By Sterling K. Berberian (auth.)
A systematic exposition of Baer *-Rings, with emphasis at the ring-theoretic and lattice-theoretic foundations of von Neumann algebras. Equivalence of projections, decompositio into kinds; connections with AW*-algebras, *-regular jewelry, non-stop geometries. particular issues contain the idea of finite Baer *-rings (dimension thought, relief conception, embedding in *-regular earrings) and matrix jewelry over Baer *-rings. Written for use as a textbook in addition to a reference, the e-book comprises greater than four hundred routines, followed by means of notes, tricks, and references to the literature. Errata and reviews from the writer were extra on the finish of the current reprint (2nd printing 2010).
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By Lemma 2, there exists a family (PI)of clopen sets such that U = UP,. As remarked following Proposition 1, the clopen sets form a complete lattice; let P be the supremum of the family (P,). Since P, c P for all 1 , we have U c P, therefore U c P ; it will suffice to show that U = P. Assume to the contrary that the opcn set P - U is nonempty. Let Q be a nonempty clopen set such that Q c P - U (Lemma 2). Since Q nPI= (ZI for all 1 , that is, P, c T - Q for all 1 , it follows that P c T - Q (because T-Q is clopen and P is the supremum of the P,).
A term consistent with 153, Def. } AW*-algebras were proposed by Kaplansky (1471, 1481, ) as an appropriate setting for certain parts of the algebraic theory of von Neumann algebras (defined below). There are two elemental examples, of which most other examples are refinements or derivatives: (I) the algebras C(T), where T is a Stonian space (these are discussed in Section 7), and (2) the following: Proposition 3. If X is a Hilbert space, then Y(Af) is an AW*-algebrn. Proof. X') is a Rickart C*-algebra [$3, Prop.
Write &=&(A, +) for the endomorphism ring of the additive group of A; each UEA determines an element La of 6 via L,x=ax; and each ~ E Kdetermines an element AI of & via (A I)x=Ax. Let A, = A@K with the *-algebra operations described in Definition 3. dl of 8, namely, the subring of d generated by the left-multiplications Lo and the homotheties I I. Defining p(L, + I to be the ring product (PI) (L,+i I), d,becomes an algebra over K, and the mapping @,A)rt Lo+ A I an algebra homomorphism of A, onto dl.