Titlebook: Similarity Problems and Completely Bounded Maps; Gilles Pisier Book 2001Latest edition Springer-Verlag Berlin Heidelberg 2001 Algebra.Comp

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0075-8434 n 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.978-3-540-41524-4978-3-540-44563-0Series ISSN 0075-8434 Series E-ISSN 1617-9692

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2. Non-unitarizable uniformly bounded group representations,this holds whenever . contains a subgroup isomorphic to .. We give several related characterizations of amenable groups. Then we extend the method to the case of semi-groups. This allows us to produce (this time for .) examples of power bounded operators which are not polynomially bounded.

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3. Completely bounded maps,and give several consequences. In this viewpoint, the underlying idea is the same in both cases (completely bounded maps or operators factoring through Hilbert space). At the end of this chapter, we give several examples of bounded linear maps which are not completely bounded, and related norm estimates.

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6. Hankelian Schur multipliers. Herz-Schur multipliers,e Hardy space .. Analogously, when . and . is the reduced .-algebra ., then the Schur multipliers which are completely bounded maps from . to . are identical to the completely bounded multipliers of . or equivalently to the (so-called) Herz-Schur multipliers of ..

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,1. Von Neumann’s inequality and Ando’s generalization,discuss the case of . > 2 mutually commuting contractions. We introduce the notion of semi-invariance. Finally, we show that Hilbert spaces are the only Banach spaces satisfying von Neumann’s inequality.

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4. Completely bounded homomorphisms and derivations,mostly concentrate on the Hilbert space case. In the latter case, we prove the Jundamental result that a unital homomorphism is completely bounded iff it is similar to a completely contractive one. Let . be a derivation on a .-algebra. We show that . is completely bounded iff it is inner. When . is

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6. Hankelian Schur multipliers. Herz-Schur multipliers,Hankel matrices. We show that the Schur multipliers which are completely bounded maps from . to . are closely related to the Fourier multipliers on the Hardy space .. Analogously, when . and . is the reduced .-algebra ., then the Schur multipliers which are completely bounded maps from . to . are id