Abstract
In this work we study the non-commutative Hardy-Littlewood maximal operator on Lorentz spaces of τ -measurable operators.Non-commutative maximal inequalities were studied, in particular, in [1][2][3].Another version of the (non-commutative) Hardy-Littlewood maximal operator was introduced by T. Bekjan [4].Later J. Shao investigated the Hardy-Littlewood maximal operator on non-commutative Lorentz spaces associated with finite atomless von Neumann algebra (see [5]).Namely, for an operator T affiliated with a semi-finite von Neumann algebra M, the Hardy-Littlewood maximal operator of T is defined byWhile the classical Hardy-Littlewood maximal operator of a Lebesgue measurable function f : R → R, denoted by M f (x), is defined aswhere m is a Lebesgue measure on (-∞, ∞) [10].In view of spectral theory, |A| is represented asand M A(|A|) is represented as M A(x).Thus, for the operator A, Bekjan's consideration is that M A(|A|) is defined as the operator analogue of the Hardy-Littlewood maximal operator in the classical case.Our purpose is to investigate the non-commutative Hardy-Littlewood maximal operator M in the sense of T. Bekjan (see [4]).In particular, we obtain boundedness of the non-commutative Hardy-Littlewood maximal operator in non-commutative Lorentz spaces.