A Radon-Nikodym theorem for completely multi-positive linear maps and applications
Abstract
052<p type="texpara" tag="Body Text" et="abstract" >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 self-dual 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
M
n
(B)
.