This is an announcement for the paper "$M$-structures in vector-valued polynomial spaces" by Veronica Dimant and Silvia Lassalle.

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

Archive classification: math.FA

Mathematics Subject Classification: 47H60, 46B04, 47L22, 46B20

Submitted from: vero@udesa.edu.ar

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1102.3850

or

http://arXiv.org/abs/1102.3850