CONCERNING THE SECOND DUAL OF THE GROUP ALGEBRA OF A LOCALLY COMPACT GROUP ANTHONY TO-MING LAU AND JOHN PYM The second dual L X (G)** of the group algebra of a locally compact group G has been studied in a series of papers recently. Most complete information has been obtained for the compact case [10]. Extensions to non-compact groups present severe difficulties, but successes have been achieved, notably in showing that the algebraic centre of L}{G)** for discrete commutative G is L l (G) [8] and that L}{G) is the object corresponding to the centre for general G [12]. The proof that the Banach algebra L\G)** determines G in [4] also uncovered some interesting structural points, notably that the algebra LUC(G)*, dual to the space of left uniformly continuous functions, must play an important role in L^G)**. This paper continues these investigations. There are two main themes. In Section 2, general results about the structure of L\G)** are obtained. Many of these relate to L X (G)** itself but we also introduce a subalgebra L™(G)* which is related to L X (G)** in much the same way that the measure algebra M(G) is related to the space offinitelyadditive measures. For LJf(G)* we can recover most of the results obtained for L\G)** in the compact case. In particular, here the set of extreme points of the positive unit ball is a locally compact semigroup with a simple algebraic structure. In the third part we consider operators from L°°(G) into certain of its subspaces X which commute with convolution on the left by elements of L\G). The relevance to our topic is that the algebras wefindin this way are subalgebras of LUC (G)*. This work is related to results in [3, 11]. Perhaps the most interesting case is when X=C(G); here, the algebra involved—which we denote by A (see (3.5))—has occurred in earlier papers as a notable feature of the structure of L\G)**. 1. Notation and introductory material As far as possible we follow [4] or [12] in our notation (though these references are not quite consistent with each other). The left Haar measure on the locally compact group G is A. Convolution of functions 0 , / i s defined by whenever the integral makes sense. Usually, (f>, y/ will be elements of the space L\G), f, g elements of L°>(G), and F, H elements of L\G)**. Received 14 February 1989. 1980 Mathematics Subject Classification (1985 Revision) 22D15. The research of the first author is supported by an NSERC grant. J. London Math. Soc. (2) 41 (1990) 445-460