Integral Representation of Continuous Operators with Respect to Strict Topologies
Let X be a completely regular Hausdorff space and B o be the σ -algebra of Borel sets in X . Let C b ( X ) (resp. B ( B o ) ) be the space of all bounded continuous (resp. bounded B o -measurable) scalar functions on X , equipped with the natural strict topology β . We develop a general integral rep...
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| Published in: | Resultate der Mathematik Vol. 72; no. 1-2; pp. 843 - 863 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cham
Springer International Publishing
01.09.2017
|
| Subjects: | |
| ISSN: | 1422-6383, 1420-9012 |
| Online Access: | Get full text |
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| Summary: | Let
X
be a completely regular Hausdorff space and
B
o
be the
σ
-algebra of Borel sets in
X
. Let
C
b
(
X
)
(resp.
B
(
B
o
)
) be the space of all bounded continuous (resp. bounded
B
o
-measurable) scalar functions on
X
, equipped with the natural strict topology
β
. We develop a general integral representation theory of
(
β
,
ξ
)
-continuous operators from
C
b
(
X
)
to a lcHs
(
E
,
ξ
)
with respect to the representing Borel measure taking values in the bidual
E
ξ
′
′
of
(
E
,
ξ
)
. It is shown that every
(
β
,
ξ
)
-continuous operator
T
:
C
b
(
X
)
→
E
possesses a
(
β
,
ξ
E
)
-continuous extension
T
^
:
B
(
B
o
)
→
E
ξ
′
′
, where
ξ
E
stands for the natural topology on
E
ξ
′
′
. If, in particular,
X
is a
k
-space and
(
E
,
ξ
)
is quasicomplete, we present equivalent conditions for a
(
β
,
ξ
)
-continuous operator
T
:
C
b
(
X
)
→
E
to be weakly compact. As an application, we have shown that if
X
is a
k
-space and a quasicomplete lcHs
(
E
,
ξ
)
contains no isomorphic copy of
c
0
, then every
(
β
,
ξ
)
-continuous operator
T
:
C
b
(
X
)
→
E
is weakly compact. |
|---|---|
| ISSN: | 1422-6383 1420-9012 |
| DOI: | 10.1007/s00025-017-0678-4 |