The Rieffel Correspondence for Equivalent Fell Bundles

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


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 -B-imprimitivity bimodule X.A summary of these properties is given in Theorem 2.1 below, 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 : E → T between two Fell bundles p B : B → H and p C : C → K over locally compact groupoids H and K. Rather than work at the level of the Fell-bundle C * -algebras C * (H; B) and C * (K; C ), we work with the Fell bundles themselves.We introduce a natural notion of an ideal J of a Fell bundle B. In the case where 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 [IW12].We can form the quotient Fell bundles B/J , and if H has a Haar system and if our Fell bundles are separable, then the main result in [IW12] pushes the analogy of Fell bundle ideals with invariant ideals in crossed products; that is, we have a short exact sequence of C * -algebras 0 C * (H; J ) C * (H; B) C * (H; B/J ) 0.
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 E is an equivalence between B and C as above, then there is a lattice isomorphism between the ideals of C and those of B. Furthermore, if K and J are corresponding ideals of B and C , respectively, then K and J are equivalent Fell bundles as are the quotients B/K and C /J .Naturally, these equivalences arise from submodules and quotients of the given equivalence E .
We start in Section 2 with a detailed collection of preliminary material which 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.
Since 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, 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.is an ideal in A, while We have and J are as in part (a), then Y is a K -J-imprimitivity bimodule with respect to the restricted actions and inner products.(d) If K, Y, and J are as in part (a), then the quotient Banach space X/Y is an A/K -B/J-imprimitivity bimodule.In particular, the quotient norm on X/Y equals the imprimitivity-bimodule norm.
Remark 2.2.Suppose that J is an ideal in a C * -algebra B, and that X is a right Hilbert B-module.
As a result, we have X • J = X • J part (a), and similarly with K • X.
(where 0 x is the zero element in B x . We say that p : B → X is separable if X is second countable and the Banach space Γ 0 (X; B) is separable.If the map in (B1) is actually continuous, we we call 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 → λb is continuous for each λ ∈ C. However since If p : B → X is a Banach bundle, we write Γ(X; B) for the vector space of continuous sections.If X is locally compact, then we write Γ c (X; B) and Γ 0 (X; B) for the continuous sections which have compact support or which vanish at infinity, respectively.We say that p : B → X has enough sections if given b ∈ B x there is an f ∈ Γ(X; B) such that f (x) = b.Note that if X is locally compact, then since Γ(X; B) is a C(X)-module by (B3), we can take f ∈ Γ c (X; B).Theorem 2.7 ([Laz18, Corollary 2.10]).If p : B → X is a Banach bundle over a locally compact space, then 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 ∈ X, we are given a Banach space B x .Let B be the disjoint union x∈X B x viewed as bundle p : B → X. Suppose that Γ is a subspace of sections such that (a) for each f ∈ Γ, x → f (x) is upper semicontinuous, and Then there is a unique topology on B such that p : B → X is a Banach bundle with Γ ⊂ Γ(X; B).Furthermore, the sets of the form Remark 2.10.Since 0 x ∈ C x for all x, we must have p(C ) = X.However some fibres can be the zero Banach space.Even if p| C is not open, we write Γ(X; C ) for the continuous functions f from X to C such that p(f (x)) = x for all x ∈ X.Of course, if p| C is not open, there is no reason that Γ(X; 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 B = X × B the trivial bundle over X. Fix x 0 ∈ X, and let Then in general, p| C : C → X is not open and admits only the zero section.
Proposition 2.13.Let p : B → X be a Banach bundle.Suppose that C x is a closed subspace of B x for all x ∈ X and let C = C x be as above.
and suppose that (x i ) is a net in X converging to x in X.It will suffice to see that Remark 2.14.In [FD88, Problem 41 in Chap.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 : B → X is a Banach bundle over a locally compact space, and if p| C : C → X is a Banach subbundle, then it has enough sections by Lazar's Theorem 2.7.
2.4.Quotient Banach Bundles.Let p : B → X be a Banach bundle over a locally compact space X, and let C ⊂ B be a Banach subbundle as in Definition 2.3.Then we can formally form the quotient B/C = x∈X B x /C x where B x /C x is the usual Banach space quotient.We let q : B → B/C be the quotient map so that if b ∈ B x , then q(b) = q x (b) where q x : B x → B x /C x is the usual Banach space quotient map.In particular, q is norm reducing.If f ∈ Γ c (X; B) then we will write q(f ) for the section of B/C given by q(f )(x) = q x (f (x)).
