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 $A$.References
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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