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

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1. CMB Online first

Godefroy, G.
 Uniqueness of preduals in spaces of operators We show that if $E$ is a separable reflexive space, and $L$ is a weak-star closed linear subspace of $L(E)$ such that $L\cap K(E)$ is weak-star dense in $L$, then $L$ has a unique isometric predual. The proof relies on basic topological arguments. Categories:46B20, 46B04

2. CMB 2011 (vol 54 pp. 411)

Davidson, Kenneth R.; Wright, Alex
 Operator Algebras with Unique Preduals We show that every free semigroup algebra has a (strongly) unique Banach space predual. We also provide a new simpler proof that a weak-$*$ closed unital operator algebra containing a weak-$*$ dense subalgebra of compact operators has a unique Banach space predual. Keywords:unique predual, free semigroup algebra, CSL algebraCategories:47L50, 46B04, 47L35

3. CMB 2011 (vol 54 pp. 680)

Jiménez-Vargas, A.; Villegas-Vallecillos, Moisés
 $2$-Local Isometries on Spaces of Lipschitz Functions Let $(X,d)$ be a metric space, and let $\mathop{\textrm{Lip}}(X)$ denote the Banach space of all scalar-valued bounded Lipschitz functions $f$ on $X$ endowed with one of the natural norms $\| f\| =\max \{\| f\| _\infty ,L(f)\}$ or $\|f\| =\| f\| _\infty +L(f),$ where $L(f)$ is the Lipschitz constant of $f.$ It is said that the isometry group of $\mathop{\textrm{Lip}}(X)$ is canonical if every surjective linear isometry of $\mathop{\textrm{Lip}}(X)$ is induced by a surjective isometry of $X$. In this paper we prove that if $X$ is bounded separable and the isometry group of $\mathop{\textrm{Lip}}(X)$ is canonical, then every $2$-local isometry of $\mathop{\textrm{Lip}}(X)$ is a surjective linear isometry. Furthermore, we give a complete description of all $2$-local isometries of $\mathop{\textrm{Lip}}(X)$ when $X$ is bounded. Keywords:isometry, local isometry, Lipschitz functionCategories:46B04, 46J10, 46E15

4. CMB 2001 (vol 44 pp. 370)

Weston, Anthony
 On Locating Isometric $\ell_{1}^{(n)}$ Motivated by a question of Per Enflo, we develop a hypercube criterion for locating linear isometric copies of $\lone$ in an arbitrary real normed space $X$. The said criterion involves finding $2^{n}$ points in $X$ that satisfy one metric equality. This contrasts nicely to the standard classical criterion wherein one seeks $n$ points that satisfy $2^{n-1}$ metric equalities. Keywords:normed spaces, hypercubesCategories:46B04, 05C10, 05B99

5. CMB 1999 (vol 42 pp. 344)

Koldobsky, Alexander
 Positive Definite Distributions and Subspaces of $L_p$ With Applications to Stable Processes We define embedding of an $n$-dimensional normed space into $L_{-p}$, $0 Categories:42A82, 46B04, 46F12, 60E07 6. CMB 1998 (vol 41 pp. 279) Acosta, María D.; Galán, Manuel Ruiz  New characterizations of the reflexivity in terms of the set of norm attaining functionals As a consequence of results due to Bourgain and Stegall, on a separable Banach space whose unit ball is not dentable, the set of norm attaining functionals has empty interior (in the norm topology). First we show that any Banach space can be renormed to fail this property. Then, our main positive result can be stated as follows: if a separable Banach space$X$is very smooth or its bidual satisfies the$w^{\ast }$-Mazur intersection property, then either$X$is reflexive or the set of norm attaining functionals has empty interior, hence the same result holds if$X$has the Mazur intersection property and so, if the norm of$X\$ is Fr\'{e}chet differentiable. However, we prove that smoothness is not a sufficient condition for the same conclusion. Categories:46B04, 46B10, 46B20

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