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Proceedings of the American Mathematical Society

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

 

 

Completely bounded homomorphisms of operator algebras


Author: Vern I. Paulsen
Journal: Proc. Amer. Math. Soc. 92 (1984), 225-228
MSC: Primary 47D25; Secondary 46L05
DOI: https://doi.org/10.1090/S0002-9939-1984-0754708-X
MathSciNet review: 754708
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Abstract: Let $ A$ be a unital operator algebra. We prove that if $ \rho $ is a completely bounded, unital homomorphism of $ A$ into the algebra of bounded operators on a Hilbert space, then there exists a similarity $ S$, with $ \left\Vert {{S^{ - 1}}} \right\Vert \cdot \left\Vert S \right\Vert = {\left\Vert \rho \right\Vert _{cb}}$, such that $ {S^{ - 1}}\rho ( \cdot )S$ is a completely contractive homomorphism. We also show how Rota's theorem on operators similar to contractions and the result of Sz.-Nagy and Foias on the similarity of $ \rho $-dilations to contractions can be deduced from this result.


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DOI: https://doi.org/10.1090/S0002-9939-1984-0754708-X
Article copyright: © Copyright 1984 American Mathematical Society