Multipliers of uniform topological algebras

TL;DR

This paper investigates the structure of multipliers in complete uniform topological algebras, establishing an isomorphism with inverse limits of multipliers of their Arens-Michael normed factors, and discusses conditions for their continuity.

Contribution

It provides a new isomorphism between the multiplier algebra of a uniform topological algebra and the inverse limit of multipliers of its normed factors, clarifying the continuity of multipliers.

Findings

Multiplier algebra is isomorphic to inverse limit of factors' multiplier algebras

Continuity of multipliers when factors are complete

Topological algebra isomorphism under certain conditions

Abstract

Let $E$ be a complete uniform topological algebra with Arens-Michael normed factors $\left(E_{\alpha}\right)_{\alpha\in\Lambda}.$ Then $M\left(E\right) \cong \varprojlim M\left(E_{\alpha}\right)$ within an algebra isomorphism $\varphi$. If each factor $E_{\alpha}$ is complete, then every multiplier of $E$ is continuous and $\varphi$ is a topological algebra isomorphism where $M\left(E\right)$ is endowed with its seminorm topology.