Irish Math. Soc. Bulletin 61 (2008), 29–63 29 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 re-lations 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 nat-ural definition of a Cartan subalgebra in a C∗-algebra and an extension of Kumjian’s theorem which covers graph algebras and some foliation algebras. 1