Spatial discretization of restricted group algebras

TL;DR

This paper studies the spatial discretization of restricted group algebras using the finite section method, focusing on stability, quasicommutator ideals, and fractality for specific groups.

Contribution

It introduces a detailed analysis of the finite section algebra for restricted group algebras, highlighting the role of discrete boundaries and fractality in certain groups.

Findings

The finite section algebra's quasicommutator ideal is characterized.

Stability of finite section sequences depends on discrete boundary properties.

Algebras are shown to be fractal for commutative and free non-commutative groups.

Abstract

We consider spatial discretizations by the finite section method of the restricted group algebra of a finitely generated discrete group, which is represented as a concrete operator algebra via its left-regular representation. Special emphasis is paid to the quasicommutator ideal of the algebra generated by the finite sections sequences and to the stability of sequences in that algebra. For both problems, the sequence of the discrete boundaries plays an essential role. Finally, for commutative groups and for free non-commutative groups, the algebras of the finite sections sequences are shown to be fractal.