2015/12/08 by Gerrit van Dijk, Karl‐Hermann Neeb, van Dijk, Gerrit +5 · 1 citation
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1512.02451
openalex publication_date 2015/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this note we collect several characterizations of unitary representations (π, H) of a finite dimensional Lie group G which are trace class, i.e., for each compactly supported smooth function f on G, the operator π(f) is trace class. In particular we derive the new result that, for some m ∈ ℕ, all operators π(f), f ∈ Cmc(G), are trace class. As a consequence the corresponding distribution character θπ is of finite order. We further show π is trace class if and only if every operator A, which is smoothing in the sense that AH⊆ H^∞, is trace class and that this in turn is equivalent to the Fréchet space H^∞ being nuclear, which in turn is equivalent to the realizability of the Gaussian measure of H on the space H-∞ of distribution vectors. Finally we show that, even for infinite dimensional Fréchet-Lie groups, A and A^* are smoothing if and only if A is a Schwartz operator, i.e., all products of A with operators from the derived representation are bounded.