an:05702386 Zbl 1200.46038 Choukri, Rachid; El Kinani, Abdellah; Oudadess, Mohamed On some von~Neumann topological algebras. EN Banach J. Math. Anal. 3, No. 2, 55-63, electronic only (2009). ISSN 1735-8787/e 2009

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*46H20 46L05 regular von Neumann algebras; topological algebras; locally $C^*$-algebras 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 ${\Bbb H}$. Wies\l aw Tadeusz \D Zelazko (Warszawa)