On some von~Neumann topological algebras (Q967144)

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 \(\mathbb R,\mathbb C\) or \({\mathbb H}\).