with respect to the pre-Hilbert \(B_e\)-module structure given by the operations
$$\begin{aligned} \langle f,g\rangle _{L^2}:=\int _G f(t)^*g(t)\,\mathrm dt,\qquad (f\cdot b)(t):=f(t)b, \end{aligned}$$
for \(f,g\in \mathrm {C_c}(\mathcal {B})\) and \(b\in B_e\).
The Banach bundle \(\mathcal {L}^2\mathcal {B}\) is, as a topological bundle, the constant fiber bundle
$$\begin{aligned} L^2(\mathcal {B})\times G\rightarrow G,\ f\delta _t\mapsto t. \end{aligned}$$
The norm is given by \(\Vert f\delta _r\Vert :=\Delta (r)^{-1/2}\Vert f\Vert _{L^2}\).

Proposition 3.1

Given \(r,s,t,p\in G\), \(f,g\in \mathrm {C_c}(\mathcal {B})\) and \(b\in B_t\), define
$$\begin{aligned} \langle f\delta _r,g\delta _s{\rangle _\mathcal {B}}&:=\int _G f(pr)^*g(ps)\, \,\mathrm dp \end{aligned}$$
(3.1)
$$\begin{aligned} f\delta _r b&:=fb\delta _{rt},\ \text{ with } fb(p):=f(p{t^{-1}})b. \end{aligned}$$
(3.2)
With these operations, \(\mathcal {L}^2\mathcal {B}\) becomes a full right Hilbert \(\mathcal {B}\)-bundle.

Proof

To simplify the notation we define \(L_r:=\mathrm {C_c}(\mathcal {B})\times \{r\}\subseteq \mathcal {L}^2\mathcal {B}\). It is clear that the function \(L_r\times B_t\rightarrow L_{rt}\), \((f\delta _r,b)\mapsto f\delta _rb\), is bilinear and that \(L_r\times L_s\rightarrow B_{{r^{-1}}s},\ g\delta _s\mapsto \langle f\delta _r,g\delta _s{\rangle _\mathcal {B}}\), is linear. Straightforward computations show that \(\langle f\delta _r,g\delta _s b{\rangle _\mathcal {B}}= \langle f\delta _r,g\delta _s{\rangle _\mathcal {B}}b\), \(\langle f\delta _r,g\delta _s b{\rangle _\mathcal {B}}^*=\langle g\delta _s b,f\delta _r{\rangle _\mathcal {B}}\) and \(\langle f\delta _r,f\delta _r{\rangle _\mathcal {B}}= \Delta (r)^{-1}\langle f,f\rangle _{L^2}\). In particular, \(\langle f\delta _r,f\delta _r{\rangle _\mathcal {B}}\ge 0\).

The canonical pre-Hilbert \(B_e\)-module structure of \(L_r\) induces the norm of \(\mathcal {L}^2\mathcal {B}\), because
$$\begin{aligned} \Vert \langle f\delta _r,f\delta _r{\rangle _\mathcal {B}}\Vert = \Vert \Delta (r)^{-1}\langle f,f\rangle _{L^2}\Vert =\Delta (r)^{-1}\Vert f\Vert _{L^2}^2. \end{aligned}$$
The action of \(\mathcal {B}\) on \(\mathrm {C_c}(\mathcal {B})\times G\) can be extended in a unique way to \(\mathcal {L}^2\mathcal {B}\) because
$$\begin{aligned} \Vert f\delta _r b\Vert ^2=\Vert b\langle f\delta _r,f\delta _r{\rangle _\mathcal {B}}b\Vert \le \Vert b\Vert ^2\Vert \langle f\delta _r,f\delta _r{\rangle _\mathcal {B}}\Vert =\Vert b\Vert ^2\Vert f\delta _r\Vert ^2. \end{aligned}$$
To see that the inner product defined on \(\mathrm {C_c}(\mathcal {B})\times G\) extends to \(\mathcal {L}^2\mathcal {B}\) it suffices to prove that
$$\begin{aligned} \Vert \langle f\delta _r,g\delta _s{\rangle _\mathcal {B}}\Vert \le \Vert f\delta _r\Vert \Vert g\delta _s\Vert . \end{aligned}$$
(3.3)
To do this take a representation \(T:\mathcal {B}\rightarrow \mathbb {B}(\mathcal {H})\) with \(T|_{B_e}\) faithful. Then \(\Vert T_b\Vert =\Vert T_{b^*b}\Vert ^{1/2}=\Vert b^*b\Vert ^{1/2}=\Vert b\Vert \) for all \(b\in \mathcal {B}\). Let \(Tf\delta _r\in \mathrm {C_c}(G,\mathbb {B}(\mathcal {H}))\) be defined as \(Tf\delta _r(t):=T_{f(tr)}\) and consider \(\mathrm {C_c}(G,\mathbb {B}(\mathcal {H}))\) as a subspace of \(L^2(G,\mathbb {B}(\mathcal {H}))\). The Cauchy–Schwarz inequality in \(L^2(G,\mathbb {B}(\mathcal {H}))\) implies
$$\begin{aligned} \Vert \langle f\delta _r,g\delta _s{\rangle _\mathcal {B}}\Vert&= \left\| \int _G \big (Tf\delta _r(t)\big )^* Tg\delta _s(t)\,\mathrm dt\right\| = \Vert \langle Tf\delta _r,Tg\delta _s\rangle _{L^2}\Vert \\&\le \Vert \langle Tf\delta _r,Tf\delta _r\rangle _{L^2}\Vert ^{1/2}\Vert \langle Tg\delta _s,Tg\delta _s\rangle _{L^2}\Vert ^{1/2}\\&\le \Vert T_{\langle f\delta _r,f\delta _r{\rangle _\mathcal {B}}}\Vert ^{1/2}\Vert T_{\langle g\delta _s,g\delta _s{\rangle _\mathcal {B}}}\Vert ^{1/2} = \Vert f\delta _r \Vert \Vert g\delta _s\Vert . \end{aligned}$$
This implies inequality (3.3).
With respect to the density of inner products note that
$$\begin{aligned} \overline{{\text {span}}}\, \langle L_r,L_r{\rangle _\mathcal {B}}= \overline{{\text {span}}}\, \Delta (r)^{-1}\langle \mathrm {C_c}(\mathcal {B}),\mathrm {C_c}(\mathcal {B}){\rangle _\mathcal {B}}=B_e \end{aligned}$$
for all \(r\in G\).
The constant section associated to \(f\in L^2(\mathcal {B})\) is \(f\delta :G\rightarrow \mathcal {L}^2\mathcal {B}\), \(t\mapsto f\delta _t\). Since for all \(r\in G\), \(\{f\delta _r:f\in \mathrm {C_c}(\mathcal {B})\}=L_r\), to show that the inner product and action are continuous it suffices to prove that the functions
$$\begin{aligned}&G\times G\rightarrow \mathcal {B},\ (r,s)\mapsto \int _G f(tr)^*g(ts)\,\mathrm dt, \quad \text{ and } \quad G\times G\rightarrow L^2(\mathcal {B}),\nonumber \\&\quad (r,s)\mapsto f[g(r)], \end{aligned}$$
are continuous for all \(f,g\in \mathrm {C_c}(\mathcal {B})\), where f[g(s)] means f acting on \(b=g(s)\) as defined in (3.2). The continuity of the first function follows adapting (Doran and Fell 1988a, II 15.19). The other function has range in \(\mathrm {C_c}(\mathcal {B})\) and is continuous in the inductive limit topology, so it is continuous as a function with codomain \(L^2(\mathcal {B})\). \(\square \)

Definition 3.2

The canonical \(L^2\)-bundle of the Fell bundle \(\mathcal {B}\) is the Hilbert \(\mathcal {B}\)-bundle \(\mathcal {L}^2\mathcal {B}\) described in the last Proposition.

We are interested in the identification of the Fell bundle of generalized compact operators \(\mathbb {K}(\mathcal {L}^2\mathcal {B})\) of \(\mathcal {L}^2\mathcal {B}\) (see end of Sect. 2), up to isomorphism of Fell bundles, because this Fell bundle is weakly equivalent to \(\mathcal {B}\). We will show that \(\mathbb {K}(\mathcal {L}^2\mathcal {B})\) is a semidirect product Fell bundle associated to an action of G on a \({\mathrm {C}}^*\)-algebra.

Following Abadie (2003), we write \(\mathbb {k}_c(\mathcal {B})\) for the space of compactly supported continuous functions \(k:G\times G\rightarrow \mathcal {B}\) with \(k(r,s)\in B_{r{s^{-1}}}\) for all \(r,s\in G\). In other words, \(\mathbb {k}_c(\mathcal {B})\) is the space of compactly supported continuous sections of the pullback of \(\mathcal {B}\) along the map \(G\times G\rightarrow G\), \((r,s)\mapsto r{s^{-1}}\). It is a normed \(^{*}\)-algebra with
$$\begin{aligned} h*k(r,s)= & {} \int _G h(r,t)k(t,s)\,\mathrm dt \qquad k^*(r,s)=k(s,r)^*\\ \Vert k\Vert _2:= & {} \left( \int _{G^2}\Vert k(r,s)\Vert ^2\,\mathrm dr\,\mathrm ds\right) ^{1/2}. \end{aligned}$$
We may also endow \(\mathbb {k}_c(\mathcal {B})\) with the inductive limit topology and in this way it becomes a topological \(^{*}\)-algebra.

Completing \(\mathbb {k}_c(\mathcal {B})\) with respect to \(\Vert \ \Vert _2\) we obtain the Banach *-algebra \(\mathcal {HS}(\mathcal {B})\) of Hilbert-Schmidt operators of \(\mathcal {B}\). The \({\mathrm {C}}^*\)-algebra of kernels of \(\mathcal {B}\) is the enveloping \({\mathrm {C}}^*\)-algebra of \(\mathcal {HS}(\mathcal {B})\); it is denoted by \({\mathbb {k}(\mathcal {B})}\). There is a canonical action of G on \({\mathbb {k}(\mathcal {B})}\) given by the formula \( \beta _t(k)(r,s) = \Delta (t)k(rt,st)\) for \(k\in \mathbb {k}_c(\mathcal {B})\) and \(r,s,t\in G\).

The \({\mathrm {C}}^*\)-algebra \(\mathbb {K}(L^2(\mathcal {B}))\) of (generalised) compact operators of the Hilbert \(B_e\)-module \(L^2(\mathcal {B})\) can be canonically identified with an ideal in \({\mathbb {k}(\mathcal {B})}\): for \(f,g\in \mathrm {C_c}(\mathcal {B})\), the usual operator \(\theta _{f,g}\in \mathbb {K}(L^2(\mathcal {B}))\) given by \(\theta _{f,g}(h)=f\langle g{\mid }h\rangle \) is identified with the element \({}_{\mathbb {k}}\langle f,g\rangle \in \mathbb {k}_c(\mathcal {B})\) defined by \({}_{\mathbb {k}}\langle f,g\rangle (r,s)=f(r)g(s)^*\). These elements span an ideal \(I_c(\mathcal {B}):={\text {span}}\{{}_{\mathbb {k}}\langle f,g\rangle :f,g\in \mathrm {C_c}(\mathcal {B})\}\) in \(\mathbb {k}_c(\mathcal {B})\). Its closure \(I(\mathcal {B})\) is therefore a \({\mathrm {C}}^*\)-ideal of \({\mathbb {k}(\mathcal {B})}\). The \(\beta \)-orbit of \(I_c(\mathcal {B})\) is dense in \(\mathbb {k}_c(\mathcal {B})\) in the inductive limit topology. Moreover, \(I_c(\mathcal {B})\) is dense in \(\mathbb {k}_c(\mathcal {B})\) in the inductive limit topology if and only if \(\mathcal {B}\) is saturated.

