Invariant approximation property for direct product with a finite group

Abstract

We will study the invariant approximation property in various con texts. An interesting question, which we will address next is the behavior of this property with respect to group extensions. To prepare for that we first study a relationship of uniform Roe algebras attached to coarsely equivalent metric spaces in the following case. Let X be a bounded geometry metric space and assume that there is a bijective coarse equivalence φ : X −→ Y × N, where N is a finite metric space. Then there is an isomorphism C ∗ u (X) ∼= C ∗ u (Y ) ⊗ C ∗ u (N) ∼= C ∗ u (Y ) ⊗ Mn(C), where n = |N|. We shall use this result to prove that the invariant approxi mation property is preserved under taking direct product with a finite group : let H be a discrete group with the IAP and K a finite group. Then the direct product G = H × K has IAP.