Similarity problems and completely bounded maps / Gilles Pisier.Material type: TextSeries: Lecture notes in mathematics (Springer-Verlag): 1618.Publisher: Berlin ; New York : Springer, 2001Copyright date: ©2001Edition: Second, expanded editionDescription: 1 online resource (vi, 198 pages) : illustrationsContent type: text Media type: computer Carrier type: online resourceISBN: 9783540445630; 3540445633Subject(s): Linear operators | C*-algebras | Representations of groups | Mappings (Mathematics) | C*-algebras | Linear operators | Mappings (Mathematics) | Representations of groupsGenre/Form: Electronic books. Additional physical formats: Print version:: Similarity problems and completely bounded maps.DDC classification: 510 s | 512/.55 LOC classification: ML3506 | .S36 1968QA3 | .L28 v. 1618Online resources: Click here to access online
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"Includes the solution to 'The Halmos problem'."
Includes bibliographical references (pages 182-193) and index.
Von Neumann's inequality and Ando's generalization -- Non-unitarizable uniformly bounded group representations -- Completely bounded maps -- Completely bounded homomorphisms and derivations -- Schur multipliers and grothendieck's inequality -- Hankelian schur multipliers. Herz-Schur multipliers -- Similarity problem for cyclic homomorphisms on a C*-algebra -- Completely bounded maps in the banach space setting -- Sz.-nagy-halmos similarity problem -- Kadison similarity problem.
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.
Print version record.