2 Preliminaries

Our groupoids will all be locally compact Hausdorff with open range and source maps. We refer to [Reference Williams6, Section 2.3] for groupoid equivalences. Recall here that if $G$ and $H$ are groupoids, then a space $Z$ is a $(G,H)$ -equivalence if there are free and proper commuting actions of $G$ and $H$ on $Z$ ( $G$ on the left and $H$ on the right) such that the moment map $\rho $ for the $G$ -action factors through a homeomorphism of $Z/H$ with $G^{(0)}$ , and the moment map $\sigma $ for the $H$ -action factors through a homeomorphism of $G\backslash Z$ with $H^{(0)}$ . We also require the moment maps to be open, which is actually automatic as we require the range and source maps on groupoids to be open. If the space $Z$ is a $(G,H)$ -equivalence, then there are continuous and open pairings ${}_{G}[\cdot ,\cdot ]:Z*_\sigma Z\to G$ and $[\cdot ,\cdot ]_H:Z*_\rho Z\to H$ which are given by $x={}_{G}[x,y]\cdot y$ for all $(x,y)\in Z*_\sigma Z$ , and $x\cdot [x,y]_H=y$ for all $(x,y)\in Z*_\rho Z$ .

We refer to [Reference Kaliszewski, Quigg and Williams5, Section 2] for the required background on Banach bundles, Fell bundles and Fell bundle equivalence. In particular, our Fell bundles will all be saturated (so that each fibre is an imprimitivity bimodule). If $p:\mathscr {B}\to G$ is a Fell bundle, $q:\mathscr {E}\to T$ is a Banach bundle and $T$ is a left $G$ -space, then $\mathscr {B}$ acts on the left of $\mathscr {E}$ if there is a continuous pairing,

$$ \begin{align*}(b,e)\mapsto b\cdot e \text{ from} \ \mathscr{B}*\mathscr{E}=\{(b,e)\in \mathscr{B}\times \mathscr{E}:s(p(b))=\rho(q(e))\}\ \text{to} \ \mathscr{E},\end{align*} $$

that covers the pairing $(\gamma ,x)\mapsto \gamma \cdot x$ from $G*T$ to $T$ , that is, $q(b\cdot e)=~p(b)\cdot q(e)$ is associative in the sense that $b\cdot (c\cdot e)=(bc)\cdot e$ and is submultiplicative for the norms in the sense that $\|b\cdot e\|\le \|b\|\,\|e\|$ . Right actions are defined similarly. If $\mathscr {B}\to G$ and $\mathscr {C}\to H$ are Fell bundles, then a Banach bundle $\mathscr {E}\to T$ over a $(G,H)$ -equivalence $T$ is a $\mathscr {B}\, \text {--}\, \mathscr {C}$ equivalence if there are continuous commuting actions of $\mathscr {B}$ and $\mathscr {C}$ on $\mathscr {E}$ ( $\mathscr {B}$ on the left and $\mathscr {C}$ on the right) such that the bimodule actions are associative in the sense that $(b\cdot e)\cdot c=b\cdot (e\cdot c)$ , there is a continuous fibre-wise sesquilinear pairing that covers the pairing ${}_{G}[\cdot ,\cdot ]$ in the sense that commutes with the pairing $\mathscr {B}*\mathscr {E}\to \mathscr {B}$ in the sense that , and similarly for the right action, and each $E_x$ is a $B_{\rho (x)}\,\text {--}\, C_{\sigma (x)}$ imprimitivity bimodule.

We now summarise from [Reference Hall, Kaliszewski, Quigg and Williams2, Reference Hall, Kaliszewski, Quigg and Williams3] the definitions of semidirect product Fell bundles and action Fell bundles, although we change the notation for semidirect products; our original notation was always meant to be temporary, and we have now settled on what seems to us to be an acceptable choice. If $G$ and $H$ are locally compact groupoids such that $G$ acts on $H$ by isomorphisms, then the semidirect product groupoid [Reference Hall, Kaliszewski, Quigg and Williams2, Definition 5.1] is the set

$$ \begin{align*} H\propto G=\{(h,x)\in H\times G:\rho(h)=r(x)\}, \end{align*} $$

with source and range

$$ \begin{align*} s(h,x)=(x^{-1}\cdot s_H(h),s_G(x))\quad\text{and}\quad r(h,x)=(r_H(h),r_G(x)), \end{align*} $$

and operations

$$ \begin{align*} (h,x)(k,y)=(h(x\cdot k),xy)\quad\text{and}\quad (h,x)^{-1}=(x^{-1}\cdot h^{-1},x^{-1}). \end{align*} $$

