×

Weakly compact operators and strict topologies. (English) Zbl 1292.46029

In this paper, \(X\) is a completely regular Hausdorff space, \(E\) a quasi-complete locally convex space with \(E'\) its topological dual, \(C_{b}(X)\) the space of all bounded, real-valued continuous functions on \(X\), \( \mathcal{B} \) the algebra generated by the zero-sets of \(X\) and \(\mathcal{B}a \) the Baire subsets of \(X\); also, \(M(X) = (C_{b}(X), \| \cdot \|)'\) and \( M_{t}(X), \; M_{\tau}(X)\), \(M_{\sigma}(X)\) are subsets of \(M(X)\) consisting of real-valued tight, \( \tau\)-smooth, and \(\sigma\)-smooth measures on \(X\).
Using the fact that \(M(X)\) is a Dedekind complete vector lattice and \( M_{t}(X), \; M_{\tau}(X)\), \(M_{\sigma}(X)\) are projective bands in \(M(X)\), the purely finitely additive measures \(M_{pfa}(X)\) are defined to be the elements of the disjoint complement of \(M_{\sigma}(X)\) in \(M(X)\), the purely \(\sigma\)-additive measures \(M_{p\sigma a}(X)\) are defined to be the elements of the disjoint complement of \(M_{\tau}(X)\) in \(M_{\sigma}(X)\), and the purely \(\tau\)-additive measures \(M_{p\tau a}(X)\) are defined to be the elements of the disjoint complement of \(M_{t}(X)\) in \(M_{\tau}(X)\). Thus \(M(X)\) can be decomposed as: \[ M(X)= M_{pfa}(X) + M_{p\sigma a}(X) + M_{p\tau a}(X) + M_{t}(X). \] In this paper, the author extends this result to \(E\)-valued measures. As proved in [S. S. Khurana, Georgian Math. J. 14, No. 4, 687–698 (2007; Zbl 1154.46025)], every weakly compact operator \(m: C_{b}(X) \to E \), just as in the real-valued case, gives a regular, finitely additive, \(E\)-valued measure on \( \mathcal{B} \) and the space of all such measures has natural subspaces of \(\sigma\)-smooth, \( \tau\)-smooth, and tight measures; these measures, coming from weakly compact operators, are also strongly bounded (they are also called exhaustive). The measure \(m\) is called purely finitely additive if \(e' \circ m \in M_{pfa}(X)\) for all \(e' \in E'\) and similar meanings for purely \(\sigma\)-additive measures and purely \(\tau\)-additive measures. The main result of the paper is:
If \(m\) is the representing measure of a weakly compact operator \(m: C_{b}(X) \to E \), then \(m\) can be uniquely decomposed as \(m=m_{1}+m_{2}+m_{3}+m_{4}\), where \(m_{1} \in M_{t}(X)\), \(m_{2}\) is a purely \(\tau\)-additive measure, \(m_{3}\) is a purely \(\sigma\)-additive measure, and \(m_{4}\) is a purely finitely additive measure.

MSC:

46G10 Vector-valued measures and integration
28A33 Spaces of measures, convergence of measures
28A25 Integration with respect to measures and other set functions
28B05 Vector-valued set functions, measures and integrals

Citations:

Zbl 1154.46025
Full Text: DOI

References:

[1] Aliprantis, C. D.; Burkinshaw, O., Positive Operators (1985), Academic Press: Academic Press New York · Zbl 0567.47037
[2] Aliprantis, C. D.; Burkinshaw, O., Locally Solid Riesz Spaces with Applications to Economics, Math. Surveys Monogr., vol. 105 (2003) · Zbl 1043.46003
[3] Aguayo, J.; Sánchez, J., Weakly compact operators and the strict topologies, Bull. Aust. Math. Soc., 39, 353-359 (1989) · Zbl 0665.46038
[4] Brooks, J. K., Decomposition theorems for vector measures, Proc. Am. Math. Soc., 21, 27-29 (1969) · Zbl 0185.21902
[5] Diestel, J.; Uhl, J. J., Vector Measures, Math. Surveys Monogr., vol. 15 (1977), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0369.46039
[6] Drewnowski, L., Decompositions of set functions, Stud. Math., 48, 23-48 (1973) · Zbl 0269.28003
[7] Edwards, R. E., Functional Analysis, Theory and Applications (1965), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York · Zbl 0182.16101
[8] Hoffmann-Jörgensen, J., Vector measures, Math. Scand., 28, 5-32 (1971) · Zbl 0217.38001
[9] Huff, R. E., The Yosida-Hewitt decomposition as an ergodic theorem, (Vector and Operator-Valued Measures and Applications. Vector and Operator-Valued Measures and Applications, Proc. Sympos., Snowbird Resort, Alta Utah, 1972 (1973), Academic Press: Academic Press New York), 133-139 · Zbl 0289.28013
[10] Khurana, S. S., Vector measures on topological spaces, Georgian Math. J., 14, 4, 687-698 (2007) · Zbl 1154.46025
[11] Kluvanek, I., The extension and closure of vector measures, (Vector and Operator Valued Measures and Applications. Vector and Operator Valued Measures and Applications, Proc. Sympos., Snowbird Resort, Alta, Utah, 1972 (1973), Academic Press: Academic Press New York), 175-198 · Zbl 0302.28009
[12] Knowles, J., Measures on topological spaces, Proc. Lond. Math. Soc. (3), 17, 139-156 (1967) · Zbl 0154.05101
[13] Lewis, D. R., Integration with respect to vector measures, Pac. J. Math., 33, 1, 157-165 (1970) · Zbl 0195.14303
[14] Nowak, M., Vector measures and strict topologies, Topol. Appl., 159, 5, 1421-1432 (2012) · Zbl 1247.46036
[15] Panchapagesan, T. V., Applications of a theorem of Grothendieck to vector measures, J. Math. Anal. Appl., 214, 89-101 (1997) · Zbl 0892.28009
[16] Ruess, W., [Weakly] compact operators and DF-spaces, Pacific J. Math., 98, 2, 419-441 (1982) · Zbl 0441.47031
[17] Sentilles, F. D., Bounded continuous functions on a completely regular spaces, Trans. Am. Math. Soc., 168, 311-336 (1972) · Zbl 0244.46027
[18] Traynor, T., A general Yosida-Hewitt decomposition, Can. J. Math., 24, 1164-1169 (1972) · Zbl 0219.46034
[19] Uhl, J. J., Extensions and decompositions of vector measures, J. Lond. Math. Soc. (2), 3, 672-676 (1971) · Zbl 0213.33901
[20] Varadarajan, V. S., Measures on topological spaces, Mat. Sb. (N. S.). Mat. Sb. (N. S.), Transl. Am. Math. Soc. Ser. 2, vol. 48, 97, 161-228 (1965), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0152.04202
[21] Wheeler, R., A survey of Baire measures and strict topologies, Expo. Math., 1, 97-190 (1983) · Zbl 0522.28009
[22] Yosida, K.; Hewitt, E., Finitely additive measures, Trans. Amer. Math. Soc., 72, 46-66 (1952) · Zbl 0046.05401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.