Let $ m , n in mathbb{N}$, $D$ be a division ring, and $M_{m times n}(D)$ denote the bimodule of all $m times n$ matrices with entries from $D$. First, we characterize one-sided submodules of $M_{m times n}(D)$ in terms of left row reduced echelon or right column reduced echelon matrices with entries from $D$. Next, we introduce the notion of a nest module of matrices with entries from $D$. We ...

let $ m , n in mathbb{n}$, $d$ be a division ring, and $m_{m times n}(d)$ denote the bimodule of all $m times n$ matrices with entries from $d$. first, we characterize one-sided submodules of $m_{m times n}(d)$ in terms of left row reduced echelon or right column reduced echelon matrices with entries from $d$. next, we introduce the notion of a nest module of matrices with entries from $d$. we ...

Following earlier work in which we provided algebraic characterizations of the right, left, and two-sided Bourgain algebras, as well as the second order Bourgain algebras, associated with a nest algebra, we herein demonstrate that a given nest algebra has (essentially) at most six different third order Bourgain algebras, and that every fourth order (or higher) Bourgain algebra of the nest algeb...

We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu’s non-commutative disc algebras and to fr...

We show that the family of nest algebras with r non-zero nest projections is stable, in the sense that an approximate containment of one such algebra within another is close to an exact containment. We use this result to give a local characterisation of limits formed from this family. We then consider quite general regular limit algebras and characterise these algebras using a local condition w...

In analogy with a construction from function theory, we herein define right, left, and two-sided Bourgain algebras associated with an operator algebra A. These algebras are defined initially in Banach space terms, using the weak-* topology on A, and our main result is to give a completely algebraic characterization of them in the case where A is a nest algebra. Specifically, if A = algN is a ne...