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THE RIEFFEL CORRESPONDENCE FOR EQUIVALENT FELL BUNDLES

Published online by Cambridge University Press:  17 January 2024

S. KALISZEWSKI
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287, USA e-mail: kaliszewski@asu.edu
JOHN QUIGG*
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287, USA
DANA P. WILLIAMS
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551, USA e-mail: dana.williams@Dartmouth.edu
*
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Abstract

We establish a generalized Rieffel correspondence for ideals in equivalent Fell bundles.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

1 Introduction

Morita equivalence is a fundamental tool in the study of ${C}^*$ -algebras. For example, Morita equivalent ${C}^*$ -algebras A and B share much of their fine structure and have equivalent representation theories. Many such properties are elucidated as the ‘Rieffel correspondence’ induced by an $A\,\text {--}\, B$ -imprimitivity bimodule $\mathsf {X}$ . A summary of these properties is given in Theorem 2.1, but the key feature is that the Rieffel correspondence gives a natural lattice isomorphism between the ideal lattices of the two ${C}^*$ -algebras. In the case of ${C}^*$ -algebras associated to dynamical systems of various sorts, perhaps the fundamental tool used to generate useful Morita equivalences is the notion of Fell-bundle equivalence. In this article, we show that there is an analogous Rieffel correspondence induced by an equivalence $q\colon \mathscr {E}\to T$ between two Fell bundles $p_{\mathscr {B}}\colon \mathscr {B}\to H$ and $p_{\mathscr {C}}\colon \mathscr {C}\to K$ over locally compact groupoids H and K. Rather than work at the level of the Fell-bundle ${C}^*$ -algebras ${C}^*(H; \mathscr {B})$ and ${C}^*(K;\mathscr {C})$ , we work with the Fell bundles themselves. We introduce a natural notion of an ideal $\mathscr {J}$ of a Fell bundle $\mathscr {B}$ . In the case where $\mathscr {B}$ is the Fell bundle corresponding to a group or groupoid G acting on a ${C}^*$ -algebra A, these Fell-bundle ideals naturally correspond to G-invariant ideals of A in the standard sense. More generally, our ideals are the same as the Fell subbundles studied in [Reference Ionescu and WilliamsIW12]. We can form the quotient Fell bundles $\mathscr {B}/\mathscr {J}$ , and if H has a Haar system and if our Fell bundles are separable, then the main result in [Reference Ionescu and WilliamsIW12] pushes the analogy of Fell bundle ideals with invariant ideals in crossed products; that is, we have a short exact sequence of ${C}^*$ -algebras

However, as we do not work with ${C}^*$ -algebras, we do not require our groupoids to have Haar systems.

In this article, our main result is that if $\mathscr {E}$ is an equivalence between $\mathscr {B}$ and $\mathscr {C}$ as above, then there is a lattice isomorphism between the ideals of $\mathscr {C}$ and those of $\mathscr {B}$ . Furthermore, if $\mathscr {K}$ and $\mathscr {J}$ are corresponding ideals of $\mathscr {B}$ and $\mathscr {C}$ , respectively, then $\mathscr {K}$ and $\mathscr {J}$ are equivalent Fell bundles as are the quotients $\mathscr {B}/\mathscr {K}$ and $\mathscr {C}/\mathscr {J}$ . Naturally, these equivalences arise from submodules and quotients of the given equivalence $\mathscr {E}$ .

We start in Section 2 with a detailed collection of preliminary material that summarizes and conveniently collects in one place the basics of Banach bundles, Fell bundles and Fell-bundle equivalence. We also introduce our notion of ideals of Fell bundles and develop some of their basic properties. In Section 3, we establish our basic Rieffel correspondence as Theorem 3.10. Then in Section 4, we establish the equivalence between corresponding ideals and their quotients.

We know that if two separable Fell bundles are equivalent, and if both of the underlying groupoids have Haar systems, then their corresponding Fell-bundle ${C}^*$ -algebras are Morita equivalent and the classical Rieffel correspondence gives an isomorphism between the ideal lattices of the two Fell-bundle ${C}^*$ -algebras. In Section 5, we confirm the natural conjecture that if two ideals correspond under our Rieffel correspondence for Fell-bundle ideals, then the corresponding ideals in the Fell-bundle ${C}^*$ -algebras also correspond under the classical Rieffel correspondence.

Conventions.

We use the standard conventions in the subject. In particular, homomorphisms between ${C}^*$ -algebras are assumed to be $*$ -preserving and ideals in ${C}^*$ -algebras are two-sided and norm closed. Locally compact is meant to mean locally compact and Hausdorff, and our groupoids are always meant to be locally compact and Hausdorff. Suppose that A is an algebra and $\mathsf {X}$ is a (left) A-module. If $S\subset A$ and $\mathsf {Y}\subset \mathsf {X}$ , then by convention, $S\cdot \mathsf {Y}=\operatorname {span}\{a\cdot x:{a\in A \text { and } x\in \mathsf {Y}}\}$ . Similarly, if ${\langle}\cdot , \cdot{\rangle}$ is an A-valued sesquilinear form on $\mathsf {X}$ , then ${\langle}\mathsf {Y}_{1},\mathsf {Y}_{2}{\rangle}=\operatorname {span}\{{\langle}x,y{\rangle}:{x\in \mathsf {Y}_{1} \text { and } y\in \mathsf {Y}_{2}}\}$ . If A is a ${C}^*$ -algebra and the A-module $\mathsf {X}$ is a Banach space, then we call $\mathsf {X}$ a Banach A-module if $\|a\cdot x\|\le \|a\|\|x\|$ for all $a\in A$ and $x\in \mathsf {X}$ . Further, we say that $\mathsf {X}$ is nondegenerate if $A\cdot \mathsf {X}$ is dense in $\mathsf {X}$ .

2 Preliminaries

2.1 The Rieffel correspondence

If A is a ${C}^*$ -algebra, then we let $\mathcal {I}(A)$ denote the lattice of ideals in A. Suppose that $\mathsf {X}$ is an $A\, {\text {--}}\, B$ -imprimitivity bimodule, and let $\mathcal C(\mathsf {X})$ be the lattice of closed $A\, {\text {--}}\, B$ -submodules of $\mathsf {X}$ . Then the Rieffel correspondence asserts that there are natural lattice isomorphisms among $\mathcal {I}(A)$ , $\mathcal C(\mathsf {X})$ and $\mathcal {I}(B)$ . Specifically, we have the following summary from [Reference Raeburn and WilliamsRW98, Section 3.3].

Theorem 2.1 (Rieffel correspondence).

Suppose that A and B are ${C}^*$ -algebras and that $\mathsf {X}$ is an $A\, {\text {--}}\, B$ -imprimitivity bimodule.

  1. (a) Suppose that $\mathsf {Y}$ is a closed $A\, {\text {--}}\, B$ -submodule of $\mathsf {X}$ . Then

    (2-1) $$ \begin{align} K=\overline{_{A}{\langle}\mathsf{Y},\mathsf{X}{\rangle}}=\overline{_{A}{\langle}\mathsf{X},\mathsf{Y}\rangle} =\overline{_{A}{\langle}\mathsf{Y},\mathsf{Y}\rangle}\end{align} $$
    is an ideal in A, while
    (2-2) $$ \begin{align} J=\overline{{\langle}\mathsf{Y},\mathsf{X}{\rangle}_{B}}=\overline{{\langle}\mathsf{X},\mathsf{Y}{\rangle}_{B}} =\overline{{\langle}\mathsf{Y},\mathsf{Y}{\rangle}_{B}} \end{align} $$
    is an ideal in B. We have
    (2-3) $$ \begin{align} K\cdot \mathsf{X}= \overline{K\cdot \mathsf{X}}=\mathsf{Y}=\overline{\mathsf{X}\cdot J}= \mathsf{X}\cdot J. \end{align} $$
  2. (b) In particular, $J\mapsto \mathsf {X}\cdot J$ is a lattice isomorphism of $\mathcal {I}(B)$ onto $\mathcal C(\mathsf {X})$ with inverse $\mathsf {Y}\mapsto \overline {{\langle}\mathsf {Y},\mathsf {Y}{\rangle}_{B}}$ and $K\mapsto K\cdot \mathsf {X}$ is a lattice isomorphism of $\mathcal {I}(A)$ onto $\mathcal C(\mathsf {X})$ with inverse $\mathsf {Y} \mapsto \overline {_{A}\langle\mathsf {Y},\mathsf {Y}\rangle}$ .

  3. (c) If K, $\mathsf {Y}$ and J are as in part (a), then $\mathsf {Y}$ is a $K\, {\text {--}}\, J$ -imprimitivity bimodule with respect to the restricted actions and inner products.

  4. (d) If K, $\mathsf {Y}$ and J are as in part (a), then the quotient Banach space $\mathsf {X}/\mathsf {Y}$ is an $A/K\, {\text {--}}\, B/J$ -imprimitivity bimodule. In particular, the quotient norm on $\mathsf {X}/\mathsf {Y}$ equals the imprimitivity-bimodule norm.

Remark 2.2. Suppose that J is an ideal in a ${C}^*$ -algebra B, and that $\mathsf {X}$ is a right Hilbert B-module. Then $\mathsf {Y}=\overline {\mathsf {X}\cdot J}$ is a nondegenerate Banach J-module. Therefore, the Cohen factorization [Reference Raeburn and WilliamsRW98, Proposition 2.33] implies that every element of $\mathsf {Y}$ is of the form $x\cdot b$ with $x\in \mathsf {Y}$ and $b\in J$ , so

$$ \begin{align*} \mathsf{Y}=\mathsf{X}\cdot J=\{x\cdot b:{x\in \mathsf{X} \text{ and } b\in J}\}. \end{align*} $$

As a result, we have $\overline {\mathsf {X}\cdot J}=\mathsf {X}\cdot J$ as in part (a), and similarly with $K\cdot \mathsf {X}$ .

Proof of Theorem 2.1.

This is just a reworking of the basic results in [Reference Raeburn and WilliamsRW98, Section 3.3]. The equalities in Equations (2-1) and (2-2) follow from [Reference Raeburn and WilliamsRW98, Lemma 3.23]. The lattice isomorphisms follow from [Reference Raeburn and WilliamsRW98, Theorem 3.22], while Equation (2-3) follows from [Reference Raeburn and WilliamsRW98, Proposition 3.24] together with Remark 2.2. The statements about imprimitivity bimodules follow from [Reference Raeburn and WilliamsRW98, Proposition 3.25].

2.2 Banach bundles

Roughly speaking, a Banach bundle is a topological bundle in which each fibre is a Banach space. More precisely, we have the following definition.

Definition 2.3. A Banach bundle over a topological space X is a topological space $\mathscr {B}$ together with a continuous, open surjection $p\colon \mathscr {B}\to X$ and complex Banach space structures on each fibre $B_{x}=p^{-1}(\{x\})$ satisfying the following axioms.

