On the Krein-Milman-Ky Fan theorem for convex compact metrizable sets

The Krein-Milman theorem (1940) states that every convex compact subset of a Hausdorfflocally convex topological space, is the closed convex hull of its extreme points. In 1963, Ky Fan extended the Krein-Milman theorem to the general framework of $\Phi$-convexity. Under general conditions on the class of functions $\Phi$, the Krein-Milman-Ky Fan theorem asserts then, that every compact $\Phi$-convex subset of a Hausdorff space, is the $\Phi$-convex hull of its $\Phi$-extremal points. We prove in this paper that, in the metrizable case the situation is rather better. Indeed, we can replace the set of $\Phi$-extremal points by the smaller subset of $\Phi$-exposed points. We establish under general conditions on the class of functions $\Phi$, that every $\Phi$-convex compact metrizable subset of a Hausdorff space, is the $\Phi$-convex hull of its $\Phi$-exposed points. As a consequence we obtain that each convex weak compact metrizable (resp. convex weak$^*$ compact metrizable) subset of a Banach space (resp. of a dual Banach space), is the closed convex hull of its exposed points (resp. the weak$^*$ closed convex hull of its weak$^*$ exposed points). This result fails in general for compact $\Phi$-convex subsets that are not metrizable

Any law of group metric invariant is an inf-convolution

In this article, we bring a new light on the concept of the inf-convolution operation $\oplus$ and provides additional informations to the work started in \cite{Ba1} and \cite{Ba2}. It is shown that any internal law of group metric invariant (even quasigroup) can be considered as an inf-convolution. Consequently, the operation of the inf-convolution of functions on a group metric invariant is in reality an extension of the internal law of $X$ to spaces of functions on $X$. We give an example of monoid $(S(X),\oplus)$ for the inf-convolution structure, (which is dense in the set of all $1$-Lipschitz bounded from bellow functions) for which, the map $\arg\min : (S(X),\oplus) \rightarrow (X,.)$ is a (single valued) monoid morphism. It is also proved that, given a group complete metric invariant $(X,d)$, the complete metric space $(\mathcal{K}(X),d_{\infty})$ of all Katetov maps from $X$ to $\R$ equiped with the inf-convolution has a natural monoid structure which provides the following fact: the group of all isometric automorphisms $Aut_{Iso}(\mathcal{K}(X))$ of the monoid $\mathcal{K}(X)$, is isomorphic to the group of all isometric automorphisms $Aut_{Iso}(X)$ of the group $X$. On the other hand, we prove that the subset $\mathcal{K}_C(X)$ of $\mathcal{K}(X)$ of convex functions on a Banach space $X$, can be endowed with a convex cone structure in which $X$ embeds isometrically as Banach space

A convex extension of lower semicontinuous functions defined on normal Hausdorff space

We prove that, any problem of minimization of proper lower semicontinuous function defined on a normal Hausdorff space, is canonically equivalent to a problem of minimization of a proper weak * lower semicontinuous convex function defined on a weak * convex compact subset of some dual Banach space. We estalish the existence of an bijective operator between the two classes of functions which preserves the problems of minimization

Chiral effective action of QCD: Precision tests, questions and electroweak extensions

This talk first discusses some aspects of the chiral expansion with three light flavours related to the (non) applicability of the OZI rule. Next, the extension of ChPT to an effective theory of the full standard model is considered. Some applications of a systematic description of the coupling constants by sum rules (e.g. to the determination of quark masses and $K_{l3}$ decays) are presented.Comment: 6 pages, plenary talk at the International Conference on QCD and Hadronic physics, Beijing 16-20 June 200

A property (T) for C*-algebras

We define a notion of Property (T) for an arbitrary $C^*$-algebra $A$ admitting a tracial state. We extend this to a notion of Property (T) for the pair $(A,B),$ where $B$ is a $C^*$-subalgebra of $A.$ Let $\Gamma$ be a discrete group and $C^*_r(\Gamma)$ its reduced algebra. We show that $C^*_r(\Gamma)$ has Property (T) if and only if the group $\Gamma$ has Property (T) . More generally, given a subgroup $\Lambda$ of $\Gamma$, the pair $(C^*_r(\Gamma),C^*_r(\Lambda))$ has Property (T) if and only if the pair of groups $(\Gamma, \Lambda)$ has Property (T).Comment: 14 page

Limited operators and differentiability

We characterize the limited operators by differentiability of convex continuous functions. Given Banach spaces $Y$ and $X$ and a linear continuous operator $T: Y \longrightarrow X$, we prove that $T$ is a limited operator if and only if, for every convex continuous function $f: X \longrightarrow \R$ and every point $y\in Y$, $f\circ T$ is Fr\'echet differentiable at $y\in Y$ whenever $f$ is G\^ateaux differentiable at $T(y)\in X$

An asymmetric Putnam-Fuglede theorem for *-paranormal operators

The well-known asymmetric form of Putnam-Fuglede theorem asserts that if $A$ and $B$ are bounded normal operators and $AX = XB^*$ for some bounded operator $X$, then $A^*X = XB$. In this paper we showed that the above theorem does not hold for paranormal operator $A$, even if we assume that $B$ has to be unitary and an operator $X$ is taken from Hilbert-Schmidt class. Additionally, we showed the similar resualt for *-paranormal operators.Comment: 5 page