On some topologies which coincide on the unit sphere of the Fourier-Stieltjes algebra $B(G)$ and of the measure algebra $M(G)$

Abstract

Introduction. Let G be an arbitrary locally compact group [^4(G)], B(G) the [Fourier] Fourier-Stieltjes algebra of G and M(G) the Banach algebra of bounded Radon measures on G (see definitions in what follows). We prove in §1 of this paper that we w*-topology z w * and the multiplier topology z M coincide on the unit sphere S = {u e B(G)\ \\u\\ = 1} of B(G\ where u a -> u in z M if and only if ||(w a -w)v|| -» 0 for each v e A(G). This result proves a conjecture of McKennon [10, p. 49]. It improves a result of Derighetti [1] and McKennon [10] (that z w * = z uc on S, where z uc is the topology of uniform convergence on compacta) which in turn improves a theorem of Raikov [13] and Yoshizawa [17] (that z w * = z uc on the positive definite face of S). Applying this result we show in theorem B x that for any compact K a G the Banach space A K (G) = {u e A(G); supp u a K} has the Radon-Nikodym property and consequently a strong Krein-Milman theorem, for closed bounded convex subsets of A K (G) 9 follows. Theorem B 2 of this section consists of a long list of topologies which coincide on S. §3 consists of a measure theoretical selfcontained proof of a result of McKennon [10] which states that the w* and the ZAmultiplier topology on S = {ju e M(G); ||//|| = 1} coincide (//« -> ft in the latter if and only if || (/ua -fi)*f\\p -• 0 for each fe LP). The reader familiar with [10, pp.21-25 and 32-33] will find, we think, that our proof is simpler, more natural and self-contained. Finally we investigate in §2, subsets of Bf(G) (the space of multipliers of A p (G)) on which the topologies z M and a(Bf, L l ) coincide. As a consequence a necessary and sufficient condition for a subset of Ap(G) to be norm compact is given (in case G is amenable). In view of [8] the results seem to be of interest even for the nonamenable case. Definitions and notations.Let G be a locally compact group with unit e. C(G) (C 00 (G)) [CQ(G)] will denote the space of complex bounded continuous functions (with compact support) [which vanish at infinity]. À or dx will denote a left Haar measure on G. \\f\\ p = ($\f\ p dx) v * will denote the

An $L^p$-version of a theorem of D.A. Raikov

“…Rota [10], first used in harmonic analysis by E.M. Stein ; our application is close to the work of M. Cowling [3]. On the other hand this paper continues the line of studies taken up by E.E.Granirer and M. Leinert in [7].…”

Decomposition of $B(G)$

“…The following lemma is contained in the proof of Theorem B 1 in Granirer and Leinert [11] (p. 464). It plays an important role in the proof of our main result.…”

Some geometric properties on the Fourier and Fourier-Stieltjes algebras of locally compact groups, Arens regularity and related problems

“…In [34] E. Granirer and M. Leinert, answering a conjecture of McKennon [52], proved that, for any locally compact group G, on the unit sphere S = {cp £ B(G) : \\<p\\ = 1} , the weak*-topology and the multiplier topology agree. The multiplier topology is the one for which a net (cpa) converges to </> if and only if, for each y/ £ A(G), \\(<f>a -cp)y/\\ -» 0.…”