Intermediate C*-algebras of Cartan Embeddings

Let $A$ be a C$^*$-algebra and let $D$ be a Cartan subalgebra of $A$. We study the following question: if $B$ is a C$^*$-algebra such that $D \subseteq B \subseteq A$, is $D$ a Cartan subalgebra of $B$? We give a positive answer in two cases: the case when there is a faithful conditional expectation from $A$ onto $B$, and the case when $A$ is nuclear and $D$ is a C$^*$-diagonal of $A$. In both cases there is a one-to-one correspondence between the intermediate C$^*$-algebras $B$, and a class of open subgroupoids of the groupoid $G$, where $\Sigma \rightarrow G$ is the twist associated with the embedding $D \subseteq A$.


Introduction
An interesting, and common, type of question in the study of operator algebras is the following. Suppose D is an algebra embedded in an algebra A in such a way that that the inclusion has some nice properties. If B is an algebra intermediate to Perhaps the most famous of these results is the Galois correspondence for the crossed products of von Neumann algebra factors by discrete groups. This says that if N is a factor von Neumann algebra, and G is a discrete group acting freely on N , then the map H → N H gives a one-to-one correspondence between subgroups of H ⊆ G and intermediate von Neumann algebras N ⊆ M ⊆ N G. This result is due to Izumi, Longo and Popa [24], building on the work of Choda [11]. An alternative, elegant proof was given by Cameron and Smith [9].
Similar Galois correspondence results have been proved for reduced crossedproducts of C * -algebras. If A is a C * -algebra and G is a discrete group acting on A by automorphisms, then Choda [12] gives a one-to-one correspondence between subgroups of G and a certain class of intermediate C * -algebras A ⊆ B ⊆ A r G. An intermediate algebra B is in the desired class if, among other things, there is a faithful conditional expectation A r G → B. Cameron and Smith [10], generalising an earlier result of Izumi [23] for finite groups, showed if A is a simple C * -algebra and G is a discrete group acting on A by outer automorphisms, then the map H → A H is a one-to-one correspondence between subgroups of H ⊆ G and C * -algebras B satisfying A ⊆ B ⊆ A G. In this case there is always a faithful conditional expectation from A r G onto an intermediate C * -algebra.
Beyond the rigid structure of crossed products, there are striking examples of these intermediate algebra type results. Suppose D is a Cartan subalgebra of a von Neumann algebra M . Aoi proved that if D ⊆ N ⊆ M , then D is also a Cartan subalgebra of N [2]. Cameron, Pitts and Zarikian have given an alternative proof of this theorem [8]. Feldman and Moore [19,20] gave a one-to-one correspondence between Cartan embeddings and measured equivalence relations. Thus, analogous to the Galois correspondence results mentioned above, Aoi's result gives a one-toone correspondence between sub-measured equivalence relations and von Neumann algebras D ⊆ N ⊆ M . Alternatively, Donsig, Fuller and Pitts [15] give a measurefree description of Cartan embeddings based on extensions of inverse semigroups. There, Aoi's theorem is used to show a one-to-one correspondence between a class of sub-inverse monoids of an inverse monoid S and intermediate von Neumann algebras D ⊆ N ⊆ M . These results are generalized beyond Cartan embeddings in [16].
In this note we address an analogous problem in the C * -algebra setting. Suppose D is a Cartan subalgebra of a C * -algebra A, in the sense of Renault [32]. Renault showed that there is a twist Σ q → G such that A can be identified with the reduced C * -algebra of the twist C * r (Σ; G) and D is identified with C(G (0) ). In Theorem 3.3, we exhibit a one-to-one correspondence between open subgroupoids H ⊆ G and C * -algebras B such that D ⊆ B ⊆ A and D is Cartan in B. We further show in Theorem 3.5 that this map gives a one-to-one correspondence between subgroupoids Stronger results can be found with some additional, very natural, hypotheses.
