By Shoichiro Sakai

ISBN-10: 3540636331

ISBN-13: 9783540636335

From the studies: "This publication is a wonderful and complete survey of the idea of von Neumann algebras. It contains all of the basic result of the topic, and is a precious reference for either the newbie and the expert." Mathematical Reviews

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This *-representation {U,K} is called the universal *-representation of sI. 6. Theorem. The universal *-representation {U,K} of d is an isometric isomorphism. Therefore, every C*-algebra is *-isomorphic to a uniformly closed self-adjoint subalgebra of B(Y() on some Hilbert space Yf. Proof. Without loss of generality, we can assume that d has an identity. 92 = sup (p(a* a). 11 a* all (cf. 2) qy E ,50' = 11a11 2 . On the other hand, Ilircp(a)11 11 all. d. 11 all = 11 U(41 for aesif. Let ,4' be a W*-algebra and let (p be a normal state on di'.

5. The u-weak operator topology. ) on B(Yf). The set of all such semi-norms will define a Hausdorff locally convex topology, the o--weak operator topology on B(Yf). Let V be the linear space of all continuous linear functionals on B(dr) with respect to the weak operator topology. Then the weak operator topology is equivalent to 0- (B (r), V). One can easily see that the unit sphere S of B(lf) is cf(B(YP), V)-compact. Let B ( 1f)* be the dual of B ( P) with the uniform topology. Since the uniform topology is stronger than the weak operator topology, V may be identified with a linear subspace of B(W)*.

E. d. 5. Proposition. Every o-(/#,A,)-closed two-sided ideal is of the form ,/h'z for some central projection z in 4'. Proof. Let f be a 4/K,Ad-closed two-sided ideal of 4'. Then f=dizi = z 2 ,4' for some projections z1 , z2 in 4', where z1 (resp. z 2) is the identity of 5 r' 5* Hence z 1 =z2 =z and so 5=5*. Therefore 5 is a W*-algebra with the identity z. --zxz, and zx=(zx)z=zxz for xe4'. Hence, z is a central projection of 4'. d. 6. Definition. Let 2' be the center of a W*-algebra 4'. Then 2' is 41i, di)-closed.

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

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