On representations associated with completely n-positive linear maps on pro-C *-algebras

Abstract

It is shown that an n × n matrix of continuous linear maps from a pro-C * -algebra A to L(H), which verifies the condition of complete positivity, is of the form, where Φ is a representation of A on a Hilbert space K, V is a bounded linear operator from H to K, and [Tij ] n i,j=1 is a positive element in the C * -algebra of all n × n matrices over the commutant of Φ(A) in L(K). This generalizes a result of C. Y. Suen in Proc. Amer. Math. Soc., 112(3), 1991, 709-712. Also, a covariant version of this construction is given.