This paper is mainly devoted to the study of the Hardy-Littlewood maximal function on noncommutative Lorentz spaces and to obtaining (p, q)-(p, q)-type inequality for the Hardy-Littlewood maximal function on noncommutative Lorentz spaces. MSC: 47A30; 47L05

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Let A be a maximal subdiagonal algebra of a finite von Neumann algebra M. For 0 < p < ∞, we define the noncommutative Hardy-Lorentz spaces and establish the Riesz and Szegö factorizations on these spaces. We also present some results of Jordan morphism on these spaces. MSC: 46L53; 46L51

In this paper, we introduce two variables norm functionals of τ-measurable operators and establish their joint log-convexity. Applications of this log-convexity will include interpolated Young, Heinz and Trace inequalities related to τ-measurable operators. Additionally, interpolated versions and their monotonicity will be presented as well.

In this paper, we prove some convexity inequalities in noncommutative L p spaces generalizing the previous result of Hiai and Zhan. Moreover, we generalize a variational inequality for positive definite matrices due to Hansen to the case of noncommutative L p spaces. MSC: 47A30; 47L05; 47L50