Proposition 2.16.Let p : B → X be a Banach bundle and C ⊂ B a Banach subbundle.Then p : B/C → X is a Banach bundle in the quotient topology.Furthermore, the quotient map q : B → B/C is continuous and open, and the quotient topology on B/C is the unique topology such that Remark 2.17.As pointed out in [Laz18], Proposition 2.16 can be sorted out of [Gie82, Chap.9].We give the short proof for completeness.
it follows that q(f )(y) < ǫ for y ∈ V .This establishes the claim.
It follows from Theorem 2.8 that there is a unique topology on B/C such that B/C is a Banach bundle with Γ : Next we claim that the quotient map [MW08,Lemma A.3] implies that q(a i ) → q(a 0 ).Thus q is continuous as claimed.
To see that q is also open, Let V be an open neighborhood of b ∈ B. Then in view of Theorem 2.8, there is a f ∈ Γ c (X; B), and open neighborhood U of p(b), and a We need to verify that q(V ) is a neighborhood of q(b).Since q(f ) ∈ Γ c (X; B/C ), it will suffice to see that 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 Hence there is an a ∈ W (f, U, ǫ) such that q(a) = q(c).This suffices show that q is open.
Since q is continuous and open, the topology on B/C is the quotient topology.Definition 2.18.Suppose that p : B → G is a Banach bundle over a second countable locally compact Hausdorff groupoid G. Let We say that p : , the fibre B u is a C * -algebra with respect to the inherited multiplication and involution on B u , and (FB5) for each g ∈ G, B g is an B r(g) -B s(g) -imprimitivity bimodule when equipped with the inherited actions and inner products given by    Fell bundle with respect to the inherited operations.In particular, C must be closed under multiplication and involution.
We will be focused on Fell subbundles that are multiplicatively absorbing as follows.
Definition 2.23.A Fell subbundle J of a Fell bundle B is called an ideal if ab ∈ J whenever (a, b) ∈ B (2) and either a ∈ J or b ∈ J .
Example 2.24.Suppose that α : G → Aut(A) is a C * -dynamical system for a group G. Let B = A × G be the associated Fell bundle over G: (a, s)(b, r) = (aα s (b), sr).Let I be an α-invariant ideal of A.
Example 2.25.Suppose that A is a C * -bundle over X so that A = Γ 0 (X; A ) is a C * -algebra.Let J be an ideal in A and for each x ∈ X let It follows from Proposition 2.13 that J is a Banach subbundle, and in fact is obviously an ideal of the Fell bundle A .Clearly, J ⊂ Γ 0 (X; J ).Since J is an ideal in the C 0 (X)-algebra A, if φ ∈ C 0 (X) and b ∈ J, then φ • b ∈ J. Now it follows from [Wil07, Proposition C.24] that J is dense in Γ 0 (X; J ). Therefore J = Γ 0 (X; J ).
Lemma 2.26.Suppose that J is an ideal in B. Then for each g ∈ G, J g is a J r(g) -J s(g) -imprimitivity bimodule.Furthermore, (2.5) where we are taking advantage of Remark 2.2.Furthermore, The first assertion is immediate since J is, by assumption, a Fell subbundle.The remaining statements follow from the Rieffel correspondence-see Theorem 2.1.Definition 2.27.Let p : B → G be a Fell bundle and J ⊂ B a Banach subbundle.We call J a weak ideal of B if whenever (a, b) ∈ B (2) then ab ∈ J whenever either a or b is in 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 : B → G is a Fell bundle, then every weak ideal in B is an ideal.
