Convolutions and products of partially ordered vector-valued positive measures. (English) Zbl 0677.46029
Let X, Y, Z be monotone complete partially ordered vector spaces. The main objective of this paper is to further investigate properties of partially ordered vector-valued positive measures using previous results of J. D. M. Wright [cf. Proc. London Math. Soc., III. Ser. 19, 107- 122 (1969; Zbl 0186.465); ibid. 25, 675-688 (1972; Zbl 0243.46049), Math. Z. 120, 193-203 (1971; Zbl 0208.157), Quart. J. Math., II. Ser. 24, 189-206 (1973; Zbl 0255.28011), J. London Math. Soc., II. Ser. 8, 699-706 (1974; Zbl 0289.46007)] and results (some of them still unpublished) concerning an order preserving bilinear integration process established by P. K. Pavlakos [cf. Can. J. Math., II. Ser. 35, 353- 372 (1983; Zbl 0487.28010)].
The derived integral has the form \(\int_{T}f(t)dm(t)\), where the integrable function f (resp. positive measure m) is defined on T (resp. on a \(\sigma\)-algebra of subsets of T) and takes values in X (resp. Y).
The integral takes values in the cut completion \(\hat V\) of \(V:=Z^+- Z^+\) via a positive bilinear order separately continuous function from \(X\times Y\) into Z.
We investigate notions of convolution of two positive quasi-regular measures defined on the Borel \(\sigma\)-algebra of a locally compact Hausdorff topological semigroup G and taking values in X, Y respectively.
For this purpose assuming moreover that X is a partially ordered orderunit-normed space we state and prove a Fubini-type theorem. The obtained results can be immediately applied in \(C^*\)-algebras (especially in \(W^*\)-algebras or \(AW^*\)-algebras of type I), in Jordan algebras, in partially ordered *-involutory \((0^*\)-) algebras, in semifields as well as in mathematical foundation of quantum probability theory.
The derived integral has the form \(\int_{T}f(t)dm(t)\), where the integrable function f (resp. positive measure m) is defined on T (resp. on a \(\sigma\)-algebra of subsets of T) and takes values in X (resp. Y).
The integral takes values in the cut completion \(\hat V\) of \(V:=Z^+- Z^+\) via a positive bilinear order separately continuous function from \(X\times Y\) into Z.
We investigate notions of convolution of two positive quasi-regular measures defined on the Borel \(\sigma\)-algebra of a locally compact Hausdorff topological semigroup G and taking values in X, Y respectively.
For this purpose assuming moreover that X is a partially ordered orderunit-normed space we state and prove a Fubini-type theorem. The obtained results can be immediately applied in \(C^*\)-algebras (especially in \(W^*\)-algebras or \(AW^*\)-algebras of type I), in Jordan algebras, in partially ordered *-involutory \((0^*\)-) algebras, in semifields as well as in mathematical foundation of quantum probability theory.
Reviewer: P.K.Pavlakos
MSC:
| 46G10 | Vector-valued measures and integration |
| 28B05 | Vector-valued set functions, measures and integrals |
| 46A40 | Ordered topological linear spaces, vector lattices |
| 44A35 | Convolution as an integral transform |
Keywords:
partially ordered vector-valued positive measures; partially ordered orderunit-normed space; Fubini-type theorem; \(C^*\)-algebras; \(W^*\)- algebras; \(AW^*\)-algebras of type I; Jordan algebras; partially ordered *-involutory \((0^*\)-) algebras; semifields; quantum probabilityCitations:
Zbl 0186.465; Zbl 0243.46049; Zbl 0208.157; Zbl 0255.28011; Zbl 0289.46007; Zbl 0487.28010References:
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