Isometric stability property of certain Banach spaces

Alexander Koldobsky

Isometric stability property of certain Banach spaces

Functional Analysis 2016-09-06 v1

Authors: Alexander Koldobsky

Abstract

Let $E$ be one of the spaces $C(K)$ and $L_1$ , $F$ be an arbitrary Banach space, $p>1,$ and $(X,\sigma)$ be a space with a finite measure. We prove that $E$ is isometric to a subspace of the Lebesgue-Bochner space $L_p(X;F)$ only if $E$ is isometric to a subspace of $F.$ Moreover, every isometry $T$ from $E$ into $L_p(X;F)$ has the form $Te(x)=h(x)U(x)e, e\in E,$ where $h:X\rightarrow R$ is a measurable function and, for every $x\in X,$ $U(x)$ is an isometry from $E$ to $F.$

Keywords

banach spaces banach space properties banach algebras

Cite

@article{arxiv.math/9306208,
  title  = {Isometric stability property of certain Banach spaces},
  author = {Alexander Koldobsky},
  journal= {arXiv preprint arXiv:math/9306208},
  year   = {2016}
}

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