Lieb functions and m-positivity of norms

Some properties of 2- or 3-positive maps on matrix spaces are proved, where special attention is given to norms. It is shown that for a locally bounded function the definition of a Lieb function can be simplified somewhat. Unitarily invariant norms are characterized by a certain monotonicity property. The set of 2-positive numerical radii is described and an example of a 3-positive norm is obtained.