Cartan Subalgebras in $C^*$-Algebras

Abstract

According to J. Feldman and C. Moore's well-known theorem on Cartan subalgebras, a variant of the group measure space construction gives an equivalence of categories between twisted countable standard measured equivalence relations and Cartan pairs, i.e. a von Neumann algebra (on a separable Hilbert space) together with a Cartan subalgebra. A. Kumjian gave a C*-algebraic analogue of this theorem in the early eighties. After a short survey of maximal abelian self-adjoint subalgebras in operator algebras, I present a natural definition of a Cartan subalgebra in a C*-algebra and an extension of Kumjian's theorem which covers graph algebras and some foliation algebras.