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Compact homomorphisms of $ C\sp *$-algebras


Author: F. Ghahramani
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
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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

DOI: https://doi.org/10.1090/S0002-9939-1988-0943066-7
Keywords: Compact homomorphism, finite rank operator, group of unitary elements, semisimple Banach algebra, simple pole, spectral idempotent
Article copyright: © Copyright 1988 American Mathematical Society

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