# On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups - Mathematics > Representation Theory

On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups - Mathematics > Representation Theory - Download this document for free, or read online. Document in PDF available to download.

Abstract: Let $G$ be a 1-connected Banach-Lie group or, more generally, a BCH-Liegroup. On the complex enveloping algebra $U \C\g$ of its Lie algebra $\g$ wedefine the concept of an analytic functional and show that every positiveanalytic functional $\lambda$ is integrable in the sense that it is of the form$\lambdaD = \la \dd\piDv, v a$ for an analytic vector $v$ of a unitaryrepresentation of $G$. On the way to this result we derive criteria for theintegrability of *-representations of infinite dimensional Lie algebras ofunbounded operators to unitary group representations.For the matrix coefficient $\pi^{v,v}g = \la \pigv,v a$ of a vector $v$in a unitary representation of an analytic Fr\-echet-Lie group $G$ we show that$v$ is an analytic vector if and only if $\pi^{v,v}$ is analytic in an identityneighborhood. Combining this insight with the results on positive analyticfunctionals, we derive that every local positive definite analytic function ona 1-connected Fr\-echet-BCH-Lie group $G$ extends to a global analyticfunction.

Author: ** Karl-Hermann Neeb**

Source: https://arxiv.org/