We continue our investigations of the representation theoretic side of reflection positivity by studying positive definite functions Ψ on the additive group (ℝ, +) 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 = Δ-1) and an integral representation. Our second main result is the existence of a Bil (V) -valued positive definite function f on the group ℝ τ = ℝ ⋊ [idℝτ] 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 Hf for which the one-parameter group on the GNS spaceH 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 (λ - d2 / dt2)-1 on the circle ℝ/β of length β.
Publication Source (Journal or Book title)
Pacific Journal of Mathematics
Neeb, K., & Ólafsson, G. (2019). KMS conditions, standard real subspaces and reflection positivity on the circle group. Pacific Journal of Mathematics, 299 (1), 117-169. https://doi.org/10.2140/pjm.2019.299.117