A smoothing operator for a unitary representation π : G → U(H) of a (possibly infinite dimensional) Lie group G is a bounded operator A : H → H whose range is contained in the space H ∞ of smooth vectors of (π, H). Our first main result characterizes smoothing operators for Fréchet-Lie groups as those for which the orbit map π A : G → B(H), g → π(g)A is smooth. For unitary representations (π, H) which are semibounded, i.e., there exists an element x0 ∈ g such that all operators idπ(x) from the derived representation, for x in a neighborhood of x0, are uniformly bounded from above, we show that H ∞ coincides with the space of smooth vectors for the one-parameter group πx 0 (t) = π(exp tx0). As the main application of our results on smoothing operators, we present a new approach to host C * -algebras for infinite dimensional Lie groups, i.e., C * -algebras whose representations are in one-to-one correspondence with certain continuous unitary representations of G. We show that smoothing operators can be used to obtain host algebras and that the class of semibounded representations can be covered completely by host algebras. In particular, the latter class permits direct integral decompositions.

A unitary representation of a, possibly infinite dimensional, Lie group G is called semi-bounded if the corresponding operators idπ(x) from the derived representations are uniformly bounded from above on some nonempty open subset of the Lie algebra g. Not every Lie group has nontrivial 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 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.MSC2000: 22E45, 22E65.

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Let (π, H) be a continuous unitary representation of the (infinite dimensional) Lie group G, and γ : R → Aut(G) be a group homomorphism which defines a continuous action of R on G by Lie group automorphisms. Let π # (g, t) = π(g)U t be a continuous unitary representation of the semidirect product group G ⋊ γ R on H. The first main theorem of the present note provides criteria for the invariance of the space H ∞ of smooth vectors of π under the operators U f = R f (t)U t dt for f ∈ L

An oscillator group G is a semidirect product of a Heisenberg group with a oneparameter group. In this article we construct Olshanski semigroups for infinite-dimensional oscillator groups. These are complex involutive semigroups which have a polar decomposition. The main application will be for representations π of G which are semibounded, i.e., there exists a non-empty open subset U of the Lie algebra g such that the operators idπ(x) from the derived representation are uniformly bounded from above for x ∈ U . More precisely we show that every semibounded representation of an oscillator group G extends to a non-degenerate holomorphic representation of such a semigroup and conversely each non-degenerate holomorphic representation of such a semigroup gives rise to a semibounded representation of G. The main application of this result is a classification of representations of the canonical commutation relations with a positive Hamiltonian, which will be obtained in a subsequent paper. Moreover it yields direct integral decomposition into irreducible ones and implies the existence of a dense subspace of analytic vectors for semibounded representations of G.