Remark 3.3

There is a canonical coaction \(\delta _\mathcal {B}\) of G on \(C^*(\mathcal {B})\), the so-called dual coaction, and it is shown in Abadie (2003) that \({\mathbb {k}(\mathcal {B})}\) is canonically isomorphic to the crossed product \(C^*(\mathcal {B})\rtimes _{\delta _\mathcal {B}}G\) by this coaction. Moreover, this isomorphism carries the canonical action of G on \({\mathbb {k}(\mathcal {B})}\) to the dual action of G on \(C^*(\mathcal {B})\rtimes _{\delta _\mathcal {B}}G\). Thus \({\mathbb {k}(\mathcal {B})}\cong C^*(\mathcal {B})\rtimes _{\delta _\mathcal {B}}G\) as G-\({\mathrm {C}}^*\)-algebras. The dual coaction on \(C^*(\mathcal {B})\) is maximal and its normalisation is the dual coaction \(\delta _\mathcal {B}^\mathrm r\) on \(C^*_\mathrm r(\mathcal {B})\) (see Buss and Echterhoff 2015). This means that the regular representation \(C^*(\mathcal {B})\twoheadrightarrow C^*_\mathrm r(\mathcal {B})\) induces an isomorphism \(C^*(\mathcal {B})\rtimes _{\delta _\mathcal {B}}G\cong C^*_\mathrm r(\mathcal {B})\rtimes _{\delta _{\mathcal {B}}^\mathrm r}G\cong {\mathbb {k}(\mathcal {B})}\) and, moreover, there exists a natural isomorphism
$$\begin{aligned} {\mathbb {k}(\mathcal {B})}\rtimes _\beta G\cong C^*(\mathcal {B})\rtimes _{\delta _\mathcal {B}} G\rtimes _{\widehat{\delta }_\mathcal {B}}G\cong C^*(\mathcal {B})\otimes \mathbb {K}(L^2(G)) \end{aligned}$$
(3.4)
which factors through an isomorphism
$$\begin{aligned} {\mathbb {k}(\mathcal {B})}\rtimes _{\beta ,\mathrm r} G\cong C^*_\mathrm r(\mathcal {B})\rtimes _{\delta _\mathcal {B}^\mathrm r} G\rtimes _{\widehat{\delta }_\mathcal {B}^\mathrm r,\mathrm r}G \cong C^*_\mathrm r(\mathcal {B})\otimes \mathbb {K}(L^2(G)). \end{aligned}$$

Before we state our next result we introduce some notation. We shall denote by \(\mathcal {B}_\beta ={\mathbb {k}(\mathcal {B})}\times _\beta G\) the semidirect product Fell bundle associated to \(\beta \) (as defined in Doran and Fell 1988b, page 798 for ordinary actions or, more generally, for twisted partial actions in Exel 1997).

Recall that \(\mathbb {K}(L^2(\mathcal {B}))\) can be identified with an ideal of \({\mathbb {k}(\mathcal {B})}\), so we have a representation of \({\mathbb {k}(\mathcal {B})}\) as adjointable operators of \(L^2(\mathcal {B})\). We use the notation Tf to represent the action of \(T\in {\mathbb {k}(\mathcal {B})}\) on \(f\in L^2(\mathcal {B})\). For every \(k\in \mathbb {k}_c(\mathcal {B})\) and \(f\in \mathrm {C_c}(\mathcal {B})\) we have \(kf\in \mathrm {C_c}(\mathcal {B})\) and \(kf(r)=\int _G k(r,s)f(s)\,\mathrm ds\).

Theorem 3.4

Let \(\mathcal {B}\) be a Fell bundle and denote by \(\mathcal {L}^2\mathcal {B}\) its canonical \(L^2\)-bundle, which is a full right Hilbert \(\mathcal {B}\)-bundle. Then \(\mathcal {L}^2\mathcal {B}\) is a full left Hilbert \(\mathcal {B}_\beta \)-bundle with the action and inner product given by
$$\begin{aligned} T\delta _t f\delta _r = \Delta (t)^{1/2} \beta _{tr}^{-1}(T)f\delta _{tr}\qquad {}_{\mathcal {B}_\beta }\langle f\delta _r,g\delta _s \rangle =\Delta (rs)^{-1/2} \beta _r({}_{\mathbb {k}}\langle f,g\rangle )\delta _{r{s^{-1}}}, \end{aligned}$$
where \(T\delta _t\in \mathcal {B}_\beta \) and \(f\delta _r,g\delta _s\in \mathcal {L}^2\mathcal {B}\). Moreover, the left and right Hilbert bundles structures of \(\mathcal {L}^2\mathcal {B}\) are compatible and therefore \(\mathcal {L}^2\mathcal {B}\) is a weak equivalence \(\mathcal {B}_\beta -\mathcal {B}\)-bundle.

Proof

The left action is clearly bilinear and the left inner product is linear in the first variable because the inner product \({}_{\mathbb {k}}\langle \ ,\ \rangle \) is linear in the first variable. Moreover,
$$\begin{aligned} {}_{\mathcal {B}_\beta }\langle f\delta _r,g\delta _s \rangle ^* = \Delta (r)^{-1/2}\Delta (s)^{-1/2} \beta _{s{r^{-1}}}(\beta _r({}_{\mathbb {k}}\langle f,g\rangle ^*))\delta _{{s^{-1}}r} = {}_{\mathcal {B}_\beta }\langle g\delta _s,f\delta _r \rangle . \end{aligned}$$
To show that the left operations are compatible, we compute
$$\begin{aligned} {}_{\mathcal {B}_\beta }\langle T\delta _t f\delta _r,g\delta _s\rangle&= \Delta (rs)^{-1/2}\beta _{tr}({}_{\mathbb {k}}\langle \beta _{tr}^{-1}(T)f,g\rangle )\delta _{tr{s^{-1}}}. \end{aligned}$$
and
$$\begin{aligned} T\delta _t {}_{\mathcal {B}_\beta }\langle f\delta _r,g\delta _s\rangle&= \Delta (rs)^{-1/2} T\beta _{tr}({}_{\mathbb {k}}\langle f,g\rangle )\delta _{tr{s^{-1}}}. \end{aligned}$$
Then the compatibility of the left operations will follow once we show that
$$\begin{aligned} \beta _{tr}({}_{\mathbb {k}}\langle \beta _{tr}^{-1}(T)f,g\rangle ) = T\beta _{tr}({}_{\mathbb {k}}\langle f,g\rangle ), \end{aligned}$$
(3.5)
for all \(T\in {\mathbb {k}(\mathcal {B})}\) and \(f,g\in L^2(\mathcal {B})\). But using linearity and continuity it suffices to consider \(T\in \mathbb {k}_c(\mathcal {B})\) and \(f,g\in \mathrm {C_c}(\mathcal {B})\). Then we can make all the computations in \(\mathbb {k}_c(\mathcal {B})\). With this assumption, the left hand side of (3.5) evaluated at \((x,y)\in G^2\) is
$$\begin{aligned} \beta _{tr}({}_{\mathbb {k}}\langle \beta _{tr}^{-1}(T)h,g\rangle )(x,y)&= \Delta (tr) [\beta _{tr}^{-1}(T)f](xtr) g(ytr)^*\\&= \Delta (tr) \int _G \beta _{tr}^{-1}(T)(xtr,z)f(z)\,\mathrm dz\, g(ytr)^* \\&= \int _G T(x,z{r^{-1}}{t^{-1}})f(z)\,\mathrm dz\, g(ytr)^*. \end{aligned}$$
The right hand side of (3.5) evaluated at \((x,y)\in G^2\) is
$$\begin{aligned} T\beta _{tr}({}_{\mathbb {k}}\langle f,g\rangle )(x,y)&= \int _G T(x,z) \Delta (tr)f(ztr)g(ytr)^*\,\mathrm dz \\&= \Delta (tr) \int _G T(x,ztr {r^{-1}}{t^{-1}}) f(ztr)\,\mathrm dz\, g(ytr)^*\\&= \int _G T(x,z {r^{-1}}{t^{-1}}) f(z)\,\mathrm dz\, g(ytr)^*. \end{aligned}$$
Thus we have shown that (3.5) holds and this implies that the left operations are compatible.
Now note that \({}_{\mathcal {B}_\beta }\langle f\delta _r,f\delta _r \rangle = \Delta (r)^{-1}\beta _r({}_{\mathbb {k}}\langle f,f\rangle )\delta _e\ge 0\) and
$$\begin{aligned} \Vert {}_{\mathcal {B}_\beta }\langle f\delta _r,f\delta _r \rangle \Vert =\Delta (r)^{-1}\Vert \beta _r({}_{\mathbb {k}}\langle f,f\rangle )\Vert = \Delta (r)^{-1}\Vert f \Vert _{L^2}^2= \Vert \langle f\delta _r,f\delta _r{\rangle _\mathcal {B}}\Vert , \end{aligned}$$
so the norms given by the left and right inner products agree.

The continuity of the left operations follows directly from their definition and from the fact that the topologies of \(\mathcal {B}_\beta ={\mathbb {k}(\mathcal {B})}\times G\) and \(\mathcal {L}^2\mathcal {B}= L^2(\mathcal {B})\times G\) are the product topologies.

Since the linear \(\beta \)-orbit of \(\mathbb {K}(L^2(\mathcal {B}))\) is dense in \({\mathbb {k}(\mathcal {B})}\), it follows from the definition of the left inner product that
$$\begin{aligned} {\mathbb {k}(\mathcal {B})}\delta _e = \overline{{\text {span}}}\, \{ {}_{\mathcal {B}_\beta }\langle f\delta _r,g\delta _r \rangle :f,g\in L^2(\mathcal {B}),\ r\in G \}. \end{aligned}$$
At this point we know that \(\mathcal {L}^2\mathcal {B}\) is a full left Hilbert \(\mathcal {B}_\beta \)-bundle. The proof will be completed once we show that the left and right operations are compatible, that is,
$$\begin{aligned} {}_{\mathcal {B}_\beta }\langle f\delta _r,g\delta _s\rangle h\delta _t = f\delta _r\langle g\delta _s,h\delta _t{\rangle _\mathcal {B}}. \end{aligned}$$
It suffices to consider \(f,g,h\in \mathrm {C_c}(\mathcal {B})\) and by computing the left and right sides we obtain the following equivalent equation (without the place marker \(\delta _{r{s^{-1}}t}\)):
$$\begin{aligned} \Delta (s)^{-1}\beta _{{t^{-1}}s}({}_{\mathbb {k}}\langle f,g\rangle )h = f\int _G g(zs)^*h(zt)\,\mathrm dz. \end{aligned}$$
The left hand side evaluated at \(x\in G\) is
$$\begin{aligned} \Delta (t)^{-1} \int _G f(x{t^{-1}}s)g(z{t^{-1}}s)^*h(z)\,\mathrm dz = f(x{t^{-1}}s) \int _G g(zs)^*h(zt)\,\mathrm dz, \end{aligned}$$
which is exactly \(f\int _G g(zs)^*h(zt)\,\mathrm dz\) evaluated at x. \(\square \)

The following result shows that every Fell bundle is strongly equivalent to the semidirect product Fell bundle of a partial action.

Theorem 3.5

Let \(\mathcal {B}\) be a Fell bundle and denote by \(\mathcal {L}^2\mathcal {B}=\{L_t\}_{t\in G}\) its canonical \(L^2\)-bundle. If \(\mathcal {X}=\{L_t B_t^*B_t\}_{t\in G}\) and \(\alpha \) is the restriction to \(\mathbb {K}:=\mathbb {K}(L^2(\mathcal {B}))\) of the canonical action on the \({\mathrm {C}}^*\)-algebra of kernels of \(\mathcal {B}\), then \(\mathcal {X}\) is a Banach subbundle of \(\mathcal {L}^2\mathcal {B}\) and it is a strong equivalence \(\mathcal {B}_\alpha -\mathcal {B}\)-bundle with the structure inherited from \(\mathcal {L}^2\mathcal {B}\).

Proof

Since \(\{B_t^*B_t\}_{t\in G}\) is a continuous family of ideals of \(B_e\), \(\mathcal {X}\) is a Banach subbundle of \(\mathcal {L}^2\mathcal {B}\).

To simplify our notation we define \(\mathbb {K}_t:=\beta _t(\mathbb {K})\cap \mathbb {K}=\beta _t(\mathbb {K})\cdot \mathbb {K}\). Recall that the fiber over t of \(\mathcal {B}_\alpha \) is \(\mathbb {K}_t\delta _t\).

