On Differentiable Vectors for Representations of Infinite Dimensional\n Lie Groups

2010/02/08 by Karl‐Hermann Neeb, Neeb, Karl-Hermann
Mathematics · #22E45 #22E65 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1002.1602

openalex publication_date 2010/02/08 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28

Abstract

In this paper we develop two types of tools to deal with differentiability\nproperties of vectors in continuous representations \π : G \→ GL(V) of an\ninfinite dimensional Lie group G on a locally convex space V. The first\nclass of results concerns the space V^\∞ of smooth vectors. If G is a\nBanach--Lie group, we define a topology on the space V^\∞ of smooth\nvectors for which the action of G on this space is smooth. If V is a Banach\nspace, then V^\∞ is a Fr 'echet space. This applies in particular to\nC^*-dynamical systems ( cA,G, \α), where G is a Banach--Lie group.\nFor unitary representations we show that a vector v is smooth if the\ncorresponding positive definite function la \π(g)v,v ra is smooth.\n The second class of results concerns criteria for C^k-vectors in terms of\noperators of the derived representation for a Banach--Lie group G acting on a\nBanach space V. In particular, we provide for each k \∈ N examples of\ncontinuous unitary representations for which the space of C^k+1-vectors is\ntrivial and the space of C^k-vectors is dense.\n

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