Families of completely positive maps associated with monotone metrics

Abstract

An operator convex function on (0,\infty) which satisfies the symmetry condition k(1/x) = x k(x) can be used to define a type of non-commutative multiplication by a positive definite matrix (or its inverse) using the primitive concepts of left and right multiplication and the functional calculus. The operators for the inverse can be used to define quadratic forms associated with Riemannian metrics which contract under the action of completely positive trace-preserving maps. We study the question of when these operators define maps which are also completely positive (CP). Although A --> D^{-1/2} A D^{-1/2} is the only case for which both the map and its inverse are CP, there are several well-known one parameter families for which either the map or its inverse is CP. We present a complete analysis of the behavior of these families, as well as the behavior of lines connecting an extreme point with the smallest one and some results for geometric bridges between these points. Our primary tool is an order relation based on the concept of positive definite functions. Although some results can be obtained from known properties, we also prove new results based on the positivity of the Fourier transforms of certain functions. Concrete computations of certain Fourier transforms not only yield new examples of positive definite functions, but also examples in the much stronger class of infinitely divisible functions.