Vol. 18, No. 1, 1966

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Commutativity and ideals in the biduals of topological algebras

Sidney (Denny) L. Gulick

Vol. 18 (1966), No. 1, 121–137

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.

Mathematical Subject Classification
Primary: 46.55
Received: 26 February 1965
Published: 1 July 1966
Sidney (Denny) L. Gulick