Introduction to Operator Space TheoryThe 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. |
Contents
II | 17 |
III | 28 |
V | 34 |
VI | 40 |
VII | 42 |
VIII | 43 |
IX | 47 |
X | 51 |
XL | 178 |
XLI | 182 |
XLII | 183 |
XLIII | 191 |
XLIV | 200 |
XLV | 210 |
XLVI | 215 |
XLVII | 217 |
XI | 52 |
XII | 59 |
XIII | 63 |
XIV | 64 |
XV | 65 |
XVI | 67 |
XVII | 68 |
XVIII | 71 |
XIX | 81 |
XX | 86 |
XXII | 92 |
XXIII | 93 |
XXV | 98 |
XXVI | 101 |
XXVII | 102 |
XXVIII | 106 |
XXIX | 109 |
XXX | 122 |
XXXI | 130 |
XXXII | 135 |
XXXIII | 138 |
XXXIV | 148 |
XXXV | 165 |
XXXVII | 172 |
XXXVIII | 173 |
XXXIX | 175 |
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Common terms and phrases
A₁ analog arbitrary assume Banach space c.b. map C*-algebra CB(E Chapter clearly commuting completely contractive completely isometric embedding completely isomorphic completely positive Consider Corollary defined definition E₁ easy to check element embeds equipped equivalent exact Exercise extends factorization finite rank finite-dimensional subspace free group free product Haagerup tensor product hence Hilbert space Hilbertian homomorphism identity implies inclusion inequality infimum injective Kirchberg Lemma linear map linear span Math matrix min(E Mn(A Mn(E Moreover morphism n-dimensional n-tuples Neumann algebra noncommutative norm Note nuclear C*-algebra obtain operator algebra operator space structure polynomial preceding projection Proof Proposition proved quotient Remark representation resp result satisfies sequence space H subalgebra suffices supremum supremum runs surjection tensor product Theorem theory ultraproduct unit ball unital operator algebra unitary von Neumann algebra
Popular passages
Page 471 - V. Peller, Estimates of functions of power bounded operators on Hilbert space, J. Operator Theory 7 (1982), 341-372.
Page 469 - E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974), 103-116.
Page 471 - ¡M 05.0400 47A65 (46L05) Completely bounded homomorphisms of operator algebras. Proc. Amer. Math. Soc. 92 (1984), no. 2, 225-228. (GA Elliott) 8Sm:47049 46L05.0380 47D25 (46L05) (with Suen, Ching Yun) Commutant representations of completely bounded maps.
Page 471 - G. Pisier. A simple proof of a theorem of Jean Bourgain. Michigan Math. J. 39 (1992), 475—484.