On uncomplemented isometric copies of \(c_0\) in spaces of continuous functions on products of the two-arrows space. II. (English) Zbl 1359.46015
Summary: Let \(\mathbb{L}\) be the two-arrows space. It is a separable, Hausdorff, compact space. Let \((\bigoplus_{n = 2}^\infty C(\mathbb{L}^n))_{c_0}\) be the \(c_0\)-sum of the sequence of Banach spaces of all continuous scalar (real or complex) functions on \(n\)-fold products of \(\mathbb{L}\). We show that for every subspace \(X\) of the Banach space \((\bigoplus_{n = 2}^\infty C(\mathbb{L}^n))_{c_0}\) isomorphic to \(c_0\) the quotient space \((\bigoplus_{n = 2}^\infty C(\mathbb{L}^n))_{c_0} / X\) does not contain any subspace isomorphic to \(c_0(\Gamma)\) for any uncountable set \(\Gamma\). The space \((\bigoplus_{n = 2}^\infty C(\mathbb{L}^n))_{c_0}\) contains an uncomplemented subspace isometric to \(c_0\).
For Part I see [ibid. 26, No. 1, 162–173 (2015; Zbl 1326.46016)].
For Part I see [ibid. 26, No. 1, 162–173 (2015; Zbl 1326.46016)].
MSC:
| 46B26 | Nonseparable Banach spaces |
| 46E15 | Banach spaces of continuous, differentiable or analytic functions |
Citations:
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