Vol. 299, No. 1, 2019

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KMS conditions, standard real subspaces and reflection positivity on the circle group

Karl-Hermann Neeb and Gestur Ólafsson

Vol. 299 (2019), No. 1, 117–169

We continue our investigations of the representation theoretic side of reflection positivity by studying positive definite functions ψ on the additive group (,+) satisfying a suitably defined KMS condition. These functions take values in the space Bil(V ) of bilinear forms on a real vector space V . As in quantum statistical mechanics, the KMS condition is defined in terms of an analytic continuation of ψ to the strip

{z 0 Imz β}

with a coupling condition ψ(iβ + t) = ψ(t)¯ on the boundary. Our first main result consists of a characterization of these functions in terms of modular objects (Δ,J) (J an antilinear involution and Δ > 0 selfadjoint with JΔJ = Δ1) and an integral representation.

Our second main result is the existence of a Bil(V )-valued positive definite function f on the group τ = {id,τ} with τ(t) = t satisfying f(t,τ) = ψ(it) for 0 t β. We thus obtain a 2β-periodic unitary one-parameter group on the GNS space f for which the one-parameter group on the GNS space ψ is obtained by Osterwalder–Schrader quantization.

Finally, we show that the building blocks of these representations arise from bundle-valued Sobolev spaces corresponding to the kernels

(λ2 d2dt2)1

on the circle β of length β.

KMS condition, reflection positivity, standard real subspace
Mathematical Subject Classification 2010
Primary: 43A35, 43A65, 47L30
Secondary: 47L90, 81T05
Received: 31 October 2016
Accepted: 13 September 2018
Published: 18 April 2019
Karl-Hermann Neeb
Department Mathematik
Friedrich-Alexander Universität Erlangen-Nürnberg
Gestur Ólafsson
Department of Mathematics
Louisiana State University
Baton Rouge, LA
United States