On topologizing invertibility
Motivated by topological zero devisors, we say that an element x in a metrizable topological algebra (A,T) is (boundedly) topologically invertible if for some (bounded) sequence \((x_ n)\) in A, \(xx_ n\to 1\) and \(x_ nx\to 1\). If A is a complete locally m-convex algebra, or a complete locally bounded algebra or is a locally convex algebra with Montel property, then such an x in A is invertible in A. The Arens' algebra \(L^ w[0,1]=\cap_{1\leq p<\infty}L^ p[0,1]\) [\textit{R. Arens}, Bull. Am. Math. Soc. 52, 931-935 (1946; Zbl 0060.269)] provides an example of a locally convex algebra admitting a topologically invertible element which is not invertible. However, a metrizable locally convex topological algebra A can be embedded in a larger metrizable locally convex algebra \(\ell^{\infty}(A)\) (of all bounded sequences in A) such that all boundedly topologically invertible elements in A are invertible in \(\ell^{\infty}(A)\).
locally convex algebra with Montel property
scientific article; zbMATH DE number 3931784