Some new characterizations of central positive elements in C⁎-algebras

Abstract

In this paper, we give several characterizations for the centrality of elements in positive definite cones of C⁎-algebras. From the results to be presented, we mention only two. The first one is a characterization of centrality which is related to the usual order and to the positive part of selfadjoint elements which then easily implies Sherman's famous result characterizing commutative C⁎-algebras. Furthermore, we give a substantially new type of characterization of central positive definite elements in terms of a triangle inequality. Namely, we show that for a certain generalized distance measure (emerging from the Kubo-Ando geometric mean), the triangle inequality is satisfied for a given positive definite element A and for all positive definite elements B,C of a C⁎-algebra exactly when A is central. In the proofs of the results, Kadison's transitivity theorem plays a fundamental role.