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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Compact homomorphisms of $C^ *$-algebras
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by F. Ghahramani PDF
Proc. Amer. Math. Soc. 103 (1988), 458-462 Request permission

Abstract:

Suppose $A$ is a ${C^*}$-algebra and $B$ is a Banach algebra such that it can be continuously imbedded in $B(H)$, the Banach algebra of bounded linear operators on some Hilbert space $H$. It is shown that if $\theta$ is a compact algebra homomorphism from $A$ into $B$, then $\theta$ is a finite rank operator, and the range of $\theta$ is spanned by a finite number of idempotents. If, moreover, $B$ is commutative, then $\theta$ has the form $\theta (x) = {\mathcal {X}_1}(x){E_1} + \cdots + {\mathcal {X}_k}(x){E_k}$, where ${E_1}, \ldots ,{E_k}$ are fixed mutually orthogonal idempotents in $B$ and ${\mathcal {X}_1}, \ldots ,{\mathcal {X}_k}$ are fixed multiplicative linear functionals on $A$.
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 103 (1988), 458-462
  • MSC: Primary 46L05; Secondary 46J05
  • DOI: https://doi.org/10.1090/S0002-9939-1988-0943066-7
  • MathSciNet review: 943066