Endomorphisms of Z-absorbing C*-algebras without conditional expectations

TL;DR

This paper constructs a specific endomorphism of the Jiang-Su algebra lacking a conditional expectation, revealing new structural properties and implications for the classification of certain nuclear C*-algebras.

Contribution

It provides the first example of a $ ext{Z}$-endomorphism without a conditional expectation, impacting the understanding of $ ext{Z}$-absorbing C*-algebras.

Findings

Constructed a $ ext{Z}$-endomorphism without a conditional expectation.

Showed that $ ext{Z}$-absorbing C*-algebras are non-transportable in $ ext{O}_2$.

Answered a question posed by Kirchberg.

Abstract

We construct an endomorphism of the Jiang-Su algebra $\mathcal{Z}$ which does not admit a conditional expectation. This answers a question in the testamentary homework by E. Kirchberg. As an application, it is shown that any unital separable nuclear $\mathcal{Z}$-absorbing C*-algebra is non-transportable in the Cuntz algebra $\mathcal{O}_2$.