Noncommutative Cartan subalgebras of C*-algebras

摘要

J. Renault has recently found ageneralization of the characterization of C*-diagonals obtained byA. Kumjian in the eighties, which in turn is a C*-algebraic version ofJ. Feldman and C. Moore's well known theorem on Cartan subalgebras ofvon Neumann algebras. Here we propose to give a version of Renault'sresult in which the Cartan subalgebra is not necessarily commutative[sic]. Instead of describing a Cartan pair as a twisted groupoidC*-algebra we use N. Sieben's notion of Fell bundles over inversesemigroups which we believe should be thought of astwisted étale groupoids with noncommutative unit space. En passant we prove a theorem on uniqueness ofconditional expectations.