To continue we identify the ideal of \(\mathbb {K}\) corresponding to \(B_t^*B_t\) through \(L^2(\mathcal {B})\); we claim this ideal is \(\mathbb {K}_{t^{-1}}\). Given \(f,g\in \mathrm {C_c}(\mathcal {B})\) and \(a,b,c,d\in B_t\) we define \(u,v\in \mathrm {C_c}(\mathcal {B})\) by \(u(r):=f(rt)a^*bd^*\) and \(v(r):=g(rt)c^*\). Then \( {}_{\mathbb {k}}\langle f a^*b ,g c^*d \rangle = \beta _{t^{-1}}({}_{\mathbb {k}}\langle u,v\rangle )\in \mathbb {K}_{t^{-1}}\). This implies \(L^2(\mathcal {B})B_t^*B_t\subseteq \mathbb {K}_{t^{-1}}L^2(\mathcal {B})\).

To prove \(\mathbb {K}_{t^{-1}}L^2(\mathcal {B}) \subseteq L^2(\mathcal {B})B_t^*B_t\) it suffices to show
$$\begin{aligned} \langle \mathbb {K}_{t^{-1}}L^2(\mathcal {B}), L^2(\mathcal {B}) \rangle _{B_e}\subseteq B_t^*B_t. \end{aligned}$$
Note that \(\mathbb {K}_{t^{-1}}L^2(\mathcal {B})=\beta _{t^{-1}}(\mathbb {K})\mathbb {K}L^2(\mathcal {B})= \beta _{t^{-1}}(\mathbb {K})L^2(\mathcal {B})\). If \(k\in \mathbb {k}_c(\mathcal {B})\) and \(f,g\in \mathrm {C_c}(\mathcal {B})\) then
$$\begin{aligned} \langle \beta _{t^{-1}}(k)f,g\rangle _{B_e}= \int _G \int _G f^*(s)k(r{t^{-1}},s{t^{-1}})^*g(r)\,\mathrm dr\,\mathrm ds. \end{aligned}$$
Now, if k represents an element of the ideal \(\mathbb {K}\), then \(k(s,t)\in \mathcal {B}_s\mathcal {B}_t^*\) for all \(s,t\in G\). Since \(f^*(s)k(r{t^{-1}},s{t^{-1}})^*g(r)\in B_{s^{-1}}B_{s{t^{-1}}} B_{t {r^{-1}}} B_r \subseteq B_{t^{-1}}B_t=B_t^*B_t\) for all \(r,s\in G\), we conclude that \(\mathbb {K}_{t^{-1}}L^2(\mathcal {B})=L^2(\mathcal {B})B_t^*B_t\).
The operations of \(\mathcal {X}\) are the ones inherited from \(\mathcal {L}^2\mathcal {B}\). To show they are well defined and satisfy (1R-3R) and (1L-3L), it suffices to show
$$\begin{aligned} \mathcal {B}_\alpha \mathcal {X}\subseteq \mathcal {X},\quad \mathcal {X}\mathcal {B}\subseteq \mathcal {B}\quad \text{ and } \quad {}_{\mathcal {B}_\beta }\langle \mathcal {X},\mathcal {X}\rangle \subseteq \mathcal {B}_\alpha . \end{aligned}$$
(3.6)
To prove the first inclusion take \(r,s\in G\), \(T\in \mathbb {K}_r\) and \(f\in L^2(\mathcal {B})B_s^*B_s\). By Cohen’s factorisation theorem, we may decompose f as \(f=T'f'\) with \(T'\in \mathbb {K}\) and \(f'\in L^2(\mathcal {B})\). Since \(L_{rs}B_{rs}^*B_{rs}= \beta _{rs}^{-1}(\mathbb {K})\mathbb {K}L^2(\mathcal {B})\delta _{rs}\) we have
$$\begin{aligned} T\delta _r f\delta _s = \Delta (r)^{1/2}\beta _{rs}^{-1}(T)f\delta _{rs} =\Delta (r)^{1/2}\beta _{rs}^{-1}(T)T'f'\delta _{rs}\in L_{rs}B_{rs}^*B_{rs}. \end{aligned}$$
This shows that \(\mathcal {B}_\alpha \mathcal {X}\subseteq \mathcal {X}\).
The second inclusion in (3.6) follows from \(L_sB_s^*B_s B_r\subseteq L_{sr}B_{sr}^*B_{sr}\), which we now show. Since \(B_s^*B_s\) and \(B_rB_r^*\) are ideals of \(B_e\) and \(B_r=B_rB_r^*B_r\), we have
$$\begin{aligned} L_sB_s^*B_s B_r&= L_sB_s^*B_s B_rB_r^*B_r = L_s B_rB_r^* B_s^*B_s B_r \subseteq L_{sr} (B_sB_r)^*(B_sB_r)\\&\subseteq L_{sr} B_{sr}^*B_{sr}. \end{aligned}$$
The inclusion \({}_{\mathcal {B}_\beta }\langle \mathcal {X},\mathcal {X}\rangle \subseteq \mathcal {B}_\alpha \) follows from the fact that
$$\begin{aligned} {}_{\mathcal {B}_\beta }\langle L_rB_r^*B_r,L_sB_s^*B_s\rangle&= \beta _r({}_{\mathbb {k}}\langle \mathbb {K}_{r^{-1}}L^2(\mathcal {B}),\mathbb {K}_{s^{-1}}L^2(\mathcal {B}) \rangle )\delta _{r{s^{-1}}}\\&= \beta _r(\mathbb {K}_{r^{-1}}\cap \mathbb {K}_{s^{-1}})\delta _{r{s^{-1}}} = \mathbb {K}_r \cap \mathbb {K}_{r{s^{-1}}} \delta _{r{s^{-1}}}\\&\subseteq \mathbb {K}_{r{s^{-1}}} \delta _{r{s^{-1}}}. \end{aligned}$$
To finish we need to show \(\mathcal {X}\) is strongly full on both sides. The strong fullness on the right follows from the computation
$$\begin{aligned} \overline{{\text {span}}}\, \langle L_t B_t^*B_t,L_t B_t^*B_t{\rangle _\mathcal {B}}&= \overline{{\text {span}}}\, \langle L^2(\mathcal {B})B_t^*B_t\delta _t,L^2(\mathcal {B})B_t^*B_t\delta _t{\rangle _\mathcal {B}}\\&=\overline{{\text {span}}}\, \{B_t^*B_t\langle L^2(\mathcal {B}),L^2(\mathcal {B})\rangle _{B_e}B_t^*B_t \} = B_t^*B_t. \end{aligned}$$
And the strong fullness on the left is implied by
$$\begin{aligned} \overline{{\text {span}}}\, {}_{\mathcal {B}_\beta }\langle L_t B_t^*B_t,L_t B_t^*B_t\rangle= & {} \overline{{\text {span}}}\, {}_{\mathcal {B}_\beta }\langle L^2(\mathcal {B})B_t^*B_t\delta _t,L^2(\mathcal {B})B_t^*B_t\delta _t\rangle \\= & {} \overline{{\text {span}}}\, \{\beta _t({}_{\mathbb {k}}\langle Tf,Tf\rangle )\delta _e:T\in \mathbb {K}_{t^{-1}}, f\in L^2(\mathcal {B})\}\\= & {} \mathbb {K}_t\delta _e =\mathbb {K}_t\delta _t (\mathbb {K}_t\delta _t)^*. \end{aligned}$$
\(\square \)

As an immediate consequence of the last result and Example 2.8, we get that every Fell bundle has a Morita enveloping Fell bundle which is the semidirect product bundle of an action on a \({\mathrm {C}}^*\)-algebra. We will see later (Corollary 4.11) that the Morita enveloping Fell bundle is unique up to strong equivalence.

4 Morita Equivalence of Actions and Fell Bundles

From the previous sections we know that the canonical action on the \({\mathrm {C}}^*\)-algebra of kernels of a Fell bundle determines the weak equivalence class of that bundle. But what can we say about the canonical actions on the kernels of two weakly equivalent Fell bundles? Of course these actions, when viewed as Fell bundles are weakly equivalent. Our goal is to show that they are (Morita) equivalent as actions (hence strongly equivalent as Fell bundles).

Assume \(\mathcal {A}\) and \(\mathcal {B}\) are Fell bundles over G and let \(\alpha \) and \(\beta \) stand for the canonical actions on the \({\mathrm {C}}^*\)-algebras of kernels of \(\mathcal {A}\) and \(\mathcal {B}\), respectively. If \(\alpha \) is Morita equivalent to \(\beta \) (as actions on \({\mathrm {C}}^*\)-algebras) then \(\mathcal {B}_\alpha \) and \(\mathcal {B}_\beta \) are equivalent as Fell bundles (via a strong equivalence: Example 2.7) and, by transitivity, \(\mathcal {A}\) is weakly equivalent to \(\mathcal {B}\). Before proving the converse we prove the following.

Lemma 4.1

If \(\mathcal {A}\) is a Fell subbundle of \(\mathcal {B}\) then the natural inclusion \(\iota :\mathbb {k}_c(\mathcal {A})\rightarrow \mathbb {k}_c(\mathcal {B})\) extends to an injective (hence isometric) \(^{*}\)-homomorphism \(\pi :\mathbb {k}(\mathcal {A})\rightarrow \mathbb {k}(\mathcal {B})\).

Proof

First note that \(\iota \) has a unique extension to a *-homomorphism \(\mathcal {HS}(\mathcal {A})\rightarrow \mathcal {HS}(\mathcal {B})\), which induces the *-homomorphism \(\pi :\mathbb {k}(\mathcal {A})\rightarrow \mathbb {k}(\mathcal {B})\) that we want to show is injective. Since \(\mathcal {A}\) is included in \(\mathcal {B}\), \(L^2(\mathcal {A})\) is contained in \(L^2(\mathcal {B})\) as a \({\mathrm {C}}^*\)-subtring and we may think of \(I:=\mathbb {K}(L^2(\mathcal {A}))\) as a \({\mathrm {C}}^*\)-subalgebra of \(\mathbb {K}(L^2(\mathcal {B}))\) (see Abadie 2003, Proposition 4.1). This implies that \(\pi |_{I}\) is injective.

Since \(\pi \) is \(\alpha -\beta \)-equivariant, the \(\beta \)-closed linear orbit of \(\pi (I)\) is \(\pi (\mathbb {k}(\mathcal {A}))\). Moreover, \(\pi |_I:I\rightarrow \pi (I)\) is an isomorphism of partial actions between \(\alpha |_I\) and \(\beta |_{\pi (I)}\). Since \(\alpha \) is an enveloping action of \(\alpha |_I\) and \(\beta |_{\pi (\mathbb {k}(\mathcal {A}))}\) one of \(\beta |_{\pi (I)}\), the uniqueness of the enveloping action (Abadie 2003, Theorem 2.1) implies the unique \(\alpha - \beta \)-equivariant extension of \(\pi |_I\)is injective. But this extension is \(\pi \). \(\square \)

Theorem 4.2

Let \(\mathcal {A}\) and \(\mathcal {B}\) be Fell bundles over G and let \(\alpha \) and \(\beta \) be the canonical actions on the \({\mathrm {C}}^*\)-algebras of kernels of \(\mathcal {A}\) and \(\mathcal {B}, \) respectively. Then \(\mathcal {A}\) is weakly equivalent to \(\mathcal {B}\) if and only if \(\alpha \) is Morita equivalent to \(\beta \).

Proof

The converse follows by the comments we made at the beginning of the present section. For the direct implication assume that \(\mathcal {X}\) is an \(\mathcal {A}-\mathcal {B}\) equivalence bundle. We may think of \(\mathcal {A}\) and \(\mathcal {B}\) as full hereditary Fell subbundles of their linking Fell bundle \(\mathbb {L}(\mathcal {X})\), so \({\mathbb {k}(\mathcal {A})}\) and \({\mathbb {k}(\mathcal {B})}\) are \({\mathrm {C}}^*\)-subalgebras of \(\mathbb {k}(\mathbb {L}(\mathcal {X}))\) by Lemma 4.1. Note that \(\mathbb {k}_c(\mathcal {A})\mathbb {k}_c(\mathbb {L}(\mathcal {X})) \mathbb {k}_c(\mathcal {A})\subseteq \mathbb {k}_c(\mathcal {A})\) and \(\mathbb {k}_c(\mathcal {B})\mathbb {k}_c(\mathbb {L}(\mathcal {X})) \mathbb {k}_c(\mathcal {B})\subseteq \mathbb {k}_c(\mathcal {B})\), so by taking completion we conclude that \({\mathbb {k}(\mathcal {A})}\) and \({\mathbb {k}(\mathcal {B})}\) are hereditary in \(\mathbb {k}(\mathbb {L}(\mathcal {X}))\).

