On approximately inner automorphisms of certain crossed product 𝐶*-algebras

Abstract

Let G G be a compact connected topological group having a dense subgroup isomorphic to Z \mathbf {Z} . Let C ( G ) ∝ ⋊ Z C(G) \stackrel {\rtimes }{\propto } \mathbf {Z} be the crossed product C ∗ C^* -algebra of C ( G ) C(G) with Z \mathbf {Z} , where Z \mathbf {Z} acts on G G by rotations. Automorphisms of C ( G ) ∝ ⋊ Z C(G) \stackrel {\rtimes }{\propto } \mathbf {Z} leaving invariant the canonical copy of C ( G ) C(G) are shown to be approximately inner iff they act trivially on K 1 ( C ( G ) ∝ ⋊ Z ) K_1(C(G) \stackrel {\rtimes }{\propto } \mathbf {Z}) .