COMPLETELY POSITIVE HYPERGROUP ACTIONS 3
coincide. Further
for p G M(S) and / G Ch(S), \\p\\ - sup
U
gdfi
s
9eCh(S),\\g\\suPl
and
1 1 sup = SUp
fdv
s
: I / G M ( 5 ) , | | I / | | 1
Therefore, Cb(S) and M(S) can be considered as closed subspaces of (M(5), || ||)d
and (Cb(S), ||
||Sup)d
respectively. By Banach Alaoglu theorem the unit ball of
(M(S), ||
\\)d
is M(S)-compact and that of (C6(5), || ||
s u p
)
d
is C6(S)-compact.
1.3.4- A Compatible Pair. By [16], Theorem 2.6, if S is paracompact then
(Cb(5),/3) is a strong Mackey space, i.e., every Cfc(5)-compact subset of M(*S) is
equicontinuous and, therefore, its C6(S)-closed convex hull is Cb(S)-compact. A
fortiori, (Cfc(S),/3) is a Mackey space. So by Krein's theorem, cf., [48], 24.5(4),
the M(S)-closed convex cover of an M(S)-compact subset of Cb(S) is M(S)-
compact. To sum up, if S is paracompact then (Cb(S),M(S)) is a compatible
pair. We note that it is not of the form (X,
Xd)
or
(Xd,
X) unless S is compact
or discrete.
1.4. Presentations and opresentations. Let (X, X*) be an admissible
pair. By [3], Proposition 1.2, for a norm-bounded weakly continuous function
ao : S X and p G M(S) there is a unique vector x G X such that
p(x) = / p(a0(s))dfjb(s) for p G X*.
We denote this x by a(^). Then a(ps) = 0:0(5) for s in 5, and thus a may
be thought of as an extension to M(5) of the function o^. We also have that a
is linear and ||a(/i)|| sup{||ao(s)|| : s G 5}||/i||; therefore, a is continuous on
(M(5), || ||) to (X, || ||). Thus a is bounded on norm-bounded subsets of M(5),
in particular, M
p
(5).
Moreover a is continuous on (M(5),r6) to X with the weak topology. Let S
be paracompact and let (M,M*) be the compatible pair (M(5),Cb(S)).
i.^.L Since the set of measures in M + (5) with finite support is dense
in (M + (5),r
6
) we have that a is the unique positive-homogeneous, additive,
weakly continuous extension to (M
+
(5),r
6
) of ao and the unique linear, weakly
continuous extension to (M(5),r6) of ao- Hence we may refer to ao or a as we
like. Accordingly, we write ao as a if no confusion arises.
1.4-2. It follows from the Duality Theory in locally convex spaces that if
Z is an X*-closed convex subset (respectively, cone, subspace) of X and ao(s) G
Z for each s in S then a(p) also is in Z for each p in Mp(5) (respectively,
3Vt+(5),M(5)).
We are now ready to define and study variants of a representation U of a
locally compact abelian group G on X and of an action 0 of G on Lw (X, Y)
determined by representations U and V of G on X and Y" respectively; cf., [3],
Definition 1.3, discussion on p. 221, and Proposition 1.6.
1.4.3. A
PRESENTATION
of S on X is a map A : 5 LW(X) such that
£ = sup{||A(s)|| : 5 G 5} 00 and for each x G X, the map A^ : 5 » X given by
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