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