We dedicate this article to the memory of William B. Arveson, who has inspired all of us with his vision and deep insights that completely changed the way we look at operator theory.

In the 1970s, Feldman and Moore classified separably acting von Neumann algebras containing Cartan MASAs using measured equivalence relations and 2-cocycles on such equivalence relations. In this paper, we give a new classification in terms of extensions of inverse semigroups. Our approach is more algebraic in character and less point-based than that of Feldman-Moore. As an application, we give a restatement of the spectral theorem for bimodules in terms of subsets of inverse semigroups. We also show how our viewpoint leads naturally to a description of maximal subdiagonal algebras.

Abstract. Let S be the semigroup S = ⊕k i=1 S i , where for each i ∈ I, S i is a countable subsemigroup of the additive semigroup R + containing 0. We consider representations of S as contractions {T s } s∈S on a Hilbert space with the Nica-covariance property: T * s T t = T t T * s whenever t ∧ s = 0. We show that all such representations have a unique minimal isometric Nica-covariant dilation.This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant representations of the action of S on an operator algebra A by completely contractive endomorphisms. We conclude by calculating the C * -envelope of the isometric nonself-adjoint semicrossed product algebra (in the sense of Kakariadis and Katsoulis).

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

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.