Finite-codimensional subspaces of Daugavet spaces: projection constants and minimal projections

Abstract

Over the real or complex field, we establish a duality formula for projection constants of finite-codimensional subspaces of Banach spaces with the Daugavet property. If \[ Y=\bigcap_{j=1}^n \ker f_j \subset X, \qquad W=\operatorname{span}\{f_1,\dots,f_n\} \subset X^*, \] then \[ λ(Y,X)=1+λ(W,X^*), \] and minimal projections onto $Y$ correspond exactly to weak${}^{\ast }$-continuous minimal projections onto $W$. This yields, in particular, a complete description of the hyperplane case: every hyperplane has projection constant $2$, and $\ker f$ admits a minimal projection if and only if $f$ attains its norm. We then specialise to the real space $X=C[0,1]$. Our second ingredient is a transfer principle from duplication-stable finite-dimensional subspaces of ${\ell }_1^N$ to piecewise-constant subspaces of $L_1[0,1]\subset M[0,1]=C[0,1{]}^{\ast }$. For the regular symmetric spaces constructed by Chalmers and the second-named author and the second named author and Prophet, respectively, the transferred subspaces retain their projection constants but admit no weak${}^{\ast }$-continuous minimal projections. Passing to annihilators yields finite-codimensional subspaces of the real space $C[0,1]$ for which the infimum defining the projection constant is not attained. As a consequence, for every $\hat{I}›\in [2,\infty )$ there exists a finite-codimensional subspace $Y$ of the real space $C[0,1]$ such that \[ λ(Y,C[0,1])=Λ, \] and the infimum defining $\hat{I}»(Y,C[0,1])$ is not attained. For each even codimension $n$ we moreover realise every value in the interval $(2,1+{\hat{I}}_n^2]$, where \[ β_n = \mathsf E_{{\mathsf P}_n}\Bigl|\sum_{j=1}^n \varepsilon_j\Bigr| = n2^{-n}\binom{n}{n/2} \sim \sqrt{\frac{2n}π}, \] $({\epsilon }_j)$ is a Rademacher family on $\hat{I}{\text{c◯}}_n=\{-1,1{\}}^n$, and ${\mathsf{P}}_n$ is the uniform probability measure.

15 pp