To identify the equivalence bimodule inside \(\mathbb {k}(\mathbb {L}(\mathcal {X}))\) implementing the Morita equivalence between \({\mathbb {k}(\mathcal {A})}\) and \({\mathbb {k}(\mathcal {B})}\) define
$$\begin{aligned} \mathbb {k}_c(\mathcal {X}):=\{ f\in \mathbb {k}_c(\mathbb {L}(\mathcal {X})):f(r,s)\in X_{r{s^{-1}}} \text{ for } \text{ all } r,s\in G\}, \end{aligned}$$
and denote by \(\mathbb {k}(\mathcal {X})\) the completion of \(\mathbb {k}_c(\mathcal {X})\) in \(\mathbb {k}(\mathbb {L}(\mathcal {X}))\).

Note that \(\mathbb {k}_c(\mathcal {A})\mathbb {k}_c(\mathcal {X})\subseteq \mathbb {k}_c(\mathcal {X})\) and \(\mathbb {k}_c(\mathcal {X})\mathbb {k}_c(\mathcal {X})^*\subseteq \mathbb {k}_c(\mathcal {A})\), so by taking closures and linear span we obtain \({\mathbb {k}(\mathcal {A})}\mathbb {k}(\mathcal {X})\subseteq \mathbb {k}(\mathcal {X})\) and \(\mathbb {k}(\mathcal {X})\mathbb {k}(\mathcal {X})^*\subseteq {\mathbb {k}(\mathcal {A})}\). By symmetry, we also have \(\mathbb {k}(\mathcal {X}){\mathbb {k}(\mathcal {B})}\subseteq \mathbb {k}(\mathcal {X})\) and \(\mathbb {k}(\mathcal {X})^*\mathbb {k}(\mathcal {X})\subseteq {\mathbb {k}(\mathcal {B})}\).

The next step is to show that \(\mathbb {k}(\mathcal {X})^*\mathbb {k}(\mathcal {X})={\mathbb {k}(\mathcal {B})}\). Set \(\Gamma :=\overline{{\text {span}}}\, \{ f^{*}*g:f,g\in \mathbb {k}_c(\mathbb {L}(\mathcal {X})) \}\), considered as a subset of the pullback \(\mathcal {C}\) of \(\mathcal {B}\) along the map \(G\times G\rightarrow G,\ (r,s)\mapsto r{s^{-1}}\). To show that \(\Gamma \) is dense in the inductive limit topology on \(\mathrm {C_c}(\mathcal {C})=\mathbb {k}_c(\mathcal {B})\) (and so dense in \({\mathbb {k}(\mathcal {B})}\)) we use (Abadie 2003, Lemma 5.1). Now consider the algebraic tensor product \(\mathrm {C_c}(G)\odot \mathrm {C_c}(G)\) as a dense subspace of \(\mathrm {C_c}(G\times G)\) for the inductive limit topology. Given \(f,g\in \mathbb {k}_c(\mathcal {B})\) and \(\phi ,\psi \in \mathrm {C_c}(G)\) define \(\phi \cdot f (r,s):=\phi (r)f(r,s)\). Then \((\phi \odot \psi )f^* *g = (\phi ^* f)^* * (\psi g)\in \Gamma \). So \(\mathrm {C_c}(G)\odot \mathrm {C_c}(G)\Gamma \subseteq \Gamma \) and it suffices to show that, for every \((r,s)\in G\), \(\{u(r,s):u\in \Gamma \}\) is dense in \(B_{r{s^{-1}}}\).

Given \(t\in G\), \(x\in X_{t{r^{-1}}}\) and \(y\in X_{t{s^{-1}}}\) take \(f,g\in \mathrm {C_c}(\mathcal {X})\) such that \(f(t{r^{-1}}) = x\) and \(g(t{s^{-1}})=y\). Consider the set of compact neighbourhoods of e, \(\mathcal {N}\), ordered by inclusion: \(U\le V\) if and only if \(V\subseteq U\) and for every \(U\in \mathcal {N}\) take \(\phi \in \mathrm {C_c}(G)^+\) with support contained in U and such that \(\int _G \phi _U^2(p)\,\mathrm dp=1\). Define \(k_f^U\in \mathbb {k}_c(\mathcal {X})\) as \(k_f^U(p,q)=\phi ({t^{-1}}p)f(pq^{-1})\). Then
$$\begin{aligned} \lim _U (k^U_f)^* * k_g^U (r,s)&=\lim _U \int _G \phi _U({t^{-1}}p)^2 \langle f(p{r^{-1}}),g(p{s^{-1}}){\rangle _\mathcal {B}}\,\mathrm dp\\&= \langle f(t{r^{-1}}),g(t{s^{-1}}){\rangle _\mathcal {B}}=\langle x,y{\rangle _\mathcal {B}}. \end{aligned}$$
We conclude that the closure C of \(\{u(r,s):u\in \Gamma \}\) contains \(\langle X_{t{r^{-1}}},X_{t{s^{-1}}}{\rangle _\mathcal {B}}\) for all \(t\in G\). By Remark 2.4 this implies \(C=B_{r{s^{-1}}}\).

Now we know that \(\mathbb {k}(\mathcal {X})\) is a \({\mathbb {k}(\mathcal {A})}-{\mathbb {k}(\mathcal {B})}\)-equivalence bimodule. To finish the proof note that if \(\gamma \) is the canonical action of G on \(\mathbb {k}(\mathbb {L}(\mathcal {X}))\) then \(\mathbb {k}(\mathcal {X})\) is \(\gamma \)-invariant, \(\alpha =(\gamma |_{\mathbb {k}(\mathcal {X})})^l\) and \(\beta = (\gamma |_{\mathbb {k}(\mathcal {X})})^r\) (recall the notation from Example 2.7). \(\square \)

Corollary 4.3

(cf. Abadie and Ferraro 2017, Proposition 4.13) If \(\mathcal {A}\) and \(\mathcal {B}\) are weakly equivalent Fell bundles over G, then their full and reduced cross-sectional \({\mathrm {C}}^*\)-algebras are (strongly) Morita equivalent. This equivalence respects the dual coactions. Conversely, if the dual coactions on the (full or reduced) cross-sectional \({\mathrm {C}}^*\)-algebras of \(\mathcal {A}\) and \(\mathcal {B}\) are equivalent (as coactions), then \(\mathcal {A}\) and \(\mathcal {B}\) are weakly equivalent as Fell bundles.

Proof

By Theorem 4.2, \(\mathcal {A}\) and \(\mathcal {B}\) are weakly equivalent if and only if their \({\mathrm {C}}^*\)-algebras of kernels \({\mathbb {k}(\mathcal {A})}\) and \({\mathbb {k}(\mathcal {B})}\) are G-equivariantly Morita equivalent. And by Remark 3.3, we have canonical isomorphisms of G-\({\mathrm {C}}^*\)-algebras \({\mathbb {k}(\mathcal {A})}\cong C^*(\mathcal {A})\rtimes _{\delta _\mathcal {A}}G\cong C^*_\mathrm r(\mathcal {A})\rtimes _{\delta _\mathcal {A}^\mathrm r}G\) and \({\mathbb {k}(\mathcal {B})}\cong C^*(\mathcal {B})\rtimes _{\delta _\mathcal {B}}G\cong C^*_\mathrm r(\mathcal {B})\rtimes _{\delta _\mathcal {B}^\mathrm r}G\), where \(\delta _\mathcal {A}^{(\mathrm r)}\) and \(\delta _\mathcal {B}^{(\mathrm r)}\) denote the dual coactions on \(C^*_{(\mathrm r)}(\mathcal {A})\) and \(C^*_{(\mathrm r)}(\mathcal {B})\), respectively. The coaction \(\delta _\mathcal {A}\) is maximal, and the coaction \(\delta _\mathcal {A}^{\mathrm r}\) is normal (the normalisation of \(\delta _\mathcal {A}\)), see Buss and Echterhoff (2015). This means that \(\delta _\mathcal {A}\) (resp. \(\delta _\mathcal {A}^\mathrm r\)) is Morita equivalent to the dual coaction on \({\mathbb {k}(\mathcal {A})}\rtimes _\alpha G\) (resp. \({\mathbb {k}(\mathcal {A})}\rtimes _{\alpha ,\mathrm r} G\)), and a similar assertion holds for \(\mathcal {B}\) in place of \(\mathcal {A}\). Combining all this and the standard result that equivalent actions or coactions have equivalent (full or reduced) crossed products, the desired result now follows. \(\square \)

Remark 4.4

The above result extends to the exotic cross-sectional \({\mathrm {C}}^*\)-algebras \(C^*_\mu (\mathcal {B})\) associated to Morita compatible cross-product functors \(\rtimes _\mu \) as defined in Buss and Echterhoff (2015). This is because, by definition, \(C^*_\mu (\mathcal {B})\) is the quotient of \(C^*(\mathcal {B})\) that turns the isomorphism (3.4) into an isomorphism
$$\begin{aligned} {\mathbb {k}(\mathcal {B})}\rtimes _{\beta ,\mu }G\cong C^*_\mu (\mathcal {B})\otimes \mathbb {K}(L^2(G)). \end{aligned}$$
And this isomorphism preserves the dual coactions whenever the exotic crossed-product admits such a coaction; this means that \(\rtimes _\mu \) is a duality cross-product functor in the language of Buss et al. (2016) (see also Abadie and Ferraro 2017). This is a big class of functors and, in particular, includes all correspondence functors (see Buss and Echterhoff 2015, Corollary 4.6).

Recall that a Fell bundle \(\mathcal {A}\) is amenable if the regular representation \(\lambda _\mathcal {A}:C^*(\mathcal {A})\rightarrow C^*_\mathrm r(\mathcal {A})\) is faithful.

Corollary 4.5

Let \(\mathcal {A}\) and \(\mathcal {B}\) be two weakly equivalent Fell bundles. Then \(\mathcal {A}\) is amenable if and only if \(\mathcal {B}\) is amenable.

Proof

As already explained in Remark 3.3, the canonical isomorphism \(C^*(\mathcal {A})\otimes \mathbb {K}(L^2(G))\cong {\mathbb {k}(\mathcal {A})}\rtimes _\alpha G\) factors (via the regular representations) through an isomorphism \(C^*_\mathrm r(\mathcal {A})\otimes \mathbb {K}(L^2(G))\cong {\mathbb {k}(\mathcal {A})}\rtimes _{\alpha ,\mathrm r} G\). This means that \(\mathcal {A}\) is amenable if and only if the G-action \(\alpha \) on \({\mathbb {k}(\mathcal {A})}\) is amenable in the sense that the regular representation \({\mathbb {k}(\mathcal {A})}\rtimes _{\alpha } G\rightarrow {\mathbb {k}(\mathcal {A})}\rtimes _{\alpha ,\mathrm r} G\) is faithful. Since \(\mathcal {A}\) is weakly equivalent to \(\mathcal {B}\) if and only if the G-actions on \({\mathbb {k}(\mathcal {A})}\) and \({\mathbb {k}(\mathcal {B})}\) are equivalent by Theorem 4.2, the assertion follows from the well-known result that amenability of actions is preserved by equivalence of actions. \(\square \)

We will have more to say about amenability in Sect. 6.

Corollary 4.6

Let \(\alpha \) and \(\beta \) be partial actions of G on \({\mathrm {C}}^*\)-algebras. Then \(\mathcal {B}_\alpha \) is weakly equivalent to \(\mathcal {B}_\beta \) if and only if \(\alpha \) and \(\beta \) have equivalent Morita enveloping actions.

