Tracially reflexive C*-algebras

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 Schröder-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.