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Similarity Problems and Completely Bounded Maps (Paperback, 2nd exp. ed. 2001)
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Similarity Problems and Completely Bounded Maps (Paperback, 2nd exp. ed. 2001)
Series: Lecture Notes in Mathematics, 1618
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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.
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