# Reflection Positivity on the Circle

• 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 the close relation between reflection positivity on the circle group $${\mathbb T}$$ and the Kubo–Martin–Schwinger (KMS) condition for states of $$C^*$$-dynamical systems. Here a crucial point is a pure representation theoretic perspective on the KMS condition formulated as a property of form-valued positive definite functions on $${\mathbb R}$$: For $$\beta > 0$$, we consider the open strip $$\mathscr {S}_\beta := \{ z \in {\mathbb {C}}: 0< \mathop {\mathrm{Im}}\nolimits z < \beta \}.$$