  1. (B1) The map $b\mapsto \|b\|$ is upper semicontinuous from $\mathscr {B}$ to $\mathbf {R}^{+}$ (this means that for all $\epsilon>0$ , $\{b\in \mathscr {B}:\|b\|<\epsilon \}$ is open).

  2. (B2) The map $(a,b)\mapsto a+b$ from $\mathscr {B}^{(2)}=\{(a,b)\in \mathscr {B}\times \mathscr {B}:p(a)=p(b)\}$ to $\mathscr {B}$ is continuous.

  3. (B3) The map $(\lambda ,b)\mapsto \lambda b$ is continuous from $\mathbf {C}\times \mathscr {B}$ to $\mathscr {B}$ .

  4. (B4) If $(b_{i})$ is a net in $\mathscr {B}$ such that $p(b_{i})\to x$ and $\|b_{i}\|\to 0$ , then $b_{i}\to 0_{x}$ in $\mathscr {B}$ (where $0_{x}$ is the zero element in $B_{x}$ ).

We say that $p:\mathscr {B}\to X$ is separable if X is second countable and the Banach space $\Gamma _{0}(X;\mathscr {B})$ is separable. If the map in axiom (B1) is actually continuous, we call $\mathscr {B}$ a continuous Banach bundle.

Remark 2.4. In some treatments, axiom (B3) in Definition 2.3 is replaced by the formally weaker axiom that $b\mapsto \lambda b$ is continuous for each $\lambda \in \mathbf {C}$ . However, since $\{b\in \mathscr {B}:\|b\|<\epsilon \}$ is open, the proof of [Reference Fell and DoranFD88, Proposition II.13.10] shows the two definitions are equivalent.

Remark 2.5 (The literature).

Continuous Banach bundles are treated in detail in Sections 13–14 in [Reference Fell and DoranFD88, Ch. II] and many of the results there apply mutatis mutandis to Banach bundles. In the past, Banach bundles as defined above were called ‘upper semicontinuous Banach bundles’. We have adopted the convention to drop the modifier in the general case. Banach bundles are discussed briefly in [Reference Muhly and WilliamsMW08, Appendix A], which is where Definition 2.3 comes from, and the corresponding notion of a ${C}^*$ -bundle is treated in detail in [Reference WilliamsWil07, Appendix C].

The topology on the total space $\mathscr {B}$ of a Banach bundle might not be well behaved. For example, it need not be Hausdorff [Reference WilliamsWil07, Example C.27]. However, we do have the following lemma.

Lemma 2.6. If $p\colon \mathscr {B}\to X$ is a Banach bundle, then the relative topology on $B_{x}$ is the (Banach space) norm topology.

Proof. In the continuous case, this is [Reference Fell and DoranFD88, Proposition II.13.11], and proof carries over to the general case; see [Reference Duwenig, Williams and ZimmermanDWZ22, Lemma 2.2].

If $p:\mathscr {B}\to X$ is a Banach bundle, we write $\Gamma (X;\mathscr {B})$ for the vector space of continuous sections. If X is locally compact, then we write $\Gamma _{c}(X;\mathscr {B})$ and $\Gamma _{0}(X;\mathscr {B})$ for the continuous sections that have compact support or that vanish at infinity, respectively. We say that $p:\mathscr {B}\to X$ has enough sections if given $b\in B_{x}$ , there is an $f\in \Gamma (X;\mathscr {B})$ such that $f(x)=b$ . Note that if X is locally compact, then since $\Gamma (X;\mathscr {B})$ is a $C(X)$ -module by axiom (B3), we can take $f\in \Gamma _{c}(X;\mathscr {B})$ .

Theorem 2.7 [Reference LazarLaz18, Corollary 2.10].

If $p:\mathscr {B}\to X$ is a Banach bundle over a locally compact space, then $\mathscr {B}$ has enough sections.

While the notion of a Banach bundle is a natural mathematical object, generally Banach bundles arise in nature from their sections as described in the following result.

Theorem 2.8 (Hofmann–Fell).

Let X be a locally compact space and suppose that for each $x\in X$ we are given a Banach space $B_{x}$ . Let $\mathscr {B}$ be the disjoint union $\coprod _{x\in X}B_{x}$ viewed as a bundle $p:\mathscr {B}\to X$ . Suppose that $\Gamma $ is a subspace of sections such that:

  1. (a) for each $f\in \Gamma $ , $x\mapsto \|f(x)\|$ is upper semicontinuous; and

  2. (b) for each $x\in X$ , $\{f(x):f\in \Gamma \}$ is dense in $B_{x}$ .

Then there is a unique topology on $\mathscr {B}$ such that $p:\mathscr {B}\to X$ is a Banach bundle with $\Gamma \subset \Gamma (X;\mathscr {B})$ . Furthermore, the sets of the form

$$ \begin{align*} W(f,U,\epsilon)=\{a\in \mathscr{B}:{p(a)\in U \text{ and } \|a-f(p(a))\|<\epsilon}\} \end{align*} $$

with $f\in \Gamma $ , U open in X and $\epsilon>0$ form a basis for this topology.

Proof. In the continuous case, this is [Reference Fell and DoranFD88, Theorem II.13.18]. In general, it is stated in [Reference Dupré and GilletteDG83, Proposition 1.3] and also follows mutatis mutandis from [Reference WilliamsWil07, Theorem C.25].

2.3 Banach subbundles

A subbundle of a Banach bundle is a Banach subbundle if it is a Banach bundle in the inherited structure.

Definition 2.9. Let $p\colon \mathscr {B}\to X$ be a Banach bundle. We say that $\mathscr {C}\subset \mathscr {B}$ is a Banach subbundle if each $C_{x}=B_{x}\cap C$ is a closed vector subspace of $B_{x}$ , and $p|_{{\mathscr {C}}}\colon \mathscr {C}\to X$ is a Banach bundle when we give $C_{x}$ the Banach-space structure coming from $B_{x}$ and we give $\mathscr {C}$ the relative topology.

Remark 2.10. Since $0_{x}\in C_{x}$ for all x, we must have $p(\mathscr {C})=X$ . However, some fibres can be the zero Banach space.

Remark 2.11. If $\{C_{x}\}$ is any collection of closed subspaces with $C_{x}\subset B_{x}$ and if we give $\mathscr {C}=\coprod C_{x}=\{b\in \mathscr {B}:b\in C_{p(b)}\}$ the relative topology, then $p\colon \mathscr {C}\to X$ is a continuous surjection satisfying axioms (B1), (B2), (B3), and (B4) of Definition 2.3. However, $p\colon \mathscr {C}\to X$ may fail to be a Banach subbundle unless we also have $p|_{{\mathscr {C}}}$ open.

Even if $p|_{{\mathscr {C}}}$ is not open, we write $\Gamma (X;\mathscr {C})$ for the continuous functions f from X to $\mathscr {C}$ such that $p(f(x))=x$ for all $x\in X$ . Of course, if $p|_{{\mathscr {C}}}$ is not open, there is no reason that $\Gamma (X;\mathscr {C})$ should contain anything other than the zero section, as shown in the next example.

Example 2.12. Let B be a Banach space and $\mathscr {B}=X\times B$ the trivial bundle over X. Fix $x_{0}\in X$ and let

$$ \begin{align*} C_{x}= \begin{cases} B&\text{if } x=x_{0}, \\ 0_{x}&\text{otherwise.} \end{cases} \end{align*} $$

Then in general, $p|_{{\mathscr {C}}}:\mathscr {C}\to X$ is not open and admits only the zero section.

Proposition 2.13. Let $p\colon \mathscr {B}\to X$ be a Banach bundle. Suppose that $C_{x}$ is a closed subspace of $B_{x}$ for all $x\in X$ and let $\mathscr {C}=\coprod C_{x}$ be as above. If $\{f(x):f\in \Gamma (X;\mathscr {C})\}$ is dense in $C_{x}$ for all $x\in X$ , then the bundle $p|_{{\mathscr {C}}}\colon \mathscr {C}\to X$ is a Banach subbundle of $\mathscr {B}$ .

Proof. Suppose that $\{f(x):f\in \Gamma (X;\mathscr {C})\}$ is dense in $C_{x}$ for all x. Let U be a nonempty (relatively) open set in $\mathscr {C}$ . In view of Remark 2.11, to show that $p|_{{\mathscr {C}}}\colon \mathscr {C}\to X$ is a Banach subbundle, it suffices to see that $p(U)$ is open in X. Let $x\in p(U)$ and suppose that $(x_{i})$ is a net in X converging to x in X. It suffices to see that $(x_{i})$ is eventually in $p(U)$ . Let $b\in U$ be such that $p(b)=x$ . Then for each n, the set $\{b'\in \mathscr {B}: \|b'-b\|<1/n\}$ is an open neighbourhood of b in $\mathscr {B}$ . Hence, there is $f_{n}\in \Gamma (X;\mathscr {C})$ such that ${\|f_{n}(x)-b\|<1/n}$ . Thus, $\|f_{n}(x)-b\|\to 0$ . By axiom (B4), $f_{n}(x)-b\to 0_{x}$ in $\mathscr {B}$ . However, by axiom (B2), $f_{n}(x) \to b$ in $\mathscr {B}$ . Since everything in sight is in $\mathscr {C}$ and $\mathscr {C}$ has the relative topology, for some N, $f_{N}(x)\in U$ . However, $f_{N}(x_{i})\to f_{N}(x)$ . So $f_{N}(x_{i})$ is eventually in U. Therefore $x_{i}$ is eventually in $p(U)$ .

Remark 2.14. In [Reference Fell and DoranFD88, Problem 41 in Ch. II], Fell and Doran call a family $\{C_{x}\}$ of subspaces as in Proposition 2.13 in a continuous Banach bundle a lower semicontinuous choice of subspaces.

Remark 2.15. If $p\colon \mathscr {B}\to X$ is a Banach bundle over a locally compact space and if $p|_{{\mathscr {C}}}\colon \mathscr {C}\to X$ is a Banach subbundle, then it has enough sections by Lazar’s Theorem 2.7. Hence, $\{f(x):f\in \Gamma (X;\mathscr {C})\}$ is not only dense, it is all of $C_{x}$ .

2.4 Quotient Banach bundles

Let $p:\mathscr {B}\to X$ be a Banach bundle over a locally compact space X and let $\mathscr {C}\subset \mathscr {B}$ be a Banach subbundle as in Definition 2.3. Then we can formally form the quotient $\mathscr {B}/\mathscr {C}=\coprod _{x\in X} B_{x}/C_{x}$ , where $B_{x}/C_{x}$ is the usual Banach space quotient. We let $q:\mathscr {B}\to \mathscr {B}/\mathscr {C}$ be the quotient map so that if $b\in B_{x}$ , then ${q(b)=q_{x}(b)}$ , where $q_{x}:B_{x}\to B_{x}/C_{x}$ is the usual Banach space quotient map. In particular, q is norm reducing. If $f\in \Gamma _{c}(X;\mathscr {B})$ , then we write $q(f)$ for the section of $\mathscr {B}/\mathscr {C}$ given by $q(f)(x)=q_{x}(f(x))$ .

