By Shoichiro Sakai

From the studies: "This booklet is a superb and finished survey of the idea of von Neumann algebras. It contains the entire primary result of the topic, and is a important reference for either the newbie and the expert." Mathematical Reviews

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**Example text**

1. General Theory 46 Let Q be the spectrum space of 4'. Then jll = C(0). Let (p be a normal state on dl. By Riesz' representation theorem, cp defines a unique positive Radon measure ,u,p on Q such that cp(a)-- a(t)d,uv (t) (ae Proof. Let a — *a(t) be the mapping of 41 into the W*-algebra 12 )(Q„ ucp). is a *-homomorphism. Moreover, for be C(Q), a(t)b(t)d p(t)=. (p(ab) f2 and C(Q) is dense in L1 (Q ,pt,p). Therefore, Ov is a W*-homomorphism. Hence 0(4') is a W*-subalgebra of Loe(Q,,u,p), and so 4)(,a)=L` (52„u,p ), since &,( W ) C(Q) in /2° (Q„uv).

Then n4,(a)=0 for all (pe,99 . So q(a)=O for all (peg,,',. 7). q. e. d. 16. 6 is due to Gelfand-Naimark [55]. They defined a C*-algebra al as a Banach *-algebra satisfying the following conditions: 1. Ilx* xll = IlxII 2 and 2. 1 +x*x has an inverse (xe 41). 6. They further asked whether axiom 2. was superfluous. This question was solved by Fukamiya, Kelley-Vaught and Kaplansky (cf. 4). Also they asked whether the axiom 1. could be replaced by lix* xll = Ilx* IIIIx. This question was solved by Glimm and Kadison [61] and Ono [136].

Similarly 7q= s((p') q, and so q= . V . Put w'*w=-- h+ ik (h,k self adjoint). Then h,k e qjh'q since wi* we q,Wq. cp(w'* w) = (p' (w'* w') = (p' (q) = 1 = (p(h)+ i cp(k). Hence, cp(h) = 1 . Since I h I < 1, h = q, and since IIW* w I <1, k = 0. Therefore w'* w = q and w = w'. q. e. d. 14. 3 is due to Grothendieck [66]. References. [152], [209]. 15. Concrete C* Algebras and W* Algebras - - Let Yf be a complex Hilbert space and let B(/€ ) be the algebra of all bounded linear operators on A'. We can define various topologies in B(Y19).