Realizations of two-sided quantum K-systems

Two-sided quantum K-systems can be considered on the C∗ and von Neumann level. On the C∗ level examples can be constructed without any obstacles on the basis of first and second quantization. On the von Neumann level, however, q-deformation in second quantization prohibits KMS-states. Nevertheless, an example can be found of a two-sided modular K-system, where the relative commutants are trivial.