?~1-bases in Banach algebras and Arens irregularities in harmonic analysis

摘要

Since the beginning when Richard Arens defined his products on thesecond dual of a Banach algebra, algebras in harmonic analysis such as ?~1 haveturned out to be Arens irregular (even in an extreme way).The present survey is an attempt to trace this story and to explain the combinatoricsin the group or in its dual object causing Arens (extreme) irregularity. Thisincludes our most recent work using ?~1-bases to study the problem.