Non-denseness of factorable matrix functions

TL;DR

This paper demonstrates that in certain function algebras on compact abelian groups, the set of factorable matrix functions is not dense among invertible matrix functions, revealing complex topological structures.

Contribution

It proves the non-denseness of factorable matrix functions in invertible groups for specific algebras, including continuous and Wiener algebras, under certain group conditions.

Findings

Factorable matrix functions do not densely populate invertible matrix functions.

Infinitely many connected components lack any factorable matrices.

These components are disjoint from the subgroup of triangularizable matrix functions.

Abstract

It is proved that for certain algebras of continuous functions on compact abelian groups, the set of factorable matrix functions with entries in the algebra is not dense in the group of invertible matrix functions with entries in the algebra, assuming that the dual abelian group contains a subgroup isomorphic to Z^3. These algebras include the algebra of all continuous functions and the Wiener algebra. More precisely, it is shown that infinitely many connected components of the group of invertible matrix functions do not contain any factorable matrix functions, again under the same assumption. Moreover, these components actually are disjoint with the subgroup generated by the triangularizable matrix functions.