Similarity problems and completely bounded maps
The lecture notes by G. Pisier are centred around the following three similarity problems: (1) Is every continuous bounded representation \(\pi\) of a locally compact group \(G\) on a Hilbert space \(H\) similar to a unitary representation? (In other words, does there exist an invertible operator \(S: H\to H\) such that the representation \(S^{- 1} \pi S\) is a unitary representation?) (2) Is every bounded unital homomorphism \(u: A\to B(H)\) mapping a \(C^*\)-algebra \(A\) into the algebra \(B(H)\) of all bounded operators acting on a Hilbert space \(H\) similar to a \(*\)-representation? (3) Is every polynomially bounded operator acting on a Hilbert space similar to a contraction? The author provides a very nice exposition of various results, concepts and counterexamples related to these problems and reflecting recent development in the theory that surrounds them. The fundamental concept of the book is the notion of complete boundedness of operators. The author shows that all similarity problems mentioned above can be placed in a common context and reformulated as a question of complete boundedness of some bounded linear operators. The text is organized into seven chapters. The first chapter is devoted to a von Neumann inequality for a contraction on a Hilbert space and its extension (due to Ando) for two commuting contractions. The interesting fact that Ando's result is not valid for three commuting contractions is commentated. Moreover, Hilbert spaces are characterized as the only Banach spases satisfying von Neumann inequality. The second chapter deals with bounded representations of discrete groups. The space \(B(G)\) of all matrix coefficients of the unitary representations of \(G\) is introduced and studied. It is shown that the free group on three generators admits a non-unitarizable uniformly bounded group representation. Besides, several characterizations of amenable groups are given in terms of function spaces defined on them. The third chapter is devoted to the key notion of the book -- complete boundedness. The fundamental factorization (extension) theorem for completely bounded maps is proved. The proof is based on an elegant author's approach. In the fourth chapter the complete boundedness of homomorphisms is studied. The main result of this chapter is the following theorem (due to Paulsen): A unital bounded homomorphism mapping a \(C^*\)-algebra into an algebra \(B(H)\) is completely bounded if and only if it is similar to a \(*\)-representation. In connection with this result it is proved that any derivation \(\delta: A\to B(H)\) is completely bounded if and only if it is inner. Applying the main theorem for the disc algebra, the author derives Paulsen's theorem saying that an operator on a Hilbert space is similar to a contraction if and only if it is completely polynomially bounded. In the fifth chapter Schur multipliers and their connection with Grothendieck's inequality are studied. The unital ball of the space of Schur multipliers is characterized and the Grothendieck's inequality is then viewed as a corollary of this result. The Grothendieck constant is also discussed. In the sixth short chapter Hankel-Schur multipliers and Herz-Schur multipliers are investigated. The seventh chapter is focused on similarity problems for bounded homomorphisms. The crucial result of this chapter is the Haagerup theorem saying that any unital cyclic homomorphism of a \(C^*\)-algebra is similar to a \(*\)-representation. The proof is based on an interesting inequality for general linear map between \(C^*\)-algebras proved by the author. Positive solutions of problem (3) for \(C^*\)-algebras without tracial states and nuclear \(C^*\)-algebras are then presented. Finally, it is shown that problem (3) is in essential equivalent to the problem whether all derivations are inner. In the concluding chapter the notion of \(p\)-complete boundedness is discussed. The book is extremely well organized and readable. It should become a part of the working library of anyone interested in operator algebras, operator theory, Banach space theory and related fields. By revealing interesting connections between various ideas of modern functional analysis the book is an inspirating addition to the literature, a quality worthy of one who has contributed so much to the development and understanding of the subject.
bounded representations of discrete groups
scientific article; zbMATH DE number 825912