Proposition 2.16. Let $p:\mathscr {B}\to X$ be a Banach bundle and $\mathscr {C}\subset \mathscr {B}$ a Banach subbundle. Then $\bar p:\mathscr {B}/\mathscr {C} \to X$ is a Banach bundle in the quotient topology. Furthermore, the quotient map $q:\mathscr {B}\to \mathscr {B}/\mathscr {C}$ is continuous and open, and the quotient topology on $\mathscr {B}/\mathscr {C}$ is the unique topology such that $\Gamma = \{q(f):f\in \Gamma _{c}(X;\mathscr {B})\} \subset \Gamma _{c}(X;\mathscr {B}/\mathscr {C})$ .

Remark 2.17. As pointed out in [Reference LazarLaz18], Proposition 2.16 can be sorted out of [Reference GierzGie82, Ch. 9]. We give the short proof for completeness.

Proof. Let $f\in \Gamma _{c}(X;\mathscr {B})$ . We claim $x\mapsto \|q_{x}(f(x))\|$ is upper semicontinuous. Fix $\epsilon>0$ . Suppose $\|q_{x}(f(x))\|<\epsilon $ . Then by definition of the quotient norm, there is a $c\in C_{x}$ such that $\|f(x)+c\|<\epsilon $ . Let $d\in \Gamma _{c}(X;\mathscr {C})$ be such that $d(x)=c$ . Then there is a neighbourhood V of x such that $\|f(y)+d(y)\|<\epsilon $ if $y\in V$ . Since $q(f)=q(f+d)$ , it follows that $\|q(f)(y)\|<\epsilon $ for $y\in V$ . This establishes the claim.

It follows from Theorem 2.8 that there is a unique topology on $\mathscr {B}/\mathscr {C}$ such that $\mathscr {B}/\mathscr {C}$ is a Banach bundle with $\Gamma :=\{q(f):f\in \Gamma _{c}(X;\mathscr {B})\} \subset \Gamma _{c}(X;\mathscr {B}/\mathscr {C})$ .

Next we claim that the quotient map $q:\mathscr {B}\to \mathscr {B}/\mathscr {C}$ is continuous. Suppose that $(a_{i})$ is a net in $\mathscr {B}$ such that $a_{i}\in B_{x_{i}}$ and $a_{i}\to a_{0}$ in $\mathscr {B}$ . Then $x_{i}\to x_{0}$ in X. Let $f\in \Gamma _{c}(X;\mathscr {B})$ be such that $f(x_{0})=a_{0}$ . Then

$$ \begin{align*} \|f(x_{i})-a_{i}\|\to 0. \end{align*} $$

Since q is norm reducing,

$$ \begin{align*} \|q(f)(x_{i})-q(a_{i})\|\to 0. \end{align*} $$

Since $q(f)\in \Gamma _{c}(X;\mathscr {B}/\mathscr {C})$ , [Reference Muhly and WilliamsMW08, Lemma A.3] implies that $q(a_{i})\to q(a_{0})$ . Thus, q is continuous as claimed.

To see that q is also open, let V be an open neighbourhood of $b\in \mathscr {B}$ . Then in view of Theorem 2.8, there is a $f\in \Gamma _{c}(X;\mathscr {B})$ , an open neighbourhood U of $p(b)$ and an $\epsilon>0$ such that

$$ \begin{align*} b\in W(f,U,\epsilon):=\{a\in B:{p(a)\in U \text{ and } \|a-f(p(a))\|<\epsilon}\}. \end{align*} $$

We need to verify that $q(V)$ is a neighbourhood of $q(b)$ . Since $q(f) \in \Gamma _{c}(X;\mathscr {B}/\mathscr {C})$ , it suffices to see that

$$ \begin{align*} q(W(f,U,\epsilon))=\{q(c):{p(c)\in U \text{ and } \|q(c)-q(f)(p(c))\|<\epsilon} \}. \end{align*} $$

Since the left-hand side is clearly a subset of the right-hand side, it suffices to consider $q(c)$ in the right-hand side. If $x=p(c)$ , then

$$ \begin{align*} \epsilon>\|q(c)-q(f)(x)\|=\|q_{x}(c-f(x))\|=\inf_{q_{x}(a)=c}\|a-f(x)\|. \end{align*} $$

Hence, there is an $a\in W(f,U,\epsilon )$ such that $q(a)=q(c)$ . This suffices to show that q is open.

Since q is continuous and open, the topology on $\mathscr {B}/\mathscr {C}$ is the quotient topology.

2.5 Fell bundles

Fell bundles are natural generalizations of Fell’s Banach $*$ -algebraic bundles from [Reference Fell and DoranFD88, Ch. VIII] and were introduced by Yamagami in [Reference YamagamiYam87]. The following definition comes from [Reference Muhly and WilliamsMW08, Definition 1.1].

Definition 2.18. Suppose that $p:\mathscr {B}\to G$ is a Banach bundle over a second countable locally compact Hausdorff groupoid G. Let

$$ \begin{align*} \mathscr{B}^{(2)}=\{(a,b)\in \mathscr{B}\times \mathscr{B}:(p(a),p(b))\in G^{(2)}\}. \end{align*} $$

We say that $p:\mathscr {B}\to G$ is a Fell bundle if there is a continuous, bilinear, associative multiplication map $(a,b)\mapsto ab$ from $\mathscr {B}^{(2)}$ to $\mathscr {B}$ and a continuous involution $b\mapsto b^{*}$ from $\mathscr {B}$ to $\mathscr {B}$ such that:

  1. (FB1) $p(ab)=p(a)p(b)$ ;

  2. (FB2) $p(a^{*})=p(a)^{-1}$ ;

  3. (FB3) $(ab)^{*}=b^{*}a^{*}$ ;

  4. (FB4) for each $u\in G^{(0)}$ , the fibre $B_{u}$ is a ${C}^*$ -algebra with respect to the inherited multiplication and involution on $B_{u}$ ; and

  5. (FB5) for each $g\in G$ , $B_{g}$ is an $B_{r(g)} \, {\text {--}}\, B_{s(g)}$ -imprimitivity bimodule when equipped with the inherited actions and inner products given by

    (2-4) $$ \begin{align} {{}_{{B}_{r(g)}}{\langle}a,b{\rangle}=ab^{*}\quad\text{and} \quad {\langle}a,b{\rangle}{}_{{B}_{s(g)}}=a^{*}b.} \end{align} $$

We say that the Fell bundle $p:\mathscr {B}\to G$ is separable if it is separable as a Banach bundle.

Remark 2.19 (Saturated).

It should be noted that our Fell bundles are saturated in that whenever $(g,h)\in G^{(2)}$ , then $B_{g}\cdot B_{h}:=\operatorname {span}\{ab:{a\in B_{g} \text { and } b\in B_{h}}\} $ is always dense in $B_{gh}$ [Reference Muhly and WilliamsMW08, Lemma 1.2]. This is a consequence of item (FB5). Some authors prefer to work with a weakened version of item (FB5) where the inner products in Equation (2-4) are not full.

Remark 2.20. If $p\colon \mathscr {B}\to G$ is a Fell bundle, then the restriction $\mathscr {B}|_{{G^{(0)}}}$ is a ${C}^*$ -bundle and $\Gamma _{0}(G^{(0)};\mathscr {B})$ is a ${C}^*$ -algebra called the associated ${C}^*$ -algebra to $\mathscr {B}$ . (The terminology is a bit challenging. If G has a Haar system, then one can also form the Fell-bundle ${C}^*$ -algebra ${C}^*(G;\mathscr {B})$ by viewing $\Gamma _{c}(G;\mathscr {B})$ as a $*$ -algebra and completing as in [Reference Muhly and WilliamsMW08].)

2.6 Equivalence of Fell bundles

Suppose that T is a left G-space. Then we say that a Fell bundle $p\colon \mathscr {B}\to G$ acts on (the left of) a Banach bundle $q:\mathscr {E}\to T$ if there is a continuous map $(b,e) \mapsto b\cdot e$ from $\mathscr {B}*\mathscr {E}:=\{(b,e)\in \mathscr {B}\times \mathscr {E}:s(b)=r(q(e))\}$ to $\mathscr {E}$ such that:

  1. (a) $q(b\cdot e)=p(b)\cdot q(e)$ ;

  2. (b) $a\cdot (b\cdot e)=(ab)\cdot e$ for appropriate $a,b\in B$ and $e\in \mathscr {E}$ ; and

  3. (c) $\|b\cdot e\|\le \|b\|\|e\|$ .

Right actions of a Fell bundle are defined similarly.

Let T be a $(G,H)$ equivalence with open moment maps $\rho \colon T\to G^{(0)}$ and ${\sigma \colon T\to H^{(0)}}$ as in [Reference WilliamsWil19, Definition 2.29]. It is shown in [Reference WilliamsWil19, Lemma 2.42] that there are open continuous maps $\tau _{G}\colon T*_{\sigma }T\to G$ and $\tau _{H}\colon T*_{\rho }T\to H$ such that $\tau _{G}(e,f)\cdot f=e$ and $e\cdot \tau _{H}(e,f)=f$ .

Definition 2.21 [Reference Muhly and WilliamsMW08, Definition 6.1].

Suppose that T is a $(G,H)$ -equivalence, and that $p_{\mathscr {B}}\cdot \mathscr {B}\to G$ and $p_{\mathscr {C}}\colon \mathscr {C}\to H$ are Fell bundles. Then a Banach bundle $q\colon \mathscr {E}\to T$ is a $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -equivalence if the following conditions hold.

  1. (E1) There is a left $\mathscr {B}$ -action and a right $\mathscr {C}$ -action on $\mathscr {E}$ such that $b{\kern-0.5pt}\cdot{\kern-0.5pt} (e{\kern-0.5pt}\cdot{\kern-0.5pt} c){\kern-0.5pt}={\kern-0.5pt}(b{\kern-0.5pt}\cdot{\kern-0.5pt} e){\kern-0.5pt}\cdot{\kern-0.5pt} c$ for composable $b\in \mathscr {B}$ , $e\in E$ and $c\in \mathscr {C}$ .

