Rafael Payá

University of Granada | UGR · Department of Mathematical Analysis

We show that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional. For instance, it has no WLUR points in its unit ball, its norm is not Frechet smooth and its dual norm is neither smooth nor strictly convex. Actually, these results also hold if the space has the (strictly weaker)...

For a bounded function f from the unit sphere of a closed subspace X of a Banach space Y, we study when the closed convex hull of its spatial numerical range W(f) is equal to its intrinsic numerical range V(f). We show that for every infinite-dimensional Banach space X there is a superspace Y and a bounded linear operator such that . We also show t...

El propósito de este trabajo es revisar el estado actual de la investigación reciente sobre el índice numérico de los espacios de Banach, presentando algunos de los resultados obtenidos en los últimos años y proponiendo un cierto número de problemas abiertos. Abstract The aim of this paper is to review the state-of-the-art of recent research concer...

We find some necessary conditions for a real Banach space to be an almost CL-space. We also discuss the stability of CL-spaces and almost CL-spaces byc 0- andl 1-sums. Finally, we address the question if a space of vector-valued continuous functions can be a CL-space or an almost CL-space.

We study the numerical index of a Banach space from the isomorphic point of view, that is, we investigate the values of the nu-merical index which can be obtained by renorming the space. The set of these values is always an interval which contains [0, 1/3[ in the real case and [e −1 , 1/2[ in the complex case. Moreover, for most Banach spaces the l...

We give a sufficient condition on a C*-algebra to ensure that every weakly compact operator into an arbitrary Banach space can be approximated by norm attaining operators and that every continuous bilinear form can be approximated by norm attaining bilinear forms. Moreover we prove that the class of C*-algebras satisfying this condition includes th...

This paper gives new sufficient conditions for the density of the set of norm attaining multilinear forms in the space of all continuous multilinear forms on a Banach space. The symmetric case is also discussed.

Given an arbitrary measure μ and a localizable measure v, we show that the set of norm attaining operators is dense in the space of all bounded linear operators from L1(μ) into L∞ (v).

Given an arbitrary measure m\mu and a localizable measure n\nu , we show that the set of norm attaining operators is dense in the space of all bounded linear operators from L1(m)L_{1}(\mu ) into L¥(n).L_{\infty }(\nu ).

We show that the numerical index of a c0-, l1-, or l∞-sum of Banach spaces is the infimum of the numerical indices of the summands. Moreover, we prove that the spaces C(K, X) and L1(μ, X) (K any compact Hausdorff space, μ any positive measure) have the same numerical index as the Banach space X. We also observe that these spaces have the so-called...

We show that an infinite-dimensional real Banach space with numerical index 1 satisfying the Radon–Nikodym property contains l1. It follows that a reflexive or quasi-reflexive real Banach space cannot be re-normed to have numerical index 1, unless it is finite-dimensional.

We show that the set of norm attaining operators is dense in the space of all bounded linear operators fromL 1 intoL ∞.

We show that continuous bilinear forms on spaces of continuous functions can be approximated by norm attaining bilinear forms.

For each natural number N , we give an example of a Banach space X such that the set of norm attaining N –linear forms is dense in the space of all continuous N –linear forms on X, but there are continuous (N + 1)–linear forms on X which cannot be approximated by norm attaining (N +1)–linear forms. Actually, X is the canonical predual of a suitable...

For each natural number N, we give an example of a Banach space X such that the set of norm attaining N-linear forms is dense in the space of all continuous N-linear forms on X, but there are continuous (N + 1)-linear forms on X which cannot be approximated by norm attaining (N + 1)-linear forms. Actually, X is the canonical predual of a suitable L...

The well known Bishop-Phelps Theorem asserts that the set of norm attaining linear forms on a Banach space is dense in the dual space [3]. This note is an outline of recent results by Y. S. Choi [5] and C. Finet and the author [7], which clarify the relation between two different ways of extending this theorem.

We answer a question posed by R. Aron, C. Finet and E. Werner, on the bilinear version of the Bishop-Phelps theorem, by exhibiting an example of a Banach spaceX such that the set of norm-attaining bilinear forms onX×X is not dense in the space of all continuous bilinear forms.

