Symmetric Hilbert Spaces and Related Topics Infinitely Divisible Pos
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261 Alain Guichardet Faculte des Sciences de Poitiers, France
Symmetric Hilbert Spaces and Related Topics Infinitely Divisible Positive Definite Functions Continuous Products and Tensor Products Gaussian and Poisson ian Stochastic Processes
Springer-Verlag Berlin· Heidelberg· New York 1972
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Zurich
261 Alain Guichardet Faculte des Sciences de Poitiers, France
Symmetric Hilbert Spaces and Related Topics Infinitely Divisible Positive Definite Functions Continuous Products and Tensor Products Gaussian and Poisson ian Stochastic Processes
Springer-Verlag Berlin· Heidelberg· New York 1972
AMS Subject Classifications (1970): 46C10, 46M05, 60G15, 81A20
ISBN 3-540-05803-6 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05803-6 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1972. Library of Congress Catalog Card Number 72-76390. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Table of Content
Introduction .
3 3
Chapter 1. The symmetric measure space of a measure space. • 1.1. The symmetric set of a set. • . . . . • . 1.2. The functor S on the category of sets. • • . 1.3. The symmetric Borel space of a Borel space. 1.4. The symmetric measure space of a measure space. 1.5. Application to linear processes and factored probability spaces. . . . . . . . • . . • •
12
Chapter 2. The symmetric Hilbert space of a Hilbert space. ••
18
2.1. Definitions and general properties • . . . 2.2. The unitary operators UA,b,c and the 2.3. Relation with sywtietric measure spaces
Po:.
18
22 28
Chapter 3. Positive definite functions of type (S) • . 3.1. Positive definite functions. • • . • • • 3.2. Positive definite functions of type (S). Definitions and first properties . • . . 3.3. The case of commutative locally compact groups
30 30
35 38
Chapter 4. Conditionally positive definite functions and infinitely divisible positive definite functions.
47
4.1. Conditionally positive definite matrices and kerne Is. . • . . • . . . • . . • . . • 4.2. Conditionally positive definite functions on groups • • . . . . . • • . • . • . • • . . • 4.3. Infinitely divisible positive definite functions Chapter
5. Boolean Algebras of tensor decompositions of Hilbert space . . .
. . .
. . . . • .
. • .
a
Chapter 6. Factorizable positive definite functions on current groups. • • •.• 6.1. Definitions. • . .•. 6.2. Results. . . • . . •.. 6.3. Study of other current groups. Chapter 7. Gaus
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