Two decades before Renault's work on Cartan subalgebras of C * -algebras [32], Kumjian introduced the stronger property of C * -diagonal [25]. In Theorem 4.5 we give a Galois-correspondence type result in this context. Suppose D is a C *diagonal of a nuclear C * -algebra A. Let Σ q → G be the twist corresponding to the embedding D ⊆ A. The map H → C * r (Σ H ; H) gives a one-to-one correspondence between open subgroupoids H ⊆ G with In contrast with the work of [10,12,23] there is not necessarily a conditional expectation onto the intermediate subalgebra.
Takeishi [34] showed nuclearity of A is equivalent to the amenability of G. Let G (0) ⊆ H ⊆ G be an open subgroupoid, and let Σ H → H be the corresponding subtwist of Σ → G. In Theorem 4.2, amenability of G is used to show that, viewing C * r (Σ H ; H) as a subalgebra of C * r (Σ; G), an element a ∈ C * r (Σ; G) is in C * r (Σ H ; H) if and only if a (when viewed as a function on G) vanishes off H. This is a key ingredient in the proof of Theorem 4.5. When the amenability condition is removed, we show in Theorem 4.4 that B is contained in the C * -algebra of functions supported on H and C * r (Σ H ; H) ⊆ B. In the final section we focus on the case of C * -algebraic crossed products, where we strengthen our results beyond the nuclear case. Let Γ be a discrete group acting on a compact Hausdorff space X by homeomorphisms. To apply Theorem 4.5 to a crossed product we would require that the algebra C(X) is a C * -diagonal in the reduced crossed-product C(X) Γ, and that C(X) Γ is nuclear. This happens if and only if Γ is an amenable group acting freely on X. We can, however, prove a version of Theorem 4.5 which does not require that Γ is amenable, and instead assume Γ has the approximation property of Cowling and Haagerup [13]. Let Γ be a group which satisfies the approximation property and acts freely on a compact Hausdorff space X. Let Γ × X be the corresponding transformation groupoid. We prove in Corollary 5.8 that the map H → C * r (H) gives a one-to-one correspondence between open subgroupoids H ⊆ Γ × X with {e} × X ⊆ H and all C * -algebras B satisfying C(X) ⊆ B ⊆ C(X) r Γ. Thus, while there is not a Galois correspondence from the subgroups of Γ, there is a Galois-type correspondence from the open subgroupoids of the transformation groupoid Γ × X. This result is further evidence of the value of the groupoid approach to C * -algebras, even in the relatively straightforward setting of crossed products by discrete group actions.
This Galois-type correspondence is a corollary of a spectral theorem for bimodules in C(X) r Γ. In Theorem 5.7 we show that there is a one-to-one correspondence between the open subsets of Γ × X and the norm-closed C(X)-bimodules in C(X) Γ. A similar spectral theorem is proved for actions of groups satisfying the approximation property on simple C * -algebras in [10]. This result is also a direct analogue of the Spectral Theorem for Bimodules for Cartan embeddings in von Neumann algebras; see [8,15,21].
In the von Neumann algebra setting, L ∞ (X, μ) is Cartan in the crossed product L ∞ (X, μ) Γ if and only if the action of Γ on the measure space (X, μ) is (measurably) free [35,Corollary V.7.7]. It has been argued, e.g. by Tomiyama [36], that topological freeness is the correct analogue of free actions on measure spaces. There is good reason for this viewpoint. However, Corollary 5.8 shows that there are settings when free actions, not topologically free actions, are needed in order to get desirable analogues of von Neumann algebra results.

