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 kinds; connections with AW*-algebras, *-regular jewelry, non-stop geometries. exact subject matters contain the idea of finite Baer *-rings (dimension concept, aid thought, embedding in *-regular jewelry) and matrix earrings over Baer *-rings. Written for use as a textbook in addition to a reference, the ebook comprises greater than four hundred routines, followed by means of notes, tricks, and references to the literature. Errata and reviews from the writer were additional on the finish of the current reprint (2nd printing 2010).

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The difficulty is avoided by choosing the right definition: Definition 3. Let A be a Baer *-ring, B a *-subring of A. We say that B is a Baer *-subring of A provided that (i) X E B implies RP(x)E B, and (ii) if S is any nonempty set of projections in B, then s u p S ~ B . It follows that B is itself a Baer *-ring, with unambiguous lattice constructs: Proposition 4. subring of A, tlzm B is also a Baer *-ring, and RP's, LP's, sups and infs in B are unambiguous. Proof. Let e = s u p ( R P ( x ) : x ~ B ) ; by hypothesis eEB, and it is clear that e is a unity element for B.

A *-ring A is said to satisfy the very weak (EP)-axiom (briefly, the (VWEP)-axiom) if for every ~ E A x#O, , there exists y e {x*x)' such that (x* x) (y*y) = e, e a nonzero projection. Obviously (EP) * (WEP) * (VWEP) => the involution is proper. Exercises 1C. A C*-algebra A is an AW*-algebra if and only if (A) in the partially ordcred set of projections of A, every nonempty set of orthogonal projections has a supremum, and (B) every masa [#I, Exer. 141 in A is the closed linear span of its projections.

Fix a unit vector x in ,it. Given any y t P,consider the operator T z = (zlx)y ( ~ € 2 In )particu. lar, Tx=(xlx)y= y, thus y t T(A2). %')), thus y t T ( . ' 5. r) (because B contains every central projection of A), but in general equality does not hold. ( F o r example, let d be 4 6 . ) Such *-subrings B are Baer *-subrings of A in the sense of [54, Def. 31; for Baer *-cubrings of the form eAe, the situation is clearer: Proposition 4. Let A he a Baer *-ring, e a projection in A, and ,f a projection in eAe (tlzat is, j' 5 e).

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