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From finite to infinite dimensional moment problems Infusino, Maria
Description
In this talk we give an introduction to infinite dimensional moment problems, i.e. for measures supported on infinite dimensional spaces. Although infinite dimensional moment problems have a long history, the theory is still not as well developed as in the finite dimensional case. We will focus on the following problem: when can a linear functional on a unital commutative real algebra A be represented as an integral w.r.t. a Radon measure on the character space X(A) of A equipped with the Borel Ã algebra generated by the weak topology Our main idea is to construct X(A) as a projective limit of the character spaces of all finitely generated subalgebras of A, so to be able to exploit the classical finite dimensional moment theory in the infinite dimensional case. Thus, we obtain existence results for representing measures defined on the cylinder Ã algebra on X(A), carried by the projective limit construction. If in addition the wellknown Prokhorov (Ã ÂµK) condition is fulfilled, then we can solve our problem by extending such representing measures from the cylinder to the Borel Ã algebra on X(A). These results allow us to establish infinite dimensional analogues of the classical RieszHaviland and Nussbaum theorems as well as a representation theorem for linear functionals nonnegative on a Ã¢partially ArchimedeanÃ¢ quadratic module of A. Our work applies to the case when A is the algebra of polynomials in infinitely many variables or the symmetric tensor algebra of a real infinite dimensional vector space, providing alternative proofs of some recent results for these instances of the moment problem and offering at the same time a unified setting which enables comparisons. (This is a joint work with Salma Kuhlmann, Tobias Kuna and Patrick Michalski)
Item Metadata
Title 
From finite to infinite dimensional moment problems

Creator  
Publisher 
Banff International Research Station for Mathematical Innovation and Discovery

Date Issued 
20190530T09:00

Description 
In this talk we give an introduction to infinite dimensional moment problems, i.e. for measures supported on infinite dimensional spaces. Although infinite dimensional moment problems have a long history, the theory is still not as well developed as in the finite dimensional case. We will focus on the following problem: when can a linear functional on a unital commutative real algebra A be represented as an integral w.r.t. a Radon measure on the character space X(A) of A equipped with the Borel Ã algebra generated by the weak topology Our main idea is to construct X(A) as a projective limit of the character spaces of all finitely generated subalgebras of A, so to be able to exploit the classical finite dimensional moment theory in the infinite dimensional case. Thus, we obtain existence results for representing measures defined on the cylinder Ã algebra on X(A), carried by the projective limit construction. If in addition the wellknown Prokhorov (Ã ÂµK) condition is fulfilled, then we can solve our problem by extending such representing measures from the cylinder to the Borel Ã algebra on X(A). These results allow us to establish infinite dimensional analogues of the classical RieszHaviland and Nussbaum theorems as well as a representation theorem for linear functionals nonnegative on a Ã¢partially ArchimedeanÃ¢ quadratic module of A. Our work applies to the case when A is the algebra of polynomials in infinitely many variables or the symmetric tensor algebra of a real infinite dimensional vector space, providing alternative proofs of some recent results for these instances of the moment problem and offering at the same time a unified setting which enables comparisons. (This is a joint work with Salma Kuhlmann, Tobias Kuna and Patrick Michalski)

Extent 
66.0 minutes

Subject  
Type  
File Format 
video/mp4

Language 
eng

Notes 
Author affiliation: University of Konstanz

Series  
Date Available 
20191127

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0385987

URI  
Affiliation  
Peer Review Status 
Unreviewed

Scholarly Level 
Postdoctoral

Rights URI  
Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International