On \(TQ\)-algebras
Let \(A\) be a topological algebra with unit \(e\). An element \(x\) in \(A\) is called left (right) topologically invertible if there is a net \((z_\alpha)\subset A\) with \(\lim_\alpha z_\alpha x=e\) (\(\lim_\alpha xz_\alpha=e\)). A left (right) \(TQ\)-algebra is a topological algebra for which the set of left (right) topologically invertible elements is open. The main result of the paper (Theorem 4.3) characterizes these algebras by providing several equivalent conditions. E.g., \(A\) is a left \(TQ\)-algebra if and only if its left topological spectral radius is continuous at \(0\), or if and only if it is upper semicontinuous. Other results concern the functional topological spectrum (when this spectrum is defined through multiplicative-linear functionals) and the question when a \(TQ\)-algebra is a \(Q\)-algebra (its set of invertible elements is open).
scientific article; zbMATH DE number 6804280