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Search: MSC category 46B04 ( Isometric theory of Banach spaces )

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1. CJM Online first

Dantas, Sheldon; García, Domingo; Maestre, Manuel; Martín, Miguel
 The Bishop-Phelps-BollobÃ¡s property for compact operators We study the Bishop-Phelps-BollobÃ¡s property (BPBp for short) for compact operators. We present some abstract techniques which allows to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$ has the BPBp for compact operators, then so do $(C_0(L),Y)$ for every locally compact Hausdorff topological space $L$ and $(X,Y)$ whenever $X^*$ is isometrically isomorphic to $\ell_1$. If $X^*$ has the Radon-NikodÃ½m property and $(\ell_1(X),Y)$ has the BPBp for compact operators, then so does $(L_1(\mu,X),Y)$ for every positive measure $\mu$; as a consequence, $(L_1(\mu,X),Y)$ has the the BPBp for compact operators when $X$ and $Y$ are finite-dimensional or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(\nu)$ for any positive measure $\nu$ and $1\lt p\lt \infty$. For $1\leq p \lt \infty$, if $(X,\ell_p(Y))$ has the BPBp for compact operators, then so does $(X,L_p(\mu,Y))$ for every positive measure $\mu$ such that $L_1(\mu)$ is infinite-dimensional. If $(X,Y)$ has the BPBp for compact operators, then so do $(X,L_\infty(\mu,Y))$ for every $\sigma$-finite positive measure $\mu$ and $(X,C(K,Y))$ for every compact Hausdorff topological space $K$. Keywords:Bishop-Phelps theorem, Bishop-Phelps-BollobÃ¡s property, norm attaining operator, compact operatorCategories:46B04, 46B20, 46B28, 46B25, 46E40

2. CJM 2012 (vol 65 pp. 331)

Kadets, Vladimir; Martín, Miguel; Merí, Javier; Werner, Dirk
 Lushness, Numerical Index 1 and the Daugavet Property in Rearrangement Invariant Spaces We show that for spaces with 1-unconditional bases lushness, the alternative Daugavet property and numerical index 1 are equivalent. In the class of rearrangement invariant (r.i.) sequence spaces the only examples of spaces with these properties are $c_0$, $\ell_1$ and $\ell_\infty$. The only lush r.i. separable function space on $[0,1]$ is $L_1[0,1]$; the same space is the only r.i. separable function space on $[0,1]$ with the Daugavet property over the reals. Keywords:lush space, numerical index, Daugavet property, KÃ¶the space, rearrangement invariant spaceCategories:46B04, 46E30

3. CJM 2004 (vol 56 pp. 472)

Fonf, Vladimir P.; Veselý, Libor
 Infinite-Dimensional Polyhedrality This paper deals with generalizations of the notion of a polytope to infinite dimensions. The most general definition is the following: a bounded closed convex subset of a Banach space is called a \emph{polytope} if each of its finite-dimensional affine sections is a (standard) polytope. We study the relationships between eight known definitions of infinite-dimensional polyhedrality. We provide a complete isometric classification of them, which gives solutions to several open problems. An almost complete isomorphic classification is given as well (only one implication remains open). Categories:46B20, 46B03, 46B04, 52B99
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