  2. (E2) There are continuous sesquilinear maps $(e,f)\mapsto {}_{\mathscr {B}}{\langle}e,f{\rangle}$ from $\mathscr {E}*_{\sigma }\mathscr {E}$ to $\mathscr {B}$ and $(e,f)\mapsto {\langle}e,f{\rangle}_{\mathscr {C}}$ from $\mathscr {E}*_{\rho }\mathscr {E}$ to $\mathscr {C}$ such that:

    1. (i) $p_{\mathscr {B}}({}_{\mathscr {B}}{\langle}e,f{\rangle}) =\tau _{G}(q(e),q(f))$ and $p_{\mathscr {C}}({\langle}e,f{\rangle}_{\mathscr {C}}) =\tau _{H}(q(e),q(f))$ ;

    2. (ii) ${}_{\mathscr {B}}{\langle}e,f{\rangle}^{*} ={}_{\mathscr {B}}{\langle}f,e{\rangle}$ and ${\langle}e,f{\rangle}_{\mathscr {B}}^{*} = {\langle}f,e{\rangle}_{\mathscr {B}}$ ;

    3. (iii) ${}_{\mathscr {B}}{\langle}b\cdot e,f{\rangle}=b{}_{\mathscr {B}}{\langle}e,f{\rangle}$ and ${\langle}e,f\cdot c{\rangle}_\mathscr {C}= {\langle}e,f>{\rangle}_\mathscr {C}$ ; and

    4. (iv) ${}_{\mathscr {B}}{\langle}e,f{\rangle}\cdot g=e\cdot {\langle}f,g{\rangle}_{\mathscr {C}}$ .

  3. (E3) With the actions and inner products coming from conditions (E1) and (E2), each $E_{t}$ is a $B_{\rho (t)} \, {\text {--}}\, C_{\sigma (t)}$ -imprimitivity bimodule.

2.7 Fell subbundles and ideals

Naturally, a subbundle of a Fell bundle is called a Fell subbundle if it is a Fell bundle in the inherited structure.

Definition 2.22. Let $p\colon \mathscr {B}\to G$ be a Fell bundle over a groupoid G. We call $\mathscr {C}\subset \mathscr {B}$ a Fell subbundle if $\mathscr {C}$ is a Banach subbundle such that $p|_{{\mathscr {C}}}\colon \mathscr {C}\to G$ is a Fell bundle with respect to the inherited operations. In particular, $\mathscr {C}$ must be closed under multiplication and involution.

We focus on Fell subbundles that are multiplicatively absorbing.

Definition 2.23. A Fell subbundle $\mathscr {J}$ of a Fell bundle $\mathscr {B}$ is called an ideal if ${ab\in \mathscr {J}}$ whenever $(a,b)\in \mathscr {B}^{(2)}$ and either $a\in \mathscr {J}$ or $b\in \mathscr {J}$ .

Example 2.24. Suppose that $\alpha \colon \mathcal {G}\to \operatorname {Aut}(A)$ is a ${C}^*$ -dynamical system for a group $\mathcal {G}$ . Let $\mathscr {B}=A\times \mathcal {G}$ be the associated Fell bundle over $\mathcal {G}$ : $(a,s)(b,r)=(a\alpha _{s}(b),sr)$ . Let I be an $\alpha $ -invariant ideal of A. Then $\mathscr {J}=I\times \mathcal {G}$ is an ideal in $\mathscr {B}$ .

Example 2.25. Suppose that $\mathscr {A}$ is a ${C}^*$ -bundle over X so that $A=\Gamma _{0}(X;\mathscr {A})$ is a ${C}^*$ -algebra. Let J be an ideal in A and for each $x\in X$ , let $J_{x}=\{a(x):a\in J\}$ so that $J_{x}$ is an ideal in $A_{x}$ . Let

$$ \begin{align*} \mathscr{J}=\coprod_{x\in X}J_{x}. \end{align*} $$

It follows from Proposition 2.13 that $\mathscr {J}$ is a Banach subbundle and in fact is obviously an ideal of the Fell bundle $\mathscr {A}$ . Clearly, $J\subset \Gamma _{0}(X;\mathscr {J})$ . Since J is an ideal in the $C_{0}(X)$ -algebra A, if $\phi \in C_{0}(X)$ and $b\in J$ , then $\phi \cdot b\in J$ . Now it follows from [Reference WilliamsWil07, Proposition C.24] that J is dense in $\Gamma _{0}(X;\mathscr {J})$ . Therefore, $J=\Gamma _{0}(X;\mathscr {J})$ .

Lemma 2.26. Suppose that $\mathscr {J}$ is an ideal in $\mathscr {B}$ . Then for each $g\in G$ , $J_{g}$ is a $J_{r(g)} \, {\text {--}}\, J_{s(g)}$ -imprimitivity bimodule. Furthermore,

(2-5) $$ \begin{align} J_{g}=B_{g}\cdot J_{s(g)} = J_{r(g)}\cdot B_{g}, \end{align} $$

where we are taking advantage of Remark 2.2. Furthermore,

$$ \begin{align*} J_{r(g)}=\overline{J_{g}B_{g}^{*}}=\overline{B_{g}J_{g}^{*}} =\overline{J_{g}J_{g}^{*}} \quad\text{and} \quad J_{s(g)} =\overline{J_{g}^{*}B_{g}}=\overline{B_{g}^{*}J_{g}} = \overline{J_{g}^{*}J_{g}}. \end{align*} $$

Proof. The first assertion is immediate since $\mathscr {J}$ is, by assumption, a Fell subbundle. The remaining statements follow from the Rieffel correspondence; see Theorem 2.1.

Definition 2.27. Let $p:\mathscr {B}\to G$ be a Fell bundle and $\mathscr {J}\subset \mathscr {B}$ a Banach subbundle. We call $\mathscr {J}$ a weak ideal of $\mathscr {B}$ if whenever $(a,b)\in \mathscr {B}^{(2)}$ , then $ab\in \mathscr {J}$ whenever either a or b is in $\mathscr {J}$ .

Remark 2.28. If I is an ideal in a ${C}^*$ -algebra, then the existence of approximate identities implies that I is $*$ -closed and hence a ${C}^*$ -subalgebra. A similar serendipity applies to weak ideals.

Proposition 2.29. If $p:\mathscr {B}\to G$ is a Fell bundle, then every weak ideal in $\mathscr {B}$ is an ideal.

Proof. Let $\mathscr {J}$ be a weak ideal in $\mathscr {B}$ . Since $J_{g}$ is closed with respect to the norm on $B_{g}$ , it is a closed $B(r(g))\, {\text {--}}\, B(s(g))$ -submodule of the $B(r(g))\, {\text {--}}\, B(s(g))$ -imprimitivity bimodule $B_{g}$ . In particular, $J_{u}$ is an ideal in the ${C}^*$ -algebra $B_{u}$ for all $u\in G^{(0)}$ . Then, applying the Rieffel correspondence, $J_{g}$ is a $K_{g}\,{\text {--}}\, I_{g}$ -imprimitivity bimodule where $I_{g}$ is the closed linear span of elements of the form $b^{*}a$ with $b\in B_{g}$ and $a\in J_{g}$ . Similarly, $K_{g}$ is the closed linear span of products $ab^{*}$ with $a\in J_{g}$ and $b\in B_{g}$ . Furthermore,

(2-6) $$ \begin{align} J_{g}=B_{g}\cdot I_{g}=K_{g}\cdot B_{g}. \end{align} $$

Note that $I_{g}$ is an ideal in $J_{s(g)}$ . Fix $c\in J_{s(g)}$ . If $b\in B_{g}$ , then $bc\in J_{g}$ by the weak ideal property. Since $B_{g}^{*}\cdot B_{g}$ is a dense ideal in $B_{s(g)}$ , we can find an approximate unit $(e_{i})$ in $B_{s(g)}$ where each $e_{i}=\sum _{k=1}^{n_{i}} b_{k}^{*}b_{k}$ with each $b_{k}\in B_{g}$ . However, then $e_{i}c$ is in $I_{g}$ and $e_{i}c\to c$ . Hence, $c\in I_{g}$ and $I_{g}=J_{s(g)}$ . A similar argument shows that $K_{g}=J_{r(g)}$ .

Since $J_{s(g)}$ is an ideal in the ${C}^*$ -algebra $B_{s(g)}$ , we have $J_{s(g)}^{*}=J_{s(g)}$ . Hence, using Equation (2-6),

$$ \begin{align*} J_{g}^{*}= (B_{g}\cdot J_{s(g)})^{*}= J_{s(g)}^{*}\cdot B_{g}^{*} = J_{r(g^{-1})} \cdot B_{g^{-1}} =J_{g^{-1}}. \end{align*} $$

In particular, $\mathscr {J}^{*}=\mathscr {J}$ and $\mathscr {J}$ is closed under taking adjoints. Since $\mathscr {J}$ is a weak ideal, it is closed under multiplication and we just showed it is also closed under the adjoint operation. Now we just have to observe that it is a Fell bundle. However, this follows from the above discussion and identification of $I_{g}$ with $J_{s(g)}$ and $K_{g}$ with $J_{r(g)}$ .

It is standard to think of a Fell bundle $p\colon \mathscr {B} \to G$ as a generalized groupoid crossed product of G acting on the associated ${C}^*$ -algebra $A:=\Gamma _{0}(G^{(0)};\mathscr {B})$ . As an example of this rubric, it is shown in [Reference Ionescu and WilliamsIW12, Proposition 2.2] that there is a natural action of G on $\operatorname {Prim} A$ given as follows. Note that $\operatorname {Prim} A$ is naturally fibred over $G^{(0)}$ . Since $B_{g}$ is a $B_{r(g)} \, {\text {--}}\, B_{s(g)}$ -imprimitivity bimodule, the Rieffel correspondence induces a homeomorphism $\phi _{g}:\operatorname {Prim}(B_{s(g)})\to \operatorname {Prim}(B_{r(g)})$ [Reference Raeburn and WilliamsRW98, Corollary 3.33]. Then the G-action is given by $g\cdot P_{s(g)}= \phi _{g}(P_{s(g)})$ . Naturally, an ideal I in A is called G-invariant if $\operatorname {hull}(I):=\{P\in \operatorname {Prim} A:P\supset I\}$ is G-invariant. If I is an ideal in A, then we let $I_{u}=q_{u}(I)$ , where $q_{u}:\Gamma _{0}(G^{(0)};\mathscr {B})\to B_{u}$ is the evaluation map.

Proposition 2.30 [Reference Ionescu and WilliamsIW12].

Suppose $p\colon \mathscr {B}\to G$ is a Fell bundle and that I is a G-invariant ideal in the associated ${C}^*$ -algebra $\Gamma _{0}(G^{(0)};\mathscr {B})$ . Then

$$ \begin{align*} \mathscr{B}_{I}:=\{b\in\mathscr{B}:b^{*}b\in I_{s(b)}\} \end{align*} $$

is an ideal in $\mathscr {B}$ . Conversely, if $\mathscr {J}$ is an ideal in $\mathscr {B}$ , then $I=\Gamma _{0}(G^{(0)};\mathscr {J})$ is a G-invariant ideal in $\Gamma _{0}(G^{(0)};\mathscr {B})$ and $\mathscr {J}=\mathscr {B}_{I}$ .

