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1991-1992 | 56 | 3 | 243-294
Tytuł artykułu

Decomposition and disintegration of positive definite kernels on convex *-semigroups

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Języki publikacji
EN
Abstrakty
EN
The paper deals with operator-valued positive definite kernels on a convex *-semigroup 𝓢 whose Kolmogorov-Aronszajn type factorizations induce *-semigroups of bounded shift operators. Any such kernel Φ has a canonical decomposition into a degenerate and a nondegenerate part. In case 𝓢 is commutative, Φ can be disintegrated with respect to some tight positive operator-valued measure defined on the characters of 𝓢 if and only if Φ is nondegenerate. It is proved that a representing measure of a positive definite holomorphic mapping on the open unit ball $𝓐_•$ of a commutative Banach *-algebra 𝓐 is supported by the holomorphic characters of $𝓐_•$. A relationship between positive definiteness and complete positivity is established in the case of commutative W*-algebras.
EN
Contents Introduction. 1. Preliminaries. 2. Predilatable kernels. 3. Criteria of predilatability. 4. Degenerate and nondegenerate predilatable kernels. 5. Canonical decomposition of predilatable kernels. 6. Weakly predilatable kernels. 7. Disintegration of nondegenerate predilatable kernels. 8. Continuity of predilatable mappings on topological *-algebras. 9. Disintegration of holomorphic positive definite mappings on commutative Banach *-algebras. 10. Holomorphic positive definite mappings on noncommutative Banach *-algebras. 11. Completely positive k-linear mappings. 12. Multiplicative k-homogeneous polynomials. 13. Positive definiteness versus complete positivity. 14. Appendix References.
Słowa kluczowe
Rocznik
Tom
56
Numer
3
Strony
243-294
Opis fizyczny
Daty
wydano
1992
otrzymano
1990-08-02
Twórcy
autor
  • Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków, Poland
Bibliografia
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Bibliografia
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