Extremal Positive Maps on M3(ℂ) and Idempotent Matrices

A new method of analysing positive bistochastic maps on the algebra of complex matrices [Formula: see text] has been proposed. By identifying the set of such maps with a convex set of linear operators on [Formula: see text], one can employ techniques from the theory of compact semigroups to obtain results concerning asymptotic properties of positive maps. It turns out that the idempotent elements play a crucial role in classifying the convex set into subsets, in which representations of extremal positive maps are to be found. It has been shown that all positive bistochastic maps, extremal in the set of all positive maps of [Formula: see text], that are not Jordan isomorphisms of [Formula: see text] are represented by matrices that fall into two possible categories, determined by the simplest idempotent matrices: one by the zero matrix, and the other by a one-dimensional orthogonal projection. Some norm conditions for matrices representing possible extremal maps have been specified and examples of maps from both categories have been brought up, based on the results published previously.

The algebraic structure of quantum partial isometries

The partial isometries of [Formula: see text] form compact semigroups [Formula: see text]. We discuss here the liberation question for these semigroups, and for their discrete versions [Formula: see text]. Our main results concern the construction of half-liberations [Formula: see text] and of liberations [Formula: see text]. We include a detailed algebraic and probabilistic study of all these objects, justifying our “half-liberation” and “liberation” claims.