Topological Stable Rank of Nest Algebras

TL;DR

This paper computes the topological stable rank of nest algebras, showing it is either 2 or infinity, with specific conditions for when it equals 2, and introduces new methods for analyzing stable ranks in Banach algebra contexts.

Contribution

It introduces a general method for extending invertible rows to invertible matrices and applies it to compute the stable rank of nest algebras, providing new insights and quantitative measures.

Findings

The topological stable rank of nest algebras is either 2 or infinity.

rtsr(T(N)) = 2 only when N has certain ordinal type and atom growth.

New inequalities and methods for analyzing stable ranks over Banach algebras.

Abstract

We establish a general result about extending a right invertible row over a Banach algebra to an invertible matrix. This is applied to the computation of right topological stable rank of a split exact sequence. We also introduce a quantitative measure of stable rank. These results are applied to compute the right (left) topological stable rank for all nest algebras. This value is either 2 or infinity, and rtsr(T(N)) = 2 occurs only when N is of ordinal type less than omega^2 and the dimensions of the atoms grows sufficiently quickly. We introduce general results on `partial matrix algebras' over a Banach algebra. This is used to obtain an inequality akin to Rieffel's formula for matrix algebras over a Banach algebra. This is used to give further insight into the nest case.