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On some von Neumann topological algebras. (English) Zbl 1200.46038

The authors consider unital algebras $A$ with the following property: for each $x$, there exists $y$ with $x=xyx$ $\left(x,y\in A\right)$. Their main result states that such a ${B}_{0}$-algebra (completely metrizable locally convex algebra) with an open group of invertible elements is finite-dimensional. Using this result, the authors show that a locally ${C}^{*}$-algebra with the above property is an inverse limit of finite-dimensional algebras. Another result states that such an $F$-algebra (completely metrizable algebra) is a finite product of division algebras of type $F$.

Reviewer’s remark. It remains open whether such a division algebra must be finite-dimensional, i.e., equal to $ℝ,ℂ$ or $ℍ$.

MSC:
 46H20 Structure and classification of topological algebras 46L05 General theory of ${C}^{*}$-algebras