Positive Maps and Operator Means

In this chapter, we first study operator convex and operator monotone functions and their fundamental properties. We then explore positive and completely positive maps in a general setting. Finally, we pay attention to the theory of operator means introduced by Kubo and Ando [1] and demonstrate their features, in particular, we seek some important properties of the operator geometric and operator harmonic means. Throughout this chapter, we denote by \(\mathbb {B}_\textrm{sa}(\mathscr {H})(J)\) the set of all self-adjoint operators in \(\mathbb {B}_\textrm{sa}(\mathscr {H})\) whose spectra are contained in an interval \(J \subseteq \mathbb {R}\) .