Proof.Let J be a weak ideal in B. Since J g is closed with respect to the norm on B g , it is a closed B(r(g)) -B(s(g))-submodule of the B(r(g)) -B(s(g))-imprimitivity bimodule B g .In particular, J u is an ideal in the C * -algebra B u for all u ∈ G (0) .Then, applying the Rieffel correspondence, J g is a K g -I g -imprimitivity bimodule where I g is the closed linear span of elements of the form b * a with b ∈ B g and a ∈ J g .Similarly, K g is the closed linear span of products ab * with a ∈ J g and b ∈ B g .Furthermore, (2.6) Note that I g is an ideal in J s(g) .Fix c ∈ J s(g) .If b ∈ B g , then bc ∈ J g by the weak ideal property.Since B * g • B g is a dense ideal in B s(g) , we can find an approximate unit (e i ) in B s(g) where each But then e i c is in I g and e i c → c.Hence c ∈ 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 (2.6), we have In particular, J * = J and J is closed under taking adjoints.Since 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.But 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 : B → G as a generalized groupoid crossed product of G acting on the associated C * -algebra A := Γ 0 (G (0) ; B).As an example of this rubric, it is shown in [IW12, Proposition 2.2] that there is a natural action of G on Prim A given as follows.Note that Prim A is naturally fibred over G (0) .Since B g is a B r(g) -B s(g) -imprimitivity bimodule, the Rieffel Correspondence induces a homeomorphism φ g : Prim(B s(g) ) → Prim(B r(g) ) [RW98, Corollary 3.33].Then the G-action is given by g • P s(g) = φ g (P s(g) ).Naturally, an ideal If I is an ideal in A then we let I u = q u (I) where q u : Γ 0 (G (0) ; B) → B u is the evaluation map.

Proposition 2.30 ([IW12]
). Suppose p : B → G is a Fell bundle and that I is a G-invariant ideal in the associated C * -algebra Γ 0 (G (0) ; B).Then Proof.It follows from the proof of [IW12, Lemma 3.1] that an ideal I ⊂ Γ 0 (G (0) ; B) is G-invariant if and only if for all g we have φ g (I s(g) ) = I r(g) .By the Rieffel correspondence, the latter is equivalent to Suppose that I is G-invariant.Then it follows from [IW12, Proposition 3.3] that J := B I is a Fell subbundle such that (2.7) holds.Suppose that a ∈ B g and b ∈ B h are composable.If a ∈ J , then Similarly, if b ∈ J , then and B I is an ideal.Now suppose that J is an ideal in B. Let Then as in Example 2.25, we have I u = J u .Now it follows from Lemma 2.26 and (2.7) that I is G-invariant.Since B I and J have the same fibres, clearly B I = J .

The Rieffel Correspondence for Fell Bundle Equivalences
In this section, we let q E : E → T be an equivalence between p B : B → H and p C : C → K.In particular, T is a (H, K)-equivalence and we let ρ : T → H (0) and σ : T → K (0) be the open moment maps.
We will need the following observation from [MW08, Lemma 6.2].
Lemma 3.1.As above, let q E : E → T be a Fell-bundle equivalence between p B : B → H and p C : C → K. Then (b, e) → b • e induces an imprimitivity bimodule isomorphism of Corollary 3.2.Let E , B, and C be as above.Let J be an ideal in C and σ(t Therefore by Lemma 2.26 we have Definition 3.3.Let q E : E → T be an equivalence between p B : B → H and Proposition 3.4.Let E , B, and C be as above.If J is an ideal in C , then Proof.We have In particular, E • J is a bundle over T with closed fibres We still need to see that On the other hand if σ(t) = r(k), then The corresponding statements for K • E are proved similarly.
Lemma 3.5.In the current set-up, the ideal R t depends only on σ(t) and the ideal L t depends only on ρ(t).Hereafter, we will denote them by R σ(t) and L ρ(t) , respectively.
On the other hand, consider Since the convergence takes place in E t , the convergence is in norm.It follows that Therefore we have equality in (3.1), and which, using (E2)(ii) and (E2)(iii), is The proof for L t is similar.
where the closure takes place in the Banach space E t•k .Lemma 3.6.In the setting above, both where we have invoked Lemma 3.5 to realize that R t depends only on r(k) = ρ(t). Proof.Clearly, This implies the first assertion.
For the second, we just need to see that Using Lemma 3.6 and (3.2), we have Lemma 3.7.In the current set-up, Proof.As in the proof of Proposition 2.13, the issue is to see that p : J M → K is open.Let U be a nonempty (relatively) open set in J M .Given k ∈ p(U) it will suffice 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 relabeling, we can assume ) is eventually in U. Therefore, we may as well assume that c ′ = n j=1 e j , m j C with each e j ∈ E t and each m j ∈ M t .Since Banach bundles have enough sections we can find f ∈ Γ(K; C ), g j ∈ Γ(T ; E ), and h j ∈ Γ(T ; M ) such that f (k) = c, g j (t) = e j , and h j (t) = m j .
