On an invariance property of the space of smooth vectors - Neeb, Karl Hermann; Salmasian, Hadi; Zellner, Christoph (Kyoto Journal of Mathematics, 2015)
Keywords: Infinite-dimensional Lie groups; unitary representation; integrated representations; smooth vectors; Arveson spectral theory

On the characterization of trace class representations and Schwartz operators - van Dijk, Gerrit; Neeb, Karl Hermann; Salmasian, Hadi; Zellner, Christoph (Journal of Lie Theory, 2016)
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 C^m_c(G), are trace class. As a consequence the corresponding distribution character θ_π 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 AHH^∞, 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 bounde
Keywords: Trace class representation; smoothing operator; Schwartz operator; unitary representation

Oscillator algebras with semi-equicontinuous coadjoint orbits - Neeb, Karl Hermann; Zellner, Christoph (Differential Geometry and its Applications, 2013)
A unitary representation of a, possibly infinite dimensional, Lie group G is called semibounded if the corresponding operators <span class="formulatext stixSupport mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224512000976&_mathId=si1.gif&_user=111111111&_pii=S0926224512000976&_rdoc=1&_issn=09262245&md5=5b978cbc6bcc9559a8559e450f5d2f67" title="Click to view the MathML source">idπ(x)idπ(x) from the derived representations are uniformly bounded from above on some non-empty open subset of the Lie algebra <span class="formulatext stixSupport mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0926224512000976&_mathId=si2.gif&_user=111111111&_pii=S0926224512000976&_rdoc=1&_issn=09262245&md5=e652da85d4bb41b281ee15ada98773fa" title="Click to view the MathML source">gg. Not every Lie group has non-trivial semibounded unitary representations, so that it becomes an important issue to decide when this is the case. In the present paper we describe a complete solution of this problem for the class of generalized oscillator groups, which are semidirect products of Heisenberg groups with a one-parameter group γ. For these groups it turns out that the existence of non-trivial semibounded representations is equivalent to the existence of the so-called semi-equicontinuous non-trivial coadjoint orbits, a purely geometric condition on the coadjoint action. This in turn can be expressed by a positivity condition on the Hamiltonian function corresponding to the infinitesimal generator D of γ. A central point of our investigations is that we make no assumption on the structure of the spectrum of D. In particular, D can be any skew-adjoint operator on a Hilbert space.
Keywords: Infinite dimensional Lie group; Semibounded representation; Oscillator group; Coadjoint orbit; Complex structure

Smoothing operators and C*-algebras for infinite dimensional Lie groups - Neeb, Karl Hermann; Salmasian, Hadi; Zellner, Christoph (International Journal of Mathematics, 2017)