By Shoichiro Sakai

ISBN-10: 3540636331

ISBN-13: 9783540636335

From the experiences: "This ebook is a wonderful and entire survey of the idea of von Neumann algebras. It contains the entire primary result of the topic, and is a priceless reference for either the newbie and the expert." Mathematical Reviews

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**Example text**

Ifi = A/ .. Henceforward, we shall call a self-adjoint subalgebra which is closed with respect to the weak operator topology, a weakly closed self-adjoint subalgebra of B(0), since there can be no confusion. 2. Proposition. Let ,Af be a weakly closed self-adjoint subalgebra of B(0), and let ,A(* be the predual of X Then 1. ilf,,A(* ) are equivalent on bounded spheres; 2. iff* ) are equivalent on bounded spheres. Proof. 1. is clear from the (7(4/,' S,;)-compactness of the unit sphere of Ar. On the other hand, x a,—*x in the strong operator topology is equivalent to (x ac —x)*(x,,— x)--0 in the weak operator topology, q.

E(A n). _(21 (uniformly). n 1 hr (1 - p) = 0, (A 1 so that p=e(1). q. e. d. 2. Lemma. A {6' ( L)- e(A)} < fite(kt)- (ill - hr } - { 2 e( 2 )-(21 - hr } < ,u{e(p)- e(1)} for u_2. Proof. (A1-h)e(2) and (j11 h) ± =(j11 -h)e(u). Hence, kte(p)-(ii 1 -h ) + = h e(p) and Ae(A)- (21 - hr = h e(A) and - A(e02)- e(A)) < h(e(,u)- e(A)) < ,u(e(p)- e(A)). q. e. d. 3. Theorem. For any self-adjoint xe,iii, there exists a system of projections { e(A)} (- co <2< co) in 4', called a resolution of the identity such that 1.

Therefore, z =1 —s((p). defines a *-isomorphism of s((p) onto If (Q„ uv). ). ). on the localizable measure space ( 221 2611 Let K be a Stonean space. A positive Radon measure /2 on K is called normal if for every uniformly bounded increasing directed set u. 1, {Lc} of continuous real valued functions on Q, 1. u. b. e. 11 imply f = 0). Then, C(K) is a W*-algebra if and only if K is hyper-Stonean (cf. [31]). Reference. [182]. Remark. 19. The C*-Algebra CVO of all Compact Linear Operators on a Hilbert Space 1' Let Yt° be a Hilbert space, C(Yf) the C*-algebra of all compact linear operators on lf , T(Yf) the Banach space of all trace-class operators, B ( r) the W*-algebra of all bounded linear operators on Yf.

### C*-Algebras and W*-Algebras (Classics in Mathematics) by Shoichiro Sakai

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