A Radon-Nikodym theorem for completely multi-positive linear maps and applications

052<p type=“texpara” tag=“Body Text” et=“abstract” >A completely $n$ -positive linear map from a locally $C^{\ast }$-algebra $A$ to another locally $C^{\ast }$-algebra $B$is an $n\times n$ 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 $C^{\ast }$-algebra$A$toa$C^{\ast}$-algebra$B$,intermsofaself-dualHilbert$C^{\ast}$-modulestructureinducedbyeachstrictcompletely$n$ -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 $M_n(B)$.