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

Jan Stochel

Annales Polonici Mathematici (1992)

  • Volume: 56, Issue: 3, page 243-294
  • ISSN: 0066-2216

Abstract

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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.

How to cite

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Jan Stochel. "Decomposition and disintegration of positive definite kernels on convex *-semigroups." Annales Polonici Mathematici 56.3 (1992): 243-294. <http://eudml.org/doc/262273>.

@article{JanStochel1992,
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.},
author = {Jan Stochel},
journal = {Annales Polonici Mathematici},
keywords = {operator-valued positive definite kernels on a convex -semigroup; Kolmogorov-Aronszajn type factorizations; -semigroups of bounded shift operators; canonical decomposition into a degenerate and a nondegenerate part; representing measure of a positive definite holomorphic mapping; holomorphic characters; complete positivity; commutative -algebras},
language = {eng},
number = {3},
pages = {243-294},
title = {Decomposition and disintegration of positive definite kernels on convex *-semigroups},
url = {http://eudml.org/doc/262273},
volume = {56},
year = {1992},
}

TY - JOUR
AU - Jan Stochel
TI - Decomposition and disintegration of positive definite kernels on convex *-semigroups
JO - Annales Polonici Mathematici
PY - 1992
VL - 56
IS - 3
SP - 243
EP - 294
AB - 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.
LA - eng
KW - operator-valued positive definite kernels on a convex -semigroup; Kolmogorov-Aronszajn type factorizations; -semigroups of bounded shift operators; canonical decomposition into a degenerate and a nondegenerate part; representing measure of a positive definite holomorphic mapping; holomorphic characters; complete positivity; commutative -algebras
UR - http://eudml.org/doc/262273
ER -

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