Spaces Of Continuous Affine Functions Research Articles

Centre of a compact convex set

We introduce the notion centre of a convex set and study the space of continuous affine functions on a compact convex set with a centre. We show that these spaces are precisely the dual of a base normed space in which the underlying base has a (unique) centre. We also characterize the corresponding base norm space. We obtain a condition on a compact, balanced, convex subset of a locally convex space, so that the corresponding space of continuous affine functions on the convex set is an absolute order unit space. Similarly, we characterize a condition on the base with a centre of a base normed space, so that the latter becomes an absolutely base normed space.

Nice operators into $$L_1$$ L 1 -preduals

A Banach space X is said to be “nice” if every extreme operator from any Banach space into X is a nice operator (that is, its adjoint preserves extreme points). We prove that a nice \(L_1\)-predual space is isometrically isomorphic to \(c_0(I)\) for some set I. In fact, by using the structure topology, we get a more general result which allows us to conclude that there exist extreme non-nice operators into certain spaces of continuous affine functions which are not \(L_1\)-preduals.

Facial topology and extreme operators

Let K be a simplex and let A0(K) denote the space of continuous affine functions on K vanishing at a fixed extreme point, denoted by 0. We prove that if any extreme operator T from a Banach space X to A0(K) is a nice operator (that is, T⁎, the adjoint of T, preserves extreme points), then the facial topology of the set of extreme points different from 0 is discrete, and so A0(K) is isometrically isomorphic to c0(I) for some set I. From here we derive the corresponding result for A(K), namely, if K is a simplex such that each extreme operator from any Banach space to A(K) is a nice operator, then the set of extreme points of K is finite.

Isomorphisms of spaces of continuous affine functions on compact convex sets with Lindelöf boundaries

Let X, Y be compact convex sets such that every extreme point of X and Y is a weak peak point and both ext X and ext Y are Lindelof spaces. We prove that if there exists an isomorphism T: U c (X) → U c (Y) with ∥T∥·∥T -1 ∥ < 2, then ext X is homeomorphic to ext Y. This generalizes results of C. H. Chu and H. B. Cohen.

Choquet like sets in function spaces

In convex analysis when studying function spaces of continuous affine functions, notions of a geometrical character like faces, split and parallel faces, exposed or Archimedean faces were investigated in detail by many authors. In this paper we transfer these notions to a more general setting of Choquet theory of abstract function spaces. We prefer a direct functional analytic approach to the treatment of problems instead of using a transfer of a function space to its state space. Methods invoked are based mainly on a measure theory and basic tools of functional analysis and are different from ones using a geometric visualization.

Isomorphisms of spaces of continuous affine functions

Let K and S be compact convex sets and let A(K) and A(S) be the corresponding Banach spaces of continuous affine functions. If the Banach-Mazur distance between A(K) and A(S) is less than 2, then under certain geometric conditions, the extreme boundaries of K and S are homeomorphic. This extends a result of Amir and Cambern and has applications to function algebras

Korovkin-type theorems for positive functionals in spaces of continuous affine functions

Si stabiliscono dei risultati concernenti la convergenza di reti di forme lineari positive verso una prefissata forma lineare positiva nel contesto di spazi di funzioni continue ed affini definite su un insieme convesso compatto. Si presentano degli esempi e delle applicazioni in spazi di funzioni continue, inC*-algebre di operatori su spazi di Hilbert e in spazi di Banach reticolati.

On a Theorem of Cambern

It is shown that Cambern’s generalized Banach-Stone theorem for spaces of continuous functions on compact sets cannot be extended to the case of spaces of continuous affine functions on simplexes.

On a theorem of Cambern

It is shown that Cambern’s generalized Banach-Stone theorem for spaces of continuous functions on compact sets cannot be extended to the case of spaces of continuous affine functions on simplexes.

Inverse limits of compact convex sets and direct limits of spaces of continuous affine functions

Inverse limits of compact convex sets and direct limits of spaces of continuous affine functions

Directed Banach spaces of affine functions

0. Introduction. Let X be a compact convex set and let F be a closed face of X. In this paper we develop a technique which yields sufficient conditions for F to be a peak-face of X (a subset of X where a continuous affine function on X attains its maximum). The theory is based on a duality between certain types of ordered Banach spaces. This duality is an extension of the results of [6] (see also [17]) and relates the directness of an ordered Banach space E to the degree to which the triangle inequality can be reversed on the positive elements of E*. A precise formulation of this is given in ?1. In ?2 we define a compact convex set X to be at an extreme point x if there is a bounded nonnegative affine function f on X such that f(x) =0 and X= conv ({x} u {y E X: f(y) > 1}). If X is at the G, extreme point x then the results of ?1 are applied to show that x is a peak-point of X. Every compact convex set Xhas a natural identification with the positive elements of norm one in A(X)*, where A(X) is the space of continuous affine functions on X. If N is the subspace of A(X)* spanned by the closed face F of X then by making use of the quotient map from A(X)* to A(X)*/N we can extend the definition of conical to the closed face F. This is then used to establish a sufficient condition for F to be a peak-face. This procedure of using the quotient map is used repeatedly throughout and as a by-product yields different (and possibly simpler) proofs of some known results. For example we use this approach (see Proposition 4.2) to reprove a result of Alfsen's [2] concerning the complementary face of a closed face of a Choquet simplex. In ?3 we define a class Y of compact convex sets X for which it turns out that (1) every closed G, face F of X is a peak-face and (2) every continuous affine function on F can be extended to a continuous affine function on X. It is known that Choquet simplexes have these two properties and we show that Y in fact contains the simplexes. In addition it is proved that Y contains the a-polytopes. (These are defined by R. Phelps [19] and he proves that they correspond exactly to the polyhedrons defined by Alfsen [1].) In [19] Phelps also defines the 3-polytopes as the intersection of a simplex S with a closed subspace of A(S)* of finite codimension. In ?4 we show that the P-polytopes are at each extreme point and that those P-polytopes which are