Continuity of positive nonlinear maps between -algebras
Volume 263 / 2022
Abstract
Let \mathscr {A} and \mathscr {B} be unital C^*-algebras acting on some Hilbert spaces. We investigate several topological properties of n-positive and n-monotone maps. It is shown that every 3-positive map \Phi : (\mathscr {A}, \|\cdot \|)\to (\mathscr {B}, {\rm SOT}) is continuous, where {\rm SOT} denotes the strong operator topology. Furthermore, we show that a 3-positive map is norm-continuous if it is norm-continuous at some positive invertible operator. Moreover, we prove that every 2-monotone map is norm-continuous. In addition, we show that in the definition of continuous Lieb function, the monotonicity condition is unnecessary. Finally, some interrelations between n-monotonicity and (n+1)-positivity of positive nonlinear maps are presented. Several counterexamples illustrate the tightness of the results.
