Ideal structures in vector-valued polynomial spaces (original) (raw)

$M$-structures in vector-valued polynomial spaces

Journal of Convex Analysis, 2011

This paper is concerned with the study of MMM-structures in spaces of polynomials. More precisely, we discuss for EEE and FFF Banach spaces, whether the class of weakly continuous on bounded sets nnn-homogeneous polynomials, mathcalPw(nE,F)\mathcal P_w(^n E, F)mathcalPw(nE,F), is an MMM-ideal in the space of continuous nnn-homogeneous polynomials mathcalP(nE,F)\mathcal P(^n E, F)mathcalP(nE,F). We show that there is some hope for this to happen only for a finite range of values of nnn. We establish sufficient conditions under which the problem has positive and negative answers and use the obtained results to study the particular cases when E=ellpE=\ell_pE=ellp and F=ellqF=\ell_qF=ellq or FFF is a Lorentz sequence space d(w,q)d(w,q)d(w,q). We extend to our setting the notion of property (M)(M)(M) introduced by Kalton which allows us to lift MMM-structures from the linear to the vector-valued polynomial context. Also, when mathcalPw(nE,F)\mathcal P_w(^n E, F)mathcalPw(nE,F) is an MMM-ideal in mathcalP(nE,F)\mathcal P(^n E, F)mathcalP(nE,F) we prove a Bishop-Phelps type result for vector-valued polynomials and relate norm-attaining polynomials with farthest points and remotal sets.

A note on a class of Banach algebra-valued polynomials

International Journal of Mathematics and Mathematical Sciences, 2002

LetFbe a Banach algebra. We give a necessary and sufficient condition forFto be finite dimensional, in terms of finite typen-homogeneousF-valued polynomials.

A class of polynomials from Banach spaces into Banach algebras

Publications of the Research Institute for Mathematical Sciences, 2001

Let E be a complex Banach space and F be a complex Banach algebra. We will be interested in the subspace IP f (n E, F) of P (n E, F) generated by the collection of functions ϕ n (n ∈ AE, ϕ ∈ L(E, F)) where ϕ n (x) = (ϕ(x)) n for each x ∈ E.

Subspaces of polynomials on Banach spaces

2002

We study some subspaces of homogeneous polynomials which are defined as dual spaces to the symmetric tensor products of Banach spaces, endowed with special cross-norms.

A Characterization of Polynomially Convex Sets in Banach Spaces

Results in Mathematics

Let E be a Banach space and △ * be the closed unit ball of the dual space E *. For a compact set K in E, we prove that K is polynomially convex in E if and only if there exist a unital commutative Banach algebra A and a continuous function f : △ * → A such that (i) A is generated by f (△ *), (ii) the character space of A is homeomorphic to K, and (iii) K = sp(f) the joint spectrum of f. In case E = C (X), where X is a compact Hausdorff space, we will see that △ * can be replaced by X. 1

To What Extent Does the Dual Banach Space E0 Determine the Polynomials Over E?

Workshop on Materialized, 2000

We show that under conditions of regularity, if E0 is isomorphic to F0 then the spaces of homogeneous polynomials over E and F are isomor- phic. Some subspaces of polynomials more closely related to the structure of dual spaces (weakly continuous, integral) are shown to be isomorphic in full generality.