Cartan-preserving *-automorphism groups: realization and obstructions for compact abelian groups

Abstract

In this paper, we investigate automorphism groups preserving Cartan subalgebras of C*-algebras. First, we describe these groups in terms of automorphisms and 1-cocycles of twisted \'etale groupoids. As a consequence, we obtain a C*-algebraic analogue of a theorem of Feldman and Moore on Cartan-preserving automorphisms of von Neumann algebras. We then study Cartan-fixing automorphism groups. We show that every UCT Kirchberg algebra admits a Cartan subalgebra whose Cartan-fixing automorphism group contains every second countable compact abelian group. In contrast, for C*-algebras arising from expansive effective groupoids, we prove that compact Cartan-fixing automorphism groups must have finitely generated Pontryagin duals. As an application, we establish the existence of inequivalent Cartan subalgebras for Kirchberg algebras arising from expansive effective groupoids.