On the structure of ideals and multipliers: A unified approach

We study the structure of one-sided ideals in a Banach algebra $\mathcal {A}$. We find very general conditions under which any left (right) ideal is of the form $\mathcal {A} q$ ( $q \mathcal {A}$) for some idempotent right (left) multiplier on $\mathcal {A}$. 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 $C^*$-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).