@article{1200.46038,
author="Choukri, Rachid and El Kinani, Abdellah and Oudadess, Mohamed",
title="{On some von~Neumann topological algebras.}",
language="English",
journal="Banach J. Math. Anal. ",
volume="3",
number="2",
pages="55-63",
year="2009",
abstract="{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 ${\Bbb H}$.}",
reviewer="{Wies\l aw Tadeusz \D Zelazko (Warszawa)}",
keywords="{regular von Neumann algebras; topological algebras; locally
    $C^*$-algebras}",
classmath="{*46H20 (Structure and classification of topological algebras)
46L05 (General theory of C*-algebras)
}",
}