It is shown in [Reference Hall, Kaliszewski, Quigg and Williams2, Lemma 5.4] that if $H$ has an invariant Haar system $\lambda _H=\{\lambda _H^v\}_{v\in H^{(0)}}$ in the sense that

$$ \begin{align*} \int_H f(x\cdot t)\,d\lambda_H^v(t)=\int_H f(t)\,d\lambda_H^{x\cdot v}(t), \end{align*} $$

then we get a Haar system on the semidirect product $H\propto G$ by

$$ \begin{align*} d\lambda^{(v,w)}(t,x)=d\lambda_H^v(t)\,d\lambda_H^w(x). \end{align*} $$

If $p:\mathscr {A}\to H$ is a Fell bundle and $G$ acts on $\mathscr {A}$ by isomorphisms, we get a Banach bundle over $H\propto G$ via the pull back $\pi _1^*\mathscr {A}=\{(a,x)\in A\times (H\propto G):\rho (p(a))=r(x)\}$ . We define Fell bundle operations by

$$ \begin{align*} (a,x)(b,y)=(a(x\cdot b),xy)\quad\text{and}\quad(a,x)^*=(x^{-1}\cdot a^*,x^{-1}), \end{align*} $$

getting the semidirect product Fell bundle $\mathscr {A}\propto G$ with bundle projection ${p'(a,x)=(p(x),x)}$ onto $H\propto G$ . We get an action $\alpha $ of $G$ on $\mathscr {A}$ via

$$ \begin{align*} \alpha_x(f)(h)=x\cdot f(x^{-1}\cdot h)\quad\text{for }f\in \Gamma_c(H_{s(x)},\mathscr{A}), \end{align*} $$

and, provided $H$ has an invariant Haar system, $\alpha $ implements an action of $G$ on the $C^{*}$ -algebra of $\mathscr {A}$ . Moreover, the $C^{*}$ -algebra of the semidirect product Fell bundle is isomorphic to the crossed product of the $C^{*}$ -algebra of $\mathscr {A}$ by the action $\alpha $ [Reference Hall, Kaliszewski, Quigg and Williams2, Theorem 6.3].

Now, we turn to action Fell bundles, defined following [Reference Hall, Kaliszewski, Quigg and Williams3, Remark 4.12]. Let $\overline p:\mathscr {B}\to M$ be a Fell bundle, assume that $M$ acts on a space $T$ and let $M\rtimes T$ be the action groupoid with unit space identified with $T$ , so that

$$ \begin{align*} r(m,t)=m\cdot t,\quad s(y,t)=t \quad\text{and}\quad(m,k\cdot t)(k,t)=(mk,t). \end{align*} $$

The action Fell bundle is

$$ \begin{align*} \mathscr{B}\rtimes T=\{(b,t):s(\overline p(b))=\rho(t)\}, \end{align*} $$

with source and range $s(b,t)=t$ , $r(b,t)=\overline p(b)\cdot t$ , and operations

$$ \begin{align*} (b,\overline p(b')\cdot t)(b',t)=(bb',t)\quad\text{and}\quad(b,t)^*=(b^*,\overline p(b)\cdot t). \end{align*} $$

The crux of the ladder technique mentioned in the introduction is a straightforward observation about maps between sets: if

is a commutative diagram of sets and maps such that $u$ and $v$ are bijections, then $f$ , $g$ and $h$ are also bijections. We refer to such a diagram as a ladder, with vertical struts $u$ and $v$ , and rungs $f$ , $g$ and $h$ . We refer to the preceding observation as the Ladder Lemma.

3 Fell bundle ideals

The notion of a Fell bundle ideal and the Rieffel correspondence for equivalent Fell bundles is central here. For details, we refer to [Reference Kaliszewski, Quigg and Williams5, Sections 2.7 and 3]. We recall some of the basics for convenience. A Fell subbundle $\mathscr {I}$ of a Fell bundle $\mathscr {B}\to G$ is an ideal of $\mathscr {B}$ if $bc\in \mathscr {I}$ whenever $(b,c)\in \mathscr {B}^{(2)}$ and either $b\in \mathscr {I}$ or $c\in \mathscr {I}$ . In particular, an ideal of a Fell bundle is itself a Fell bundle and is therefore closed under the adjoint operation.

