C*-Algebras and W*-Algebras
From the reviews: "This book is an excellent and comprehensive survey of the theory of von Neumann algebras. It includes all the fundamental results of the subject, and is a valuable reference for both the beginner and the expert." (Math. Reviews) "In theory, this book can be read by a well-trained third-year graduate student - but the reader had better have a great deal of mathematical sophistication. The specialist in this and allied areas will find the wealth of recent results and new approaches throughout the text especially rewarding." (American Scientist) "The title of this book at once suggests comparison with the two volumes of Dixmier and the fact that one can seriously make this comparison indicates that it is a far more substantial work that others on this subject which have recently appeared"(BLMSoc)
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aesi arbitrary automorphism B(jf Banach algebra Banach space bounded linear operators C*-algebra with identity C*-norm C*-subalgebra central sequence commutative W*-algebra complex numbers Consider contradiction converges Corollary countably decomposable define Definition denoted directed set ergodic exists a positive exists an element factor faithful normal family of mutually finite W*-algebra following conditions Hence there exists Hilbert space homomorphism implies increasing directed set infinite irreducible isomorphism Lemma Let G Let Jt Let Q linear mapping linear subspace maximal commutative Moreover mutually orthogonal n(si non-zero projection partial isometry polar decomposition positive element positive integer positive linear functional positive number predual Proof Proposition R.Then Radon measure representation resp s-topology self-adjoint element semi-finite separable Hilbert space separating vector strongly residual suffices to assume Suppose tensor product Theorem theory two-sided ideal uniform closure uniquely extended unit sphere unitary elements W*-algebra W*-algebra and let W*-representation weak operator topology xeJt
Page 243 - R. The adjoint of a bilinear operation. Proc. Amer. Math. Soc. 2 (1951), 839-848.