On approximation properties of group C* - algebras

Abstract

In this thesis we study analytic techniques from operator theory that encapsulate geometric properties of a group. Rapid Decay Property (Property RD) provides estimates for the operator norm of elements of the group ring (in the left-regular representation) in terms of the Sobolev norm. Roughly, property RD is the noncommutative analogue of the fact that smooth functions are continuous. Our work then concentrates on a particular form of an approximation property for the reduced C*- algebra of a group: the invariant approximation property. This statement captures a particular relationship between three important operator algebras associated with a group: the reduced C*- algebra, the von Neumann algebra, and the uniform Roe algebra. The main result is the proof of the invariant approximation property for groups equipped with a conditionally negative length function. We prove also that the invariant approximation property passes to sub- groups and then discuss the behaviour of the invariant approximation property with the respect to certain classes of extensions. We show that the invariant approximation property passes to direct products with finite group. We show that the invariant approximation property passes to extensions of the following form. If G is a discrete group and H is a finite index normal subgroup of G with IAP, then G has IAP

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ORCID for Jacek Brodzki: orcid.org/0000-0002-4524-1081

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