Abstract
We generalise the Riesz representation theorems for positive linear functionals on \(\text {C}_{\text {c}}(X)\) and \(\text {C}_{\text {0}}(X)\), where X is a locally compact Hausdorff space, to positive linear operators from these spaces into a partially ordered vector space E. The representing measures are defined on the Borel \(\sigma \)-algebra of X and take their values in the extended positive cone of E. The corresponding integrals are order integrals. We give explicit formulas for the values of the representing measures at open and at compact subsets of X. Results are included where the space E need not be a vector lattice, nor a normed space. Representing measures exist, for example, for positive linear operators into Banach lattices with order continuous norms, into the regular operators on KB-spaces, into the self-adjoint linear operators on complex Hilbert spaces, and into JBW-algebras.
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Notes
In the course of the present paper, its prequel [10], and its sequels [
When \(E\) and F are vector lattices, where F is Dedekind complete, then the notions of order continuous and \(\sigma \)-order continuous operators in Definition 2.2 agree with the usual ones in the literature as on [27, p. 123]; see [10, Remark 3.5] for this.
When \(E\) is a vector lattice, then the notion of normality in Definition 2.3 coincides with the usual one in the literature (see [1, p. 21], for example) that \({E_{\text {oc}}^{\thicksim }}\) separates the points of \(E\); see [10, Lemma 3.7] for this. It also follows from [10, Lemma 3.7] that a vector lattice \(E\) is \(\sigma \)-normal as in Definition 2.3 if and only if \({E_{\sigma \text {oc}}^{\thicksim }}\) separates the points of \(E\).
As in [10, Section 6], we say that a partially ordered vector space \(E\) has the countable sup property when, for every net \(\{{x}_\lambda \}_{\lambda \in \Lambda }\subseteq {E^+}\) and \(x\in {E^+}\) such that \(x_\lambda \uparrow x\), there exists an at most countably infinite set of indices \(\{\lambda _n: n\ge 1 \}\) such that \(x=\sup _{n\ge 1} x_{\lambda _n}\). In this case, there also always exist indices \(\lambda _1\le \lambda _2\le \cdots \) such that \(x_{\lambda _n}\uparrow x\). For vector lattices, this definition coincides with the usual one as can be found on, e.g., [5, p. 103].
References
Abramovich, Y.A., Aliprantis, C.D.: An Invitation to Operator Theory. Graduate Studies in Mathematics, vol. 50. American Mathematical Society, Providence (2002)
Alfsen, E.M., Shultz, F.W.: State Spaces of Operator Algebras. Mathematics: Theory & Applications. Basic Theory, Orientations, and \(\text{C}{\ast }\)-products. Birkhäuser Boston Inc., Boston (2001)
Alfsen, E.M., Shultz, F.W.: Geometry of State Spaces of Operator Algebras. Mathematics: Theory & Applications. Birkhäuser Boston Inc, Boston (2003)
Aliprantis, C.D., Burkinshaw, O.: Principles of Real Analysis, 3rd edn. Academic Press Inc, San Diego (1998)
Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Dordrecht (2006) (Reprint of the 1985 original)
Cohn, D.L.: Measure Theory. Birkhäuser Boston Inc., Boston (1993) (Reprint of the 1980 original)
Conway, J.B.: A Course in Functional Analysis, Graduate Texts in Mathematics, second edition, vol. 96. Springer, New York (1990)
Coquand, T.: A note on measures with values in a partially ordered vector space. Positivity 8(4), 395–400 (2004)
Davidson, K.R.: \({\text{ C }}{\ast }\)-algebras by Example. Fields InstituteMonographs, vol. 6.American Mathematical Society, Providence (1996)
de Jeu, M., Jiang, X.: Order integrals. Positivity 26, no. 2, Paper No. 32 (2022)
de Jeu, M., Jiang, X.: Riesz representation theorems for lattices of regular operators. In preparation; to be made available on ar**v (2022)
de Jeu, M., Jiang, X.: Riesz Representation Theorems for Positive Algebra Homomorphisms. (2021). ar**v:2109.10690
de Jeu, M., Ruoff, F.: Positive representations of \(C_0(X)\). I. Ann. Funct. Anal. 7(1), 180–205 (2016)
Folland, G.B.: Real Analysis. Modern Techniques and Their Applications. Pure and Applied Mathematics, 2nd edn. Wiley, New York (1999)
Hanche-Olsen, H., Størmer, E.: Jordan Operator Algebras, Volume 21 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston (1984)
Khurana, S.S.: Lattice-valued Borel measures. Rocky Mt. J. Math. 6(2), 377–382 (1976)
Khurana, S.S.: Lattice-valued Borel measures. II. Trans. Am. Math. Soc. 235, 205–211 (1978)
Khurana, S.S.: Lattice-valued Borel measures. III. Arch. Math. (Brno) 44(4), 307–316 (2008)
Lipecki, Z.: Riesz type representation theorems for positive operators. Math. Nachr. 131, 351–356 (1987)
Rudin, W.: Principles of Mathematical Analysis, 3rd edn. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York-Auckland-Düsseldorf (1976)
Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill Book Co., New York (1987)
Wnuk, W.: Banach Lattices with Order Continuous Norms. Polish Scientific Publishers PWN, Warsaw (1999)
Wright, J.D.M.: Stone-algebra-valued measures and integrals. Proc. Lond. Math. Soc. 3(19), 107–122 (1969)
Wright, J.D.M.: The measure extension problem for vector lattices. Ann. Inst. Fourier (Grenoble) 21(4), 65–85 (1971)
Wright, J.D.M.: Vector lattice measures on locally compact spaces. Math. Z. 120, 193–203 (1971)
Wright, J.D.M.: Measures with values in a partially ordered vector space. Proc. Lond. Math. Soc. 3(25), 675–688 (1972)
Zaanen, A.C.: Riesz Spaces. II, Volume 30 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam (1983)
Zaanen, A.C.: Introduction to Operator Theory in Riesz Spaces. Springer, Berlin (1997)
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The authors thank Marten Wortel for helpful discussions on JBW-algebras.
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Communicated by Maria Joita.
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de Jeu, M., Jiang, X. Riesz representation theorems for positive linear operators. Banach J. Math. Anal. 16, 44 (2022). https://doi.org/10.1007/s43037-022-00177-7
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DOI: https://doi.org/10.1007/s43037-022-00177-7