# Reflection Positive Representations

• Karl-Hermann Neeb
• Gestur Ólafsson
Chapter
Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 32)

## Abstract

In this chapter we turn to operators on reflection positive (real or complex) Hilbert spaces and introduce the Osterwalder–Schrader transform to pass from operators on $$\mathscr {E}_{+}$$ to operators on $$\widehat{\mathscr {E}}$$ (Sect. 3.1). The objects represented in reflection positive Hilbert spaces $$(\mathscr {E},\mathscr {E}_+,\theta )$$ are symmetric Lie groups $$(G,\tau )$$, i.e., a Lie group G, endowed with an involutive automorphism $$\tau$$. A typical example in physics arises from the euclidean motion group and time reversal. There are several ways to specify compatibility of a unitary representation $$(U,\mathscr {E})$$ of $$(G,\tau )$$ with $$\mathscr {E}_+$$ and $$\theta$$ and thus to define reflection positive representations (Sect. 3.3). One is to specify a subset $$G_{+}\subseteq G$$ and assume that $$\mathscr {E}_{+}$$ is generated by applying $$G_{+}^{-1}$$ to a suitable subspace of $$\mathscr {E}_{+}$$. The other simpler one applies if $$S := G_{+}^{-1}$$ is a subsemigroup of G invariant under the involution $$s \mapsto s^{\sharp } = \tau (s)^{-1}$$. Then we simply require $$\mathscr {E}_{+}$$ to be S-invariant. In both cases we can use the integrability results in Chap.  to obtain unitary representations of the 1-connected Lie group $$G^{c}$$ with Lie algebra $${\mathfrak g}^{c} = {\mathfrak h}+ i {\mathfrak q}$$ on $$\widehat{\mathscr {E}}$$. As reflection positive unitary representations are mostly constructed by applying a suitable Gelfand–Naimark–Segal (GNS) construction to reflection positive functions, we discuss this correspondence in some detail in Sect. 3.4. In particular, we discuss the Markov condition in this context (Proposition 3.4.9).

## Keywords

Positive Reflection Unitary Representation Suitable Subspace Subsemigroup Euclidean Motion Group
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