标题： KMS条件，标准实子空间和圆群上的反射正性 显示英文标题

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

摘要： 在本文中，我们继续研究反射正性表示理论方面的问题，通过研究定义在加法群 (R,+) 上并满足适当定义的KMS条件的正定函数\psi 。这些函数取值于实向量空间 V 上的双线性形式空间 Bil(V)。如同量子统计力学中一样，KMS条件是通过将\psi 解析延拓到条带{z \in C\: 0 \leq Im z \leq b}并在边界上满足耦合条件\psi (ib + t) =\oline{\psi (t)}来定义的。 我们的第一个主要结果包括这些函数在模对象（\Delta ，J）（J 为反线性对合，\Delta > 0 自伴且 J\Delta J =\Delta ^{-1}）和积分表示方面的表征。 我们的第二个主要结果是存在一个取值于 Bil(V) 的正定函数 f 在群 R_\tau = R \rtimes {\id_\R,\tau} 上，满足 \tau (t) = -t，使得 f(t,\tau ) = \psi (it) 对 t \in R 成立。因此，我们在 GNS 空间 H_f 上得到一个 2b 周期的酉一参数群，而 GNS 空间 H_\psi 上的一参数群则是通过 Osterwalder--Schrader 量子化得到的。 最后，我们证明这些表示的构建块来源于对应于圆 R/bZ 上核 1/(\lambda ^2 - (d^2)/(dt^2\}) 的向量丛Sobolev空间。 显示英文摘要

摘要： In the present paper we continue our investigations of the representation theoretic side of reflection positivity by studying positive definite functions \psi 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 \psi to the strip { z \in C\: 0 \leq Im z \leq b} with a coupling condition \psi (ib + t) = \oline{\psi (t)} on the boundary. Our first main result consists of a characterization of these functions in terms of modular objects (\Delta, J) (J an antilinear involution and \Delta > 0 selfadjoint with J\Delta J = \Delta^{-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_\tau = R \rtimes {\id_\R,\tau} with \tau(t) = -t satisfying f(t,\tau) = \psi(it) for t \in R. We thus obtain a 2b-periodic unitary one-parameter group on the GNS space H_f for which the one-parameter group on the GNS space H_\psi 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 1/(\lambda^2 - (d^2)/(dt^2}) on the circle R/bZ of length b.