Proof

Recall from Abadie (2003) that the canonical action on the \({\mathrm {C}}^*\)-algebra of kernels of \(\mathcal {B}_\alpha \) is a Morita enveloping action for \(\alpha \) and that Morita enveloping actions are unique up to Morita equivalence of actions. Then the proof follows directly using transitivity of Morita equivalence of Fell bundles and the last theorem. \(\square \)

Our next theorem will show that strong equivalence of Fell bundles corresponds to Morita equivalence of partial actions in the ordinary sense (recall Example 2.7 and the notation used there). First we need the following auxiliary result.

Lemma 4.7

Suppose X is an \(A-B\)-equivalence bimodule, \(\gamma \) is an action of G on X and that I and J are \({\mathrm {C}}^*\)-ideals of A and B (respectively) such that \(IX=XJ\). Then \(\gamma ^l|_I=(\gamma |_{IX})^l\) and \(\gamma ^r|_J=(\gamma |_{XJ})^r\). In particular \(\gamma ^l|_I\) is Morita equivalent to \(\gamma ^r|_J\) (as partial actions).

Proof

To simplify the notation, we denote \(\alpha :=\gamma ^l\) and \(\beta =\gamma ^r\). Since for all \(t\in G\) we have \(\alpha _t(I)X = \gamma _t(IX)\), the ideal of X corresponding to \(I_t=I\cap \alpha _t(I)\) is \(IX\cap \gamma _t(IX)\). By symmetry we obtain \(I_t X=XJ_t\). Since \(I_{t^{-1}}=\overline{{\text {span}}}\, {}_A\langle X\cap \gamma _t(IX),X\cap \gamma _t(IX)\rangle \) and for \(x,y\in X\cap \gamma _t(IX)\) we have
$$\begin{aligned} \alpha _t({}_A\langle x,y\rangle )={}_A\langle \gamma _t(x),\gamma _t(y)\rangle = (\gamma |_{IX})^l({}_A\langle x,y\rangle ). \end{aligned}$$
We conclude that \(\alpha |_I = (\gamma |_{IX})^l\). The rest follows by symmetry. \(\square \)

Theorem 4.8

Let \(\mathcal {A}\) and \(\mathcal {B}\) be Fell bundles over G and denote by \(\alpha \) and \(\beta \) the restrictions of the canonical action on the \({\mathrm {C}}^*\)-algebras of kernels, \({\mathbb {k}(\mathcal {A})}\) and \({\mathbb {k}(\mathcal {B})}\), to \(\mathbb {K}_\mathcal {A}:=\mathbb {K}(L^2(\mathcal {A}))\) and \(\mathbb {K}_\mathcal {B}:=\mathbb {K}(L^2(\mathcal {B}))\), respectively. Then \(\mathcal {A}\) is strongly equivalent to \(\mathcal {B}\) if and only if \(\alpha \) is Morita equivalent to \(\beta \).

Proof

If \(\alpha \) is equivalent to \(\beta \) in the usual sense (as defined in Abadie (2003)), then their associated Fell bundles \(\mathcal {B}_\alpha \) and \(\mathcal {B}_\beta \) are strongly equivalent, as shown in Example 2.7. Since \(\mathcal {A}\) (resp. \(\mathcal {B}\)) is strongly equivalent to \(\mathcal {B}_\alpha \) (resp. \(\mathcal {B}_\beta \)) by Theorem 3.5, the strong equivalence between \(\mathcal {A}\) and \(\mathcal {B}\) follows by transitivity (Theorem A.1).

For the converse assume that \(\mathcal {X}\) is a strong equivalence \(\mathcal {A}-\mathcal {B}\)-bundle. Let \(\mathbb {k}(\mathcal {X})\) be the \({\mathbb {k}(\mathcal {A})}-{\mathbb {k}(\mathcal {B})}\)-equivalence bimodule constructed in the proof of Theorem 4.2, which was constructed inside \(\mathbb {k}(\mathbb {L}(\mathcal {X}))\).

By the previous Lemma and the proof of Theorem 4.2 it suffices to show that \(\mathbb {K}_\mathcal {A}\mathbb {k}(\mathcal {X})=\mathbb {k}(\mathcal {X})\mathbb {K}_\mathcal {B}\). Let \(\mathcal {C}\) be the pullback of \(\mathcal {B}\) along the map \(\rho :G^2\rightarrow G,\ (r,s)\mapsto r{s^{-1}}\). We think of \(\mathbb {k}_c(\mathcal {B})\) as \(\mathrm {C_c}(\mathcal {C})\).

Note that the bundle \(\mathcal {D}^\mathcal {B}:=\{B_rB_{{s^{-1}}}\}_{(r,s)\in G^2}\) is a Banach subbundle of \(\mathcal {C}\) (recall Notation 2.1). Moreover, using Abadie (2003, Lemma 5.1) (as in the proof of Theorem 4.2) one shows that \({\text {span}}\{ {}_{\mathbb {k}}\langle f,g\rangle :f,g\in \mathrm {C_c}(\mathcal {B}) \}\) is dense in \(\mathrm {C_c}(\mathcal {D}^\mathcal {B})\) for the inductive limit topology. Thus \(\mathbb {K}_\mathcal {B}\) is the closure of \(\mathrm {C_c}(\mathcal {D}^\mathcal {B})\) in \({\mathbb {k}(\mathcal {B})}\).

In a similar way define \(\mathcal {D}^{\mathcal {X}\mathcal {B}}:=\{X_rB_{s^{-1}}\}_{(r,s)\in G^2}\), which is a Banach subbundle of the pullback of \(\mathbb {L}(\mathcal {X})\) along \(\rho \). Note that \(\mathbb {k}_c(\mathcal {X}){\text {span}}{}_{\mathbb {k}}\langle \mathrm {C_c}(\mathcal {B}),\mathrm {C_c}(\mathcal {B})\rangle \subseteq \mathrm {C_c}(\mathcal {D}^\mathcal {X})\) and that, by arguments similar to those in the proof of Theorem 4.2, \(\mathbb {k}_c(\mathcal {X}){\text {span}}{}_{\mathbb {k}}\langle \mathrm {C_c}(\mathcal {B}),\mathrm {C_c}(\mathcal {B})\rangle \) is dense in \(\mathrm {C_c}(\mathcal {D}^{\mathcal {X}\mathcal {B}})\) for the inductive limit topology. Thus \(\mathbb {k}(\mathcal {X}){\mathbb {k}(\mathcal {B})}\) is the closure of \(\mathrm {C_c}(\mathcal {D}^{\mathcal {X}\mathcal {B}})\) in \(\mathbb {k}(\mathbb {L}(\mathcal {X}))\).

By symmetry we obtain that \({\mathbb {k}(\mathcal {A})}\mathbb {k}(\mathcal {X})\) is the closure in \(\mathbb {k}(\mathbb {L}(\mathcal {X}))\) of the space of compactly supported continuous sections of the bundle \(\mathcal {D}^{\mathcal {A}\mathcal {X}}:=\{ A_rX_{s^{-1}}\}_{(r,s)\in G^2}\). Thus all we need to show is that \(\mathcal {D}^{\mathcal {A}\mathcal {X}}=\mathcal {D}^{\mathcal {X}\mathcal {B}}\). It suffices to prove that \(A_rX_s=X_rB_s\) for all \((r,s)\in G^2\). If we think of \(B_s\) as a right full \(B_s^*B_s\)-Hilbert module and we use that \(B_s^*B_s=\overline{{\text {span}}}\, \langle X_s,X_s{\rangle _\mathcal {B}}\), then we conclude that
$$\begin{aligned} X_rB_s =\overline{{\text {span}}}\, X_r B_s \langle X_s,X_s{\rangle _\mathcal {B}}= \overline{{\text {span}}}\, {}_\mathcal {A}\langle X_r,X_sB_{s^{-1}}{\rangle }X_s \subseteq A_rX_s. \end{aligned}$$
Conversely,
$$\begin{aligned} A_rX_s = \overline{{\text {span}}}\, {}_\mathcal {A}\langle X_r,X_r{\rangle }A_r X_s = \overline{{\text {span}}}\, X_r{}_\mathcal {A}\langle X_r,A_rX_s{\rangle }\subseteq X_rB_s. \end{aligned}$$
\(\square \)

The next result shows that our notion of strong equivalence of Fell bundles recovers exactly the notion of equivalence of partial actions (introduced in Abadie (2003)).

Corollary 4.9

Let \(\alpha \) and \(\beta \) be partial actions of G on \({\mathrm {C}}^*\)-algebras. Then \(\alpha \) is Morita equivalent to \(\beta \) if and only if \(\mathcal {B}_\alpha \) is strongly equivalent to \(\mathcal {B}_\beta \).

Proof

This follows from Theorem 4.8 together with the fact (proved in Abadie (2003)) that the restricted partial action to \(\mathbb {K}(L^2(\mathcal {B}_\alpha ))\) of the canonical global action on \(\mathbb {k}(\mathcal {B}_\alpha )\) is Morita equivalent to \(\alpha \). \(\square \)

Corollary 4.10

Two saturated Fell bundles are weakly equivalent if and only they are strongly equivalent.

Proof

If \(\mathcal {A}\) is a saturated Fell bundle, then its \({\mathrm {C}}^*\)-algebra of kernels \({\mathbb {k}(\mathcal {A})}\) coincides with \(\mathbb {K}(L^2(\mathcal {A}))\) and the canonical partial action is already global (see Abadie 2003, Proposition 5.4). The result now follows as a direct combination of Theorems 4.2 and 4.8. \(\square \)

Now, combining Corollary 4.10, Example 2.8, Remark 2.12, and Theorem 3.5, we get:

Corollary 4.11

Every Fell bundle has a Morita enveloping Fell bundle which is the semidirect product bundle of an action on a C*-algebra. This action is unique up to (strong) Morita equivalence of actions on \({\mathrm {C}}^*\)-algebras. Any two Morita enveloping Fell bundles of a Fell bundle are strongly equivalent.

5 Partial Actions Associated with Fell Bundles

By the spectrum of a \({\mathrm {C}}^*\)-algebra A we mean the space \(\hat{A}\) of unitary equivalence classes \([\pi ]\) of irreducible representations \(\pi \) of A with the Jacobson topology induced from the primitive ideal space \({{\mathrm{Prim}}}(A)=\{\ker (\pi ): [\pi ]\in \hat{A}\}\). Open subsets of \(\hat{A}\) or \({{\mathrm{Prim}}}(A)\) correspond bijectively to ideals of A: the open subset of \({{\mathrm{Prim}}}(A)\) (resp. \(\hat{A}\)) associated with an ideal \(I\subseteq A\) is \(\{p\in {{\mathrm{Prim}}}(A): I\varsubsetneq p\}\) (resp. \(\{[\pi ]: \pi |_I\not =0\}\)).

As shown in Abadie and Abadie (2017), every Fell bundle \(\mathcal {B}=(B_t)_{t\in G}\) over a discrete group G induces a partial action \(\hat{\alpha }^\mathcal {B}\) of G on the spectrum \(\hat{B}_e\) of \(B_e\). We briefly recall how \(\hat{\alpha }^\mathcal {B}\) is defined. For each \(t\in G\) we let \(D^\mathcal {B}_t:=B_tB_t^*\); then \(D^\mathcal {B}_t\) is an ideal of \(B_e\) and \(B_t\) can be viewed as a \(D^\mathcal {B}_t-D^\mathcal {B}_{t^{-1}}\) imprimitivity bimodule. Let \(\mathcal {V}_t^{\mathcal {B}}:=\{[\pi ]\in \hat{B}_e:\pi |_{D^\mathcal {B}_t}\ne 0 \}\) be the open subset of \(\hat{B_e}\) associated with \(D^\mathcal {B}_t\). If \([\pi ]\in \mathcal {V}_{t^{-1}}^{\mathcal {B}}\), then \([\pi |_{D^\mathcal {B}_{t^{-1}}}]\in \hat{D}^\mathcal {B}_{t^{-1}}\), and therefore \([\text {Ind}_{B_t}(\pi |_{D^\mathcal {B}_{t^{-1}}})]\in \hat{D}^\mathcal {B}_{t}\). Then \(\hat{\alpha }_t([\pi ])\in \mathcal {V}_t^\mathcal {B}\) is defined to be the class of the unique extension of \(\text {Ind}_{B_t}(\pi |_{D^\mathcal {B}_{t^{-1}}})\) to all of \(B_e\) (see the second statement of Lemma 5.2). In fact one can give a more direct definition: \(\hat{\alpha }_t([\pi ])=[\text {Ind}_{B_t}\pi ]\) (this is a consequence of our Lemma 5.2(3)). Here and throughout, if E is a Hilbert \(C'-C\)-bimodule for \({\mathrm {C}}^*\)-algebras \(C',C\), we write \({{\mathrm{Ind}}}_E(\pi )\) for the representation of \(C'\) induced via E from a representation \(\pi :C\rightarrow \mathbb {B}(H)\). Recall that \({{\mathrm{Ind}}}_E(\pi )\) acts on the (balanced tensor product) Hilbert space \(E\otimes _\pi H\) by the formula \({{\mathrm{Ind}}}_E\pi (x)(y\otimes _\pi h):=x\cdot y\otimes _\pi h\) for all \(x\in C'\), \(y\in E\) and \(h\in H\).