Preliminaries
We recall the key details of Cartan embeddings in C * -algebras and their relation to twists. Recall that if A is a C * -algebra and D is a subalgebra, then n ∈ A normalizes D if nDn * ∪ n * Dn ⊆ D. We denote the normalizers of D in A by (iv) every pure state on D extends uniquely to a pure state on A, i.e D has the unique extension property in A.
It is worth noting that the conditional expectation E above is unique [32, Proposition 4.2]. Archbold, Bunce and Gregson [3, Corollary 2.7] classify precisely when an abelian subalgebra D of a C * -algebra A satisfies the unique extension property of Definition 2.1(iv) in terms of properties of the conditional expectation. This allows us to give the following alternative differentiation between C * -diagonals and Cartan embeddings when A is unital.

Theorem 2.2 (c.f. [3, Corollary 2.7]). Let D be a Cartan subalgebra of a unital C * -algebra A, with faithful conditional expectation
Suppose D is a Cartan subalgebra of A, with associated conditional expectation E. We will briefly recap how to construct the groupoid twist from the embedding D ⊆ A. As D is abelian, D C 0 (X) for a locally compact Hausdorff space X. If n ∈ A normalizes D then n * n ∈ D, and thus n * n can be viewed as a continuous positive function on X. Let By [25,Proposition 6] the normalizer n defines a partial homeomorphism β n on X, with domain s(n) and range s(n * ) satisfying Further, as a collection of linear functionals, Σ inherits the relative weak * -topology. Under this topology Σ is a topological groupoid. There is an action of the unit circle T on Σ by The groupoid G is given the quotient topology. Equivalently, the topology on G is generated by the basic sets D). Under this topology G is a topologically principal,étale groupoid [32]. If D is a C * -diagonal of A then G is a principal,étale groupoid [25]. This twist will usually be abbreviated to Σ → G, or Σ q → G when we wish to emphasize the quotient map.
To construct the associated line bundle L over G, we put the following equiv- The collection of continuous compactly supported cross-sections C c (Σ; G) of L becomes a * -algebra under the following operations. For f, g ∈ C c (Σ; G) the convolution product on C c (Σ; G) is defined by The adjoint is given by . A detailed description of the construction of the reduced C * -algebra C * r (Σ; G) can be found in [5, Section 2]. We recall main theorems of [25] and [32]. [25] and [32]). Let D be a Cartan subalgebra of a C * -algebra A.
). In the sequel we will do this without comment.
Remark 2.5. In both [25] and [32] only separable C * -algebras are studied. Theorem 2.3, however, does hold in the non-separable case if effective groupoids are considered in place of topologically principal groupoids [26,Corollary 7.6].
Our main result, Theorem 4.5, applies when the groupoid is amenable. Amenability of groupoids was introduced by Renault [31, Chapter II.3]. For our purposes we will use a characterization of amenability in terms of positive-type functions. Let G be a groupoid. A function h : We note the following theorem of Takeishi [34] which classifies when C * r (Σ; G) is nuclear. The case when the twist is trivial can be found in [7, Theorem 5.6.18], and for more general groupoids in [1, Corollary 6.2.14].
Theorem 2.7 (c.f. [34,Theorem 5.4]). Let Σ → G be a twist, with G anétale locally compact groupoid. Then C * r (Σ; G) is nuclear if and only if G is amenable. One can also construct the full C * -algebra C * (Σ; G) of the twist Σ → G by completing C c (Σ; G) with respect to the supremum norm over all * -representations We note the following result due to Sims and Williams [33]. . Let Σ → G be a twist, with G anétale, Hausdorff, amenable groupoid. Then the reduced C * -algebra C * r (Σ; G) is isomorphic to the full C * -algebra C * (Σ; G).

Intermediate subalgebras of Cartan embeddings
Our primary goal is to study C * -algebras B satisfying D ⊆ B ⊆ A, when D is Cartan in A. In particular, does the (twisted) groupoid C * -algebra structure of A force B to have a similar structure? The main result of this section, Theorem 3.5, gives a positive answer in the case when there is a faithful conditional expectation from A to B.
We recall the following result.
Here is a sketch of the proof. That Σ H → H is a twist is straightforward. The second claim requires some care. Define ι : That it is an isomorphism follows from the faithfulness of the conditional expectation of C * r (Σ; G) onto C * (G (0) ).
We now describe the intermediate subalgebras of Cartan embeddings. In the Galois correspondence results in [10,12,23]   Let U ⊆ G be an open bisection containing γ and choose f ∈ C c (Σ; G) supported on U such that |f (γ)| = 1. As γ n → γ, we may assume that each γ n ∈ U . Since H is open, there are open sets V n ⊆ H ∩ U such that γ n ∈ V n . For each n we choose g n ∈ C c (G (0) ) with g n supported on r(V n ) and g n (r(γ n )) = 1.
Since for each n, g n ∈ C c (G (0) ) ⊆ C c (Σ H ; H), it follows that Further, note g n * f is supported on V n ⊆ H. Thus g n * f ∈ C c (Σ H ; H) and F (g n * f ) = g n * f . Finally, note that for any h ∈ C * r (Σ; G), g n * h(γ n ) = h(γ n ), since g n (r(γ n )) = 1. Hence  N (B, D). Since D is regular in A, if b ∈ B there is a sequence (a n ) n ∈ span N (A, D) such that a n → b. Hence F (a n ) → F (b) = b. It follows that span N (B, D) is dense in B, so D is Cartan in B. By

Nuclear C * -algebras and intermediate algebras of C * -diagonals
In this section we consider C * -algebras A containing a C * -diagonal D, with associated twist Σ → G. We will see that if D ⊆ B ⊆ A and A is nuclear, then D is necessarily a C * -diagonal in B. This will give a one-to-one correspondence between open subgroupoids of G and the intermediate C * -algebras B, Theorem 4.5.
The following theorem tells us that, in the amenable case, determining whether or not an element a ∈ C * r (Σ; G) lies in C * r (Σ H ; H) or not, depends solely on the support of a as a function in C 0 (Σ; G).
We suspect this containment may be strict in some circumstances.  [34,Lemma 4.2], m h i extends to a contractive completely positive map on C * r (Σ; G). As G is amenable, C * r (Σ; G) = C * (Σ; G), by Theorem 2.8. Note that m h i (f ) converges to f in C * (Σ; G) = C * r (Σ; G), by Equation (1). We further note that, for (Σ H ; H). It follows that f ∈ C * r (Σ H ; H), concluding the proof. We recall the following corollary to Theorem 2.2. It was observed in the unital case by Muhly, Qiu and Solel in the proof of Proposition 4.4 of [29].

