COMPLETELY POSITIVE HYPERGROUP ACTIONS 5

fi has compact support. For the general case, in view of [3], p. 219, line —2,

the argument demands the completeness of X* or equivalently, Hypothesis 1.5(i)

of [3], viz., 1.1.2, (Hi) above along with the already imposed Hypothesis 1.5(H)

of [3], viz., 1.1.2, (iv) above. Parts (b) and (c) are now immediate from the

definitions.

1.4.5. (i) A will be called the

OPRESENTATION

of M(S) on X induced by

A.

(ii) A* will be called the

DUAL PRESENTATION

of A or simply the

DUAL OF

A.

A* will be called the DUAL

OPRESENTATION

or simply the DUAL OF A.

1.4.6. For subsets £ of M(5) and Z of X let A(£)Z be the linear span of

{A(fj)x : /i G £, x G Z}. The pair (£, Z) will be called CANONICAL if A(£)Z = X

and

WEAKLY

CANONICAL if A(E)Z is weakly dense in X. We call £ CANONICAL

(respectively, WEAKLY CANONICAL) if {£,X) is so. For a subset E of S the

condition will obviously refer to £ = {ps : s G E}.

For a point s of S let Us be a base of compact neighborhoods of s directed by

inclusion. Let m be a regular Borel measure on 5 whose support contains s. For

U G Us, let fu — ^ j & / , where £u is the characteristic function of U. Further,

let mu be the measure on S given by mu(F) =

m

j ^ for any Borel set F,

i.e., drrijj = fjjdm. Then the net {mjj)ueus converges to ps in (M

p

(5),r

6

). The

following proposition, which is now immediate from §1.3 and Proposition 1.4.4

above, may be thought of as a compensation for not being able to parrot the

discussion following [3], Proposition 1.4; cf., also [40], §22.

1.4.7.

PROPOSITION.

Let m be a regular Borel measure on S. Let Ma{S, m)

denote the subspace ofM(S) consisting of measures that are absolutely contin-

uous with respect to m.

(i) Let E C the support of m and Z C X be such that (E, Z) is weakly

canonical. Then £\ — {rrijj : U eUs,s E E} is contained in the unit ball

of M

a

(5, m) and (£\,Z) is weakly canonical.

(ii) Suppose that the support of m is S. Then a norm-bounded weakly

continuous function a : S — X and a presentation A : S — LW(X) are

fully determined by their counterparts on Ma(S,m) or on any rb-dense

subspace of M

a

(5, m).

We now imitate the process in [3], p. 221, to obtain presentations on spaces

of operators from presentations on spaces.

1.4.8. For a presentation A of S on X we introduce the following conditions;

cf., [3], 1.5 (ii) and Remark 1.7.

(i) For each p G l * , the function s —* \(s)*p is continuous with respect to

the norm of X*.

(ii) For each x G X, the function s — X(s)x is continuous with respect to

the norm of X.

Our first observation is that the condition 1.1.2, (iv), as a part of the compat-

ibility condition on the pair (X,X*) in Proposition 1.4.4 (vii) above, can then

be replaced by the condition 1.4.8, (i) on A, since under the latter condition,

for each p in X*, the vector-valued integral fs \(s)*pd/u,(s) exists in the norm

topology of

Xd

as the limit of a sum and equals the element A(//)*p of

Xd.