A Radon-nikodym Theorem for Completely Multi-positive Linear Maps and Its Applications

A completely n -positive linear map from a locally C∗-algebra A to another locally C∗-algebra B is an n × n matrix whose elements are continuous linear maps from A to B and which verifies the condition of completely positivity. In this paper we prove a Radon-Nikodym type theorem for strict completely n-positive linear maps which describes the order relation on the set of all strict completely n -positive linear maps from a locally C∗-algebra A to a C∗-algebra B, in terms of a selfdual Hilbert C∗-module structure induced by each strict completely n -positive linear map. As applications of this result we characterize the pure completely n-positive linear maps from A to B and the extreme elements in the set of all identity preserving completely n-positive linear maps from A to B. Also we determine a certain class of extreme elements in the set of all identity preserving completely positive linear maps from A to Mn(B). MSC: 46L05; 46L08