Since r(k i ) → r(k) = σ(t), and since σ is open, we can pass to a subsequence, relabel, and assume that there are t i ∈ T such that t i → t and σ(t i ) = r(k i ).Then is eventually in p(U) which contradicts our assumptions on (k i ) and completes the proof.Proposition 3.8.In the current set-up, . Then the corresponding ideals J M in C and K M in B are given by The following is a generalization of [RW98, Proposition 3.24].
Corollary 3.11.Suppose that q E : E → T is a Fell bundle equivalence between p B : B → H and p C : Proof.If J is an ideal in C , then according to Theorem 3.10(a), the corresponding full Banach B -C -submodule is M = E • J .Then using Theorem 3.10(b), the corresponding ideal K in B is h∈H L r(h) • B h where L r(h) is given by the right-hand side of (3.4) for any t ∈ T such that ρ(t) = r(h).This gives the result.

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 E : E → T is a Fell-bundle equivalence between p B : B → H and p C : C → K. Suppose that J is an ideal in C and that M and K are the corresponding full Banach B -C -submodule in E and ideal in B. Then M is a Fell-bundle equivalence between K and J .
Proof.Since M is a B -C -submodule of E , we clearly have a left K -action and a right J -action satisfying (E1).
For (E2), we claim that it suffices to let K e , f = B e , f and e , f J = e , f C .To see this, note that if (e, f ) ∈ M * σ M , then we can assume (e, f ) • , • J is J -valued.The rest of (E2) follows from the given properties of B • , • and • , • C .
For (E3), the fact that M t is a K ρ(t) -J σ(t) -imprimitivity bimodule follows from the Rieffel Correspondence (part (c) of Theorem 2.1).Proposition 4.2.Let q E : E → T be an equivalence between p B : B → H and p C : C → K. Suppose that J is an ideal in C and that M and K are the corresponding full Banach B -C -submodule in E and ideal in B, respectively.Then the quotient Banach bundle E /M is an equivalence between B/K and C /J .Proof.We let q K : B → B/K and q J : C → C /J be the quotient maps.Then the given left and right actions of B and C on E induce left and right actions of B/K and C /J on E /M in the expected way: q K (b) • q(e) = q(b • e) and q(e) • q J (c) = q(e • c) assuming that b • e and e • c are defined.
To see that these actions are continuous, we use the fact that q, q K , and q J are open as well as continuous (Proposition 2.16).Suppose that q(e i ) → q(e) while q K (b i ) → q K (b) with b i • e i defined for all i.We need to verify that q(b i • e i ) → q(b • e).For this, it suffices to see that every subnet has a subnet converging to q(b • e).But after passing to a subnet and relabeling, the openness of the quotient maps means we can pass to another subnet and assume that e ′ i → e and b ′ i → b with q(e ′ i ) = q(e) and q K (b ′ i ) = q K (b i ).Then the continuity of the quotient maps implies that q(b i • e i ) = q(b ′ i • e ′ i ) → q(b • e) as required.We also have q(b • e) ≤ inf{ b ′ • e ′ : q K (b ′ ) = q K (b) and q(e ′ ) = q(e) } ≤ inf{ b ′ e ′ : q K (b ′ ) = q K (b) and q(e ′ ) = q(e) } = q K (b) q(e) .
Therefore B/K acts on the left of E /M .The argument for the right action is similar.
Now we need to verify the axioms in Definition 2.21.(E1) is immediate since E is an equivalence.For Axiom (E2), we define q(e) , q(f ) C /J := q J e , f C and B/K q(e) , q(f ) = q K B e , f .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 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.

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.In order to apply the Equivalence Theorem-that is, [MW08, Theorem 6.4]-we also need our Fell bundles to be separable.
We return to the set-up in Section 3: we let q E : E → T be an equivalence between the separable Fell bundles p B : B → H and p C : C → K.In particular, T is a (H, K)-equivalence3 and we let ρ : T → H (0) and σ : T → K (0) be the open moment maps.
Then the Equivalence Theorem implies that C * (H (0) ; B) and C * (K (0) ; C ) are Morita equivalent via an imprimitivity bimodule X which is the completion of X 0 := Γ c (T, E ) with the actions and inner products given in [MW08,Theorem 6.4].Then we can let X-Ind : I C * (K (0) ; C ) → I C * (H (0) ; B) be the classical Rieffel lattice isomorphism.
