Mahmoud Filali, University of Oulu
"Arens irregularity in harmonic analysis"
Arens irregularity of a Banach algebra is due to elements in its Banach dual which are not weakly almost periodic.
Unlike C*-algebras, the usual algebras in harmonic analysis, such as group algebras or Fourier algebras, when known, all turn out to be Arens irregular (even in an extreme way). At the very beginning, about seventy years ago, Richard Arens himself proved that many group algebras are not regular, then Mahlon Day proved few years later the same result for many discrete groups including the abelian ones. Since then, a long exciting story followed; some of it is already told by various authors in a couple surveys, books and memoirs.
In the present talk, we attempt to trace this story again, but this time we try to explain the combinatorial reason (on the group or its dual object) causing such (extreme) irregularity. There are two different types of extreme Arens irregularities, arising naturally from the way the algebras are decided to be Arens irregular.
The talk is based partly on some recent joint work with Jorge Galindo.