Introduction to Operator Space Theory
Cambridge University Press, 25.08.2003 - 478 Seiten
The theory of operator spaces is very recent and can be described as a non-commutative Banach space theory. An 'operator space' is simply a Banach space with an embedding into the space B(H) of all bounded operators on a Hilbert space H. The first part of this book is an introduction with emphasis on examples that illustrate various aspects of the theory. The second part is devoted to applications to C*-algebras, with a systematic exposition of tensor products of C*-algebras. The third (and shorter) part of the book describes applications to non self-adjoint operator algebras, and similarity problems. In particular the author's counterexample to the 'Halmos problem' is presented, as well as work on the new concept of 'length' of an operator algebra. Graduate students and professional mathematicians interested in functional analysis, operator algebras and theoretical physics will find that this book has much to offer.
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analog assume Banach space bounded c.b. map C*-algebra Chapter clearly commuting completely contractive completely isometric completely isometric embedding completely isomorphic completely positive Consider converse Corollary definition direct sum E C B(H easy to check element embeds equipped equivalent exact Exercise extends factorization finite-dimensional subspace fortiori free group free product Haagerup tensor product hence Hilbert space Hilbertian homomorphism implies inclusion inequality infimum infimum runs injective Kirchberg Lemma Let us denote linear map linear span locally reflexive matrix metric surjection Mn(A Mn(E Moreover morphism n-dimensional n-tuple Neumann algebra noncommutative norm notation Note nuclear OAU(E obtain operator algebra operator space structure polynomial preceding projection Proposition proved quotient Remark resp restriction result satisfies sequence space H subalgebra subset supremum supremum runs surjection tensor product ultraproduct unit ball unital homomorphism unital operator algebra unitary von Neumann algebra
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