Rapid Decay and Invariant Approximation Property

Abstract

Analytic properties of invariant approximation property, studies analytic tech niques from operator theory that encapsulate geometric properties of a group. We show that the following theorem holds: For a discrete group G satisfying the rapid decay property with respect to a conditionally negative length function `, the reduced C ∗ -algebra C ∗ r (G) has the invariant approximation property. We then use this to show that some groups have invariant approximation property. We also show that if G is a free product group satisfying the rapid decay prop erty with respect to a conditionally negative length function `, then the reduced C ∗ -algebra C ∗ r (G) has the invariant approximation property.