On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

TL;DR

This paper develops a theory of analytic vectors and positive analytic functionals for unitary representations of infinite-dimensional Lie groups, establishing criteria for integrability and extension of local functions.

Contribution

It introduces the concept of analytic functionals on the complex enveloping algebra and proves their integrability, connecting local analyticity of vectors to global analytic functions.

Findings

Positive analytic functionals are integrable and correspond to vectors in unitary representations.

A vector is analytic if and only if its matrix coefficient is analytic near the identity.

Every local positive definite analytic function extends to a global analytic function on the group.

Abstract

Let $G$ be a 1-connected Banach-Lie group or, more generally, a BCH--Lie group. On the complex enveloping algebra $U_\C(\g)$ of its Lie algebra $\g$ we define the concept of an analytic functional and show that every positive analytic functional $\lambda$ is integrable in the sense that it is of the form $\lambda(D) = \la \dd\pi(D)v, v\ra$ for an analytic vector $v$ of a unitary representation of $G$. On the way to this result we derive criteria for the integrability of *-representations of infinite dimensional Lie algebras of unbounded operators to unitary group representations. For the matrix coefficient $\pi^{v,v}(g) = \la \pi(g)v,v\ra$ 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 identity neighborhood. Combining this insight with the results on positive analytic functionals, we derive that every local positive definite analytic function on a 1-connected Fr\'echet--BCH--Lie group $G$ extends to a global analytic function.