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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,yA). 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 , or .

MSC:
46H20Structure and classification of topological algebras
46L05General theory of C * -algebras