On the structure of ideals and multipliers: A unified approach

Abstract

We study the structure of one-sided ideals in a Banach algebra A \mathcal {A} . We find very general conditions under which any left (right) ideal is of the form A q \mathcal {A} q ( q A q \mathcal {A} ) for some idempotent right (left) multiplier on A \mathcal {A} . We further show that a large class of one-sided multipliers can be realized as a product of an invertible and an idempotent multiplier. Applying our results to algebras over locally compact quantum groups and C ∗ C^* -algebras, we demonstrate that our approach generalizes and unifies various theorems from abstract harmonic analysis and operator algebra theory. In particular, we generalize results of Bekka (and Reiter), Berglund, Forrest, and Lau–Losert. We also deduce the Choquet–Deny theorem for compact groups as an application of our approach. Moreover, we answer, for a certain class of measures on a compact group, a question of Ülger which, in the abelian case, goes back to Beurling (1938).