Tracially reflexive C*-algebras

TL;DR

This paper introduces the concept of tracially reflexive C*-algebras, explores their properties, and provides criteria for their identification, advancing understanding of their structure and relation to existing algebra classes.

Contribution

It defines tracially reflexive C*-algebras, proves key properties, and offers criteria for their recognition, addressing a question posed by L. Robert.

Findings

Commutative C*-algebras are tracially reflexive.

Tracial reflexiveness is preserved under inductive limits.

Separable topological dimension zero C*-algebras are tracially reflexive.

Abstract

Motivated by a question of L. Robert, asking whether $\rm L(T(A)) = Lsc_{C}(T(A))$ for any separable C*-algebra A, we introduce and initiate the study of \emph{tracially reflexive C*-algebras}. We first prove that commutative C*-algebras are tracially reflexive. We also prove that tracial reflexiveness satisfies permanence properties, such as being preserved under inductive limits. Subsequently, we expose two criteria for tracial reflexiveness, using the Cuntz semigroup and a weak version of the Schr\"{o}der-Simpson theorem, respectively. In particular, separable topological dimension zero C*-algebras are tracially reflexive. We end the manuscript by closing remarks that could lead to further lines of investigation involving tracial reflexiveness.