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Title

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

Authors 1

Affiliations

  1. Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków, Poland

Abstract

The paper deals with operator-valued positive definite kernels on a convex *-semigroup 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 is commutative, Φ can be disintegrated with respect to some tight positive operator-valued measure defined on the characters of 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 _• of a commutative Banach *-algebra is supported by the holomorphic characters of _•. A relationship between positive definiteness and complete positivity is established in the case of commutative W*-algebras.

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Additional information

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.

Annales Polonici Mathematici
1991-1992/56/3
Export to PBN, Opens in new tab
Pages:
243-294
Main language of publication
English
Received
1990-08-02
Published
1992
Exact and natural sciences