A note on operators fixing cotype subspaces of 𝐶[0,1]

Gasparis, I.

doi:10.1090/S0002-9939-2014-11913-7

A note on operators fixing cotype subspaces of $C[0,1]$
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Proc. Amer. Math. Soc. 142 (2014), 1633-1639 Request permission

Abstract:

Let $K$ be a compact, metrizable space. Let $X$ be a closed, linear subspace of $C(K)$ spanned by a normalized weakly null sequence $(f_n)$ such that $(|f_n|)$ satisfies a lower $q$ estimate on disjoint blocks with positive coefficients for some $1 < q < \infty$. It is proved that every $w^*$-compact subset of $B_{C(K)^*}$ which norms $X$ is non-separable in norm. This provides an alternative proof of Bourgain’s result that every $w^*$-compact subset of $B_{C(K)^*}$ which norms a subspace with non-trivial cotype is non-separable in norm.

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