Baer ∗-Rings by Sterling K. Berberian (auth.)

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 forms; connections with AW*-algebras, *-regular jewelry, non-stop geometries. detailed themes contain the speculation of finite Baer *-rings (dimension conception, aid conception, embedding in *-regular earrings) and matrix jewelry over Baer *-rings. Written for use as a textbook in addition to a reference, the ebook comprises greater than four hundred workouts, followed via 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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Exercise 11). If A admits a unitification A, in the sense of Definition 3, then A, cannot be a Baer *-ring. 14A. Let B be a Boolean algebra [$3, Exer. 141 containing more than two elements. In order that there exist a Boolean ring A without unity, such that A, =B, it is necessary and sufficient that B be infinite. 15A. Does there exist a weakly Rickart *-ring A # (0) without unity, possessing a unitification A, in thc sense of Definition 3, such that A, is a Baer *-ring? 16A. If A is a *-ring in which x* xi= 0 implies x, = ..

Suppose A is a *-ring such that, for each nonzero x in A, there exists YEA with xy=g, g a nonzero projection. Let U be a ubiquitous set of projections in A (see Exercise 9). If e is any nonzero projection in A, then there exists an orthogonal family (f,) with , f ,U~and e = sup f,. 11A. Let A be a Rickart *-ring and suppose e, f a r e projections in A that are algebraically equivalent (see [tj 1, Exer. 61). Then thc projection lattices of eAe and 6 3. Rickart *-Rings 19 fAf are isomorphic. Explicitly, if x ~ , f A eand y ~ e A fsatisfy y x = e , x y = l ; then the formula cp(g)=RP(xgy) defines an order-preserving bijection cp of the projection lattice of eAe onto the projection lattice of fAj: Moreover, if g, I g, e, then the projections g, -g, and cp(g2)- cp(g,) are algebraically equivalent.

I) With the coordinatewise operations, A is a commutative algebra over the field of real numbers. (ii) Setting x*=(l,*), A is a *-ring with proper involution. (iii) The projections in A are the sequences of 0's and 1's. (iv) Every orthogonal family of nonzero projections in A is countable. (v) A is a Baer *-ring. (vi) Setting llxl(= sup ll,l, A is a real Banach algebra, such that ((x*xll=Ilxl12. Thus A is a real AW*-algebra. (vii) A is symmetric [$ 1, Exer. 71. (viii) Write x 2 0 in case 1,2 0 for all n (equivalently, x = y * y for some y t A).

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Baer ∗-Rings by Sterling K. Berberian (auth.)
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