Topological algebras of random elements (Q2809357)

The paper is an interesting approach to defining the terms (like spectrum, Jacobson radical, spectral radius, hull and kernel) known in the theory of topological algebras in the context of the space of Bochner-measurable random variables with values in a unital complex Banach algebra. Using this approach, the authors prove analogues of many known results (e.g., continuity of a modular unital homomorphism, etc.) for topological algebras for the case of the unital topological algebra of all equivalence classes of almost-sure limits of measurable functions with values in a unital Banach algebra. The methods used in the investigation come from the theory of topological (mainly Banach) algebras. It seems that there is also a possibility to continue the same kind of work in the context of more general (for example, non-normed) topological algebras by considering measurable functions with values in more general topological algebras and replacing the ``language of norms'' by the ``language of neighbourhoods'' and using results about general topological algebras instead of results about Banach algebras.