## Theory of Operator Algebras I (Google eBook)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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a-finite a-strongly abelian abelian C*-algebra abelian von Neumann Ah)m approximate identity assume Banach algebra Banach space Borel set C*-algebra C*-subalgebra center valued trace central projection closed ideal closure commutes completely positive conjugate space contains continuous function converges strongly countable cross-norm cyclic representation define denote direct sum enveloping von Neumann equivalent faithful semifinite normal finite dimensional follows Hilbert space homomorphism invariant involutive Banach algebra isomorphism lower semicontinuous maximal abelian minimal projection Neumann algebra nonzero projection norm operator algebras polar decomposition Polish space positive linear functional predual Proof Q.E.D. Corollary Q.E.D. Definition Q.E.D. Lemma Q.E.D. Proposition Q.E.D. Theorem resp respectively self-adjoint semifinite normal trace sequence Show Souslin spectrum subalgebra Suppose tensor product theory topology unit ball unital C*-algebra universal enveloping vector von Neumann algebra x,y e

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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