  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.
