  In this note we characterize those unitary one-parameter groups U^c which
admit euclidean realizations in the sense that they are obtained by the
analytic continuation process corresponding to reflection positivity from a
unitary representation $U$ of the circle group. These are precisely the ones
for which there exists an anti-unitary involution $J$ commuting with $U^c$.
This provides an interesting link with the modular data arising in
Tomita--Takesaki theory. Introducing the concept of a positive definite
function with values in the space of sesquilinear forms, we further establish a
link between KMS states and reflection positivity on the circle.
