The Krein-Milman theorem in operator convexity

We generalize the Krein-Milman theorem to the setting of matrix convex sets of Effros-Winkler, extending the work of Farenick-Morenz on compact C-convex sets of complex matrices and the matrix state spaces of C-algebras. An essential ingredient is to prove the non-commutative analogue of the fact that a compact convex set may be thought of as the state space of the space of continuous affine functions on .

     

THE ANALYTIC KREIN-MILMAN PROPERTY IN BANACH SPACES

In[1],A.V.BuknvalovandA.A.Danilevichhaveintroducedtheanalyticcordon--NikodylllpropertyincompleXBanacllspacesastheanalyticallalogueoftilewellknowllRadoll-NikodylllpropertyconcerningtherealgeometricalstrllcturesofBanachspaces.LetXbeacomplexBanachspace,XissaidtohavetheanalyticRadon--Nikodymproperty(analyticRNP,illshort)ifforeveryuniformlyboundedanalyticfunctiollffromtheopenunitdiscDofCwitllvaluesinX,f:D~X,fhasradiallimitsa.e.onthetornsT=={e*':0E[0,Zx]}illX,tllisllleallsthatfora.e.e6[0,…     

Extreme harmonic functions on a ball

In order to show that one can recapture the Riesz-Herglotz theorem from the Krein-Milman theorem, we determine directly the set of extreme points of the convex set of positive harmonic functions on the unit ball (normalized by 1 at the origin). The characterization is obtained using standard facts from abstract analysis combined with a minimum of very basic results on harmonic functions.     

THE ISOMETRIC EXTENSION OF AN INTO MAPPING FROM THE UNIT SPHERE S[e(Г)] TO THE UNIT SPHERE S(E) Dedicated to Professor Wu Wenjun on the occasion of his 90th birthday

This is such a article to consider an "into" isometric mapping between two unit spheres of two infinite dimensional spaces of different types.In this article, we find a useful condition (using the Krein-Milman property) under which an into-isometric mapping from extended.