Extremal properties of section spaces
59
used the characterization of C(X, E)∗ very strongly. However, to the knowledge of
the authors, there is no such concise characterization of Γ(π)∗. The investigation
in [10] was handicapped by this lack of nice characterization. Indeed, contrast the
two situations:
1) If X is a compact Hausdorff space, and if E is a Banach space, then C(X, E)∗
can be isometrically identified with the space M(X, E∗) of all countably additive
E∗-valued Borel measures on X, with the variation norm, and action hf, µi =
R
X fdµ.
2) Let π : E → X be a separable real bundle and let φ ∈ Γ(π)∗. Then there is
•
S
a regular Borel measure µ on X and a choice function η : X → {Ex∗ : x ∈ X}
such that (among other properties) a) kη(x)k ≤ 1 for all x ∈ X and equality holds
µ-almost everywhere; b) the function x → hσ(x), η(x)i is Borel measurable for each
R
σ ∈ Γ(π); and c) for all σ ∈ Γ(π) we have hσ, φi =
hσ(x), η(x)i dµ. (See [4].)
X
In particular, whereas in [6] a necessary and sufficient condition for a functional
φ ∈ C(X, E)∗ to be a weak-* point of continuity (or a sequential weak-* point of
continuity, in case X is metric) of the unit ball was obtained, in [10] only a sufficient
description for φ ∈ Γ(π)∗ to be a weak-* point of continuity of the unit ball could
be proven. The purpose of this paper is to complete the characterization of weak-*
∗
points of continuity and sequential continuity in BΓ(π) for a class of bundles which
includes the trivial ones. Our proofs are actually shorter than those of the more
special cases addressed in [6], and do not employ vector measures.
∗
Recall that a point φ ∈ BZ is called a weak-* point of continuity provided that
∗
whenever {φλ} is a net in BZ which converges weak-* to φ, then {φλ} converges in
norm to φ. The definition of a sequential weak-* point of continuity replaces “nets”
by “sequences”. Recall also that if π : E → X is a Banach bundle, and if x ∈ X,
then there is an isomorphic injection jx : Ex∗ → Γ(π)∗ given by (for σ ∈ Γ(π))
hσ, jx(f)i = hσ(x), fi = hσ, f ◦ evxi , where evx : Γ(π) → Ex, σ → σ(x) ∈ Ex, is
the evaluation map. In addition, for each closed set C ⊂ X and φ ∈ Γ(π)∗, there
is an L-projection PC : Γ(π)∗ → Γ(π)∗, PC(φ) = φC. The action of φC on Γ(π) is
defined as follows: let W run through the system of open neighborhoods of C, and
for each such W, let iW ∈ C(X), iW : X → [0, 1], be such that iW (C) = 0 and
iW (X \ C) = 1 (i.e. {iW } is an approximate identity for the ideal of functions in
C(X) which vanish on C). Then
hσ, φCi = lim h(1 − iW )σ, φi .
W →C
(See [4] or [9].) That is, φC = weak-* limW →C(1 − iW )φ. If C ⊂ X is closed, we
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write φX\C = φ − φC; thus we have kφk = kφCk + φX\C , and φA makes sense
if A ⊂ X is either open or closed. With a little work, it can also be shown that if
C ⊂ X is closed, and if U ⊂ C is open, then we can also write φC = φC\U + φU ,
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with kφCk = φC\U + kφU k .
In particular, if x ∈ X and φ ∈ Γ(π)∗, we write φx = φ , and we note that if
{x}
{x1, ..., xm} ⊂ C, then
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m
m
X
X
X
X
φx
=
kφx k ≤
kφ k =
φx ≤ kφ k ≤ kφk .
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x
C
k
k
°
k=1
k=1
x∈C
x∈C