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arXiv · 1502.03791

Separable elastic Banach spaces are universal

Dale E. Alspach·Bunyamin Sari
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Abstract

A Banach space $X$ is elastic if there is a constant $K$ so that whenever a Banach space $Y$ embeds into $X$, then there is an embedding of $Y$ into $X$ with constant $K$. We prove that $C[0,1]$ embeds into separable infinite dimensional elastic Banach spaces, and therefore they are universal for all separable Banach spaces. This confirms a conjecture of Johnson and Odell. The proof uses incremental embeddings into $X$ of $C(K)$ spaces for countable compact $K$ of increasing complexity. To achieve this we develop a generalization of Bourgain's basis index that applies to unconditional sums of Banach spaces and prove a strengthening of the weak injectivity property of these $C(K)$ that is realized on special reproducible bases.

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Dale E. Alspach, Bunyamin Sari. 2015-02-12. Separable elastic Banach spaces are universal. https://arxiv.org/abs/1502.03791

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