EXTREME N-POSITIVE LINEAR-MAPS

In this article we prove that if a completely positive linear map PHI of a unital C*-algebra A into another B with only finite dimensional irreducible representations is pure, then we have N(PHI)=(PHI)ker+ker(PHI), where N(PHI)={x is-an-element-of A\PHI(x)=0},(PHI)ker={x is-an-element-of A\PHI(x*x)=0}, and ker(PHI)={x is-an-element-of A\PHI(xx*) = 0}. We also prove that for every unital strongly positive and n-positive linear map PHI of a C*-algebra A onto another B with n greater-than-or-equal-to 2, if N(PHI)=PHI(ker)+ker(PHI), then PHI is extreme in P(n)(A,B,I(B)). By this null-kernel condition, many new extreme n-positive linear maps are identified. A general procedure for constructing extreme n-positive linear maps is suggested and discussed.