Extreme non-Arens regularity of the group algebra

TL;DR

This paper proves that the group algebra of any infinite locally compact group exhibits extreme non-Arens regularity, with a stronger isometric embedding result for non-discrete groups.

Contribution

It establishes that L^1(G) is always extremely non-Arens regular for any infinite locally compact group, extending previous understanding of algebra regularity.

Findings

L^1(G) is extremely non-Arens regular for all infinite locally compact groups

For non-discrete G, there is an isometric copy of L^(G) in the quotient space

The result generalizes previous regularity properties of group algebras

Abstract

Following Granirer, a Banach algebra A is extremely non-Arens regular when the quotient space A*/WAP(A) contains a closed linear subspace which has A* as a continuous linear image. We prove that the group algebra L^1(G) of any infinite locally compact group is always extremely non-Arens regular. When G is not discrete, this result is deduced from the much stronger property that, in fact, there is a linear isometric copy of L^\infty(G) in the quotient space L^\infty(G)/CB(G), where CB(G) stands for the algebra of all continuous and bounded functions on G.