KMS conditions, standard real subspaces and reflection positivity on the circle group
type: Item
instance of: scholarly article
title: KMS conditions, standard real subspaces and reflection positivity on the circle group
zbMATH Open document ID: 1422.43009
published in: Pacific Journal of Mathematics
publication date: 2019-05-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1611.00080
review text: The authors continue their investigations of the representation theoretic side of reflection positivity by studying positive definite functions \(\psi\) on \(\mathbb R\) satisfying a \(KMS\) (Kubo-Martin-Schwinger) condition. Let \(V\) be a real vector space, \(\mathrm{Bil} (V)\) the space of real bilinear forms on \(V\) with complex values. A function \(\psi: \mathbb{R} \to \mathrm{Bil} (V)\) is called \textit{positive definite} if the kernel \(\psi(t-s)(v,w)\) is positive definite. For \(\beta >0\), let \(\mathcal{S}_\beta\) be the open strip \(\{z\in\mathbb{C}: 0< \mathrm{Im} \, z < \beta\}\). A positive definite function \(\psi\) \textit{satisfies the \(KMS\) condition for} \(\beta >0\) if \(\psi\) extends to a function \({\overline{\mathcal{S}}_\beta} \to \mathrm{Bil} (V)\) which is pointwise continuous and pointwise holomorphic on \(\mathcal{S}_\beta\) and satisfies \(\psi(t+i\beta)=\overline {\psi (t)}\) for \(t\in \mathbb{R}\). The first main result relates such functions to \textit{standard real subspaces} \(V\) of a (complex) Hilbert space \(\mathcal{H}\): these are closed real subspaces such that \(V\cap iV=\{0\}\) and \(V+iV\) is dense. Any such subspace determines a pair \((\Delta, J)\) where \(\Delta\) is a positive selfadjoint operator and \(J\) is an antilinear involution satisfying \(J\Delta J=\Delta^{-1}\). The space \(V\) consists of vectors \(v\) in the domain of \(\Delta^{1/2}\) fixed for \(J\Delta^{1/2}\). The result is: the function \(\psi\) satisfies the \(KMS\) condition for \(\beta >0\) if and only if there exists a standard real subspace \(V_1\) in \(\mathcal{H}\) and a linear map \(j: V\to V_1\) such that \[ \psi (t) (v,w) = \langle j(v), \Delta^{-it/\beta} \, j(w)\rangle \text{ for } t\in\mathbb{R}, \ v,w \in V. \] Besides that, an integral representation for \(\psi\) is given. Further, a function \(\psi\) satisfying the \(\beta\)-\(KMS\) condition generates the operator-valued function \(\varphi: [0,\beta]\to B(V_{\mathbb{C}})\) by \[ \…
Mathematics Subject Classification ID: 43A35
Mathematics Subject Classification ID: 43A65
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 47L90
Mathematics Subject Classification ID: 81T05
zbMATH DE Number: 7054789
zbMATH Keywords: KMS condition
zbMATH Keywords: reflection positivity
zbMATH Keywords: standard real subspace
author: Karl-Hermann Neeb
author: Gestur Ólafsson
reviewed by: V. F. Molchanov
MaRDI profile type: Publication
OpenAlex ID: W3105797133
arXiv ID: 1611.00080
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DOI: 10.2140/PJM.2019.299.117
description: scientific article