Download e-book for kindle: C*-Algebras and W*-Algebras (Classics in Mathematics) by Shoichiro Sakai
By Shoichiro Sakai
From the experiences: "This booklet is a superb and entire survey of the speculation of von Neumann algebras. It comprises all of the basic result of the topic, and is a useful reference for either the newbie and the expert." Mathematical Reviews
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Extra resources for C*-Algebras and W*-Algebras (Classics in Mathematics)
Then 5 is self-adjoint and the quotient algebra si If is also a C*-algebra. Proof. Letr ° be the bipolar off in si'. _21**, si*)closure of f in si**. Hence, it is a u(s1**,s1*)-closed ideal of d**. Therefore, there exists a central projection z in si** such that I" = al** z. Since 5J00 n si = d**z n si, 5 is self-adjoint. 91(1 — z). d. 4. Corollary. 4 be C*-algebras and let 0 be a *-homomorphism of si into M. Then the image 0(21) of si is a C*-subalgebra of M. Therefore, the image of a *-representation of a C*-algebra is also a C*-algebra.
B. — P41 ) for cp e T. u. b. b. Pj J is a projection (cf. 12). We define E e„ = 1. v. b. P. 2, we can easily aell see that normality is equivalent to complete additivity (i. 5. Definition. We shall call a o-(dI,A )-continuous linear functional on a normal linear functional. 13. 1. 2 is due to Dixmier . 2. In this section, the reader learns a quite new class of Banach spaces—that is, the existence of a dual Banach space E which has a unique Banach space F such that F* =E. Of course, the reflexivity implies such the property, but one can easily see that a reflexive W*algebra is finite-dimensional (cf.
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). = dlz for some projection z Let 5,0 be the kernel of Ov • Then in dl.
C*-Algebras and W*-Algebras (Classics in Mathematics) by Shoichiro Sakai