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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 $x=xyx$ $(x,y\in A)$. 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 $\Bbb R,\Bbb C$ or ${\Bbb H}$.
46H20Structure and classification of topological algebras
46L05General theory of $C^*$-algebras
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