On ${}^*$-similarity in $C^*$-algebras
Volume 252 / 2020
Studia Mathematica 252 (2020), 93-103
MSC: Primary 46L05; Secondary 47A05.
DOI: 10.4064/sm190102-29-4
Published online: 11 December 2019
Abstract
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.
