Classification of homomorphisms into simple Z-stable C^*-algebras

TL;DR

This paper classifies unital monomorphisms into certain simple Z-stable C*-algebras up to approximate unitary equivalence, expanding the understanding of structure and classification in the realm of operator algebras.

Contribution

It provides a classification framework for monomorphisms into a broad class of simple Z-stable C*-algebras, including new cases with specific tracial rank conditions.

Findings

Classification of monomorphisms up to approximate unitary equivalence.

Applicable to a wide class of Z-stable C*-algebras with tracial rank zero.

Extends previous classification results to more general algebraic settings.

Abstract

We classify unital monomorphisms into certain simple Z-stable C^*-algebras up to approximate unitary equivalence. The domain algebra C is allowed to be any unital separable commutative C^*-algebra, or any unital simple separable nuclear Z-stable C^*-algebra satisfying the UCT such that C\otimes B is of tracial rank zero for a UHF algebra B. The target algebra A is allowed to be any unital simple separable Z-stable C^*-algebra such that A\otimes B has tracial rank zero for a UHF algebra B, or any unital simple separable exact Z-stable C^*-algebra whose projections separate traces and whose extremal traces are finitely many.