Proof. It follows from [Reference Ionescu and WilliamsIW12, proof of Lemma 3.1] that an ideal $I\subset \Gamma _{0}(G^{(0)};\mathscr {B})$ is G-invariant if and only if for all g, we have $\phi _{g}(I_{s(g)})=I_{r(g)}$ . By the Rieffel correspondence, the latter is equivalent to

(2-7) $$ \begin{align} \mathscr{B}_{g}\cdot I_{s(b)}=I_{r(b)}\cdot B_{g}\quad\text{for all } g\in G. \end{align} $$

Suppose that I is G-invariant. Then it follows from [Reference Ionescu and WilliamsIW12, Proposition 3.3] that $\mathscr {J}:=\mathscr {B}_{I}$ is a Fell subbundle such that Equation (2-7) holds. Suppose that $a\in B_{g}$ and $b\in B_{h}$ are composable. If $a\in \mathscr {J}$ , then

$$ \begin{align*} ab\in I_{r(g)}\cdot B_{g}B_{h} \subset I_{r(gh)} \cdot B_{gh}=J_{gh}. \end{align*} $$

Similarly, if $b\in \mathscr {J}$ , then

$$ \begin{align*} ab\in B_{g}B_{h}\cdot I_{s(h)} \subset B_{gh}\cdot I_{s(gh)} =J_{gh} \end{align*} $$

and $B_{I}$ is an ideal.

Now suppose that $\mathscr {J}$ is an ideal in $\mathscr {B}$ . Let

$$ \begin{align*} I=\Gamma_{0}(X;\mathscr{J}). \end{align*} $$

Then as in Example 2.25, we have $I_{u}=J_{u}$ . Now it follows from Lemma 2.26 and Equation (2-7) that I is G-invariant. Since $\mathscr {B}_{I}$ and $\mathscr {J}$ have the same fibres, clearly $\mathscr {B}_{I}=\mathscr {J}$ .

3 The Rieffel correspondence for Fell-bundle equivalences

In this section, we let $q_{\mathscr {E}}\colon \mathscr {E}\to T$ be an equivalence between $p_{\mathscr {B}}\colon \mathscr {B} \to H$ and $p_{\mathscr {C}}\colon \mathscr {C}\to K$ . In particular, T is an $(H,K)$ -equivalence and we let $\rho \colon T\to H^{(0)}$ and $\sigma \colon T\to K^{(0)}$ be the open moment maps.

We need the following observation from [Reference Muhly and WilliamsMW08, Lemma 6.2].

Lemma 3.1. As above, let $q_{\mathscr {E}}\colon \mathscr {E}\to T$ be a Fell-bundle equivalence between $p_{\mathscr {B}}\colon \mathscr {B}\to H$ and $p_{\mathscr {C}}\colon \mathscr {C}\to K$ . Then $(b,e) \mapsto b\cdot e$ induces an imprimitivity bimodule isomorphism of $B_{h} {\otimes }_{B_{\rho (t)}} E_{t}$ onto $E_{h\cdot t}$ . Similarly, $(e,c)\mapsto e\cdot c$ induces an isomorphism between $E_{t}{\otimes }_{C_{\sigma (t)}}C_{k}$ and $E_{t\cdot k}$ .

Corollary 3.2. Let $\mathscr {E}$ , $\mathscr {B}$ and $\mathscr {C}$ be as above. Let $\mathscr {J}$ be an ideal in $\mathscr {C}$ and ${\sigma (t)=r(k)}$ . Then $\overline {E_{t}\cdot J_{k}}=E_{t\cdot k}\cdot J_{s(k)}$ .

Proof. Lemma 3.1 implies that $\overline {E_{t}\cdot C_{k}}=E_{t\cdot k}$ . Therefore, by Lemma 2.26,

$$ \begin{align*} \overline{E_{t}\cdot J_{k}} = \overline{E_{t}\cdot C_{k}\cdot J_{s(k)}}=E_{t\cdot k}\cdot J_{s(k)}.\\[-35pt] \end{align*} $$

Definition 3.3. Let $q_{\mathscr {E}}\colon \mathscr {E}\to T$ be an equivalence between $p_{\mathscr {B}}\colon \mathscr {B}\to H$ and $p_{\mathscr {C}}\colon \mathscr {C}\to K$ . Then a Banach submodule $\mathscr {M}$ of $\mathscr {E}$ is called a Banach $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodule if $B_{h}\cdot M_{t}\subset M_{h\cdot t}$ whenever $s(h)=\rho (t)$ and $M_{t}\cdot C_{k} \subset M_{t\cdot k}$ whenever $\sigma (t)=r(k)$ . We say that $\mathscr {M}$ is full if $\overline {B_{h}\cdot M_{t}}=M_{h\cdot t}$ and $\overline {M_{t}\cdot C_{k}}=M_{t\cdot k}$ .

Proposition 3.4. Let $\mathscr {E}$ , $\mathscr {B}$ and $\mathscr {C}$ be as above. If $\mathscr {J}$ is an ideal in $\mathscr {C}$ , then

$$ \begin{align*} \mathscr{E}\cdot \mathscr{J}:= \bigcup_{\{(t,k):\sigma(t)=r(k)\}} E_{t}\cdot J_{k} \end{align*} $$

is a full Banach $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodule of $\mathscr {E}$ with $(\mathscr {E}\cdot \mathscr {J})_{t}= E_{t}\cdot J_{\sigma (t)}$ . Similarly, if $\mathscr {K}$ is an ideal in $\mathscr {B}$ , then $\mathscr {K}\cdot \mathscr {E}$ is a full Banach $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodule with $(\mathscr {K}\cdot \mathscr {E})_{t} =K_{\rho (t)} \cdot E_{t}$ .

Proof. We have

$$ \begin{align*} \mathscr{E}\cdot \mathscr{J} &= \bigcup_{\{(t,k):\sigma(t)=r(k)\}} E_{t}\cdot J_{k} \subset \bigcup_{\{(t,k):\sigma(t)=r(k)\}} \overline{E_{t}\cdot J_{k}} \end{align*} $$

which, by Corollary 3.2, is

$$ \begin{align*} &= \bigcup_{\{(t,k):\sigma(t)=r(k)\}} E_{t\cdot k}\cdot J_{s(k)} =\bigcup_{t\in T} E_{t}\cdot J_{\sigma(t)}\subset \mathscr{E}\cdot \mathscr{J}. \end{align*} $$

Therefore, $\mathscr {E}\cdot \mathscr {J}=\bigcup _{t\in T} E_{t}\cdot J_{\sigma (t)}$ and $(\mathscr {E}\cdot \mathscr {J})_{t}=E_{t} \cdot J_{\sigma (t)}$ as claimed.

In particular, $\mathscr {E}\cdot \mathscr {J}$ is a bundle over T with closed fibres $E_{t}\cdot J_{\sigma (t)}$ . However, if $f\in \Gamma _{c}(T;\mathscr {E})$ and $\phi \in \Gamma _{c}(K^{(0)}; \mathscr {J})$ , then $\phi \cdot f$ given by $\phi \cdot f(t)= f(t)\cdot \phi (\sigma (t))$ is a section in $\Gamma _{c}(T;\mathscr {E}\cdot \mathscr {J})$ . Now it follows from Proposition 2.13 that $\mathscr {E}\cdot \mathscr {J}$ is a Banach subbundle of $\mathscr {E}$ .

We still need to see that $\mathscr {E}\cdot \mathscr {J}$ is a full $\mathscr {B}\, {\text {--}}\, \mathscr {J}$ -submodule. However, if $s(h)=\rho (t)$ , then

$$ \begin{align*} \overline{B_{h}\cdot (\mathscr{E}\cdot \mathscr{J})_{t}} &= \overline{B_{h}\cdot (E_{t}\cdot J_{\sigma(t)})} \\ &= \overline{(B_{h}\cdot E_{t})\cdot J_{\sigma(t)}} \\ &= E_{h\cdot t}\cdot J_{\sigma(t)}= (\mathscr{E}\cdot \mathscr{J})_{h\cdot t}. \end{align*} $$

However, if $\sigma (t)=r(k)$ , then

$$ \begin{align*} \overline{(\mathscr{E}\cdot \mathscr{J})_{t}\cdot C_{k}} &= \overline{E_{t}\cdot (J_{r(k)}\cdot C_{k})} \end{align*} $$

which, by Lemma 2.26, is

$$ \begin{align*} &= \overline{E_{t}\cdot (C_{k} \cdot J_{s(k)})} \\ &= E_{t\cdot k}\cdot J_{s(k)} = (\mathscr{E}\cdot\mathscr{J})_{t\cdot k}. \end{align*} $$

Thus, $\mathscr {E}\cdot \mathscr {J}$ is a full Banach $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodule as claimed.

The corresponding statements for $\mathscr {K}\cdot \mathscr {E}$ are proved similarly.

Now suppose that $\mathscr {M}$ is a full Banach $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodule of $\mathscr {E}$ . Then $M_{t}$ is a closed $B_{\rho (t)}\, {\text {--}}\, C_{\sigma (t)}$ -submodule of $E_{t}$ . By the Rieffel correspondence, $M_{t}$ is a $L_{t}\, {\text {--}}\, R_{t}$ -imprimitivity bimodule for the ideals $L_{t}=\overline {{}_{\mathscr {B}}{\langle}M_{t},E_{t}{\rangle}}$ in $B_{\rho (t)}$ and ${R_{t}=\overline { {\langle}E_{t},M_{t}{\rangle}_{\mathscr {C}}}}$ in $C_{\sigma (t)}$ . Furthermore, $L_{t}\cdot E_{t}=M_{t}=E_{t}\cdot R_{t}$ .

Lemma 3.5. In the current set-up, the ideal $R_{t}$ depends only on $\sigma (t)$ and the ideal $L_{t}$ depends only on $\rho (t)$ . Hereafter, we denote them by $R_{\sigma (t)}$ and $L_{\rho (t)}$ , respectively.

