n-Positivity and Complete Positivity of *-Representations

Abstract

This chapter is concerned with *-representations of a *-algebra 𝗔 (or more generally with linear maps of 𝗔 into the space of sesquilinear forms on a vector space) which map a distinguished wedge of “positive” matrices over 𝗔 into the positive matrices of operators (or sesquilinear forms). The general study of such order properties leads to applications which are all formulated according to the following pattern: The *-representation admits an extension to a “well-behaved” *-representation in a larger Hilbert space if and only if it satisfies a certain additional positivity condition of the above form. In order to explain this idea by a simple pertinent example, let ω be a positive linear functional on the polynomial algebra ℂ[x₁, x₂]. Then ω is non-negative on non-negative polynomials if and only if it can be represented by a positive measure (see Example 2.6.11), or equivalently, if π _ω has an integrable extension.