Multiplier algebras of normed spaces of continuous functions

TL;DR

This paper explores the properties of multiplier algebras of normed spaces of continuous functions, demonstrating how they inherit properties from the original spaces and constructing examples with only constant multipliers.

Contribution

It introduces methods to realize any separable Banach space as a NSCF over separable metrizable spaces and provides conditions for non-separability of multiplier algebras.

Findings

Multiplier algebras inherit properties from NSCFs.

Existence of NSCFs with only constant multipliers.

Conditions for non-separability of multiplier algebras.

Abstract

In this article we investigate some general properties of the multiplier algebras of normed spaces of continuous functions (NSCF). In particular, we prove that the multiplier algebra inherits some of the properties of the NSCF. We show that it is often possible to construct NSCF's which only admit constant multipliers. In order to do that, using a method from [23], we prove that any separable Banach space can be realized as a NSCF over any separable metrizable space. On the other hand, we give a sufficient condition for non-separability of a multiplier algebra.