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On ${}^*$-similarity in $C^*$-algebras

Volume 252 / 2020

L. W. Marcoux, H. Radjavi, B. R. Yahaghi Studia Mathematica 252 (2020), 93-103 MSC: Primary 46L05; Secondary 47A05. DOI: 10.4064/sm190102-29-4 Published online: 11 December 2019


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.


  • L. W. MarcouxDepartment of Pure Mathematics
    University of Waterloo
    Waterloo, Ontario, Canada N2L 3G1
  • H. RadjaviDepartment of Pure Mathematics
    University of Waterloo
    Waterloo, Ontario, Canada, N2L 3G1
  • B. R. YahaghiDepartment of Mathematics
    Faculty of Sciences
    Golestan University
    Gorgan 49138-15759, Iran

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