OS Reelle Geometrie und Algebra: Projective limit techniques for the infinite dimensional moment problem

Wann
Freitag, 25. Januar 2019
13:30 – 15:00 Uhr

Wo
F 426

Veranstalter
S. Kuhlmann, C. Scheiderer, M. Schweighofer

ReferentIn:
Patrick Michalski

Diese Veranstaltung ist Teil der Veranstaltungsreihe „Oberseminar Reelle Geometrie und Algebra“.

Consider the following general version of the classical moment problem for a linear functional $L$ on a unital commutative $\mathbb{R}$-algebra $A$: When can $L$ be represented as an integral w.r.t. a Radon measure on the character space $X(A)$ of $A$ equipped with the Borel $\sigma$-algebra generated by the weak topology $\tau_A$?
In this talk, we show that this problem is solvable if for each finitely generated subalgebra $S$ of $A$ there exists a (unique) representing Radon measure for $L\restriction_S$. The crucial step here is to construct $X(A)$ as a projective limit of the character spaces of finitely generated subalgebras of $A$. Our result allows us to generalize to infinitely (even uncountably) generated $\mathbb{R}$-algebras some of the classical theorems for the moment problem, e.g. the ones by Nussbaum and Putinar.

This is joint work in progress with Maria Infusino, Salma Kuhlmann and Tobias Kuna.