Theory of Operator Algebras I (Google eBook)

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Springer Science & Business Media, 2002 - Mathematics - 415 pages
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Since its inception by von Neumann 70 years ago, the theory of operator algebras has become a rapidly developing area of importance for the understanding of many areas of mathematics and theoretical physics. Accessible to the non-specialist, this first part of a three-volume treatise provides a clear, carefully written survey that emphasizes the theory's analytical and topological aspects.

  

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Contents

II
1
III
2
IV
6
V
13
VI
17
VII
21
VIII
23
IX
25
XXIX
179
XXX
181
XXXI
182
XXXII
188
XXXIII
192
XXXIV
203
XXXV
220
XXXVI
229

X
31
XI
35
XII
47
XIII
50
XIV
54
XV
55
XVI
58
XVII
59
XVIII
67
XIX
71
XX
79
XXI
99
XXII
101
XXIII
102
XXIV
120
XXV
130
XXVI
139
XXVII
147
XXVIII
157
XXXVII
230
XXXVIII
253
XXXIX
264
XL
289
XLI
290
XLII
309
XLIII
335
XLIV
336
XLV
344
XLVI
352
XLVII
359
XLVIII
362
XLIX
374
L
375
LI
387
LII
389
LIII
409
LIV
411
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Page xvii - the theory of unitary group-representations essentially beyond their classical frame have always been blocked by the unsolved questions connected with these problems. Third, various aspects of the quantum mechanical formalism suggest strongly the elucidation of this subject. Fourth, the knowledge obtained in these investigations gives an approach to a class of abstract algebras without a finite basis, which seems to
Page vi - At the same time, examples of algebras were increasingly studied that codify data from differential geometry or from topological dynamical systems. On the other hand, a little earlier in the seventies, the theory of von Neumann algebras underwent a vigorous growth after the discovery of a natural infinite family of pairwise nonisomorphic factors of type
Page xvii - seems to be essential for the further advance of abstract operator theory in Hilbert space under several aspects. First, the formal calculus with operator-rings leads to them. Second, our attempts to
Page vi - became apparent. Thanks to recent progress, both on the cyclic homology side as well as on the K-theory side, there is now a well developed bivariant K-theory and cyclic theory for a natural class of topological algebras as well as a bivariant character taking K-theory to cyclic theory. The 1990's also saw huge progress in the classification theory of nuclear
Page v - also found immediate applications in the study of many new examples of C*¿algebras that arose in the end of the seventies. These examples include for instance “the noncommutative tori” or other crossed products of abelian C*¿algebras by groups of homeomorphisms and abstract
Page vi - which would lead to index theorems for foliations incorporating techniques and ideas from many branches of mathematics hitherto unconnected with operator algebras. Many of the new developments began in the decade following the Kingston meeting. On the
Page vi - These ideas and many others were integrated into Connes' vast Noncommutative Geometry program. In cyclic theory and in connection with many other aspects of noncommutative geometry, the need for going beyond the class of
Page vi - bivariant form of K-theory for which operator algebraic methods are absolutely essential. Cyclic cohomology was discovered through an analysis of the fine structure of extensions of
Page vi - But the meeting also contained a preview of what was to be an explosive growth in the field. The study of the von Neumann

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