We give a new sufficient condition for a Banach space Y to satisfy Lindenstrauss’s property B, namely the set of norm-attaining operators from any other Banach space X into Y is dense. Even in the finite-dimensional case, our result gives new examples of Banach spaces with property B.

We give new sufficient conditions for a Banach space to be an Asplund (or reflexive) space in terms of certain upper semicontinuity of the duality mapping.

We prove that, for any Banach space X, the set of operators on X whose adjoints attain their numerical radii is dense in the space of all operators. We also show the denseness of the set of numerical radius attaining operators on a Banach space with the Radon-Nikodym property.

We answer a question posed by B. Sims in 1972, by exhibiting an example of a Banach spaceX such that the numerical radius attaining operators onX are not dense. Actually,X is an old example used by J. Lindenstrauss to solve the analogous problem for norm attaining operators, but the proof for the numerical radius seems to be much more difficult. Ou...

We use a Banach space recently considered by W. Gowers to improve some results on norm attaining operators. In fact we show that the norm attaining operators from this space to a strictly convex Banach space are finite-rank. The same Banach space is also used to get a new example of a space which does not satisfy the denseness of the numerical radi...

Let us say that a subspace M of a Banach space X is absolutely proximinal if it is proximinal and, for each xϵX, ∥x∥ can be expressed as a function of d(x, M), the distance from x to M, and d(0, PM(x)), the distance from the origin to the best approximant set. Then this functional dependence must be given by a suitable norm on R2. This defines a na...

We investigate a variant of the compact metric approximation property which, for subspaces X of c0, is known to be equivalent to K(X), the space of compact operators on X, being an M-ideal in the space of bounded operators on X, L(X). Among other things, it is shown that an arbitrary Banach space X has this property iff K(Y, X) is an M-ideal in L(Y...

We investigate a variant of the compact metric approximation property which, for subspaces X of c0, is known to be equivalent to K(X), the space of compact operators on X, being an M-ideal in the space of bounded operators on X, L(X). Among other things, it is shown that an arbitrary Banach space X has this property iff K(Y, X) is an Af-ideal in L(...

We show that for any Banach space the set of (bounded linear) operators whose second adjoints attain their numerical radii is norm-dense in the space of all operators. In particular, the numerical radius attaining operators on a reflexive space are dense.

We discuss some results on numerical radius attaining operators paralleling earlier results on norm attaining operators. For arbitrary Banach spaces X and Y, the set of (bounded, linear) operators from X to Y whose adjoints attain their norms is norm-dense in the space of all operators. This theorem, due to W. Zizler, improves an earlier result by...

We show that for any Banach space the set of (bounded linear) operators whose second adjoints attain their numerical radii is norm-dense in the space of all operators. In particular, the numerical radius attaining operators on a reflexive space are dense.

The 2-ball property is shown to be transitive. Combining this with some results on the decomposability of convex bodies, we produce new examples of Banach spaces which contain proper semi-M-ideals. These semi-M-ideals are not hyperplanes, nor are they the direct sums of examples which are hyperplanes.

The following result in the theory of numerical ranges in Banach algebras is well known (see [ 3 , Theorem 12.2]). The numerical range of an element F in the bidual of a unital Banach algebra A is the closure of the set of values at F of the w *-continuous states of . As a consequence of the results in this paper the following

Let |·| be a fixed absolute norm onR 2. We introduce semi-|·|-summands (resp. |·|-summands) as a natural extension of semi-L-summands (resp.L-summands). We prove that the following statements are equivalent. (i) Every semi-|·|-summand is a |·|-summand, (ii) (1, 0) is not a vertex of the closed unit ball ofR 2 with the norm |·|. In particular semi-L...

We introduce noncommutative JB*-algebras which generalize both B*-algebras and JB*-algebras and set up the bases for a representation theory of noncommutative JB*-algebras. To this end we define noncommutative JB*-factors and study the factor representations of a noncommutative JB*-algebra. The particular case of alternative B*-factors is discussed...