Only discrete groups are considered in Abadie and Abadie (2017). But in this section we prove that the partial action \(\hat{\alpha }^\mathcal {B}\) is always continuous if G is a locally compact group. We also show that strongly equivalent Fell bundles have isomorphic partial actions, and that the action of a saturated Fell bundle is the enveloping action of the partial action of any Fell bundle weakly equivalent with the former.

We begin with some preliminary results about induced representations via Hilbert bimodules; most of them are certainly well known, but we include the proofs here for convenience. Let A be a \({\mathrm {C}}^*\)-subalgebra of the \({\mathrm {C}}^*\)-algebra C, and suppose \(\pi :C\rightarrow \mathbb {B}(H)\) is a nondegenerate representation of C. We denote by \(\pi _A\) the nondegenerate part of the restriction \(\pi |_A\), that is, \(\pi _A:A\rightarrow \mathbb {B}(H_A)\) is given by \(\pi _A(a)h:=\pi (a)h\) for all \(a\in A\) and \(h\in H_A\), where \(H_A:=\overline{\text {span}}\{\pi (a)h:a\in A,h\in H\}\), the essential space of \(\pi |_A\).

Lemma 5.1

Let E be a Hilbert \(C'-C\)-bimodule, \(A'\) and A\({\mathrm {C}}^*\)-subalgebras of \(C'\) and C respectively, and \(F\subseteq E\) such that F is a Hilbert \(A'-A\)-bimodule with the structure inherited from E. Suppose \(\pi :C\rightarrow \mathbb {B}(H)\) is a representation and \(K\subseteq H\) is a closed subspace which is invariant under \(\pi _A\). Then:
  1. (1)

    There exists a unique isometry \(V:F\otimes _{\pi _A}K\rightarrow E\otimes _\pi H\) that satisfies \(V(x\otimes _{\pi _A}k)=x\otimes _{\pi }k\) for all \(x\in F\), \(k\in K\).

     
  2. (2)
    The isometry V intertwines \((\text {Ind}_E\pi )_{A'}\) and \(\text {Ind}_F\pi _A\), that is,
    $$\begin{aligned} \text {Ind}_E\pi (a')V=V\text {Ind}_F\pi _A(a')\quad \text{ for } \text{ all } a'\in A'. \end{aligned}$$
     

Proof

For a finite sum of elementary tensors \(\sum _ix_i\otimes _{\pi _A}k_i\in F\otimes _{\pi _A}K\) we compute:
$$\begin{aligned} \left\| \sum _ix_i\otimes _{\pi _A}k_i\right\| ^2= & {} \sum _{i,j}\langle x_i\otimes _{\pi _A}k_i,x_j\otimes _{\pi _A}k_j\rangle _K =\sum _{i,j}\langle k_i,\pi _A(\langle x_i,x_j\rangle _C) k_j\rangle _K\\= & {} \sum _{i,j}\langle k_i,\pi (\langle x_i,x_j\rangle _C) k_j\rangle _H =\left\| \sum _ix_i\otimes _{\pi }k_i\right\| ^2 \end{aligned}$$
Then there exists an isometry \(V:F\otimes _{\pi _A}K\rightarrow E\otimes _{\pi }H\) such that \(V(x\otimes _{\pi _A}k)=x\otimes _{\pi }k\) for all \(x\in F\), \(k\in K\), as claimed in (1). Now, if \(a'\in A'\), \(x\in F\) and \(k\in K\):
$$\begin{aligned} \text {Ind}_E\pi (a')V(x\otimes _{\pi _A}k) =a'x\otimes _{\pi }k =V(a'x\otimes _{\pi _A}k) =V\text {Ind}_F\pi _A(a')(x\otimes _{\pi _A}k), \end{aligned}$$
which proves (2). \(\square \)

Lemma 5.2

Let \(\pi :C\rightarrow \mathbb {B}(H)\) be a nondegenerate representation of a \({\mathrm {C}}^*\)-algebra C.
  1. (1)

    Let Y be a closed right ideal of C and \(A:=YY^*\) (recall Notation 2.1), which is a hereditary \({\mathrm {C}}^*\)-subalgebra of C. Consider Y as a Hilbert \(A-C\)-bimodule. Then \(\text {Ind}_{Y}\pi \) is equivalent to \(\pi _A\).

     
  2. (2)

    Let I be a closed two-sided ideal of C and let \(F_I\) be I with its natural structure of Hilbert \(C-I\)-bimodule. If \(\rho :I\rightarrow \mathbb {B}(K)\) is a nondegenerate representation, let \(\tilde{\rho }:C\rightarrow \mathbb {B}(K)\) be the unique extension of \(\rho \) to a representation of C on K, which is determined by \(\tilde{\rho }(c)(\rho (x)k)=\rho (cx)k\) for all \(c\in C\), \(x\in I\) and \(k\in K\). Then \(\text {Ind}_{F_I}\rho \) is equivalent to \(\tilde{\rho }\).

     
  3. (3)

    Suppose \(\pi \) is irreducible. If \(C'\) is a \({\mathrm {C}}^*\)-algebra and E is a Hilbert \(C'-C\)-bimodule such that \(\pi \) does not vanish on the ideal \(J:=\overline{{\text {span}}}\, \langle E,E\rangle _C\), then \(\text {Ind}_E\pi \) is irreducible and equivalent to \(\text {Ind}_{F_I}(\text {Ind}_E\pi _J)\), where \(I:=\overline{{\text {span}}}\, _{C'}\langle E,E\rangle \).

     

Proof

Let \(K':=\overline{\text {span}}\{\pi (y)h:\,y\in Y,\, h\in H\}\). Note that \(K'\) agrees with the essential space \(H_A\) of \(\pi |_A\). In fact we have \(H_A\subseteq K'\) because \(A\subseteq Y\), and the reverse inclusion follows from the fact that \(y=\lim _\lambda e_\lambda y\) and hence \(\pi (y)=\lim _\lambda \pi (e_\lambda )\pi (y)\) for every \(y\in Y\) and every approximate unit \((e_\lambda )\) of A. Now, if \(\sum _iy_i\otimes h_i\) is a finite sum of elementary tensors in \(Y\otimes _\pi H\), then
$$\begin{aligned} \left\| \sum _iy_i\otimes h_i\right\| ^2 =\sum _{i,j}\langle y_i\otimes h_i,y_j\otimes h_j\rangle =\sum _{i,j}\langle h_i,\pi (y_i^*y_j)h_j\rangle _H =\left\| \sum _i\pi (y_i)h_i\right\| ^2. \end{aligned}$$
Thus we have an isometry \(U:Y\otimes _\pi H\rightarrow H_A\) such that \(y\otimes h\mapsto \pi (y)h\). This isometry is surjective by the previous observation, hence U is a unitary. Finally, if \(a\in A\), \(y\otimes h\in Y\otimes _\pi H\):
$$\begin{aligned} U\text {Ind}_{Y}\pi (a)(y\otimes h) =U(ay\otimes h) =\pi (ay)h =\pi (a)\pi (y)h =\pi (a)U(y\otimes h), \end{aligned}$$
which proves our first statement.

To prove (2) we observe that exactly the same argument used in the proof of (1) shows that there is a unitary operator \(U:F_I\otimes _\rho K\rightarrow K\) that intertwines \(\text {Ind}_{F_I}\rho \) with \(\tilde{\rho }\).

As for (3), since I and J are the ideals generated by the left and right inner products of the bimodule E, we may view E as an \(I-J\)-imprimitivity bimodule. Since \(\pi \) does not vanish on J and \(\pi \) is irreducible, the essential space of \(\pi _J\) is H and \(\pi _J=\pi |_J\) is also irreducible. Since E is an \(I-J\)-imprimitivity bimodule, \(\text {Ind}_E{\pi _J}:I\rightarrow \mathbb {B}(E\otimes _{\pi _J}H)\) also is irreducible. Now let \(V:E\otimes _{\pi _J}H\rightarrow E\otimes _{\pi }H\) be the isometry provided by (1) of Lemma 5.1, which in this case is obviously surjective, thus a unitary operator. Since, according to Lemma 5.1(2), V intertwines \((\text {Ind}_E\pi )_I\) and \(\text {Ind}_E\pi _J\), and the latter is irreducible, then so is \((\text {Ind}_E\pi )_I\). Therefore \(\text {Ind}_E\pi \) is irreducible. Moreover, if \(\rho :=\text {Ind}_E\pi _J\), it is an easy task to show that \(V\tilde{\rho }(c')=V\text {Ind}_E\pi (c')\) for all \(c'\in C'\). This ends the proof, for \(\text {Ind}_{F_I}(\text {Ind}_E\pi _J)\) and \(\tilde{\rho }\) are equivalent by (2). \(\square \)

Lemma 5.3

Suppose, in the conditions of Lemma 5.1, that \(\pi \) is irreducible, \(A'\) is a hereditary \({\mathrm {C}}^*\)-subalgebra of \(C'\), and \(\pi |_{\langle E,E\rangle _C}\ne 0\). Then the isometry V is a unitary operator, and \([(\text {Ind}_E\pi )_{A'}]=[\text {Ind}_F\pi _A]\).

Proof

Since \(\pi \) is irreducible, then so is \(\text {Ind}_E\pi \) by Lemma 5.2(3). Moreover, if \(A'\) is a hereditary \({\mathrm {C}}^*\)-subalgebra of \(C'\), then \((\text {Ind}_E\pi )_{A'}\) is either zero or an irreducible representation of \(A'\) (Murphy 1990, Theorem 5.5.2). But \((\text {Ind}_E\pi )_{A'}\) cannot be zero because of Lemma 5.1(2) and the fact that \(\pi _A\), and therefore \(\text {Ind}_F\pi _A\), are non-zero representations. Now it follows from Lemma 5.1(2) that \(V(F\otimes _{\pi _A}K)\) is a non-zero \((\text {Ind}_E\pi )_{A'}\)-invariant subspace of \(E\otimes _{\pi }H\) and, since \((\text {Ind}_E\pi )_{A'}\) is irreducible, we must have \(V(F\otimes _{\pi _A}K)=E\otimes _{\pi }H\). That is, V is a surjective isometry, which ends the proof. \(\square \)

We show next that two strongly equivalent Fell bundles give rise to isomorphic partial actions on spectra level.

Theorem 5.4

Let \(\mathcal {A}\) and \(\mathcal {B}\) be Fell bundles over a discrete group G, and suppose that \(\mathcal {X}\) is a strong \(\mathcal {A}-\mathcal {B}\) equivalence. Let \(\mathsf {h}_{X_e}:\hat{B}_e\rightarrow \hat{A}_e\) be the Rieffel homeomorphism associated to the \(A_e-B_e\) imprimitivity bimodule \(X_e\). Then \(\mathsf {h}_{X_e}:\hat{\alpha }^\mathcal {B}\rightarrow \hat{\alpha }^\mathcal {A}\) is an isomorphism of partial actions.

