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Cell-free communication has the potential to significantly improve grant-free transmission in massive machine-type communication, wherein multiple access points jointly serve a large number of user equipments to improve coverage and spectral efficiency. In this paper, we propose a novel framework for joint active user detection (AUD), channel estimation (CE), and data detection (DD) for massive grant-free transmission in cell-free systems. We formulate an optimization problem for joint AUD, CE, and DD by considering both the sparsity of the data matrix, which arises from intermittent user activity, and the sparsity of the effective channel matrix, which arises from intermittent user activity and large-scale fading. We approximately solve this optimization problem with a box-constrained forward-backward splitting algorithm, which significantly improves AUD, CE, and DD performance. We demonstrate the effectiveness …