On minimal interpolating projections and trace duality (Q1177069)

Let \(K\) be a compact Hausdorff space and \(C(K)\) be the space of all continuous real-valued functions on \(K\). For a finite-dimensional subspace \(V\subset C(K)\), let \(\lambda(V)=\inf\{\| P\|:\;P\) projection from \(C(K)\) onto \(V\}\). A projection \(P\) from \(C(K)\) onto \(V\) is said to be minimal or interpolating if \(\| P\|=\lambda(V)\) or \(Pf=\sum_{j=1}^ n f(k_ j)v_ j\), where \((k_ j)\subset K\), \((v_ j)\subset V\) is a basis of \(V\), \(v_ i(k_ j)=\delta_{ij}\), \(i,j=1,\dots,n\), and \(n=\dim V\), respectively. In this paper the authors construct a two-dimensional subspace \(V\subset C(K)\), (\(C(K)=\ell_ \infty^ 6\)), and an interpolating projection \(P\) from \(C(K)\) onto \(V\) such that \(P\) is minimal, \(\| P\|>1\), \(V\) is a Chebyshev subspace of \(C(K)\) and \(P^*\) is not a minimal projection. Finally, they prove that if \(P\) is a minimal interpolating projection from \(C(K)=\ell_ \infty^{n+k}\) onto an \(n\)-dimensional Chebyshev subspace \(V\) and \(\| P\|>1\), then \(\sharp\{k:\;\Lambda_ p(k)=\| p\|\}>n+(\| p\|-1)^ 2\), where \(\Lambda_ p(k)=\sup\{|(Pf)(k)|:\;\| f\|\leq1\}\) and \(\sharp\) stands for the cardinality of the set.