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The Bost–Connes system is a   $C^*$  -dynamical system related to class field theory. The purpose of this paper is to construct Bost–Connes systems for local fields of characteristic zero. The notable phenomenon in the case of local fields is the absence of the phase transition. We also investigate the relation between the global and local Bost–Connes systems.

Bost-Connes Systems for Local Fields of Characteristic Zero