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Register a new userUnconditional basic sequences in spaces of large density
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Pandelis Dodos
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We study the problem of the existence of unconditional basic sequences in Banach spaces of high density. We show, in particular, the relative consistency with GCH of the statement that every Banach space of density $aleph_omega$ contains an unconditional basic sequence.
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We study density requirements on a given Banach space that guarantee the existence of subsymmetric basic sequences by extending Tsirelsons well-known space to larger index sets. We prove that for every cardinal $kappa$ smaller than the first Mahlo cardinal there is a reflexive Banach space of density $kappa$ without subsymmetric basic sequences. As for Tsirelsons space, our construction is based on the existence of a rich collection of homogeneous families on large index sets for which one can estimate the complexity on any given infinite set. This is used to describe detailedly the asymptotic structure of the spaces. The collections of families are of independent interest and their existence is proved inductively. The fundamental stepping up argument is the analysis of such collections of families on trees.
Let $X$ be a sequence space and denote by $Z(X)$ the subset of $X$ formed by sequences having only a finite number of zero coordinates. We study algebraic properties of $Z(X)$ and show (among other results) that (for $p in [1,infty]$) $Z(ell_p)$ does not contain infinite dimensional closed subspaces. This solves an open question originally posed by R. M. Aron and V. I. Gurariy in 2003 on the linear structure of $Z(ell_infty)$. In addition to this, we also give a thorough analysis of the existing algebraic structures within the set $X setminus Z(X)$ and its algebraic genericity.
We prove that, unless assuming additional set theoretical axioms, there are no reflexive space without unconditional sequences of density the continuum. We give for every integer $n$ there are normalized weakly-null sequences of length $om_n$ without unconditional subsequences. This together with a result of cite{Do-Lo-To} shows that $om_omega$ is the minimal cardinal $kappa$ that could possibly have the property that every weakly null $kappa$-sequence has an infinite unconditional basic subsequence . We also prove that for every cardinal number $ka$ which is smaller than the first $om$-Erdos cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either $c_0$ or $ell_p$, with $pge 1$.
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