  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.