Proof. If $\sigma (t')=\sigma (t)$ , then $t'=h\cdot t$ . Since $E_{t}$ is a $B_{\rho (t)} \, {\text {--}}\, C_{\sigma (t)}$ -imprimitivity bimodule, and since $R_{t}$ and $R_{h\cdot t}$ are both ideals in $C_{\sigma (t)}$ , to see that $R_{t}=R_{h\cdot t}$ , it suffices, by the Rieffel correspondence, to see that $E_{t}\cdot R_{t}=E_{t}\cdot R_{h\cdot t}$ . Since $\mathscr {M}$ is full,

$$ \begin{align*} E_{t}\cdot R_{h\cdot t} = E_{t} \cdot \overline{{\langle}E_{h\cdot t}, M_{h\cdot t}{\rangle}_{\mathscr{C}}} = \overline{E_{t}\cdot {\langle}\overline{B_{h}\cdot E_{t}},\overline{B_{h}\cdot M_{t}}{\rangle}_{\mathscr{C}}}. \end{align*} $$

Clearly,

(3-1) $$ \begin{align} \overline{E_{t}\cdot {\langle}B_{h}\cdot E_{t},B_{h}\cdot M_{t}{\rangle}_{\mathscr{C}}} \subset \overline{E_{t}\cdot {\langle}\overline{B_{h}\cdot E_{t}},\overline{B_{h}\cdot M_{t}}{\rangle}_{\mathscr{C}}}. \end{align} $$

However, consider

$$ \begin{align*} e\cdot {\langle}f,g{\rangle}_{\mathscr{C}}\end{align*} $$

with $e\in E_{t}$ , $f\in \overline {B_{h}\cdot E_{t}}$ and $g\in \overline {B_{h}\cdot M_{t}}$ . Then there are sequences $(f_{i})\subset B_{h}\cdot E_{t}$ and $(g_{i})\subset B_{h}\cdot M_{t}$ such that $f_{i}\to f$ and $g_{i}\to g$ in norm in $E_{h\cdot t}$ and hence in $\mathscr {E}$ . Therefore, $e\cdot {\langle}f_{i},g_{i}{\rangle}_{\mathscr{C}}\to e\cdot {\langle}f,g{\rangle}_{\mathscr{C}}$ in $\mathscr {E}$ . Since the convergence takes place in $E_{t}$ , the convergence is in norm. It follows that

$$ \begin{align*} E_{t}\cdot {\langle}\overline{B_{h}\cdot E_{t}},\overline{B_{h}\cdot M_{t}}{\rangle}_{\mathscr{C}} \subset \overline{E_{t}\cdot {\langle}B_{h}\cdot E_{t},B_{h}\cdot M_{t}{\rangle}_{\mathscr{C}}}. \end{align*} $$

Therefore, we have equality in Equation (3-1), and

$$ \begin{align*} E_{t}\cdot R_{h\cdot t} &= \overline{E_{t}\cdot {\langle}B_{h}\cdot E_{t},B_{h}\cdot M_{t}{\rangle}_{\mathscr{C}}} \end{align*} $$

which, using condition (E2)(iv), is

$$ \begin{align*} & \kern31pt = \overline{{}_{\mathscr {B}}{\langle}E_{t},B_{h}\cdot E_{t}{\rangle}\cdot B_{h} \cdot M_{t}} \end{align*} $$

which, using conditions (E2)(ii) and (E2)(iii), is

$$ \begin{align*} & \kern24pt = \overline{{}_{\mathscr {B}}{\langle}E_{t},E_{t}{\rangle}\cdot B_{h}^{*}B_{h}\cdot M_{t}} \end{align*} $$

which, since $\overline {B_{h}^{*}B_{h}}=B_{s(h)}=B_{\rho (t)}=\overline { {}_{\mathscr {B}}{\langle}E_{t},E_{t}{\rangle}}$ and since $\overline {B_{s(h)}\cdot M_{t}}=M_{t}$ , is

$$ \begin{align*} & \kern-9pt =M_{t}=E_{t}\cdot R_{t}. \end{align*} $$

Thus, $R_{t}=R_{h\cdot t}$ as required.

The proof for $L_{t}$ is similar.

If $\sigma (t)=r(k)$ , then ${\langle}M_{t},E_{t\cdot k}{\rangle}_{\mathscr{C}}$ is the subspace of $C_{k}$ spanned by inner products of elements in $M_{t}$ with elements of $E_{t\cdot k}$ . Then given $k\in K$ , we let $\bigoplus _{\sigma (t)=r(k)} {\langle}M_{t},E_{t\cdot k}{\rangle}_{\mathscr{C}}$ denote the subspace of $C_{k}$ generated by the summands. Then

(3-2) $$ \begin{align} {\langle}\mathscr{M},\mathscr{E}{\rangle}_{\mathscr{C}} := \coprod_{k\in K} \bigoplus_{\sigma(t)=r(k)} {\langle}M_{t},E_{t\cdot k}{\rangle}_{\mathscr{C}}= \coprod_{k\in K} \bigoplus_{\sigma(t)=r(k)} {\langle}M_{t},\overline{E_{t}\cdot C_{k}}{\rangle}_{\mathscr{C}}\end{align} $$

where the closure takes place in the Banach space $E_{t\cdot k}$ .

Lemma 3.6. In the setting above, both ${\langle}M_{t},E_{t}\cdot C_{k}{\rangle}_{\mathscr{C}}$ and ${\langle}M_{t},\overline {E_{t}\cdot C_{k}}{\rangle}_{\mathscr{C}}$ are norm dense in $R_{r(k)}\cdot C_{k}$ , where we have invoked Lemma 3.5 to realize that $R_{t}$ depends only on $r(k)=\rho (t)$ .

Proof. Clearly,

$$ \begin{align*} {\langle}M_{t},E_{t}\cdot C_{k}{\rangle}_{\mathscr{C}}= {\langle}M_{t},E_{t}{\rangle}_{\mathscr{C}}\cdot C_{k}\subset R_{r(k)}\cdot C_{k}. \end{align*} $$

Moreover,

$$ \begin{align*} \overline{{\langle}M_{t},E_{t}{\rangle}_{\mathscr{C}}\cdot C_{k}} =\overline{R_{r(k)}\cdot C_{k}} =R_{r(k)}\cdot C_{k}. \end{align*} $$

This implies the first assertion.

For the second, we just need to see that ${\langle}M_{t},\overline {E_{t}\cdot C_{k}}{\rangle}_{\mathscr{C}}\subset \overline {{\langle}M_{t},E_{t}{\rangle}_{\mathscr{C}}\cdot C_{k}}$ . To this end, suppose that $(c_{i})$ is a sequence in $E_{t}\cdot C_{k}$ converging to c in $E_{t\cdot k}$ . Then for any $m\in M_{t}$ , the sequence $({\langle}m,c_{i}{\rangle}_{\mathscr{C}})$ converges to ${\langle}m,c{\rangle}_{\mathscr{C}}$ in $\mathscr {C}$ since ${\langle}\cdot ,\cdot{\rangle}_{\mathscr{C}}$ is continuous on $\mathscr {E}*_{\rho }\mathscr {E}$ . Since the convergence takes place in $C_{k}$ , the convergence is in norm by Lemma 2.6. Since each ${\langle}m,c_{i}{\rangle}_{\mathscr{C}}\in {\langle}M_{t},E_{t}{\rangle}_{\mathscr{C}}\cdot C_{k}$ , the result follows.

Using Lemma 3.6 and Equation (3-2),

(3-3) $$ \begin{align} {\langle}\mathscr{M},\mathscr{E}{\rangle}_{\mathscr{C}}\subset \coprod_{k\in K} R_{r(k)}\cdot C_{k}. \end{align} $$

Moreover, ${\langle}\mathscr {M},\mathscr {E}{\rangle}_{\mathscr{C}}\cap C_{k}$ is norm dense in $R_{r(k)}\cdot C_{k}$ .

Lemma 3.7. In the current set-up, $\mathscr {J}_{\mathscr {M}}:=\coprod _{k\in K} R_{r(k)}\cdot C_{k}$ is a Banach subbundle of  $\mathscr {C}$ .

Proof. As in the proof of Proposition 2.13, the issue is to see that $p\colon \mathscr {J}_{\mathscr {M}}\to K$ is open. Let U be a nonempty (relatively) open set in $\mathscr {J}_{\mathscr {M}}$ . Given $k\in p(U)$ , it suffices to show that given a sequence $(k_{i})$ converging to k in K, $(k_{i})$ is eventually in U. If this fails, then after passing to a subsequence and relabelling, we can assume $k_{i}\notin U$ for all i.

Since $(\mathscr {J}_{\mathscr {M}})_{k}$ is $R_{r(k)}\cdot C_{k}$ , we can find $c\in C_{r}$ and $c'\in R_{r(k)}$ such that $c'c\in U$ and $p(c'c)=k$ . Let $t\in T$ be such that $\sigma (t)=r(k)$ . Since $R_{r(k)}$ is the closure of ${\langle}E_{t},M_{t}{\rangle}_{\mathscr{C}}$ in $C_{r(k)}$ , there is a sequence $(c_{i}')\subset {\langle}E_{t},M_{t}{\rangle}_{\mathscr{C}}$ converging to $c'$ in norm. However, then $c_{i}'c\to c'c$ in norm. Then $(c_{i}'c)$ is eventually in U. Therefore, we may as well assume that $c' = \sum _{j=1}^{n} {\langle}e_{j},m_{j}{\rangle}_{\mathscr{C}}$ with each $e_{j}\in E_{t}$ and each $m_{j}\in M_{t}$ .

Since Banach bundles have enough sections, we can find $f\in \Gamma (K;\mathscr {C})$ , $g_{j}\in \Gamma (T;\mathscr {E})$ and $h_{j}\in \Gamma (T;\mathscr {M})$ such that $f(k)=c$ , $g_{j}(t)=e_{j}$ and $h_{j}(t)=m_{j}$ .

Since $r(k_{i})\to r(k)=\sigma (t)$ and since $\sigma $ is open, we can pass to a subsequence, relabel and assume that there are $t_{i}\in T$ such that $t_{i}\to t$ and $\sigma (t_{i})=r(k_{i})$ . Then

$$ \begin{align*} d(t_{i}):=\sum_{j=1}^{n}{\langle}g(t_{i}),h(t_{i}){\rangle}_{\mathscr{C}}\in R_{\sigma(t_{i})} \quad\text{and} \quad d(t_{i})f(k_{i}) \in R_{\sigma(t_{i})}\cdot C_{k_{i}} = (\mathscr{J}_{\mathscr{M}})_{k_{i}}. \end{align*} $$

Furthermore, $d(t_{i})f(k_{i})\to c'c$ in $\mathscr {J}_{\mathscr {M}}$ . Hence, $(d(t_{i})f(k_{i}))$ is eventually in U. Since p is continuous, $p(d(t_{i})f(k_{i}))=k_{i}$ is eventually in $p(U)$ , which contradicts our assumptions on $(k_{i})$ and completes the proof.

Proposition 3.8. In the current set-up, $\mathscr {J}_{\mathscr {M}}:=\coprod _{k\in K} R_{r(k)}\cdot C_{k}$ is an ideal in $\mathscr {C}$ .

Proof. In view of Proposition 2.29, we just have to show that $\mathscr {J}_{\mathscr {M}}$ is a weak ideal. For convenience, let $J_{k}=R_{r(k)}\cdot C_{k}$ .

Suppose that $(c,m)\in C_{l}\times J_{k}$ with $s(l)=r(k)$ . By Lemma 3.6, there is a sequence $(m_{i})$ in ${\langle}M_{t},E_{t}\cdot C_{k}{\rangle}_{\mathscr{C}}$ converging to m in norm (and hence in $\mathscr {C}$ ). However, then $c m_{i}\to c m$ in $\mathscr {C}\cap C_{lk}$ in $\mathscr {C}$ and hence in norm. Since $M_{t}c^{*}\subset M_{t\cdot l^{-1}}$ , $c m_{i}\in {\langle}M_{t\cdot l^{-1}},E_{t\cdot k}{\rangle}_{\mathscr{C}}\subset J_{lk}$ (using Equations (3-2) and (3-3)). Since $J_{kl}$ is closed in norm in $C_{kl}$ , it follows that $cm\in \mathscr {J}_{\mathscr {M}}$ .

