Session 36. Topology in Functional Analysis

On uncomplemented isometric copies of \(c_0\) in spaces of continuous functions on products of the two-arrows space

Artur Michalak, Faculty of Mathematics and Computer Science , Poland
Phillips in [4] proved that \(c_0\) is an uncomplemented subspace of \(l_\infty\). We do not find in the literature many classes of separable Hausdorff compact spaces \(K\) such that there exists a subspace \(X\) isomorphic to \(c_0\) and uncomplemented in \(C(K)\). Except \(\beta \mathbb N\) appears essentially only the class of Mrówka spaces (see [3]). The reason is simple, usually it is quite hard to show the uncomplementability. There is one general method to do it, it is a modification of the Whitley proof of the Phillips theorem (see [6]). The method based on the facts that any \(C(K)\) space, when \(K\) is separable and compact, does not contain any isomorphic copy of \(c_0(\Gamma)\) for any uncountable set \(\Gamma\) but the quotient space \(C(K)/X\) contains such a copy (see [2], [6], [1]).

We construct for every \(n\geqslant 2\) a subspace \(X_n\) isometric to \(c_0\) and complemented in \(C(\mathbb L^n)\), the \(n\)-fold product of two arrows space \(\mathbb L\), such that \( \inf\{\|P\|:P:C(\mathbb L^n)\to X_n\text{ is a projection}\}\geqslant n+2 \) and the quotient space \(C(\mathbb L^n)/X_n\) has a \((3+ 4\sqrt{2})\) norming sequence of norm one functionals. The inequality together with the last fact enables us to find an isometric to \(c_0\) and uncomplemented subspace \(Y\) of \(C(\mathbb L^\mathbb N)\) such that the quotient space \(C(\mathbb L^\mathbb N)/Y\) is isomorphic to a subspace of \(l_\infty\).

References
  1. J. Ferrer, J. Kąkol, M. López Pellicer, M. Wójtowicz, On a three-space property for Lindelöf \(\Sigma\)-spaces, (WCG)-spaces and the Sobczyk property , Funct. Approx. Comment. Math., 44 (2011), 289-306.
  2. S. Molto, On a theorem of Sobczyk , Bull Austral. Math. Soc. 43 (1991), 123-130.
  3. S. Mrówka, On completely regular spaces , Fund. Math. 41 (1954), 105-106.
  4. R. S. Phillips, On linear transformations , Trans. Amer. Math. Soc., 48 (1940), 516-541.
  5. R. Whitley, Projecting \(m\) onto \(c_0\) , Amer. Math. Monthly, 73 (1965), 285-286.
  6. D. Yost, The Jonhson-Lindenstrauss space , Extracta. Math. 12 (1997), 185-192.
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