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In millimeter-wave (mmWave) systems, beamforming is needed to overcome harsh channel environments. As a promising beamforming solution, lens antenna array (LAA) implementation can provide a cost-effective solution without notable performance degradation compared to its counterpart. However, an appropriate beam selection is a challenge since it requires efficient channel estimation via an extensive beam training process for perfect beam alignment. In this paper, we propose a high mobility beam and channel tracking algorithm based on the unscented Kalman filter (UKF) to address this challenge, where the channel changes can be monitored over a certain time. The proposed algorithm tracks the channel changes after establishing a connection with an appropriate beam. The algorithm is first introduced in a multi-user beamspace multiple-input multiple-output (MIMO) system with LAA where a single beam is tracked at the user side at downlink transmission. Then, it is employed for multi-beam joint-tracking at the base station side in the uplink transmission. The analysis indicates that under different channel variation conditions, the proposed algorithm outmatches the popular extended Kalman filter (EKF) in both single-beam and multi-beam tracking systems. While it is common to individually track the beams in a multi-beam system scenario, the proposed joint tracking approach can provide around 62% performance enhancement compared to individual beam tracking using the conventional EKF method.