location:  Publications → journals → CJM
Abstract view

# $1$-complemented subspaces of spaces with $1$-unconditional bases

Published:1997-12-01
Printed: Dec 1997
• Beata Randrianantoanina
 Format: HTML LaTeX MathJax PDF PostScript

## Abstract

We prove that if $X$ is a complex strictly monotone sequence space with $1$-un\-con\-di\-tion\-al basis, $Y \subseteq X$ has no bands isometric to $\ell_2^2$ and $Y$ is the range of norm-one projection from $X$, then $Y$ is a closed linear span a family of mutually disjoint vectors in $X$. We completely characterize $1$-complemented subspaces and norm-one projections in complex spaces $\ell_p(\ell_q)$ for $1 \leq p, q < \infty$. Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are $1$-complemented in (real or complex) Orlicz or Lorentz sequence spaces. In particular if an Orlicz or Lorentz space $X$ is not isomorphic to $\ell_p$ for some $1 \leq p < \infty$ then the only subspaces of $X$ which are $1$-complemented and disjointly supported are the closed linear spans of block bases with constant coefficients.
 MSC Classifications: 46B20 - Geometry and structure of normed linear spaces 46B45 - Banach sequence spaces [See also 46A45] 41A65 - Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)