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M.González
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Since $\ell_\infty(\mathcal k)$ is a dual space, a result by Rosenthal implies that if $X^*$ contains $c_0(\Gamma)$ then it contains $\ell_\infty(\Gamma)$.

See H.P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory. Studia Math. 37 (1970), 13-36. (Correction, ibid., 311-313.)

M.González
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