Continuity of positive nonlinear maps between $C^*$-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.
