On Projections in the Noncommutative 2-Torus Algebra

TL;DR

This paper explores the structure of projections in the noncommutative 2-torus algebra by solving functional equations, generalizing known projections, and analyzing their K-theoretic classes to understand their classification.

Contribution

It provides exact solutions to functional equations defining projections, generalizes the Powers-Rieffel projection, and analyzes their K-theory classes in the noncommutative 2-torus.

Findings

Derived new classes of projections in $A_{\theta}$

Connected projections to their K-theory classes

Enhanced understanding of the structure of projections in noncommutative tori

Abstract

We investigate a set of functional equations defining a projection in the noncommutative 2-torus algebra $A_{\theta}$. The exact solutions of these provide various generalisations of the Powers-Rieffel projection. By identifying the corresponding $K_0(A_{\theta})$ classes we get an insight into the structure of projections in $A_{\theta}$.