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.
