Extremal Subsets Research Articles

Some properties of maximal measures on compact convex sets

AbstractLet be a maximal measure on a compact convex set K, K* be the state space of the space of all continuous functions f: KK ℝ which are affine in the first variable, 1 be the -algebra on K generated by the Baire sets and the compact extremal subsets of K, and = {BeK1}. Then(i) For any fixed continuous function g:K ℝ and -almost all x in K, there is a closed face of K containing x on which g is constant.(ii) The image of under the map :KK* defined by f, (x) = f(x, x) is the unique maximal measure on K* representing its barycentre(iii) induces a measure on (eK) satisfying certain regularity conditions.

Preoperative Estimation of Risk in Cardiac Surgery

In order to determine the usefulness of the preoperative subjective estimation of risk and compare it with preoperative estimation of risk based upon objective data, two groups of patients subjected to open heart surgery at the Catholic University of Chile Hospital were studied prospectively. Group I comprised 227 consecutive patients operated on in 1975 and group II comprised 181 consecutive patients operated on in 1979. There were several important differences in management and techniques between the two groups, which resulted in different factors perioperative mortality. Mortality in group I was related to extreme age subsets (p less than 0.01) and to duration an anoxic arrest (p less than 0.001); mortality in group II correlated only to preoperative functional class (NYHA) (p less than 0.02). Despite these differences, subjective risk estimation as preoperatively recorded by the anesthesiologist was accurate in both groups (r = 0.969, p less than 0.05 in group I and r = 0.998, p less than 0.01 in group II). It was concluded that in the absence of a universally valid objective risk index, subjective risk estimation provides a clinical index as useful as reliance upon presently available objective data.

Sidon sets with extremal Sidon constants

A finitely supported measure yt on an l.c.a. group is said to be extremal if II = 11 1 (# SUpp t,) 12. If yt is an extremal measure and E is the support of ,t, it follows that the Sidon constant of E is (# E)'/2, in which case E is also said to be extremal. Our results are these. (1) An independent union of m cosets of a finite subgroup H of G is extremal if and only if (essentially) m divides #H. (2) Not all extremal subsets of abelian groups have the form described in (1). (3) For any group (abelian or not), the Sidon constant of that group is at least (.8)(# G)'/3.

Facial characterizations of complex Lindenstrauss spaces

We characterize complex Banach spaces $A$ whose Banach dual spaces are ${L^1}(\mu )$ spaces in terms of $L$-ideals generated by certain extremal subsets of the closed unit ball $K$ of ${A^{\ast }}$. Our treatment covers the case of spaces $A$ containing constant functions and also spaces not containing constants. Separable spaces are characterized in terms of ${w^{\ast }}$-compact sets of extreme points of $K$, whereas the nonseparable spaces necessitate usage of the ${w^{\ast }}$-closed faces of $K$. Our results represent natural extensions of known characterizations of Choquet simplexes. We obtain also a characterization of complex Lindenstrauss spaces in terms of boundary annihilating measures, and this leads to a characterization of the closed subalgebras of ${C_{\mathbf {C}}}(X)$ which are complex Lindenstrauss spaces.

Weak ∗ Compact Convex Sets with Separable Extremal Subsets have the Radon-Nikodym Property

Weak ∗ Compact Convex Sets with Separable Extremal Subsets have the Radon-Nikodym Property

Doubly stochastic measures with prescribed support

Denote the set of doubly stochastic measures on the unit square X ×X that are supported on the graphs of measurable maps L,H∶X→X by ℳ(L, H). Conditions are given that imply that ℳ(L, H) is a singleton. Since ℳ(L, H) is in any event a (possibly empty) extremal subset of the set ℳ of all doubly stochastic measures on X×X, our results are intimately related to the problem of describing the supports of the extreme points of ℳ.

Weak* compact convex sets with separable extremal subsets have the Radon-Nikodým property

The result stated in the title, as well as several other related results, is derived.

Extreme operators on H∞

Let A1 and A2 be sup-norm algebras, each containing the constant functions. Let P(A1, A2) denote the set of bounded linear operators from A1 to A2 which carry 1 into 1 and have norm 1. Several authors have considered the problem of describing the extreme points of P(A1, A2). In the case where A1 is the algebra of continuous complex functions on some compact Hausdorff space, and A2 is the algebra of complex scalars, Arens and Kelley proved that the extreme operators in P(A1, A2) are exactly the multiplicative ones (see [1]). It was shown by Phelps in [6] that if A1 is self-adjoint, then every extreme point of P(A1, A1) is multiplicative. In [4], Lindenstrauss, Phelps, and Ryff exhibited non-multiplicative extreme points of P(A, A) and P(H∞, H∞), where A and H∞ are, respectively, the disk algebra, and the algebra of bounded analytic functions on the open unit disk D. The extreme multiplicative operators in P(A, A) were described in [6]. Rochberg proved in [8] that, if T is a member of P(A, A) which carries the identity on D into an extreme point of the unit ball of A, then T is multiplicative and is an extreme point of P(A, A). Rochberg's paper [9] is a study of certain extremal subsets of P(A, A), namely, those of the form K(F, G) = {T∊P(A, A):TF=G}, where F and G are inner functions in A. We proved in [5] that, if F is non-constant, then K(F, G) contains an extreme point of P(A, A).

Operator of best approximation on finite-dimensional subspaces

The modulus of continuity of the operator of best approximation on a subspace tends towards zero uniformly on the class of all subspaces of an n-dimensional space only if the unit ball of the space contains no extremal subsets of dimensionality k (0 < k < n-1).

Minimal extremal subsets of the unit sphere

Minimal extremal subsets of the unit sphere