Similarities of subspace lattices in Banach spaces

TL;DR

This paper investigates the structure of collineations of subspace lattices in Banach spaces, characterizing their groups and subgroups, and exploring conditions under which certain subgroups are complemented, with specific examples and partial results.

Contribution

It provides a detailed analysis of collineation groups of subspace lattices in Banach spaces, including their normality, normalizers, and conditions for subgroup complementarity, with explicit examples.

Findings

The subgroup of operators fixing all subspaces is normal in the collineation group.

In reflexive lattices, the collineation group is the normalizer of this subgroup.

Certain subspace lattices have complemented subgroups, while for others, only partial results are obtained.

Abstract

A collineation of a subspace lattice $\fL$ in a complex Banach space $\eX$ is an invertible operator $S$ on $\eX$ with the property that the image $S\eM$ of a subspace $\eM$ belongs to $\fL$ if and and only if $\eM$ belongs to it. Hence, $S$ is a collineation of $\fL$ if and only if it implements an order automorphism of $\fL$. We study the group $\Col(\fL)$ of all collineations of $\fL$ and its subgroup $\Grp(\Alg(\fL))$ of all invertible operators that fix every subspace in $\fL$. We show that $\Grp(\Alg(\fL))$ is a normal subgroup of $\Col(\fL)$; moreover, if $\fL$ is a reflexive subspace lattice, then $\Col(\fL)$ is the normalizer of $\Grp(\Alg(\fL))$ in the group of all invertible operators on $\eX$. One of the main questions that we consider is whether $\Grp(\Alg(\fL))$ is a complemented subgroup in $\Col(\fL)$. For certain subspace lattices $\fL$, such as some realizations of the diamond or the double triangle, some nests in the space of continuous functions on $[0,1]$, and the classical Volterra nest in $L^1[0,1]$, we characterize the complement of $\Grp(\Alg(\fL))$ in $\Col(\fL)$. On the other hand, for the Volterra nests in $L^p[0,1]$, where $1<p<\infty$, a further study is needed, and we prove only some partial results.