an: Zbl 1200.46038
au: Choukri, Rachid; El Kinani, Abdellah; Oudadess, Mohamed
ti: On some von~Neumann topological algebras.
la: EN
so: Banach J. Math. Anal. 3, No. 2, 55-63, electronic only (2009).
py: 2009
dt: J
cc: *46H20 46L05
ut: regular von Neumann algebras; topological algebras; locally $C^*$-algebras
ab: 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 ${\Bbb H}$.
rv: Wies\l aw Tadeusz \D Zelazko (Warszawa)