Recent results and upgrade of the ALICE muon spectrometer

The ALICE experiment at the CERN Large Hadron Collider (LHC) is a multi-purpose particle detector, mainly focused on the study of quark-gluon plasma (QGP) in heavy-ion collisions. In the forward rapidity region, 2.5 $<$ y $<$ 4, ALICE is equipped with a muon spectrometer (MS), which allows to study quarkonia and open heavy-flavor particles, both key probes to investigate QGP properties. Although in LHC Run 1 and 2 many important results were achieved, the front absorber of the MS represented a limit to the physics program, due to the multiple scattering and energy loss in the material. To assess this limitation, a new forward vertex tracker (Muon Forward Tracker, MFT) was installed between the inner tracking system (ITS) and the front absorber. This has enhanced the MS physics performance, enabling the separation of prompt/non-prompt charmonium production at forward rapidity. It will also allow one to reduce the combinatorial background from semi-leptonic decays of kaons and pions. Finally, it will greatly improve the invariant-mass resolution of the low-mass dimuon pairs. Moreover, during the ongoing LHC Run 3, the rate of Pb\textendash{}Pb collisions has been increased from 10~kHz (in Run 2) up to 50~kHz, allowing to collect a data sample about 5 times larger than the one recorded in Run 2. This contribution will provide a brief overview of the MS upgrades and it will focus on the expected physics performance during the LHC Run 3. Some of the preliminary results already obtained will also be shown.