A similar argument shows that $mc\in \mathscr {J}_{\mathscr {M}}$ if $(m,c)\in J_{k}\times C_{l}$ with $s(k)=r(l)$ . Hence, $\mathscr {J}_{\mathscr {M}}$ is a weak ideal as claimed.

Proposition 3.9. We retain the current set-up. Let $\mathscr {J}$ be an ideal in $\mathscr {C}$ . Then ${\mathscr {M}=\mathscr {E}\cdot \mathscr {J}}$ is a Banach $\mathscr {B}\, {\text {--}}\,\mathscr {J}$ -submodule of $\mathscr {E}$ and $\mathscr {J}_{\mathscr {M}}=\mathscr {J}$ . Hence, ${\mathscr {J}\mapsto \mathscr {E}\cdot \mathscr {J}}$ is a lattice isomorphism of the collection of ideals in $\mathscr {C}$ to the collection of closed $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodules of $\mathscr {E}$ .

Proof. By Proposition 3.4, $\mathscr {M}:=\mathscr {E}\cdot \mathscr {J}$ is a full Banach $\mathscr {B}\, {\text {--}}\,\mathscr {J}$ -submodule and $M_{t}=(\mathscr {E}\cdot \mathscr {J})_{t}=E_{t}\cdot J_{\sigma (t)}$ . Thus, applying the Rieffel correspondence to $E_{t}$ ,

$$ \begin{align*} R_{\sigma(t)}=\overline{\text{span}}{{\langle}E_{t},M_{t}{\rangle}_{\mathscr{C}}}= \overline{\text{span}}{{\langle}E_{t},E_{t}\cdot J_{\sigma(t)}{\rangle}_{\mathscr{C}}} =J_{\sigma(t)}. \end{align*} $$

Thus, in Equation (3-3), $R_{r(k)}=J_{r(k)}$ . Therefore,

$$ \begin{align*} \mathscr{J}_{\mathscr{M}}=\coprod_{k\in K} R_{r(k)}\cdot C_{k}=\coprod_{k\in K} J_{r(k)}\cdot C_{k}=\mathscr{J}, \end{align*} $$

where the last equality comes from Equation (2-5) of Lemma 2.26.

Now suppose that $\mathscr {M}$ is a full closed $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodule. Then Proposition 3.8 implies that $\mathscr {J}_{\mathscr {M}}$ is an ideal in $\mathscr {C}$ with $(\mathscr {J}_{\mathscr {M}})_{k}=R_{r(k)}\cdot C_{k}$ . Let $\mathscr {M}':=\mathscr {E}\cdot \mathscr {J}_{\mathscr {M}}$ . In particular, if $u\in K^{(0)}$ , $(\mathscr {J}_{\mathscr {M}})_{u}=R_{u}\cdot C_{u}=R_{u}$ . Thus, $(\mathscr {J}_{\mathscr {M}})_{k}=R_{r(k)}\cdot C_{k} = C_{k}\cdot R_{s(k)}$ by Lemma 2.26. Then $\mathscr {M}^{\prime }_{t}=(\mathscr {E}\cdot \mathscr {J}_{\mathscr {M}})_{t} =E_{t}\cdot R_{\sigma (t)}$ . This means

$$ \begin{align*} R^{\prime}_{\sigma(t)} &=\overline{{\langle}E_{t},\mathscr{M}^{\prime}_{t}{\rangle}_{\mathscr{C}}} \\ &= \overline{{\langle}E_{t},E_{t}{\rangle}_{\mathscr{C}}\cdot R_{\sigma(t)}} \\ &= \overline{{\langle}E_{t},E_{t}{\rangle}_{\mathscr{C}}\cdot R_{\sigma(t)}} \\ &= \overline{C_{\sigma(t)}\cdot R_{\sigma(t)}}=R_{\sigma(t)}. \end{align*} $$

Therefore, $\ M^{\prime }_{t}=E_{t}\cdot R_{\sigma (t)}=E_{t}\cdot R^{\prime }_{\sigma (t)} =M_{t}$ . Hence, $\mathscr {E}\cdot \mathscr {J}_{\mathscr {M}}=\mathscr {M}$ .

By symmetry, we have a lattice isomorphism $\mathscr {K}\mapsto \mathscr {K}\cdot \mathscr {E}$ between the ideals in $\mathscr {B}$ and the full Banach $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodules of $\mathscr {E}$ . Then we have the following Rieffel correspondence for Fell bundle equivalence.

Theorem 3.10. Suppose that $q_{\mathscr {E}}\colon \mathscr {E}\to T$ is a Fell-bundle equivalence between $p_{\mathscr {B}}\colon \mathscr {B}\to H$ and $p_{\mathscr {C}}\colon \mathscr {C}\to K$ . Then there are lattice isomorphisms among the ideals of $\mathscr {B}$ , the full Banach $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodules of $\mathscr {E}$ and the ideals of $\mathscr {C}$ . The correspondences are given as follows.

  1. (a) If $\mathscr {J}$ is an ideal in $\mathscr {C}$ , then the corresponding full Banach $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodule is

    $$ \begin{align*} \mathscr{E}\cdot\mathscr{J}=\bigcup_{t\in T}E_{t}\cdot J_{\sigma(t)}. \end{align*} $$
  2. (b) If $\mathscr {M}$ is a full Banach $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodule, then for each $t\in T$ , $M_{t}$ is a $L_{\rho (t)}\, {\text {--}}\, R_{\sigma (t)}$ -imprimitivity bimodule for ideals $L_{\rho (t)} = \overline {{}_{\mathscr {B}}{\langle}M_{t},E_{t}{\rangle}}$ in $B_{\rho (t)}$ and $R_{\sigma (t)}= \overline {{\langle}E_{t},M_{t}{\rangle}_{\mathscr{C}}}$ in $\mathscr {C}_{\sigma (t)}$ . Then the corresponding ideals $\mathscr {J}_{\mathscr {M}}$ in $\mathscr {C}$ and $\mathscr {K}^{\mathscr {M}}$ in $\mathscr {B}$ are given by

    $$ \begin{align*} \mathscr{J}_{\mathscr{M}} &=\bigcup_{k\in K} R_{r(k)}\cdot C_{k} \quad\text{and} \quad \mathscr{K}^{\mathscr{M}}= \bigcup_{h\in H} L_{r(h)} \cdot B_{h}. \end{align*} $$
  3. (c) If $\mathscr {K}$ is an ideal in $\mathscr {B}$ , then the corresponding $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodule is

    $$ \begin{align*} \mathscr{K}\cdot \mathscr{E}= \bigcup_{t\in T} K_{\rho(t)}\cdot E_{t}. \end{align*} $$

The following is a generalization of [Reference Raeburn and WilliamsRW98, Proposition 3.24].

Corollary 3.11. Suppose that $q_{\mathscr {E}}\colon \mathscr {E}\to T$ is a Fell-bundle equivalence between $p_{\mathscr {B}}\colon \mathscr {B}\to H$ and $p_{\mathscr {C}}\colon \mathscr {C}\to K$ . If $\mathscr {J}$ is an ideal in $\mathscr {C}$ , then the corresponding ideal $\mathscr {K}$ in $\mathscr {B}$ is $\coprod _{h\in H} H_{r(h)}\cdot B_{h}$ , where

(3.4) $$ \begin{align} H_{\rho(t)}=\overline{{}_{\mathscr {B}}{\langle}E_{t}\cdot J_{\sigma(t)},E_{t}{\rangle}}. \end{align} $$

Proof. If $\mathscr {J}$ is an ideal in $\mathscr {C}$ , then according to Theorem 3.10(a), the corresponding full Banach $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodule is $\mathscr {M}=\mathscr {E}\cdot \mathscr {J}$ . Then using Theorem 3.10(b), the corresponding ideal $\mathscr {K}$ in $\mathscr {B}$ is $\bigcup _{h\in H} L_{r(h)}\cdot B_{h}$ , where $L_{r(h)}$ is given by the right-hand side of Equation (3-4) for any $t\in T$ such that $\rho (t)=r(h)$ . This gives the result.

4 Extending the Rieffel correspondence

Now we want to state and prove the analogues for Fell bundles of parts (c) and (d) of Theorem 2.1.

Proposition 4.1. Suppose that $q_{\mathscr {E}}\colon \mathscr {E}\to T$ is a Fell-bundle equivalence between $p_{\mathscr {B}}\colon \mathscr {B}\to H$ and $p_{\mathscr {C}}\colon \mathscr {C}\to K$ . Suppose that $\mathscr {J}$ is an ideal in $\mathscr {C}$ , and that $\mathscr {M}$ and $\mathscr {K}$ are the corresponding full Banach $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodule in $\mathscr {E}$ and ideal in $\mathscr {B}$ . Then $\mathscr {M}$ is a Fell-bundle equivalence between $\mathscr {K}$ and $\mathscr {J}$ .

Proof. Since $\mathscr {M}$ is a $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodule of $\mathscr {E}$ , we clearly have a left $\mathscr {K}$ -action and a right $\mathscr {J}$ -action satisfying condition (E1).

For condition (E2), we claim that it suffices to let ${}_{\mathscr {K}}{\langle}e,f{\rangle}={}_{\mathscr {B}}{\langle}e,f{\rangle}$ and ${\langle}e,f{\rangle}_{\mathscr{J}}= {\langle}e,f{\rangle}_{\mathscr{C}}$ . To see this, note that if $(e,f)\in \mathscr {M}*_{\sigma }\mathscr {M}$ , then we can assume $(e,f) \in M_{t}\times M_{h^{-1}\cdot t}$ for some $h\in H$ and $t\in T$ . However, $M_{t}=K_{\rho (t)}\cdot E_{t}$ . Additionally, we have ${}_{\mathscr {B}}{\langle}K_{\rho (t)}\cdot E_{t},M_{h^{-1}\cdot t}{\rangle}=K_{\rho (t)}\cdot {}_{\mathscr {B}}{\langle}E_{t},M_{h^{-1}\cdot t}{\rangle}\subset K_{\rho (t)}\cdot B_{h}=K_{h}$ . Therefore, ${}_{\mathscr {K}}{\langle}\cdot ,\cdot{\rangle}$ is $\mathscr {K}$ -valued. Similarly, ${\langle}\cdot ,\cdot{\rangle}_{\mathscr{J}}$ is $\mathscr {J}$ -valued. The rest of condition (E2) follows from the given properties of ${}_{\mathscr {B}}{\langle}\cdot ,\cdot{\rangle}$ and ${\langle}\cdot ,\cdot{\rangle}_{\mathscr{C}}$ .

For condition (E3), the fact that $M_{t}$ is a $K_{\rho (t)}\, {\text {--}}\, J_{\sigma (t)}$ -imprimitivity bimodule follows from the Rieffel correspondence (part (c) of Theorem 2.1).

