K-theory of cluster C*-algebras

TL;DR

This paper establishes an isomorphism between the K_0-group of a cluster C*-algebra and the associated cluster algebra, providing a new proof of the positivity conjecture and illustrating with a specific example.

Contribution

It demonstrates that the K_0-group of a cluster C*-algebra is isomorphic to the cluster algebra, offering a novel proof of the positivity conjecture.

Findings

K_0-group is isomorphic to the cluster algebra

Provided a shorter proof of the positivity conjecture

Analyzed a specific example from annulus triangulation

Abstract

It is proved that the K_0-group of a cluster C*-algebra is isomorphic to the corresponding cluster algebra. As a corollary, one gets a shorter proof of the positivity conjecture for cluster algebras. As an example, we consider a cluster C*-algebra A(1,1) coming from triangulation of an annulus with one marked point on each boundary component.