Uniqueness of KMS States for Quantum and Classical Models within the Framework of Berezin Quantization

Strict (C*-algebraic) deformation quantization is a rigorous approach to both the quantization process of classical theories and to the precise description of the classical limit problem for
quantum theories. In this talk, we will show how this framework can be adapted to the study of algebras associated with infinite lattices, which are used for the description of thermodynamic limits of physical theories. Subsequently, we will see that, within this setting, quantum KMS states can have a definite classical limit in terms of classical KMS states. Finally, we will focus on proving, in the specific case of a Berezin-Toeplitz quantization on the two sphere, a uniqueness condition for quantum KMS states, which only depends on the properties of the, quantized, classical theory. The proof of the latter result is partly inspired by the works of Bratteli and Robinson, but adapted for our setting with the use of the SU(2) group structure. (Joint work with N. Drago and C. van de Ven)