PhD Defense Rasmus Sylvester Bryder

Title: Boundaries, injective envelopes, and reduced crossed products

Abstract:

We study boundary actions, equivariant injective envelopes, as well as the ideal structure of reduced crossed products. These topics have recently been linked to the study of C*-simple groups, that is, groups with simple reduced group C*-algebras.

In joint work with Matthew Kennedy, we consider reduced twisted crossed products over C*-simple groups. For any twisted C*-dynamical system over a C*-simple group, we prove that there is a one-to-one correspondence between maximal invariant ideals in the underlying C*-algebra and maximal ideals in the reduced crossed product. When the amenable radical of the underlying group is trivial, we verify a one-to-one correspondence between invariant tracial states on the underlying C*-algebra and tracial states on the reduced crossed product.

In subsequent joint work with Tron Omland, we give criteria ensuring C*-simplicity and the unique trace property for a non-ascending countable HNN extension. This is done by both purely algebraic and dynamical methods. Moreover, we also characterize C*-simplicity of such an HNN extension in terms of the boundary action on its Bass-Serre tree.

We finally consider equivariant injective envelopes of unital C*-algebras, and relate the intersection property for group actions on unital C*-algebras to the intersection property for the equivariant injective envelope. We show that there is a C*-algebraic inclusion of the equivariant injective envelope of the centre of the injective envelope of a unital C*-algebra A in the centre of the equivariant injective envelope of A.

 Supervisor: Assoc. Prof. Magdalena Musat, MATH, University of Copenhagen

 Assessment committee:

 Prof. Henrik Schlichtkrull (Chairman), MATH, University of Copenhagen

 Prof. Marius Dadarlat, Purdue University

 Prof. Erik Bedos, University of Oslo