Reflection Positive Hilbert Spaces

Abstract

In this chapter we discuss the basic framework of reflection positivity: reflection positive Hilbert spaces. These are triples $(\mathscr {E},\mathscr {E}_+, \theta )$, consisting of a Hilbert space $\mathscr {E}$, a unitary involution $\theta$ on $\mathscr {E}$ and a closed subspace $\mathscr {E}_+$ which is $\theta$-positive in the sense that $\langle \xi ,\theta \xi \rangle \ge 0$ for $\xi \in \mathscr {E}_+$. This structure immediately leads to a new Hilbert space $\widehat{\mathscr {E}}$ and a linear map $q : \mathscr {E}_+ \rightarrow \widehat{\mathscr {E}}$ with dense range. When the so-called Markov condition is satisfied, there even exists a closed subspace $\mathscr {E}_0 \subseteq \mathscr {E}_+$ mapped isometrically onto $\widehat{\mathscr {E}}$ (Sect. 2.3). Reflection positive Hilbert spaces arise naturally in many different contexts: as graphs of contractions (Sect. 2.2), from reflection positive distribution kernels on manifolds (Sect. 2.4) and in particular from dissecting reflections of complete Riemannian manifolds and resolvents of the Laplacian (Sect. 2.5). This motivates the short discussion of an abstract operator theoretic context of reflection positivity in Sect. 2.6.