# When is Every Quasi-Multiplier a Multiplier?

## TL;DR

This paper characterizes when every quasi-multiplier of a sigma-unital C*-algebra is a multiplier, showing it occurs precisely when the algebra is a direct sum of a dual algebra and a locally unital algebra.

## Contribution

It provides a complete characterization of sigma-unital C*-algebras where all quasi-multipliers are multipliers, extending the analysis to bimodules and centralizers.

## Key findings

- Characterization of sigma-unital C*-algebras with all quasi-multipliers as multipliers
- Identification of the algebra as a direct sum of a dual and a locally unital algebra
- Extension of results to Hilbert C*-bimodules and Pedersen's ideal centralizers

## Abstract

We answer the title question for sigma-unital C*-algebras. The answer is that the algebra must be the direct sum of a dual C*-algebra and a C*-algebra satisfying a certain local unitality condition. We also discuss similar problems in the context of Hilbert C*-bimodules and imprimitivity bimodules and in the context of centralizers of Pedersen's ideal.