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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

##### MSC:
 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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