Let A be a Banach algebra. In [1,2] R. Arens showed how to construct two multiplications on A** which make A** into a Banach algebra. The two multiplications are, in general, distinct; Arens said that A was regular if they coincided. In this paper we are concerned with the regularity of Banach algebras of the form l^S) and ^(S.eu), where S is a (discrete) semigroup and co is a weight function on S 1 . Earlier results are to be found in [3,8]. In [8, Theorem 2], Young obtains necessary and sufficient conditions for / x (5) to be regular; in [3] are given some conditions for the regularity of l^S, co). In this paper we obtain new conditions for each of these; we can then obtain our main result that l^S, co) is regular whenever US) is.In [5], H. A. M. Dzinotyiweyi introduces the notion of the spaces AP (S,co) and WAP (S, co) of weighted (weakly) almost periodic functions on a semigroup S. As is clear from [5], these two definitions have a number of shortcomings. In this paper we shall offer alternative definitions of these spaces. With these definitions we can show that WAP (S, co) = l w (S, co) if and only if l^S, co) is regular and obtain a condition for AP (S,co) to equal /^OS, co).
Definitions and elementary resultsThroughout this paper, S is an arbitrary (discrete) semigroup and co is a weight function on S; that is, co:S->U + (= {teM:t > 0}) and co(st) ^ co(s)co(t) for all s,teS.(1.1) DEFINITIONS. Let X and Y be sets and/: ^x Y-> I R be bounded, (i) We say that/is cluster on I x 7 if for each pair of sequences (x n ), (y m ) of distinct elements of X, Y, respectively, limlimy(x n ,3;J = limlim/(x n ,3;J(1) n m m n whenever both sides of (1) exist.(ii) If/ is cluster and both sides of (1) are zero (respectively positive) in all cases, we say that/is O-cluster (respectively positive-cluster). (Of course if/is O-cluster then both iterated limits in (1) will always exist.)
