Algebraic structure in the second dual of ideals of group algebras and related Banach algebras

Functional analisys seminar.

Fecha: 15 de febrero de 2023.
Hora: 11:30h - 12:30h.
Lugar: Seminario 2, IMAG.
Ponente: Prof. Jorge Galindo Pastor (Universitat Jaume I).

Abstract: It is well-known that when the second dual L1 (G)** of the group algebra L1(G) of a locally compact group G is furnished with one of the Arens multiplications, the only elements p in L1(G)** for which both multiplication operators q →pq and q → qp are continuous are the elements of L1 (G). In short, Zt (L1(G)** ) = L1(G), the topological center of L1(G)** is L1(G), i.e., it is as small as it gets. One says in this case that L1(G) is strongly Arens irregular. It is also known (at least, since Ülger's 2011 paper [Characterizations of Riesz sets] that infinite dimensional ideals of L1(G) can be Arens regular, i.e., that multiplication on their second duals can even be (separately) continuous. In this talk, we will discuss the Arens regularity properties of ideals of L1(G) with G compact and Abelian and will show that all sorts of behaviour are possible and actually occur. We will see that there is a correlation between these properties and the thinness of the subset of the dual group G’ where the Fourier transforms of the elements of the ideal are supported. On our way, we will be stressing those aspects that can be replicated on a wide family of Banach algebras that include the algebra L1(G) for G compact (not necessarily Abelian) or the Fourier algebra A(G) with G amenable and discrete.