Suppose that $q:\mathscr {E}\to T$ is a $\mathscr {B}\, \text {--}\, \mathscr {C}$ Fell bundle equivalence. The Rieffel correspondence for Fell bundle equivalences [Reference Kaliszewski, Quigg and Williams5, Theorem 3.10] says that 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 $\mathscr {C}$ . Specifically, if $\mathscr {I}$ is an ideal in $\mathscr {B}$ , then the corresponding full Banach $\mathscr {B}\, \text {--}\, \mathscr {C}$ submodule, $\mathscr {M}_{\mathscr {L}}$ , is

$$ \begin{align*} \mathscr{I}\cdot \mathscr{E}:= \bigcup_{s(h)=\rho(t)} J_{h}\cdot E_{t}. \end{align*} $$

Describing the ideal $\mathscr {I}_{\mathscr {M}}$ in $\mathscr {B}$ corresponding to a full Banach $\mathscr {B}\, \text {--}\, \mathscr {C}$ submodule, $\mathscr {M}$ , takes a bit more fussing. Extending the usual conventions for imprimitivity bimodules, we let denote the span of all possible left-inner products with $m\in \mathscr {M}$ and $e\in \mathscr {E}$ as in [Reference Kaliszewski, Quigg and Williams5, (3.2)]. Then,

(3.1)

where the closure is taken in the norm topology on $B_{h}$ .

We can describe (3.1) a bit more elegantly using a simple observation (which probably should have been noted in [Reference Kaliszewski, Quigg and Williams5]).

It is not hard to see that

4 The vertical struts

Let $H$ be a wide closed subgroupoid of $M$ . (Here, ‘wide’ simply means that $H^{(0)}=M^{(0)}$ .) Since our standing assumptions are that both $H$ and $M$ have open range (and source) maps, it follows from [Reference Williams6, Corollary 2.50] that $M$ is an $(M\rtimes M/H, H)$ equivalence. Specifically, the moment map for the left-action is given by $\rho (m)=m H$ , where we have identified the unit space of $M\rtimes M/H$ with the orbit space $M/H$ . Then, $(m,nH)\cdot n=mn$ . The moment map for the right action is $\sigma (m)=s(m)$ and then $m\cdot h=~mh$ . Note that if $\rho (m)=\rho (n)$ , then $nH=mH$ and there is a unique $h\in H$ such that $mh=n$ and $[m,n]_{H}=h $ . However, if $\sigma (m)=\sigma (n)$ , then $(mn^{-1},nH)\cdot n=m$ and ${}_{M\rtimes M/H}[m,n] = (mn^{-1},nH)$ .

Proof. The left-action defined in (a) clearly makes sense on $(\mathscr {B}\rtimes M/H)*\mathscr {B}$ and covers the $M\rtimes M/H$ action on $M$ . Also, on the one hand,

$$ \begin{align*} ((b,mH)(c,nH))\cdot d= (bc,nH)\cdot d= bcd, \end{align*} $$

while on the other hand,

$$ \begin{align*} (b,mH)\cdot ( (c,nH)\cdot d) = (b,mH)\cdot cd =bcd. \end{align*} $$

Therefore, the pairing is associative. Since

$$ \begin{align*} \| (b,mH)\cdot c\|=\|bc\|\le \|b\|\,\|c\|=\|(b,mH)\|\,\|c\|, \end{align*} $$

the pairing is submultiplicative. Therefore, we have an action of the semidirect product Fell bundle $\mathscr {B}\rtimes M/H$ on the Banach bundle $\mathscr {B}$ as required.

For the associativity of the bimodule actions,

$$ \begin{align*} ((b,mH)\cdot c)\cdot d &=(bc)\cdot d =bcd =(b,mH)\cdot (c\cdot d). \end{align*} $$

The left pairing $\mathscr {B}*_\sigma \mathscr {B}\to \mathscr {B}\rtimes M/H$ covers the pairing $M*_\sigma M\to M\rtimes M/H$ because

For the adjoint property of the left pairing,

Next,

where the operations are valid because

$$ \begin{align*} mH=p(c)H=p(cd^*)p(d)H. \end{align*} $$

The verifications of the remaining axioms involving the right pairing are similar and easier.

