Topologically and Algebraically Finitely-Generated Closed Left Ideals in Banach Algebras

Jared White (Lancaster University)

Thursday 18th January 16:00-17:00 *** ROOM CHANGE *** Boyd Orr 411


A Theorem of Sinclair and Tullo from 1974 states that a Banach algebra which is (algebraically) left Noetherian is finite dimensional. Seeking to weaken the hypothesis of Sinclair and Tullo’s result, Dales and Żelazko conjectured in a 2012 paper that a unital Banach algebra in which every maximal left ideal is algebraically finitely-generated is finite dimensional. The conjecture has attracted some attention since, and is known to hold, for example, for commutative Banach algebras, and C*-algebras.

The first part of the talk will concern algebraically finitely-generated left ideals in Banach algebras. We shall discuss results that imply that Dales and Żelazko’s conjecture holds for the group algebra and measure algebra of a locally compact group, as well as for a large class of Beurling algebras.

The second part of the talk will discuss topologically finitely-generated ideals in Banach algebras. We see that, in contrast to Sinclair and Tullo’s result, there are many natural examples of infinite dimensional Banach algebras which are topologically left Noetherian. We discuss both Banach algebras arising in harmonic analysis, and the algebra of compact operators on a Banach space with the approximation property. Our results include a classification of the closed left and right ideals of the latter Banach algebra that appears to be new.

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