On *-similarity in C*-algebras

Abstract

Two subsets X and Y of a unital C -algebra A are said to be -similar via s 2 A􀀀1 if Y = s􀀀1Xs and Y = s􀀀1X s. We show that this relation imposes a certain structure on the sets X and Y, and that under certain natural conditions (for example, if X is bounded), -similar sets must be unitarily equivalent. As a consequence of our main results, we present a generalized version of a well-known theorem of W. Specht.