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.
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