Open Access
Some Topological Invariants and Biorthogonal Systems in Banach Spaces
Piotr Koszmider
- Vol. 26, Iss: 2, pp 271-294
TLDR
In this article, the authors consider topological invariants on compact spaces related to the sizes of discrete subspaces (spread), densities of sub-spaces, Lindelof degree, irredundant families of clopen sets and others.Abstract:
We consider topological invariants on compact spaces related to the sizes of discrete subspaces (spread), densities of subspaces, Lindelof degree of subspaces, irredundant families of clopen sets and others and look at the following associations between compact topological spaces and Banach spaces: a compact K induces a Banach space C(K) of real valued continuous functions on K with the supremum norm; a Banach space X induces a compact space BX , the dual ball with the weak topology. We inquire on how topological invariants on K and BX are linked to the sizes of biorthogonal systems and their versions in C(K) and X respectively. We gather folkloric facts and survey recent results like that of Abad-Lopez and Todorcevic that it is consistent that there is a Banach space X without uncountable biorthogonal systems such that the spread of BX is uncountable or that of Brech and Koszmider that it is consistent that there is a compact space where spread of K 2 is countable but C(K) has uncountable biorthogonal systems.read more
Citations
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Compact lines and the Sobczyk property
Claudia Correa,Daniel V. Tausk +1 more
TL;DR: In this article, it was shown that Sobczyk's theorem holds for a new class of Banach spaces, namely spaces of continuous functions on linearly ordered compacta, which is a special case of the Banach space.
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On the problem of compact totally disconnected reflection of nonmetrizability
TL;DR: In this article, a ZFC example of a non-metrizable compact space K such that every totally disconnected closed subspace L ⊆ K is metrizable is presented.
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Large irredundant sets in operator algebras
TL;DR: In this article, it was shown that there is no non-commutative commutative non-separable C*-algebra with an uncountable irredundant set.
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Two Cardinal inequalities about bidiscrete systems
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Posted Content
Large Banach spaces with no infinite equilateral sets
Piotr Koszmider,Hugh Wark +1 more
Abstract: A subset of a Banach space is called equilateral if the distances between any two of its distinct elements are the same. It is proved that there exist non-separable Banach spaces (in fact of density continuum) with no infinite equilateral subset. These examples are strictly convex renormings of $\ell_1([0,1])$. A wider class of renormings of $\ell_1([0,1])$ which admit no uncountable equilateral sets is also considered.
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