KMS conditions, standard real subspaces and reflection positivity on the circle group
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 \[ \langle v, \varphi (t) w \rangle=\psi (it)(v,w). \] Let \(\tau\) be the reflection \(\tau(t)=-\, t\) on \(\mathbb{R}\). Denote by \(\mathbb{R}_\tau\) the extension of \(\mathbb{R}\) by means of \(\tau\). The function \(f\) on \(\mathbb{R}_\tau\) defined by \(f(t,\tau)=\varphi(t)\) turns out to be a \(\mathrm{Bil} (V)\)-valued positive definite function, and also \(2\beta\)-periodic. Hence it factors through a function on \(\mathbb{T}_{2\beta,\tau}=\mathbb{R}_\tau/2\beta\mathbb{Z}\), which is isomorphic to \(\mathrm{O}(2,\mathbb{R})\). This is the second main result. Thus, the positive definite functions give unitary representations of the group \(\mathrm{O}(2,\mathbb{R})\), by means of the Gelfand-Naimark-Segal construction. These representations act on spaces on the circle \(\mathbb{R}/\beta\mathbb{Z}\) corresponding to the kernels \((\lambda^2-d^2/dt^2)^{-1}\).
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