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

TL;DR

This paper establishes a duality formula for projection constants of finite-codimensional subspaces in Daugavet spaces, describes minimal projections, and constructs subspaces of C[0,1] with prescribed projection constants and non-attainment properties.

Contribution

It introduces a duality formula linking projection constants of subspaces and their annihilators, and constructs subspaces of C[0,1] with specific projection constants and non-attainment of infima.

Findings

Projection constant of hyperplanes is always 2.

Minimal projections exist iff the defining functional attains its norm.

Constructed subspaces of C[0,1] realize any projection constant ≥ 2.

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 \[ \lambda(Y,X)=1+\lambda(W,X^*), \] and minimal projections onto $Y$ correspond exactly to weak$^*$-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 $L_1[0,1]\subset M[0,1]=C[0,1]^*$. 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$^*$-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 $\Lambda\in[2,\infty)$ there exists a finite-codimensional subspace $Y$ of the real space $C[0,1]$ such that \[ \lambda(Y,C[0,1])=\Lambda, \] and the infimum defining $\lambda(Y,C[0,1])$ is not attained. For each even codimension $n$ we moreover realise every value in the interval $(2,1+\beta_n]$, where \[ \beta_n = \mathsf E_{{\mathsf P}_n}\Bigl|\sum_{j=1}^n \varepsilon_j\Bigr| = n2^{-n}\binom{n}{n/2} \sim \sqrt{\frac{2n}{\pi}}, \] $(\varepsilon_j)$ is a Rademacher family on $\Omega_n=\{-1,1\}^n$, and $\mathsf{P}_n$ is the uniform probability measure.