Purely infinite crossed products by endomorphisms

Abstract

We study the crossed product $C^{\ast }$-algebra associated to injective endomorphisms, which turns out to be equivalent to study the crossed product by the dilated autormorphism. We prove that the dilation of the Bernoulli $p$-shift endomorphism is topologically free. As a consequence, we have a way to twist any endomorphism of a $\D$-absorbing $C^{\ast }$-algebra into one whose dilated automorphism is essentially free and have the same $K$-theory map than the original one. This allows us to construct purely infinite crossed products $C^{\ast }$-algebras with diverse ideal structures.

Corrected misprints. Clarified some points in Section 3. Added a reference. Re-submitted to JMAA

References in corpus (1)

• The ideal structure of reduced crossed products