Earlier there has been described
on the bidual (second conjugate space) of each member of a large class of
topological algebras an Arens multiplication which makes it again into a
topological algebra and which extends the multiplication of the original
algebra. This paper centers around three questions concerning this Arens
multiplication.
In the first place, we characterize those commutative algebras whose biduals are
also commutative. We then discuss the extent to which the passing of commutativity
from the base algebra to the bidual is preserved under certain operations of algebra,
and we show that if S is a compact Hausdorff space, and if C(S) has the supremum
norm, then any multiplication which makes C(S) a commutative Banach
algebra renders its bidual commutative. We also give a constructive proof
of the fact that if G is an infinite locally compact abelian group, then the
bidual of L1(G) is not commutative. In the second place, we prove that the
projection to the base algebra of the radical in the bidual of a Banach algebra is
precisely the radical of the base algebra. In addition, we determine that
the weak*-closed maximal regular ideals in the biduals of a large class of
commutative topological algebras (including commutative Banach algebras)
are the weak*-closures in the biduals of closed maximal regular ideals in
the base algebras. Furthermore, we show that if T is a locally compact,
noncompact Hausdorff space, then C0(T) is an ideal in its bidual if and only if T is
discrete; nevertheless, C0(T) can always be embedded in a regular ideal of its
bidual. In the third place, we show that if G is a locally compact abelian
group, then the radical in the bidual of L1(G) is separable if and only if G is
finite.
