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In this paper, we investigate noncommutative symmetric and asymmetric maximal inequalities associated with martingale transforms and fractional integrals. Our proofs depend on some recent advances on algebraic atomic decompositions and the noncommutative Gundy decomposition. We also prove several fractional maximal inequalities. 2010 Mathematics Subject Classification. Primary: 46L53, 60G42. Secondary: 46L52, 60G46.doi:10.4153/s0008414x19000580 fatcat:rz3pgnooabgi5pz5szbfs4wphe