Field-Theoretic Weyl Deformation Quantization of Enlarged Poisson Algebras

$C^*$-algebraic Weyl quantization is extended by allowing also degenerate pre-symplectic forms for the Weyl relations with infinitely many degrees of freedom, and by starting out from enlarged classical Poisson algebras. A powerful tool is found in the construction of Poisson algebras and non-commutative twisted Banach-$*$-algebras on the stage of measures on the not locally compact test function space. Already within this frame strict deformation quantization is obtained, but in terms of Banach-$*$-algebras instead of $C^*$-algebras. Fourier transformation and representation theory of the measure Banach-$*$-algebras are combined with the theory of continuous projective group representations to arrive at the genuine $C^*$-algebraic strict deformation quantization in the sense of Rieffel and Landsman. Weyl quantization is recognized to depend in the first step functorially on the (in general) infinite dimensional, pre-symplectic test function space; but in the second step one has to select a family of representations, indexed by the deformation parameter $\hbar$. The latter ambiguity is in the present investigation connected with the choice of a folium of states, a structure, which does not necessarily require a Hilbert space representation.Comment: This is a contribution to the Special Issue on Deformation Quantization, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Weak Minimizers, Minimizers and Variational Inequalities for set valued Functions. A blooming wreath?

In the literature, necessary and sufficient conditions in terms of variational inequalities are introduced to characterize minimizers of convex set valued functions with values in a conlinear space. Similar results are proved for a weaker concept of minimizers and weaker variational inequalities. The implications are proved using scalarization techniques that eventually provide original problems, not fully equivalent to the set-valued counterparts. Therefore, we try, in the course of this note, to close the network among the various notions proposed. More specifically, we prove that a minimizer is always a weak minimizer, and a solution to the stronger variational inequality always also a solution to the weak variational inequality of the same type. As a special case we obtain a complete characterization of efficiency and weak efficiency in vector optimization by set-valued variational inequalities and their scalarizations. Indeed this might eventually prove the usefulness of the set-optimization approach to renew the study of vector optimization

Fraisse Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups

We study in this paper some connections between the Fraisse theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures.Comment: 73 pages, LaTeX 2e, to appear in Geom. Funct. Ana