Finally, for $m\in M$ , we verify that $B_m$ is a $(\mathscr {B}\rtimes M/H)_{\rho (m)}\, \text {--}\, (\mathscr {B}|_H)_{\sigma (m)}$ imprimitivity bimodule. Unravelling the definitions, we must show that $B_m$ is a $(B_{r(m)}\times \{mH\})\, \text {--}\, B_{s(m)}$ imprimitivity bimodule. Since the left-module action of $B_{r(m)}\times \{mH\}$ on $B_m$ is essentially just the left action of $B_{r(m)}$ , the axiom follows immediately from the fact that $B_m$ is a $B_{r(m)}\, \text {--}\, B_{s(m)}$ imprimitivity bimodule.

Recall that if $G$ acts on $H$ by isomorphisms, then we can form the semidirect product $H\propto G$ as in [Reference Hall, Kaliszewski, Quigg and Williams2, Definition 5.1]. Then, if $p:\mathscr {A}\to H$ is a Fell bundle and $G$ acts on $\mathscr {A}$ by isomorphisms, we can form the semidirect product Fell bundle $\mathscr {A}\propto G$ as defined prior to [Reference Hall, Kaliszewski, Quigg and Williams2, Lemma 5.5].

Proof. We appeal to Theorem 4.1, with $M=H\propto G$ and $\mathscr {B}=\mathscr {A}\propto G$ . Then, we can identify $H$ with the wide subgroupoid $H*G^{(0)}=\{(h,u)\in H\times G^{(0)}:\rho (h)=u\}$ of the semidirect product $H\propto G$ . Then, $H$ acts on the right of $M=H\propto G$ and $[h,x]\mapsto x$ is a homeomorphism of $(H\propto G)/H$ onto $G$ , where as usual, we have written $[h,x]$ for the $H$ -orbit of $(h,x)\in H\propto G$ . Furthermore, this map intertwines the natural $H\propto G$ -action on the quotient $(H\propto G)/H$ with the composition of the natural map on $H\propto G$ onto $G$ with left translation. Therefore, the horizontal maps induced by $[h,x]\mapsto x$ induce a diagram

which gives a Fell bundle isomorphism of $\mathscr {B}\rtimes M/H = (\mathscr {A}\propto G)\rtimes (H\propto G)/H$ with $(\mathscr {A}\propto G)\rtimes G$ .

Since $(\mathscr {A}\propto G)|_{H}$ is isomorphic (as a Fell bundle) to $\mathscr {A}$ , we can realise $\mathscr {A}\propto G$ as an $(\mathscr {A}\propto G)\rtimes G\, \text {--}\, \mathscr {A}$ -equivalence. Then, the formulae for the actions and inner products follow from routine computations.

For convenience and easy reference, we record some results from [Reference Hall, Kaliszewski, Quigg and Williams3]. Recall from [Reference Hall, Kaliszewski, Quigg and Williams3, Proposition 6.3] that if $G$ acts freely and properly on $H$ by isomorphisms, then $H$ becomes an $H\propto G\, \text {--}\, G\backslash H$ equivalence, where the left-action of $H\propto G$ on $H$ is given by $(h,x)\cdot k=h(x\cdot k)$ and the right action of $G\backslash H$ on $H$ is given by $h\cdot (G\cdot k)=hk$ provided $s(h)=r(k)$ . Then, we have the following result.

Concretely, we can suppose that $K$ and $G$ act on (the left) of a space $T$ such that the actions commute and the moment map $\rho :T\to K^{(0)}$ is a free and proper $G$ -bundle. Then, $G$ acts freely and properly by isomorphisms on the action groupoid $K\rtimes T$ via $x\cdot (k,t)=(k,x\cdot t)$ . We can identify $G\backslash (H\rtimes T)$ with $K$ and $K\rtimes T$ is a $(K\rtimes T)\propto G\, \text {--}\, K$ equivalence. This gives the following result.

Note that we have commuting left actions of $G$ on itself—the first just left translation and the second given by $x\cdot y=yx^{-1}$ . Furthermore, the moment map ${r:G\to G^{(0)}}$ for left translation is a free and proper $G$ -bundle for the action by right translation. Thus, if $p:\mathscr {A}\to G$ is a Fell bundle, then the corresponding $G$ -action on the action bundle $\mathscr {A}\rtimes G$ is given by $x\cdot (b,y)=(b,yx^{-1})$ . Consequently, we have the following result.

Now that we have equivalent Fell bundles, by [Reference Kaliszewski, Quigg and Williams5, Theorem 3.10], we have an associated Rieffel correspondence, giving a lattice isomorphism between the sets of Fell-bundle ideals. In the case in which a groupoid $G$ acts consistently on the Fell bundles, the Rieffel correspondence restricts to the invariant ideals. More precisely, we have the following result.

