Almost Commuting Unitaries and Classification of Purely Infinite SimpleC*-Algebras

We show that, for any irrational rotational algebraAθ,Aθ⊗O2≅O2. This is proved by combining recently established results forC*-algebras of real rank zero with the following result: For anyε>0, there isδ>0, such that for any pair of unitariesu,vin any purely infinite simpleC*-algebraAwithK1(A)=0 satisfies‖uv−vu‖<δandκ(u, v)=0.There is a pair of commuting unitariesU,V∈Awith finite spectrum such that‖u−U‖<εand‖v−V‖<ε.More general results concerning almost commuting unitaries are also obtained.