Purely infinite crossed products by endomorphisms

TL;DR

This paper investigates the structure of crossed product $C^*$-algebras generated by injective endomorphisms, demonstrating how to modify endomorphisms to achieve purely infinite properties while preserving $K$-theory.

Contribution

It introduces a method to twist endomorphisms of $ ext{D}$-absorbing $C^*$-algebras to obtain purely infinite crossed products with diverse ideal structures.

Findings

Dilation of Bernoulli $p$-shift endomorphism is topologically free.

Constructs purely infinite crossed products with varied ideal structures.

Preserves $K$-theory during the twisting process.

Abstract

We study the crossed product $C^*$-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 $C^*$-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 $C^*$-algebras with diverse ideal structures.