Intermediate C*-algebras of Cartan Embeddings

Abstract

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 ⊆ B ⊆ 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 Σ → G is the twist associated with the embedding D ⊆ A.

A correspondence between inverse subsemigroups, open wide subgroupoids and Cartan intermediate C*-subalgebras

“…Our main theorem, Theorem 2.1.10, states that wide open subgroupoids of associated groupoids with strongly tight actions corresponds to certain subsemigroups of the inverse semigroups. Combining with the work in [1], we obtain a correspondence between Cartan intermediate subalgebras in groupoid C*-algebras and certain subsemigroups of inverse semigroups. As an application, we compute all Cartan intermediate subalgebras of the Cuntz algebras which contains the fixed point algebras.…”

“…Proof. First we show (1). One can see that ker σ is a wide subsemigroup of S in a straightforward way.…”

“…[1, Lemma 3.2] states that Σ H • • = q −1 (H) naturally becomes a twist over H and there exists a natural inclusion C * λ (Σ H ) ⊂ C * λ (Σ). The authors in [1] showed this map H → C * λ (Σ H ) gives a certain correspondence as follows. Corollary 4.1.4.…”

“…A subsemigroup T ⊂ I(G) is said to be join closed if all joins of compatible pair of T also belongs to T . 1) A map from a groupoid to a group is called a cocycle if it preserves the products.…”

“…We define Renault's cerebrated theorem states that a Cartan pair arises from a twisted groupoid. We refer to [1] and [11] for twists of étale groupoids. A twisted groupoid over G is a topological groupoid Σ with the central extension…”