We define for any locally compact group G, the space of bounded uniformly continuous functionals on Ĝ,
U
C
B
(
G
^
)
UCB(\hat G)
, in the context of P. Eymard [Bull. Soc. Math. France 92 (1964), 181-236. MR 37 #4208] (for notations see next section). For
u
∈
A
(
G
)
u \in A(G)
let
u
⊥
=
{
ϕ
∈
V
N
(
G
)
;
ϕ
[
A
(
G
)
u
]
=
0
}
{u^ \bot } = \{ \phi \in VN(G);\phi [A(G)u] = 0\}
. Theorem. If for some norm separable subspace
X
⊂
V
N
(
G
)
X \subset VN(G)
and some positive definite
0
≠
u
∈
A
(
G
)
,
U
C
B
(
G
^
)
⊂
0 \ne u \in A(G),UCB(\hat G) \subset
norm closure
[
W
(
G
^
)
+
X
+
u
⊥
]
[W(\hat G) + X + {u^ \bot }]
then G is discrete. If G is discrete then
U
C
B
(
G
^
)
⊂
A
P
(
G
^
)
⊂
W
(
G
^
)
UCB(\hat G) \subset AP(\hat G) \subset W(\hat G)
.
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
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