# Similarity Problems and Completely Bounded Maps

## Second, Expanded Edition

Part of the Lecture Notes in Mathematics book series (LNM, volume 1618)

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Part of the Lecture Notes in Mathematics book series (LNM, volume 1618)

These notes revolve around three similarity problems, appearing in three different contexts, but all dealing with the space B(H) of all bounded operators on a complex Hilbert space H. The first one deals with group representations, the second one with C* -algebras and the third one with the disc algebra. We describe them in detail in the introduction which follows. This volume is devoted to the background necessary to understand these three problems, to the solutions that are known in some special cases and to numerous related concepts, results, counterexamples or extensions which their investigation has generated. While the three problems seem different, it is possible to place them in a common framework using the key concept of "complete boundedness", which we present in detail. Using this notion, the three problems can all be formulated as asking whether "boundedness" implies "complete boundedness" for linear maps satisfying certain additional algebraic identities. Two chapters have been added on the HALMOS and KADISON similarity problems.

Algebra Completely bounded representation H (p) spaces Harmonic analysis Hilbert space Polynomially bounded operator homomorphism operator algebras similar to contraction similar to unitary uniformly bounded repre uniformly bounded representation

- DOI https://doi.org/10.1007/b55674
- Copyright Information Springer-Verlag Berlin Heidelberg 2001
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Springer Book Archive
- Print ISBN 978-3-540-41524-4
- Online ISBN 978-3-540-44563-0
- Series Print ISSN 0075-8434
- Series Online ISSN 1617-9692
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