Proof

Let \(\mathcal {C}=(C_t)_{t\in G}\) stand for the linking bundle of \(\mathcal {X}\). Then \(C_e=\mathbb {L}(X_e)\) (the linking algebra of \(X_e\)), and \(Y=\left( {\begin{matrix} A_e &{} X_e\\ 0 &{} 0 \end{matrix}}\right) \) is an \(A_e-C_e\) imprimitivity bimodule, so it defines the Rieffel homeomorphism \(\mathsf {h}_Y:\hat{C}_e\rightarrow \hat{A}_e\), which in our case is given by \(\mathsf {h}_Y([\pi ])=[\pi _{A_e}]\) (Lemma 5.2 (1) ). Therefore the Rieffel correspondence \(\mathsf {R}:\mathcal {I}(C_e)\rightarrow \mathcal {I}(A_e)\) between the ideals of \(C_e\) and \(A_e\), defined by \(Y\cong A_e\oplus X_e\), is given by \(\mathsf {R}(I)=A_e\cap I\).

We claim that \(\mathsf {h}_Y\) is an isomorphism between \(\hat{\alpha }^{\mathcal {C}}\) and \(\hat{\alpha }^{\mathcal {A}}\). First note that \(\mathsf {R}(D^\mathcal {C}_t) = D^\mathcal {A}_t\), that is: \(D^\mathcal {C}_t\cap A_e=C_tC_t^*\cap A_e=A_tA_t^*=D^\mathcal {A}_t\). In fact, a simple computation gives
$$\begin{aligned} C_tC_t^* =\begin{pmatrix} A_t^*A_t+\langle X_{t^{-1}},X_{t^{-1}}\rangle _\mathcal {A}&{}\quad A_t^*X_t+X_{t^{-1}}B_t\\ \widetilde{(A_t^*X_t+X_{t^{-1}}B_t)} &{}\quad \langle X_{t},X_{t}\rangle _\mathcal {B}+B_t^*B_t \end{pmatrix}. \end{aligned}$$
Thus we have \(\mathsf {h}_Y(\mathcal {V}^\mathcal {C}_t)=\mathcal {V}^\mathcal {A}_t\) for all \(t\in G\). Now take \([\pi ]\in \mathcal {V}_{t^{-1}}\). We must show \(\mathsf {h}_Y(\alpha _t^{\mathcal {C}}([\pi ])) = \alpha _t^\mathcal {A}(\mathsf {h}_Y[\pi ])\), that is: \([\text {Ind}_Y\text {Ind}_{C_t}\pi ]=[\text {Ind}_{A_t}\text {Ind}_Y\pi ]\). Since for every representation \(\rho \) of \(C_e\) we have \([\text {Ind}_Y\rho ]=[\rho _{A_e}]\), it is enough to show that \([(\text {Ind}_{C_t}\pi )_{A_e}]=[\text {Ind}_{A_t}\pi _{A_e}]\). But this follows at once from Lemma 5.3 by taking \(C=C'=C_e\), \(A=A'=A_e\), \(E=C_t\) and \(F=A_t\).

Similarly, if we now consider the \(C_e-B_e\) imprimitivity bimodule \(Z:=\left( {\begin{matrix} 0 &{} X_e\\ 0 &{} B_e \end{matrix}}\right) \), then the Rieffel homeomorphism \(\mathsf {h}_Z:\hat{B}_e\rightarrow \hat{C}_e\) is also an isomorphism \(\mathsf {h}_Z:\hat{\alpha }^\mathcal {B}\rightarrow \hat{\alpha }^\mathcal {C}\). Then \(\mathsf {h}_Y\circ \mathsf {h}_Z:\hat{\alpha }^\mathcal {B}\rightarrow \hat{\alpha }^\mathcal {A}\) is an isomorphism of partial actions. Besides, since \(Y\otimes _{C_e}Z=\left( {\begin{matrix} A_e &{} X_e\\ 0 &{} 0 \end{matrix}}\right) \otimes _{C_e}\left( {\begin{matrix} 0 &{} X_e\\ 0 &{} B_e \end{matrix}}\right) = X_e\), and \(\mathsf {h}_Y\circ \mathsf {h}_Z=\mathsf {h}_{\,Y\otimes _{C_e}Z}\), we conclude that \(\mathsf {h}_{X_e}:\hat{\alpha }^\mathcal {B}\rightarrow \hat{\alpha }^\mathcal {A}\) is an isomorphism of partial actions. \(\square \)

Given a locally compact Hausdorff group G, let \(G_d\) denote the group G when considered with the discrete topology. Similarly, if \(\mathcal {B}\) is a Fell bundle over G, let \(\mathcal {B}_d\) be the Fell bundle over \(G_d\) instead of G.

Proposition 5.5

Let \(\mathcal {B}\) be a Fell bundle over G, and let \(G_d\) and \(\mathcal {B}_d\) as above. Then the partial action \(\hat{\alpha }^{\mathcal {B}_d}\) of \(G_d\) on \(\hat{B}_e\) is a continuous partial action of G on \(\hat{B}_e\).

Proof

Consider the canonical action \(\beta \) of G on \(\mathbb {k}(\mathcal {B})\), and let \(\gamma \) be its restriction to \(\mathbb {K}(L^2(\mathcal {B}))\). Let \(\mathcal {A}:=\mathcal {B}_{\gamma }\). Since, by Theorem 3.5, \(\mathcal {A}\) is strongly equivalent to \(\mathcal {B}\), then also \(\mathcal {A}_d\) is strongly equivalent to \(\mathcal {B}_d\). Note that if we forget the topology of G, then \(\hat{\gamma }=\hat{\alpha }^{\mathcal {A}_d}\), so the latter is a continuous partial action of G on the spectrum of \(\mathbb {K}(L^2(\mathcal {B}))\). On the other hand \(\hat{\alpha }^{\mathcal {A}_d}\) and \(\hat{\alpha }^{\mathcal {B}_d}\) are isomorphic partial actions by Theorem 5.4 and, since \(\hat{\alpha }^{\mathcal {A}_d}\) is continuous, so must be \(\hat{\alpha }^{\mathcal {B}_d}\). \(\square \)

The previous result allows us to associate a partial action to every Fell bundle, not only to those over discrete groups:

Definition 5.6

Let \(\mathcal {B}\) be a Fell bundle over G, and denote by \(\hat{\alpha }^{\mathcal {B}}\) the partial action \(\hat{\alpha }^{\mathcal {B}_d}\) considered as a partial action of G on \(\hat{B}_e\). We say that \(\hat{\alpha }^{\mathcal {B}}\) is the partial action associated to \(\mathcal {B}\).

Now Theorem 5.4 can be stated for Fell bundles over arbitrary groups:

Corollary 5.7

Suppose \(\mathcal {X}\) is a strong \(\mathcal {A}-\mathcal {B}\)-equivalence bundle. If \(\mathsf {h}:\hat{B}_e\rightarrow \hat{A}_e\) is the Rieffel homeomorphism induced by the \(A_e-B_e\)-equivalence bimodule \(X_e\), then \(\mathsf {h}\) is an isomorphism between \(\hat{\alpha }^{\mathcal {A}}\) and \(\hat{\alpha }^{\mathcal {B}}\).

Corollary 5.8

Let \(\mathcal {B}\) be a Fell bundle, and \(\beta :G\times \mathbb {k}(\mathcal {B})\rightarrow \mathbb {k}(\mathcal {B})\) the canonical action. Then \(\hat{\beta }\) is the enveloping action of \(\hat{\alpha }^\mathcal {B}\).

In previous sections we have decomposed a weak equivalence between Fell bundles as strong equivalence followed by globalization of partial actions and Morita equivalence of enveloping actions (Theorems 3.4, 3.5 and 4.2). Combining this decomposition with the previous Corollary we obtain the following result.

Corollary 5.9

If \(\mathcal {A}\) and \(\mathcal {B}\) are weakly equivalent Fell bundles, then \(\hat{\alpha }^{\mathcal {A}}\) and \(\hat{\alpha }^{\mathcal {B}}\) have the same enveloping action.

Proof

Let \(\mu \) and \(\nu \) be the canonical actions on \(\mathbb {k}(\mathcal {A})\) and \(\mathbb {k}(\mathcal {B})\), respectively. The Fell bundle associated to \(\mu |_{\mathbb {K}(L^2(\mathcal {A}))}\) is strongly Morita equivalent to \(\mathcal {A}\). Hence the partial action on the spectrum of \(\mathbb {K}(L^2(\mathcal {A}))\) induced by \(\mu \) is isomorphic to \(\alpha \) and its enveloping action is the one induced by \(\mu \) on \(\hat{\mathbb {k}(\mathcal {A})}\), \(\hat{\mu }\). For the same reasons \(\hat{\nu }\) is an enveloping action of \(\beta \). We also know \(\mu \) and \(\nu \) are Morita equivalent, so \(\hat{\mu }\) is isomorphic to \(\hat{\nu }\) and this implies \(\hat{\mu }\) is an enveloping action of \(\beta \). \(\square \)

Proposition 5.10

A Fell bundle \(\mathcal {B}\) is saturated if and only if its associated partial action \(\hat{\alpha }^{\mathcal {B}}\) is global.

Proof

If \(\mathcal {B}\) is saturated then \(D^\mathcal {B}_t = B_t B_{t^{-1}}=B_e\) for all \(t\in G\). Thus the open set of \(\hat{B_e}\) corresponding to \(D^\mathcal {B}_t\), \(U_t\), is \(\hat{B_e}\) itself for all \(t\in G\). In other words, \(\hat{\alpha }^\mathcal {B}\) is global.

Conversely, in case \(\hat{\alpha }^\mathcal {B}\) is global we have \(U_t=\hat{B_e}\) for all \(t\in G\). Since the correspondence between \({\mathrm {C}}^*\)-ideals of \(B_e\) and open sets of \(\hat{B_e}\) is bijective, we conclude that \( B_tB_{t^{-1}}= D^\mathcal {B}_t=B_e\) for all \(t\in G\). Then for every \(r,s\in G\), considering \(B_{rs}\) as a left \(B_e\)-module, we deduce that \( B_{rs} = B_eB_{rs} = B_rB_{r^{-1}}B_{rs} \subset B_r B_s \subset B_{rs}\), so \(\mathcal {B}\) is saturated. \(\square \)

The last proposition implies that saturation is an invariant of strong equivalence:

Corollary 5.11

Let \(\mathcal {X}\) be an \(\mathcal {A}-\mathcal {B}\) strong equivalence bundle. Then \(\mathcal {A}\) is saturated if and only if \(\mathcal {B}\) is saturated.