Proof. First, we show that

(4.1) $$ \begin{align} (\mathscr{A}\propto G)\cdot\mathscr{I}=\mathscr{I}\propto G. \end{align} $$

If $(a,x)\in \mathscr {A}\propto G$ and $i\in \mathscr {I}$ , then

$$ \begin{align*} (a,x)\cdot i &=(a(x\cdot i),x). \end{align*} $$

Putting $j=a(x\cdot i)$ , we have $j\in \mathscr {I}$ , because $\mathscr {I}$ is a $G$ -invariant ideal, and so ${(j,x)\in \mathscr {I}\propto G}$ .

For the opposite containment, consider $(i,x)\in \mathscr {I}\propto G$ . Let $h=p(i)$ . Then, ${i\in J_{h}=~A_{h}\cdot J_{s(h)}}$ by [Reference Kaliszewski, Quigg and Williams5, Lemma 2.26]. Then, $i=a j$ with $a\in A_{h}$ and $j\in J_{s(h)}$ by [Reference Kaliszewski, Quigg and Williams5, Remark 2.2]. Note that left multiplication by $x$ is a bijection of $A|_{{H_{s(x)}}}$ onto $A|_{{H_{r(x)}}}$ . (Recall that if $u\in G^{(0)}$ , then we let $H_{u}=\rho ^{-1}(u)$ , where $\rho :H\to G^{(0)}$ is the moment map; see [Reference Kaliszewski, Quigg and Williams5, Definition 4.9].) Since $\mathscr {I}$ is $G$ -invariant, left-multiplication by $x$ also induces a bijection of $\mathscr {I}|_{{H_{s(x)}}}$ onto $\mathscr {I}|_{{H_{r(x)}}}$ . In particular, $j=x\cdot j'$ for some $j'\in J_{s(h)}$ . Then,

$$ \begin{align*} (i,x)=(a j,x)= (a, x)\cdot j'\in (\mathscr{A}\propto G)\cdot \mathscr{I}. \end{align*} $$

Hence, (4.1) holds as claimed.

Now, let $\mathscr {L}$ be the ideal of $\mathscr {A}\propto G\rtimes G$ associated to $\mathscr {I}$ via the Rieffel correspondence. By the argument above, to show that

$$ \begin{align*} \mathscr{L}\subset (\mathscr{I}\propto G)\rtimes G, \end{align*} $$

it suffices to consider an element of the form with $(a,x)\in \mathscr {A}\propto G$ and $(i,y)\in \mathscr {I}\propto G$ . (Here, we have written in place of the more cumbersome .) Since the inner product is

$$ \begin{align*} (a(x y^{-1}\cdot i^*),x y^{-1},y), \end{align*} $$

and since

$$ \begin{align*} a(x y^{-1}\cdot i^*)\in \mathscr{I} \end{align*} $$

because $\mathscr {I}$ is a $G$ -invariant ideal and ideals are self-adjoint, we see that

For the opposite containment, consider $(i,z,y)\in \mathscr {I}\propto G\rtimes G$ . Let $h=p(i)$ . Then, $i\in J_{h}=A_{h}\cdot J_{s(h)}$ . Let $i=a\cdot j$ with $a\in A_{h}$ and $j\in J_{s(h)}$ as above. We can assume ${j=z\cdot j_{1}}$ . Note that $(a,zy)\in \mathscr {A}\propto G$ , while $(j_{1}^{*},y)\in \mathscr {I}\propto G$ . Hence,

Hence, $\mathscr {L}=\mathscr {I}\propto G\rtimes G$ as required.

Now, suppose that $\mathscr {L}$ is a $G$ -invariant ideal in $\mathscr {A}\propto G\rtimes G$ . Then, the corresponding ideal, $\mathscr {I}$ , in $\mathscr {A}$ is generated by pairings of the form with $(a,x,y)\in \mathscr {L}$ and $(b,y),(c,xy)\in \mathscr {A}\propto G$ . That is, $\mathscr {I}$ is generated by elements of the form

$$ \begin{align*} (xy)^{-1}\cdot [(a(x\cdot b))^{*}c]. \end{align*} $$

However,

Since the $G$ -action is isometric and bijective on the fibres, it now follows from (3.1) that $\mathscr {I}$ is $G$ -invariant in $\mathscr {A}$ .

