Journal of Operator Theory

Volume 91, Issue 2, Spring 2024 pp. 373-398.

The ideal structure of measure algebras and asymptotic properties of group representations

Authors: Jared T. White
Author institution: School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, U.K.

Summary: We classify the weak*-closed maximal left ideals of the measure algebra $M(G)$ for certain Hermitian locally compact groups $G$ in terms of the irreducible representations of $G$ and their asymptotic properties. This includes a classification for connected nilpotent Lie groups, and the Euclidean rigid motion groups. We then give several applications of this result to representation theory and to questions about finitely generated left ideals in Banach algebras. In particular, for certain groups we obtain an analogue of Barnes' Theorem on integrable representations for representations vanishing at infinity.

DOI: http://dx.doi.org/10.7900/jot.2022apr13.2389
Keywords: locally compact group, measure algebra, irreducible representation, vanishing at infinity, dual Banach algebra

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