Isomorphic spectrum and isomorphic length of a Banach space

  • O. Fotiy Yuriy Fedkovych Chernivtsi National University, 2 Kotsjubynskyi str., 58012, Chernivtsi, Ukraine
  • M. Ostrovskii St. John's University, 11439, New York, USA https://orcid.org/0000-0002-7164-196X
  • M. Popov Pomeranian University in Słupsk, 76-200, Słupsk, Poland, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine https://orcid.org/0000-0002-3165-5822
Keywords: Banach space, isomorphic embedding, Martin axiom
Published online: 2020-06-12

Abstract


We prove that, given any ordinal $\delta < \omega_2$, there exists a transfinite $\delta$-sequence of separable Banach spaces $(X_\alpha)_{\alpha < \delta}$ such that $X_\alpha$ embeds isomorphically into $X_\beta$ and contains no subspace isomorphic to $X_\beta$ for all $\alpha < \beta < \delta$. All these spaces are subspaces of the Banach space $E_p = \bigl( \bigoplus_{n=1}^\infty \ell_p \bigr)_2$, where $1 \leq p < 2$. Moreover, assuming Martin's axiom, we prove the same for all ordinals $\delta$ of continuum cardinality.

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How to Cite
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Fotiy O., Ostrovskii M., Popov M. Isomorphic Spectrum and Isomorphic Length of a Banach Space. Carpathian Math. Publ. 2020, 12 (1), 88-93.