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 [V ^* T _ijΦ( · )V] _i,j=1 ⁿ, where Φ is a representation of A on a Hilbert space K, V is a bounded linear operator from H to K, and [T _ij] _{n i,j=1} ⁿ 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.

Keywords

Pro-C ^*-Algebra / Completely n-positive linear maps / Covariant completely n-positive linear maps

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