The Compressed Baryonic Matter (CBM) experiment at the future Facility for Antiproton and Ion Research (FAIR) in Darmstadt is designed to investigate the properties of high-density QCD matter with multi-differential measurements of hadrons and leptons, including rare probes like multi-strange anti-hyperons and charmed particles. The research program covers the study of the high-density equation-of-state of nuclear matter, and the exploration of the QCD phase diagram at large baryon-chemical potentials, including the search for quark matter and the critical endpoint of a hypothetical 1st order phase transition. The CBM setup comprises detector systems for the identification of charged hadrons, electrons, and muons, for the determination of collision centrality and the orientation of the reaction plane, and a free-streaming data read-out and acquisition system, which allows online reconstruction and selection of events up to reaction rates of 10 MHz. In this article, emphasis is placed on the measurement of muon pairs in Au-Au collisions at FAIR beam energies, which are unique probes to determine the temperature of the fireball, and, hence, to search for a caloric curve of QCD matter. Simultaneously, the subthreshold production of charmonium can be studied via its dimuon decay, in order to shed light on the microscopic structure of QCD matter at high baryon densities. The CBM setup with focus on dimuon measurements and results of the corresponding physics performance studies will be presented.
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