Proposition 4.2. Let $q_{\mathscr {E}}\colon \mathscr {E}\to T$ be an equivalence between $p_{\mathscr {B}}\colon \mathscr {B}\to H$ and $p_{\mathscr {C}}\colon \mathscr {C}\to K$ . Suppose that $\mathscr {J}$ is an ideal in $\mathscr {C}$ and that $\mathscr {M}$ and $\mathscr {K}$ are the corresponding full Banach $\mathscr {B}\, {\text {--}}\, \mathscr {C}$ -submodule in $\mathscr {E}$ and ideal in $\mathscr {B}$ , respectively. Then the quotient Banach bundle $\mathscr {E}/\mathscr {M}$ is an equivalence between $\mathscr {B}/\mathscr {K}$ and $\mathscr {C}/\mathscr {J}$ .

Proof. We let $q^{\mathscr {K}}\colon \mathscr {B}\to \mathscr {B}/\mathscr {K}$ and $q^{\mathscr {J}}\colon \mathscr {C}\to \mathscr {C}/\mathscr {J}$ be the quotient maps. Then the given left and right actions of $\mathscr {B}$ and $\mathscr {C}$ on $\mathscr {E}$ induce left and right actions of $\mathscr {B}/\mathscr {K}$ and $\mathscr {C}/\mathscr {J}$ on $\mathscr {E}/\mathscr {M}$ in the expected way:

$$ \begin{align*} q^{\mathscr{K}}(b)\cdot q(e)=q(b\cdot e)\quad\text{and} \quad q(e)\cdot q^{\mathscr{J}}(c)=q(e\cdot c) \end{align*} $$

assuming that $b\cdot e$ and $e\cdot c$ are defined.

To see that these actions are continuous, we use the fact that q, $q^{\mathscr {K}}$ and $q^{\mathscr {J}}$ are open as well as continuous (Proposition 2.16). Suppose that $q(e_{i})\to q(e)$ while $q^{\mathscr {K}}(b_{i})\to q^{\mathscr {K}}(b)$ with $b_{i}\cdot e_{i}$ defined for all i. We need to verify that $q(b_{i}\cdot e_{i})\to q(b\cdot e)$ . For this, it suffices to see that every subnet has a subnet converging to $q(b\cdot e)$ . However, after passing to a subnet and relabelling, the openness of the quotient maps means we can pass to another subnet and assume that $e_{i}'\to e$ and $b_{i}'\to b$ with $q(e_{i}')=q(e)$ and $q^{\mathscr {K}}(b_{i}')=q^{\mathscr {K}}(b_{i})$ . Then the continuity of the quotient maps implies that $q(b_{i}\cdot e_{i}) = q(b_{i}'\cdot e_{i}')\to q(b\cdot e)$ as required.

We also have

$$ \begin{align*} \|q(b\cdot e)\| &\le \inf\{\|b'\cdot e'\|:{q^{\mathscr{K}}(b')=q^{\mathscr{K}}(b) \text{ and } q(e')=q(e)}\} \\ &\le \inf\{\|b'\|\|e'\|:{q^{\mathscr{K}}(b')=q^{\mathscr{K}}(b) \text{ and } q(e')=q(e)}\} \\ &= \|q^{\mathscr{K}}(b)\|\|q(e)\|. \end{align*} $$

Therefore, $\mathscr {B}/\mathscr {K}$ acts on the left of $\mathscr {E}/\mathscr {M}$ . The argument for the right action is similar.

Now we need to verify the axioms in Definition 2.21. Axiom (E1) is immediate since $\mathscr {E}$ is an equivalence. For Axiom (E2), we define

$$ \begin{align*} {\langle}q(e),q(f){\rangle}_{\mathscr{C}/\mathscr{J}}:= q^{\mathscr{J}}({\langle}e,f{\rangle}_{\mathscr{C}}) \quad \text{and}\quad {}_{\mathscr{B}/\mathscr{K}}{\langle}q(e),q(f)\rangle= q^{\mathscr{K}}(_{\mathscr{B}}{\langle}e,f{\rangle}). \end{align*} $$

It is not hard to check that these pairings are well defined. Then properties (i), (ii), (iii) and (iv) follow from the corresponding properties for $\mathscr {E}$ and the observation that the quotient maps are multiplicative. The continuity follows using the continuity and openness of the quotient maps as we did above for the left and right actions.

Of course, Axiom (E3) is clear.

5 At the ${C}^*$ -level

Since the previous exposition did not require it, we have purposely avoided discussing the Fell-bundle ${C}^*$ -algebras that are associated to a Fell bundle. However, there is an obvious question: how is our Rieffel correspondence for ideals in equivalent Fell bundles related to the standard Rieffel correspondence for ideals in Morita equivalent ${C}^*$ -algebras? In order that there be ${C}^*$ -algebras, we now have to assume our groupoids have Haar systems. To apply the equivalence theorem, that is, [Reference Muhly and WilliamsMW08, Theorem 6.4], we also need our Fell bundles to be separable.

We return to the set-up in Section 3: we let $q_{\mathscr {E}}\colon \mathscr {E}\to T$ be an equivalence between the separable Fell bundles $p_{\mathscr {B}}\colon \mathscr {B} \to H$ and $p_{\mathscr {C}}\colon \mathscr {C}\to K$ . In particular, T is a $(H,K)$ -equivalence (although it is not required, we note that T must be second countable since H and K are [Reference WilliamsWil19, Proposition 2.53]) and we let $\rho \colon T\to H^{(0)}$ and $\sigma \colon T\to K^{(0)}$ be the open moment maps.

Then the equivalence theorem implies that ${C}^*(H^{(0)};\mathscr {B})$ and ${C}^*(K^{(0)};\mathscr {C})$ are Morita equivalent via an imprimitivity bimodule $\mathsf {X}$ , which is the completion of $\mathsf {X}_{0}:=\Gamma _{c}(T,\mathscr {E})$ with the actions and inner products given in [Reference Muhly and WilliamsMW08, Theorem 6.4]. Then we can let

$$ \begin{align*} \mathop{\mathsf{X}\mathord{\mathop{\text{--}}}}\!\operatorname{Ind}\nolimits:\mathcal{I}({C}^*(K^{(0)};\mathscr{C}))\to \mathcal{I}({C}^*(H^{(0)};\mathscr{B})) \end{align*} $$

be the classical Rieffel lattice isomorphism.

If $\mathscr {J}$ is an ideal in $\mathscr {C}$ , then as shown in [Reference Ionescu and WilliamsIW12, Lemma 3.5], the identity map $\iota $ induces an isomorphism of ${C}^*(K^{(0)};\mathscr {J})$ onto the ideal $\operatorname {Ex}(\mathscr {J})$ , which is the closure of $\iota (\Gamma _{c}(K^{(0)};\mathscr {C}))$ in ${C}^*(K^{(0)};\mathscr {C})$ .

Let $\mathscr {J}$ be an ideal in $\mathscr {C}$ and $\mathscr {K}$ the corresponding ideal in $\mathscr {B}$ as in Theorem 3.10. The goal here is to establish that the two Rieffel correspondences are compatible in that

(5-1) $$ \begin{align} \mathop{\mathsf{X}\mathord{\mathop{\text{--}}}}\!\operatorname{Ind}\nolimits(\operatorname{Ex}(\mathscr{J}))=\operatorname{Ex}(\mathscr{K}). \end{align} $$

By [Reference Raeburn and WilliamsRW98, Proposition 3.24], the left-hand side of Equation (5-1) is

$$ \begin{align*} \overline{\operatorname{span}}\{_{*}{\langle\langle}x\cdot b,y{\rangle\rangle}:{x,y\in \mathsf{X} \text{ and } b\in \operatorname{Ex}(\mathscr{J})}\}, \end{align*} $$

where ${\langle}\cdot ,\cdot{\rangle}$ is the $\Gamma _{c}(H^{(0)};\mathscr {B})$ -valued inner product from [Reference Muhly and WilliamsMW08, Equation (6.3) in Theorem 6.4]. In particular, if $x,y\in \mathsf {X}_{0}$ and $b\in \Gamma _{c}(K^{(0)};\mathscr {J})$ , then provided $\rho (t)=s(h)$ ,

(5-2) $$ \begin{align} _{*}{\langle\langle}x\cdot b,y{\rangle\rangle}(h) = \int_{K}{}_{\mathscr{B}}{\langle}x\cdot b (h \cdot t\cdot k),y(t\cdot k){\rangle}\,d\lambda_{K}^{\sigma(t)} (k), \end{align} $$

where according to [Reference Muhly and WilliamsMW08, Equation (6.2) in Theorem 6.4],

(5-3) $$ \begin{align} x\cdot b(h\cdot t \cdot k)=\int_{K}x(h\cdot t \cdot k l)b(l^{-1}) \,d\lambda_{K}^{s(k)} (l). \end{align} $$

Let $\mathscr {M}=\mathscr {E}\cdot \mathscr {J}=\mathscr {K}\cdot \mathscr {E}$ . Note that $\mathscr {M}$ is a $\mathscr {K}\, {\text {--}}\, \mathscr {J}$ -equivalence. Then the integrand in Equation (5-3) is in the Banach space $M_{h\cdot t\cdot k}$ for all l. Hence,

$$ \begin{align*} x\cdot b(h\cdot t \cdot k)\in M_{h\cdot t\cdot k}= K_{r(h)} \cdot E_{h\cdot t\cdot k}. \end{align*} $$

Plugging into Equation (5-2), and using condition (E2)(iii) of Definition 2.21, we clearly have

$$ \begin{align*} _{*}{\langle\langle}x\cdot b,y{\rangle\rangle}(h)\in K_{r(h)}\cdot B_{h}=K_{h}. \end{align*} $$

It follows that

$$ \begin{align*} \mathop{\mathsf{X}\mathord{\mathop{\text{--}}}}\!\operatorname{Ind}\nolimits(\operatorname{Ex}(\mathscr{J})) \subset \operatorname{Ex}(\mathscr{K}). \end{align*} $$

However, we can also work with $\mathop {\mathsf {X}\mathord {\mathop {\text {--}}}}\!\operatorname {Ind}\nolimits ^{-1}$ . Then

$$ \begin{align*} \mathop{\mathsf{X}\mathord{\mathop{\text{--}}}}\!\operatorname{Ind}\nolimits^{-1}(\operatorname{Ex}(\mathscr{K}))=\overline{\operatorname{span}}\{{\langle\langle}x,c\cdot y{\rangle\rangle}_{*}:{x,y\in \mathsf{X} \text{ and } c\in\operatorname{Ex}(\mathscr{K})}\}. \end{align*} $$

A similar argument to the above shows that

(5-4) $$ \begin{align} \mathop{\mathsf{X}\mathord{\mathop{\text{--}}}}\!\operatorname{Ind}\nolimits^{-1}(\operatorname{Ex}(\mathscr{K}))\subset \operatorname{Ex}(\mathscr{J}). \end{align} $$

Now Equation (5-1) follows by applying $\mathop {\mathsf {X}\mathord {\mathop {\text {--}}}}\!\operatorname {Ind}\nolimits $ to both sides of Display (5-4).

Footnotes

Communicated by Aidan Sims

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