2014/06/10 by Karl-Hermann Neeb, Karl‐Hermann Neeb, Stephane Merigon +6
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1406.2681
openalex publication_date 2014/06/10 · arxiv created 2014/07/11 · arxiv updated 2014/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let g be a Banach Lie algebra and τ: g ---> g an involution. Write g=h+q for the eigenspace decomposition of g with respect to τand gc := h+iq for the dual Lie algebra. In this article we show the integrability of two types of infinitesimally unitary representations of gc. The first class of representation is determined by a smooth positive definite kernel K on a locally convex manifold M. The kernel is assumed to satisfying a natural invariance condition with respect to an infinitesimal action β: g → V(M) by locally integrable vector fields that is compatible with a smooth action of a connected Lie group H with Lie algebra h. The second class is constructed from a positive definite kernel corresponding to a positive definite distribution K ∈ C-∞(M × M) on a finite dimensional smooth manifold M which satisfies a similar invariance condition with respect to a homomorphism β: g → V(M). As a consequence, we get a generalization of the Luscher--Mack Theorem which applies to a class of semigroups that need not have a polar decomposition. Our integrability results also apply naturally to local representations and representations arising in the context of reflection positivity.