Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach

We study the continuity of an abstract generalization of the maximum-entropy inference - a maximizer. It is defined as a right-inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body. Using convex geometry we prove, amongst others, the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations. Further, we use numerical range methods in the case of quantum inference which refers to two observables. One result is a complete characterization of points of discontinuity for $3\times 3$ matrices.Comment: 27 page

On equal-input and monotone Markov matrices

The classes of equal-input and of monotone Markov matrices are revisited, with special focus on embeddability, infinite divisibility, and mutual relations. Several uniqueness results for the embedding problem are obtained in the process. We employ various algebraic and geometric tools, including commutativity, permutation invariance and convexity. Of particular relevance in several demarcation results are Markov matrices that are idempotents.Comment: 30 page

Partially ordered Grothendieck groups

Motivation for the study of partially ordered abelian groups has come from many different parts of mathematics, for mathematical systems with compatible order and additive (or linear) strlk'ctures are quite common. This is particularly evident in functional analysis, where spaces of various kinds of real-valued functions provide impetus for investigating partially ordered real vector spaces. In the past decade, the observation that a Grothendieck group (such as KD of a ring or algebra) often possesses a natural partially ordered abelian group structure has led to new directions of investigation, whose goals have been to develop structure theories for certain types of partially ordered abelian groups to the point where effective application to various Grothendieck groups is possible. Such recent developments in the area of partially ordered abelian groups are the subject of this note. We present a sketch of the construction of Grothendieck groups as abelian groups equipped with pre-orderings tha

The Shilov boundary of an operator space - and the characterization theorems

We study operator spaces, operator algebras, and operator modules, from the point of view of the `noncommutative Shilov boundary'. In this attempt to utilize some `noncommutative Choquet theory', we find that Hilbert C$^*-$modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We introduce certain multiplier operator algebras and C$^*-$algebras of an operator space, which generalize the algebras of adjointable operators on a C$^*-$module, and the `imprimitivity C$^*-$algebra'. It also generalizes a classical Banach space notion. This multiplier algebra plays a key role here. As applications of this perspective, we unify, and strengthen several theorems characterizing operator algebras and modules, in a way that seems to give more information than other current proofs. We also include some general notes on the `commutative case' of some of the topics we discuss, coming in part from joint work with Christian Le Merdy, about `function modules'.Comment: This is the final revised versio