Dilation of generalized Toeplitz kernels on ordered groups

We introduce the concept of Toeplitz–Kreĭn–Cotlar triplet on ordered groups and we prove a general dilation result and a general representation result for the positive definite case.This general result includes and extends previous generalizations of the Kreĭn extension theorem, the Sz.-Nagy and Foias commutant lifting theorem and the generalized Herglotz–Bochner–Weil theorem.