DILATIONS AND COMMUTANT LIFTING FOR SUBALGEBRAS OF GROUPOID C*-ALGEBRAS

Abstract

Let B be a nuclear C*-algebra that has a diagonal subalgebra D in the sense of Kumjian and let A be a closed, not necessarily self-adjoint subalgebra of B that contains D such that A + A* is dense in B. We show that every contractive representation of A has an essentially unique minimal dilation to a C*-representation of B and that the commutant of the representation of A can be lifted to the commutant of the dilation without increasing norms.