Reflection Positivity on the Circle

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 $\mathscr {S}_\beta := \{ z \in {\mathbb {C}}: 0< \mathop {\mathrm{Im}}\nolimits z < \beta \}.$