Simultaneous averaging to zero by unitary mixing operators

TL;DR

This paper proves that subspaces of a C*-algebra can be simultaneously averaged to zero using unitary mixing operators if individual elements can, and characterizes these subspaces via commutators and state kernels, with applications to center-valued expectations.

Contribution

It establishes a criterion for simultaneous averaging to zero in C*-algebras and provides a description of such subspaces, extending previous understanding of unitary mixing operators.

Findings

Subspaces can be averaged to zero simultaneously if individual elements can.

Characterization of subspaces via commutators and kernels of states.

Application to center-valued expectations in C*-algebras with the Dixmier property.

Abstract

We show that if every element a vector subspace of a C*-algebra can be averaged to zero by means of unitary mixing operators, then all the elements of the subspace can be simultaneously averaged to zero by a net of unitary mixing operators. Moreover, such subspaces admit a simple description in terms of commutators and kernels of states on the C*-algebra. We apply this result to center-valued expectations in C*-algebras with the Dixmier property.