The Radical of L ∞ (G) ∗

Abstract

THEOREM. Let G be any locally compact nondiscrete group (or any infinite discrete amenable group). Then the radical of the (complex, noncommutative) Banach algebra LOO(G)* is not norm separable. Introduction. Let G be a locally, compact group and denote by 9 the radical of the complex Banach algebra Lm(G)* with Arens multiplication induced from L'(G) (see next section). It has been proved by Civin and Yood [2] that if G is nondiscrete locally compact abelian (or G is the group of additive integers) then R is infinite dimensional. The conjecture of [2] that this holds true for all discrete abelian G has been proved by this author in [5, pp. 48, 58]; and, in fact, for any discrete infinite amenable group G, M. is infinite dimensional [5]. It has been later proved by S. Gulick in [8, p. 136], among other results, that if G is any locally compact abelian group then 9 is not even norm separable. The following is posed as an unsolved question in [8, p. 136]: G is an infinite nonabelian locally compact group is the radical of L>(G)* nonzero? The theorem quoted in the abstract gives an incomplete answer to this question. The case it leaves out is for G an infinite discrete nonamenable group, for which the methods used in this paper fail. The method of proof resembles the one used by Gulick. We keep, however, a much closer account of the items involved. Definitions and notations. Unless otherwise stated we adhere to the definitions of Hewitt-Ross [9]. C(G) [UCBr(G)] will denote the space of bounded [right uniformly] continuous, complex, bounded functions with usual sup norm (f e UCBr(G) iff f e C(G) and the map x-df, 1fJ(y)=f (xy), from G to C(G) is continuous). Thus UCBr(G)C C(G)c L (G). The Arens multiplication is defined by: If u, v c LX (G), f e LI(G), g EL1(G)thenf o g EL'(G),v o fE LO(G), 0o vE L(G)* aredefined by: (f og) ()=(f,g* ), (vof) ()=v(f 0) for all eL1(G) and Received by the editors October 6, 1972. AMS (MOS) subject class.fications (1970). Primary 22D 15, 22D05, 43A 10; Secondary 46H05, 43A15, 43A07. I Research done while the author held a Canada Council Award. The support of the Canada Council is gratefully acknowledged. ? American Mathematical Society 1973