On the construction of fuzzy spaces and modules over shift algebras

TL;DR

This paper introduces shift algebras, studies their modules depending on function spaces, and constructs fuzzy spaces as finite and infinite dimensional modules related to compact and non-compact manifolds.

Contribution

It develops a framework for shift algebras and provides a method to construct fuzzy spaces from these algebras, linking algebraic modules to geometric spaces.

Findings

Modules depend on properties of function spaces

Constructed fuzzy spaces from shift algebras

Connected finite and infinite dimensional modules to geometric spaces

Abstract

We introduce shift algebras as certain crossed product algebras based on general function spaces and study properties, as well as the classification, of a particular class of modules depending on a set of matrix parameters. It turns out that the structure of these modules depends in a crucial way on the properties of the function spaces. Moreover, for a class of subalgebras related to compact manifolds, we provide a construction procedure for the corresponding fuzzy spaces, i.e. sequences of finite dimensional modules of increasing dimension as the deformation parameter tends to zero, as well as infinite dimensional modules related to fuzzy non-compact spaces.