Proof

If two partial actions are isomorphic, then one of them is global if and only if so is the other. Therefore our claim follows from Proposition 5.10. \(\square \)

5.1 Partial Actions on Primitive Ideal Spaces

Consider a Fell bundle \(\mathcal {B}=(B_t)_{t\in G}\), and let \(\beta \) be the canonical action of G on \({\mathbb {k}(\mathcal {B})}\). Let \(\mathcal {A}\) be the Fell bundle associated to the partial action \(\alpha :=\beta |_{\mathbb {K}(L^2(\mathcal {B}))}\). In particular \(A_e=\mathbb {K}(L^2(\mathcal {B}))\), and \(\hat{\alpha }=\hat{\alpha }^\mathcal {A}\). By Theorem 3.5 we know that \(\mathcal {L}^2\mathcal {B}\) is a strong \(\mathcal {A}-\mathcal {B}\) equivalence. In particular we have the Rieffel homeomorphisms \(\mathsf {h}:\hat{B}_e\rightarrow \hat{A}_e\) and \(\tilde{\mathsf {h}}:{{\mathrm{Prim}}}(B_e)\rightarrow {{\mathrm{Prim}}}(A_e)\). Given a \({\mathrm {C}}^*\)-algebra A, let \(\kappa :\hat{A}\rightarrow {{\mathrm{Prim}}}(A)\) be the map given by \(\kappa ([\pi ])=\ker \pi \). Then the Rieffel homeomorphisms satisfy \(\tilde{\mathsf {h}}\kappa =\kappa \mathsf {h}\). According to Abadie (2003), \(\alpha \) induces a partial action \(\tilde{\alpha }\) of G on \({{\mathrm{Prim}}}{A_e}\), which is determined by \(\tilde{\alpha }_t(\kappa ([\pi ]))=\kappa (\hat{\alpha }_t([\pi ]))\). Here \(\tilde{\alpha }_t:\mathcal {O}_{t^{-1}}\rightarrow \mathcal {O}_t\), where \(\mathcal {O}_t:=\{P\in {{\mathrm{Prim}}}{A_e}:P\not \supseteq D_t^\mathcal {A}\}\). This partial action \(\tilde{\alpha }\) is continuous, because it is a restriction of the global action \(\tilde{\beta }\) induced by \(\beta \) on \({{\mathrm{Prim}}}({\mathbb {k}(\mathcal {B})})\). Now, conjugating \(\tilde{\alpha }\) by \(\tilde{\mathsf {h}}\), we obtain a continuous partial action \(\tilde{\alpha }^\mathcal {B}\) of G on \({{\mathrm{Prim}}}(B_e)\), which satisfies \(\kappa \hat{\alpha }_t^\mathcal {B}([\pi ])=\tilde{\alpha }_t^\mathcal {B}\kappa ([\pi ])\) for all \([\pi ]\in \mathcal {V}_{t^{-1}}\). Thus we have:

Theorem 5.12

Every Fell bundle \(\mathcal {B}\) over the locally compact Hausdorff group G induces a continuous partial action \(\tilde{\alpha }^\mathcal {B}=(\{\mathcal {O}_t\}_{t\in G},\{\tilde{\alpha }^\mathcal {B}_t\}_{t\in G})\) of G on \({{\mathrm{Prim}}}(B_e)\), which is given by \(\tilde{\alpha }_t(P)=B_tPB_t^{*}\) for all \(P\in \mathcal {O}_{t^{-1}}\); hence the following diagram is commutative for all \(t\in G\):

Moreover,
  1. (1)

    if \(\mathcal {X}=(X_t)\) is a strong \(\mathcal {A}-\mathcal {B}\) equivalence bundle, the Rieffel homeomorphism \(\tilde{\mathsf {h}}_{X_e}:{{\mathrm{Prim}}}(B_e)\rightarrow {{\mathrm{Prim}}}(A_e)\) is an isomorphism between \(\tilde{\alpha }_t^\mathcal {B}\) and \(\tilde{\alpha }_t^\mathcal {A}\).

     
  2. (2)

    If \(\beta \) is the canonical action of G on \({\mathbb {k}(\mathcal {B})}\), then \(\tilde{\beta }:G\times {{\mathrm{Prim}}}({\mathbb {k}(\mathcal {B})})\rightarrow {{\mathrm{Prim}}}({\mathbb {k}(\mathcal {B})})\) is the enveloping action of \(\tilde{\alpha }^\mathcal {B}\).

     

Proof

We only need to prove (2), since the remaining statements follow at once from the definition of \(\tilde{\alpha }^\mathcal {B}\). Now assertion (2) is a direct consequence of (1) and Abadie (2003, Proposition 7.4). \(\square \)

The moral of the preceding section is that a Fell bundle is essentially the same object that a semidirect product Fell bundle for a partial action, in the sense that it is always strongly equivalent to such a product. With this in mind, one should be able to translate results from semidirect product Fell bundles to arbitrary Fell bundles.

As an example, we have the following generalization of Abadie (2003, Corollary 7.2):

Corollary 5.13

Let \(\mathcal {B}\) be a Fell bundle over the locally compact Hausdorff group G. If \({{\mathrm{Prim}}}(B_e)\) is compact, then there exists an open subgroup H of G such that the reduction of \(\mathcal {B}\) to H is a saturated Fell bundle. In particular, if G is a connected group, then \(\mathcal {B}\) is a saturated Fell bundle.

Proof

Since \({{\mathrm{Prim}}}(B_e)\) is compact (Abadie 2003, Proposition 1.1) shows there exists an open subgroup H of G for which the restriction of \(\tilde{\alpha }^\mathcal {B}\) to H is a global action, that is \(D_t^\mathcal {B}=B_e\) for all \(t\in H\). Thus the reduction of \(\mathcal {B}\) to H is a saturated Fell bundle. Since the only open subgroup of a connected group is the group itself, the proof is finished. \(\square \)

6 \(\mathrm {C_0}(X)\)-Fell Bundles and Amenability

Given a locally compact Hausdorff space X, a \({\mathrm {C}}^*\)-algebra C is a \(\mathrm {C_0}(X)\)-algebra if there exists a nondegenerate \(^{*}\)-homomorphism \(\phi :\mathrm {C_0}(X)\rightarrow ZM(C)\). In this situation there exists a unique continuous function \(f_\phi :\hat{C}\rightarrow X\) such that
$$\begin{aligned} \overline{\pi }(a) = a(f_\phi ([\pi ]))1_\pi \quad \text{ for } \text{ all } \ a\in \mathrm {C_0}(X) \text{ and } [\pi ]\in \hat{C}, \end{aligned}$$
where \(\overline{\pi }\) is the natural extension of the irreducible representation \(\pi :C\rightarrow \mathbb {B}(\mathcal {H})\) to M(C) and \(1_\pi \) is the identity operator of \(\mathcal {H}\).

Assume now that \(\theta \) is an action of G on \(\mathrm {C_0}(X)\), \(\beta \) an action of G on C and that \(\phi \) is equivariant in the sense that, for all \(t\in G\), \(a\in \mathrm {C_0}(X)\) and \(c\in C:\)\(\beta _t(\phi (a)c)=\phi (\theta _t(a))\beta _t(c)\). In this situation \(f_\phi \) is \(\hat{\beta }-\hat{\theta }\)-equivariant and the Fell bundle \(\mathcal {B}_\beta \) is a \(\hat{\theta }\)-Fell bundle in the following sense.

Definition 6.1

Let \(\sigma \) be an action of the locally compact Hausdorff group G on the locally compact Hausdorff space X. A \(\sigma \)-Fell bundle is a Fell bundle over G, \(\mathcal {B}\), for which there exists a continuous function \(f:\hat{B_e}\rightarrow X\) which is a morphism of partial actions between \(\hat{\alpha }\) and \(\sigma \).

The example that motivated this definition has a converse. Suppose \(\beta \) is an action of G on the \({\mathrm {C}}^*\)-algebra B and that \(\mathcal {B}_\beta \) is a \(\sigma \)-Fell bundle. Then the unit fiber of \(\mathcal {B}\) is B and the action defined by \(\mathcal {B}_\beta \) on \(\hat{B}\) is the action defined by \(\beta \), \(\hat{\beta }\). By hypothesis there exists a \(\hat{\beta }-\sigma \)-equivariant continuous function \(f:\hat{B} \rightarrow X\). Since the points of X are closed, there exists (by Williams 2007, Lemma C.6) a unique continuous function \(g:{\text {Prim}}(B)\rightarrow X\) such that \(g\circ \kappa =f\), where \(\kappa :\hat{B}\rightarrow {\text {Prim}}(B)\) is given by \(\kappa ([\pi ])=\ker (\pi )\), as in the preceding section. The condition \(g\circ \kappa =f\) ensures that \(g:{\text {Prim}}(B)\rightarrow X\) is equivariant, considering on \({\text {Prim}}(B)\) the action induced by \(\beta \). Using Dauns–Hofmann Theorem we conclude that there exists a unique nondegenerate and equivariant \(^{*}\)-homomorphism \(\phi :\mathrm {C_0}(X)\rightarrow ZM(B)\), where the action considered on \(\mathrm {C_0}(X)\) is the one defined by \(\sigma \).

Theorem 6.2

Let \(\mathcal {B}\) be a Fell bundle over G and \(\sigma \) an action of G on the locally compact Hausdorff space X. If \(\beta \) is the canonical action of G on \({\mathbb {k}(\mathcal {B})}\) and \(\theta \) is the action on \(\mathrm {C_0}(X)\) defined by \(\sigma \), then the following are equivalent:
  1. (1)

    \(\mathcal {B}\) is a \(\sigma \)-Fell bundle.

     
  2. (2)

    There exists a nondegenerate \(^{*}\)-homomorphism \(\phi :\mathrm {C_0}(X)\rightarrow ZM({\mathbb {k}(\mathcal {B})})\) such that for all \(t\in G\), \(a\in \mathrm {C_0}(X)\) and \(k\in {\mathbb {k}(\mathcal {B})}\), \(\beta _t(\phi (a)k)=\phi (\theta _t(a))\beta (k)\).

     

Proof

By the comments preceding the statement, the implication (1)\(\Rightarrow \)(2) will follow after we show that \(\mathcal {B}_\beta \) is a \(\sigma \)-Fell bundle. Assume that \(f:\hat{B_e}\rightarrow X\) is an equivariant continuous function. If we denote by \(\alpha \) the restriction of \(\beta \) to \(A:=\mathbb {K}(L^2(\mathcal {B}))\) and \(\mathsf {h}:\hat{B_e}\rightarrow \hat{A}\) is the Rieffel homeomorphism given by the equivalence bimodule \(L^2(\mathcal {B})\), then \(f\circ \mathsf {h}^{-1} :\hat{A}\rightarrow X\) is equivariant. By Abadie (2003, Proposition 7.4) \(\hat{\beta }\) is the enveloping action of \(\hat{\alpha }\) and by Abadie (2003, Theorem 1.1) there exists a unique \(\hat{\beta }-\sigma \)-equivariant continuous extension of \(f\circ \mathsf {h}^{-1}\). \(\square \)

The next result is an extension of Anantharaman-Delaroche (2002, Theorem 5.3) to Fell bundles.

Theorem 6.3

Let \(\sigma \) be an action of G on the locally compact Hausdorff space X. Consider the conditions:
  1. (1)

    \(\sigma \) is amenable.

     
  2. (2)

    Every \(\sigma \)-Fell bundle is amenable, that is, \(C^*(\mathcal {B})=C^*_r(\mathcal {B})\).

     
  3. (3)

    For every \(\sigma \)-Fell bundle \(\mathcal {B}\) with \(B_e\) nuclear, \(C^*_r(\mathcal {B})\) is nuclear.

     
  4. (4)

    \(\mathrm {C_0}(X)\rtimes _r G\) is nuclear.

     
Then \((1)\Rightarrow (2)\Rightarrow (3)\Rightarrow (4)\) and if G is discrete \((4)\Rightarrow (1)\).

Proof

Name \(\beta \) the canonical action on \({\mathbb {k}(\mathcal {B})}\). Since \(\mathcal {B}\) is equivalent to \(\mathcal {B}_\beta \), \(\mathcal {B}\) is amenable if and only if \(\mathcal {B}_\beta \) is amenable. Moreover, since \(C^*_r(\mathcal {B})\) and \(C^*_r(\mathcal {B}_\beta )\) are Morita equivalent, one is nuclear if and only if the other one is.

Assume (1) holds. By Anantharaman-Delaroche (2002, Theorem 5.3) and Theorem 6.2, \(\mathcal {B}_\beta \) is amenable and so \(\mathcal {B}\) is amenable. Now assume that (2) holds, then (2) from Anantharaman-Delaroche (2002, Theorem 5.3) holds and it suffices to show that \({\mathbb {k}(\mathcal {B})}\) is nuclear. We know \(\mathbb {K}(L^2(\mathcal {B}))\) is nuclear because \(B_e\) is nuclear. Then Abadie (2003, Proposition 2.2) implies that \({\mathbb {k}(\mathcal {B})}\) is nuclear.

The rest of the proof follows directly from Anantharaman-Delaroche (2002, Example (3) of 4.4. together with Theorem 5.8). \(\square \)

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Copyright information

© Sociedade Brasileira de Matemática 2018

Authors and Affiliations

  • Fernando Abadie
    • 1
  • Alcides Buss
    • 2
    Email author
  • Damián Ferraro
    • 3
  1. 1.Centro de Matemática, Facultad de CienciasUniversidad de la RepúblicaMontevideoUruguay
  2. 2.Departamento de MatemáticaUniversidade Federal de Santa CatarinaFlorianópolisBrazil
  3. 3.Departamento de Matemática y Estadística del LitoralUniversidad de la RepúblicaSaltoUruguay

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