Support theorems in abstract settings

In this paper we establish a general framework in which the verification of support theorems for generalized convex functions acting between an algebraic structure and an ordered algebraic structure is still possible. As for the domain space, we allow algebraic structures equipped with families of algebraic operations whose operations are mutually distributive with respect to each other. We introduce several new concepts in such algebraic structures, the notions of convex set, extreme set, and interior point with respect to a given family of operations, furthermore, we describe their most basic and required properties. In the context of the range space, we introduce the notion of completeness of a partially ordered set with respect to the existence of the infimum of lower bounded chains, we also offer several sufficient condition which imply this property. For instance, the order generated by a sharp cone in a vector space turns out to possess this completeness property. By taking several particular cases, we deduce support and extension theorems in various classical and important settings

Non-Archimedean Preferences Over Countable Lotteries

We prove a representation theorem for preference relations over countably infinite lotteries that satisfy a generalized form of the Independence axiom, without assuming Continuity. The representing space consists of lexicographically ordered transfinite sequences of bounded real numbers. This result is generalized to preference orders on abstract superconvex spaces

Duality and separation theorems in idempotent semimodules

We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert's projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings. We illustrate the results by showing polyhedra and half-spaces over the max-plus semiring.Comment: 24 pages, 5 Postscript figures, revised (v2

Characterizations of Ordered Self-adjoint Operator Spaces

In this paper, we generalize the work of Werner and others to develop two abstract characterizations for self-adjoint operator spaces. The corresponding abstract objects can be represented as self-adjoint subspaces of $B(H)$ in such a way that both a metric structure and an order structure are preserved at each matrix level. We demonstrate a generalization of the Arveson Extension Theorem in this context. We also show that quotients of self-adjoint operator spaces can be endowed with a compatible operator space structure and characterize the kernels of completely positive completely bounded maps on self-adjoint operator spaces.Comment: 20 pages. Updated references and corrected typos. The statement of Corollary 3.17 has been strengthene

Weak∗^* closures and derived sets in dual Banach spaces

The main results of the paper: {\bf (1)} The dual Banach space $X^*$ contains a linear subspace $A\subset X^*$ such that the set $A^{(1)}$ of all limits of weak$^*$ convergent bounded nets in $A$ is a proper norm-dense subset of $X^*$ if and only if $X$ is a non-quasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual. {\bf (2)} Let $X$ be a non-reflexive Banach space. Then there exists a convex subset $A\subset X^*$ such that $A^{(1)}\neq {\bar{A}\,}^*$ (the latter denotes the weak$^*$ closure of $A$). {\bf (3)} Let $X$ be a quasi-reflexive Banach space and $A\subset X^*$ be an absolutely convex subset. Then $A^{(1)}={\bar{A}\,}^*$

The Expectation Monad in Quantum Foundations

The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures. It turns out to sit in between known monads: on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. This expectation monad is used in two probabilistic analogues of fundamental results of Manes and Gelfand for the ultrafilter monad: algebras of the expectation monad are convex compact Hausdorff spaces, and are dually equivalent to so-called Banach effect algebras. These structures capture states and effects in quantum foundations, and also the duality between them. Moreover, the approach leads to a new re-formulation of Gleason's theorem, expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval.Comment: In Proceedings QPL 2011, arXiv:1210.029