On Lindenstrauss subspaces of \(C(Q)\) (Q5932606)

Let \(Q\) be a Hausdorff compact space. A subspace \(L\subset C(Q)\) is called a Grothendieck space (\(G\)-space) if \(L=\{f\in C(Q) : f(q'_{\alpha})=h_{\alpha}f(q''_{\alpha})\), \(h_{\alpha}\in \mathbb{R}\), \(q'_{\alpha},q''_{\alpha}\in Q\), \(\alpha \in A\}\) (\(A\) is an index set). \textit{J. Blatter} [Mem. Am. Math. Soc. 120, 1-121 (1972; Zbl 0236.46027)] posed the question of describing the set of extreme points of the unit ball \(B_{L^{\bot}}\), where \(L^{\bot}\subset C^{*}(Q)\) is the annihilator of \(L\). The author answers this question in the case when \(Q\) is a metrizable compact space and the set-valued mapping \(\tau\: Q\to 2^Q\), \(\tau (q)=\{t\in Q: \exists \lambda_t\in \mathbb{R}\setminus\{0\}\), \(f(t)=\lambda_tf(q)\) \(\forall f\in L\}\) is upper semicontinuous. Some approximation properties are also obtained for Lindenstrauss spaces \(X\) with \(X^{*}=\ell^1\).