Inequalities for C-S seminorms and Lieb functions

Let M_n be the space of n × n complex matrices. A seminorm ∥ · ∥ on M_n is said to be a C-S seminorm if ∥A* A∥ = ∥AA*∥ for all A ∈ M_n and ∥A∥ ≤ ∥B∥ whenever A, B, and B-A are positive semidefinite. If ∥ · ∥ is any nontrivial C-S seminorm on M_n, we show that ∥|A∥| is a unitarily invariant norm on M_n, which permits many known inequalities for unitarily invariant norms to be generalized to the setting of C-S seminorms. We prove a new inequality for C-S seminorms that includes as special cases inequalities of Bhatia et al., for unitarily invariant norms. Finally, we observe that every C-S seminorm belongs to the larger class of Lieb functions, and we prove some new inequalities for this larger class.