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A new axiomatic approach in the theory of operator algebras. (English. Russian original) Zbl 0598.46036
Russ. Math. Surv. 40, No. 4, 213-214 (1985); translation from Usp. Mat. Nauk 40, No. 4(244), 193-194 (1985).
Let E be a Baer *-algebra over \({\mathbb{C}}\), let K be the subset of elements of the form \(\sum^{n}_{i=1}a^*_ ia_ i\) \((a_ i\in E)\) and suppose that for each \(x\in K\) there is a unique y in \(K\{\) \(x\}\) ” such that \(y^ 2=x\). Then K is a cone generating \(E_ h=\{x\in E:\) \(x^*=x\}\) and so induces a partial order on \(E_ h.\)
E is defined to be a Baer ordered *-algebra (BO\({}^*\)-algebra) if given a sequence \(\{x_ n\}\) in \(E_ h\) such that \(0\leq x_ n\leq \epsilon_ n1\) (1 is the identity of E) with \(\epsilon_ n\) a summable positive sequence, then \(\sup_{n\geq 1}\sum^{n}_{i=1}x_ i\) exists in \(E_ h\). Examples of \(BO^*\)-algebras: \(W^*\)- and \(AW^*\)-algebras; *- algebras of measurable and locally measurable operators associated with \(W^*\)- and AW\({}^*\)-algebras. The set of bounded elements in a \(BO^*\)-algebra is its smallest solid \(BO^*\)-subalgebra and admits a norm with respect to which it is an \(AW^*\)-algebra.
Let E be a unital Jordan algebra over \({\mathbb{R}}\). We say that E is an ordered Jordan algebra (OJ-algebra) if E admits a partial order compatible with the algebraic operations and with respect to which E is monotone complete and each maximal associative subalgebra of E is a lattice. Example of OJ-algebras: JBW-algebras; JW-algebras; the algebra S(A) is selfadjoint operators locally measurable with respect to the JW- algebra A; the algebras \(S(X,M^ 8_ 3)\) of continuous admissible functions from a compact Stonean space X to the algebra \(M^ 8_ 3\) of \(3\times 3\) matrices over the Cayley numbers.
Reviewer: P.G.Spain

46K05 General theory of topological algebras with involution
46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
06F25 Ordered rings, algebras, modules
17C65 Jordan structures on Banach spaces and algebras
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