If J is an ideal in C , then as shown in [IW12, Lemma 3.5], the identity map ι induces an isomorphism of C * (K (0) ; J ) onto the ideal Ex(J ) which is the closure of ι(Γ c (K (0) ; C )) in C * (K (0) ; C ).
Let J be an ideal in C and K the corresponding ideal in B as in Theorem 3.10.The goal here is to establish that the two Rieffel correspondences are compatible in that (5.1) X-Ind Ex(J ) = Ex(K ).(5.3) Let M = E • J = K • E .Note that M is a K -J -equivalence.Then the integrand in (5.3) is in the Banach space M h•t•k for all l.Hence Plugging into (5.2), and using (E2)(iii) of Definition 2.21, we clearly have * x • b , y (h) ∈ K r(h) • B h = K h .It follows that X-Ind Ex(J ) ⊂ Ex(K ).

Remark 2 .
11.If { C x } is any collection of closed subspaces with C x ⊂ B x , and if we give C = C x = { b ∈ B : b ∈ C p(b) } the relative topology, then p : C → X is a continuous surjection satisfying (B1), (B2), (B3), and (B4) of Definition 2.3.But p : C → X may fail to be a Banach subbundle unless we also have p| C open.
(2.4) B r(g) a , b = ab * and a , b B s(g) = a * b.
a • (b • e) = (ab) • e for appropriate a, b ∈ B and e ∈ E , and (c) b • e ≤ b e .Right actions of a Fell bundle are defined similarly.Let T be a (G, H) equivalence with open moment maps ρ : T → G (0) and σ : T → H (0) as in [Wil19, Definition 2.29].It is shown in [Wil19, Lemma 2.42] that there are open continuous maps τ G : T * σ T → G and τ H : T * ρ T → H such that τ G (e, f ) • f = e and e • τ H (e, f ) = f .Definition 2.21 ([MW08, Definition 6.1]).Suppose the T is a (G, H)-equivalence, and that p B • B → G and p C : C → H are Fell bundles.Then a Banach bundle q : E → T is a B -C -equivalence if the following conditions hold.(E1) There is a left B-action and a right C -action on E such that b • (e • c) = (b • e) • c for composable b ∈ B, e ∈ E, and c ∈ C .(E2) There are continuous sesquilinear maps (e, f ) → B e , f from E * σ E to B and (e, f ) → e , f C from E * ρ E to C such that (i) p B B e , f = τ G (q(e), q(f )) and p C e , f C = τ H (q(e), q(f )), (ii) B e , f * = B f , e and e , f * C = f , e C , (iii) B b • e , f = b B e , f and e , f • c C = e , f C c, and (iv) B e , f • g = e • f , g C .(E3) With the actions and inner products coming from (E1) and (E2
FD88, Proposition II.13.10] shows the two definitions are equivalent.Remark 2.5 (The Literature).Continuous Banach bundles are treated in detail in § §13-14 in [FD88, Chap.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.If p : B → X is a Banach bundle, then the relative topology on B x is the (Banach space) norm topology.
and ǫ > 0 form a basis for this topology.Let p : B → X be a Banach bundle.We say that C ⊂ B is a Banach subbundle if each C x = B x ∩ C is a closed vector subspace of B x , and p| C : C → X is a Banach bundle when we give C x the Banach-space structure coming from B x and we give C the relative topology.
We say that the Fell bundle p : B → 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) ∈ G (2) , thenB g • B h := span{ ab : a ∈ B g and b ∈ B h } is always dense in B gh [MW08, Lemma 1.2].This is a consequence of (FB5).Some authors prefer to work with a weakened version of (FB5) where the inner products in (2.4) are not full.Remark 2.20.If p : B → G is a Fell bundle, then the restriction B| G (0) is a C * -bundle and Γ 0 (G (0) ; B) is a C * -algebra called the associated C * -algebra to B. 2 2.6.Equivalence of Fell Bundles.Suppose that T is a left G-space.Then we say that a Fell bundle p : B → G acts on (the left of) a Banach bundle q ), each E t is a B ρ(t) -C σ(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 : B → G be a Fell bundle over a groupoid G.We call C ⊂ B a Fell subbundle if C is a Banach subbundle such that p| C : C → G is a School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287 Email address: kaliszewski@asu.eduSchool of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287 Email address: quigg@asu.eduDepartment of Mathematics, Dartmouth College, Hanover, NH 03755-3551 USA Email address: dana.williams@Dartmouth.edu