Abstract. Let G be a locally compact group. A weakened version of Grothendieck's double limit criterion [6; p. 183] is shown to characterize those q~ e ~ ~ (G) that are locally almost everywhere equal to a continuous weakly almost periodic function in the sense of Eberlein. Additional measure theoretic conditions guarantee continuity of such ~. As a by-product, we obtain a short proof of the classical result that q~ is continuous when almost periodic.Let 2 be some Haar measure on G and q~ ~ Y ~ (G). By C (G) we denote the C*-algebra of all bounded continuous functions G --* C, normed by the sup norm II tl. Recall [1]
that ~eC(G) is (weakly)almost periodic iff the set {g~01 g e G} of left translates of ~ is relatively (weakly) compact in C (G). The collection (W (G)) A (G) of all (weakly) almost periodic functions on G is a translation invariant C*-subalgebra of C (G). (Cf.[1] for a systematic exposition of the theory.) Given ~0 e C ~ and A, B ___ G, one says that ~o satisfies the double limit condition (DLC) on A x B iff limn limm ~o (an bm) = limm limn ~o (an bin) for any sequences (an, bn) in A • B for which the limits involved exist. For ~0 e C (G), the classical criterion of GROTHENDIECK [6; p. 18 3] states that ~.W(G) iff ~0 satisfies DLC on G x G. We will show that this criterion, in a weakened and suitably modified form, still captures the essence of weak almost periodicity in the measurable setting. The version we are going to use is the following condition:DLC~: There is a dense subset D of G such that,for every countable C c_ D, there is a locally 2-negligible N c__ G so that q~ satisfies DLC on
Cx(G\N).The integration theoretic terminology we use is in accordance with Bourbaki and the functional analytic terminology with [11]. The discrete version of a space X is denoted by Xd. By N we denote the set of all non-2-negligible, ,~-integrable subsets of G.
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