Since $\mathscr {I}\propto G\rtimes G$ is clearly $G$ -invariant in $\mathscr {A}\propto G\rtimes G$ and since the Rieffel correspondence is a bijection, the result follows.

Similarly, starting with action-product Fell bundles gives the next result.

Proof. First, we show that in analogy with (4.1),

$$ \begin{align*} (\mathscr{A}\rtimes G)\cdot\mathscr{I}=\mathscr{I}\rtimes G. \end{align*} $$

If $(a,z)\in \mathscr {A}\rtimes G$ and $i\in \mathscr {I}$ , then

$$ \begin{align*} (a,z)\cdot i &=(ai,p(i)^{-1}z ), \end{align*} $$

which is in $\mathscr {I}\rtimes G$ because $\mathscr {I}$ is an ideal.

For the opposite containment, let $(i,z)\in \mathscr {I}\rtimes G$ with $p(i)=x$ . Note that, $s(x)=r(z)$ . As in the proof of Theorem 4.6, we can factor $i=aj$ with $a\in A_{x}$ and $j\in J_{s(x)}$ . Then,

$$ \begin{align*} (i,z)=(aj,z)= (a,z)\cdot j\in (\mathscr{A}\rtimes G)\cdot \mathscr{I}. \end{align*} $$

Now, let $\mathscr {L}$ be the ideal of $\mathscr {A}\rtimes G\propto G$ associated to $\mathscr {I}$ via the Rieffel correspondence. For the containment

$$ \begin{align*} \mathscr{L}\subset \mathscr{I}\rtimes G\propto G, \end{align*} $$

let $(a,z)\in \mathscr {A}\rtimes G$ and $(i,y)\in \mathscr {I}\rtimes G$ . Then,

because $\mathscr {I}$ is an ideal and, hence, self-adjoint.

For the opposite containment, given $(i,z,y)\in \mathscr {I}\rtimes G\propto G$ , we can factor $i=aj$ with $a\in A_{x}$ and $j\in J_{s(x)}$ , where $x=p(i)$ . Since $s(x)=r(z)$ ,

because $(j^{*},zy)\in \mathscr {I}\rtimes G$ .

5 Ladders

If $p:\mathscr {A}\to H$ is a Fell bundle on which $G$ acts by isomorphisms, then we let $\mathscr {I}_G(\mathscr {A})$ be the set of $G$ -invariant ideals of the Fell bundle $\mathscr {A}$ and $\mathscr {I}_{G}(\mathscr {A}\propto G\rtimes G)$ be the $G$ -invariant ideals of the action Fell bundle $\mathscr {A}\propto G\rtimes G$ as above. Consider the diagram

(5.1)

This diagram commutes and constitutes our basic ladder diagram. Moreover, the vertical arrows coincide with the Rieffel correspondences, by Theorems 4.6 and 4.7. Therefore, by the Ladder Lemma, all the arrows are bijections.

We record part of this formally in the following result.

Theorem 5.1. If a groupoid $G$ acts on a Fell bundle $\mathscr {A}\to H$ , then every ideal of $\mathscr {A}\propto G$ is of the form $\mathscr {I}\propto G$ for a $G$ -invariant ideal of $\mathscr {A}$ and the map

$$ \begin{align*} \mathscr{I}\mapsto \mathscr{I}\propto G \end{align*} $$

gives a lattice isomorphism between the $G$ -invariant ideals of $\mathscr {A}$ and all ideals in $\mathscr {A}\propto G$ .

Note that the ladder (5.1) could be continued indefinitely upward. Additionally, it could be started with the structure on the right-hand side, so that the first rung would be an action product (rather than a semidirect product).

To see how that would look, we consider a Fell bundle $p:\mathscr {A}\to G$ over a groupoid $G$ (to continue the ladder (5.1), just replace $G$ with $G\propto G$ .) We can form the following ladder:

The argument that the right strut is a bijection between $G$ -invariant ideals is a special case of Theorem 4.6. Hence, we get a companion result to Theorem 5.1.

Theorem 5.2. If $\mathscr {A}\to G$ is a Fell bundle, then the map

$$ \begin{align*}\mathscr{I}\mapsto\mathscr{I}\rtimes G\end{align*} $$

gives a lattice isomorphism between the ideals of $\mathscr {A}$ and the $G$ -invariant ideals of $\mathscr {A}\rtimes G$ .