On -similarity in C^*-algebras
Tom 252 / 2020
Studia Mathematica 252 (2020), 93-103
MSC: Primary 46L05; Secondary 47A05.
DOI: 10.4064/sm190102-29-4
Opublikowany online: 11 December 2019
Streszczenie
Two subsets \mathcal X and \mathcal Y of a unital C^*-algebra \mathcal A are said to be {}^*-similar via s \in \mathcal A ^{-1} if \mathcal Y = s^{-1} \mathcal X s and \mathcal Y ^* = s^{-1} \mathcal X ^* s. We show that this relation imposes a certain structure on the sets \mathcal X and \mathcal Y , and that under certain natural conditions (for example, if \mathcal 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.