Proposition 4.3. Suppose D ⊆ A is a C * -diagonal. Then, for any a ∈ A and normalizer n ∈ N (A, D), nE(n * a) is a normalizer of D in the norm-closed Dbimodule generated by a.
Proof. If A is unital the result follows from Theorem 2.2. Details can be found in [17,Proposition 3.10].
Suppose now that A is not unital. LetÃ be the unitization of A. It can be shown that D +CI is a maximal abelian subalgebra ofÃ and D +CI has the unique extension property inÃ. By Note that if n ∈ N (A, D) then n ∈ N (Ã, D + CI). It follows now as in the unital case, that for any n ∈ N (A, D), nE(n * a) normalizes D and is in the norm-closed D-bimodule generated by A.
We now have all the ingredients for the main theorems of this section.
Thus E(n * b) = 0, and hence nE( Thus q([m, x]) = q(σ). Thus Corollary 4.6. Let A be a nuclear C * -algebra, and let D be a C * -diagonal of A.

Crossed products
Let Γ be a discrete group acting on a compact Hausdorff space X by homeomorphisms. The set Γ × X becomes the transformation groupoid under the partial multiplication (g 1 , g 2 · x)(g 2 , x) = ((g 1 g 2 ), x) and inversion (g, x) −1 = (g −1 , g · x). With the product topology the transformation groupoid Γ × X isétale, and clearly Hausdorff. Let C(X) r Γ be the reduced crossed product. One can show that C(X) r Γ = C * r (Γ × X). We view C(X) as a subalgebra of C(X) r Γ and denote the canonical unitary representation of Γ in C(X) r Γ by {u g } g∈Γ . Denote by E the usual faithful conditional expectation from C(X) r Γ to C(X); see [7,Proposition 4.1.9]. Recall that the action of Γ on X is free if for each g = e in Γ the set {g · x = x : x ∈ X} is empty. The action is topologically free if for each g = e in Γ the set {g·x = x : x ∈ X} has empty interior. Zeller-Meier [37] showed that C(X) is maximal abelian in C(X) r Γ if and only if the action of Γ on X is topologically free. Thus, C(X) is Cartan in C(X) r Γ if and only if the action of Γ on X is topologically free. Further C(X) is a C * -diagonal in C(X) r Γ if and only if the action of Γ on X is free. This can be seen by the fact that the transformation groupoid Γ × X is principal if and only if the action of Γ is free. One can alternatively show that freeness of the action of Γ implies that E(a) ∈ conv{uau * : u ∈ C(X), u unitary}, see [22,Proposition 11.1.19]. Thus C(X) has the unique extension property in C(X) r Γ by Theorem 2.2 if and only if the action of Γ on X is free. It follows that if Γ acts by a topologically free action on X then C(X) is Cartan in C(X) Γ with associated twist being simply the trivial twist on the transformation groupoid Γ × X.
Consider a reduced crossed product C(X) r Γ, where Γ is a discrete group acting on a compact Hausdorff space X. Recall that every element a ∈ C(X) r Γ is uniquely determined by its Fourier series. We write To be wholly consistent with the general theory of C * -diagonals discussed above one would consider the elements u g E(u * g a), as in Proposition 4.3, but we will stick the usual convention of setting a g = E(au * g ) when working with crossed products. Note that, if the action of Γ is free then, as in Proposition 4.3, a g u g is in the norm-closed C(X)-bimodule generated by a.
The elements of the transformation groupoid Γ × X act as linear functionals on C(X) r Γ. As our Fourier coefficients are of the form E(au * g )u g and not u g E(u * g a), we define the linear functionals slightly differently than in the general Cartan/C *diagonal case, with the normalizers acting on the other side. Thus, for a ∈ C(X) r Γ and (g, x) ∈ Γ × X we have , when a has Fourier series a ∼ g∈Γ a g u g .
The following example shows that Theorem 4.5 need not hold for Cartan embeddings even when C(X) r Γ is nuclear.
Example 5.1. Let D be the closed unit disk in C. Let Z act on D by an irrational rotation. That is, we fix α ∈ T such that {α n : n ∈ N} is infinite and for n ∈ Z and z ∈ D, we define n · z = α n z.
Then the action of Z on D is topologically free. It is not a free action since 0 is fixed. indeed, for N ∈ N, For any x ∈ D and r ∈ Z, a calculation shows that for f ∈ C(D) and n ∈ Z, On the other hand, when r is even, u r ∈ B, so for x ∈ D, (x, r)(u r ) = 0. Hence, Example 5.1 shows that in general there is not a nice correspondence between subgroupoids and intermediate C * -algebras of a Cartan embedding. It may be that Theorem 3.5 is the best we can hope for in that generality. The following corollary to Theorem 3.5 recovers a special case of Choda's results [12]. Proof. Since X is connected the only subgroupoids of Γ × X which are both closed and open are of the form H × X where H ⊆ Γ is a subgroup. Thus, the result follows from Theorem 3.5.
We now turn our attention to free actions. To apply Theorem 4.5 directly we also need to know when C(X) r Γ is nuclear. This happens precisely when the action of Γ on X is amenable, which in turn happens precisely when the transformation groupoid Γ × X is amenable. Thus, for C(X) to be a C * -diagonal in a nuclear C(X) r Γ we need the action of Γ on X to be free and amenable. However, if the action of Γ on X is both free and amenable, then Γ itself must be amenable [28,Corollary 4.3]. Hence Theorem 4.5, when applied to crossed products, applies only to free actions of amenable groups. We can loosen this condition.
If a ∈ C(X) Γ with Fourier series a ∼ g∈Γ a g u g . One cannot expect the series g∈Γ a g u g to converge in norm (though it does converge in the Bures-topology in the enveloping von Neumann algebra [27]). With the right class of groups Γ we can, however, recover any element a ∈ C(X) r Γ from its Fourier series.

