Topological algebras of rapidly decreasing matrices and generalizations

TL;DR

This paper generalizes the properties of rapidly decreasing matrices forming topological algebras to broader classes of weighted matrix algebras with entries in Banach algebras, extending known K-theory results.

Contribution

It extends the known topological and algebraic properties of rapidly decreasing matrices to more general weighted matrix algebras with Banach algebra entries.

Findings

Weighted matrix algebras are associative topological algebras.

The set of quasi-invertible elements is open in these algebras.

The quasi-inversion map is continuous in the generalized setting.

Abstract

It is a well-known fact in K-theory that the rapidly decreasing matrices of countable size form an associative topological algebra whose set of quasi-invertible elements is open, and such that the quasi-inversion map is continuous. We generalize these conclusions to further algebras of weighted matrices with entries in a Banach algebra.