Definition 5.3.
A discrete group Γ has the approximation property if 1 is in the weak * -closure of the space of finitely supported functions on Γ. 1 The approximation property was introduced by Cowling and Haagerup in [13], where they showed that both amenable and weakly amenable groups satisfy the approximation property. The following result was proved by Bédos and Conti [4,Section 5], generalizing earlier work of Exel [18].
Theorem 5.4. [4] Let Γ be a discrete group satisfying the approximation property and let C(X) r Γ be a reduced crossed product. If a ∈ C(X) r Γ has Fourier series a ∼ g∈Γ a g u g , then a ∈ span{a g u g : g ∈ Γ}.
Remark 5.5. Theorem 5.4 applies to twisted crossed products. Thus, the results which follow will also apply in that generality. Corollary 5.6. Let Γ be a discrete group satisfying the approximation property which acts freely on a compact Hausdorff space X. If M ⊆ C(X) r Γ is a normclosed C(X)-bimodule then M = span{n ∈ N (M, C(X))} = span{fu g : f ∈ C(X), fu g ∈ M }.
Proof. Since Γ acts freely on X, if a ∈ M with a ∼ g∈Γ a g u g , then a g u g ∈ M , by Proposition 4.3. The result follows now from Theorem 5.4.
In [10], Cameron and Smith prove a spectral theorem for A-bimodules in A r Γ, where Γ is a discrete group satisfying the approximation property, and A is a simple C * -algebra. There, the bimodules are determined by subsets of Γ. When Γ acts freely on a compact Hausdorff space X, we show that the C(X)-bimodules are determined by open subsets of the transformation groupoid Γ × X.
For g ∈ Γ we denote by π g the projection map on subsets U ⊆ Γ × X defined by π g (U ) = {x ∈ X : (g, x) ∈ U }. Theorem 5.7 (Spectral Theorem for Bimodules). Let Γ be a discrete group satisfying the approximation property, which acts freely on a compact Hausdorff space X. The map U → span{fu g : f ∈ C(X), supp(f ) ⊆ π g (U )}, defines a one-to-one correspondence between open subsets U ⊆ Γ × X and closed C(X)-bimodules in C(X) r Γ.
Proof. Let U ⊆ Γ × X be open. Let Then N U is a norm-closed C(X)-bimodule. To show that the map Φ : U → N U is a surjective map from open subsets of Γ×X and closed C(X)-bimodules in C(X) r Γ we construct the inverse map.
Let N ⊆ C(X) r Γ be a norm-closed C(X)-bimodule. For each g ∈ Γ define the set U g = {supp(f ) : f ∈ C(X), fu g ∈ N }.
Note that since N is a C(X)-bimodule the set is an ideal in C(X). Indeed J g = C 0 (U g ). Let Since each U g is open, U N is an open subset of Γ × X. Denote by Ψ the map Ψ: N → U N . That Ψ is the inverse of Φ follows from Corollary 5.6.
Since, if B is a C * -algebra satisfying C(X) ⊆ B ⊆ C(X) r Γ, then B is a C(X)-bimodule, we get the following Corollary, which extends Theorem 4.5.