A class of Exel--Laca algebras reciprocal to Cuntz--Krieger algebras

TL;DR

This paper explores a duality between certain Exel--Laca algebras and Cuntz--Krieger algebras, focusing on their K-theory and extension groups, and identifies classes where this duality holds.

Contribution

It identifies a class of unital simple Exel--Laca algebras whose reciprocal duals are simple Cuntz--Krieger algebras, and computes their strong extension groups.

Findings

Established duality between Exel--Laca and Cuntz--Krieger algebras for specific classes.

Computed strong extension groups for the identified class of Exel--Laca algebras.

Connected algebraic structures via underlying infinite matrices.

Abstract

The reciprocality means a duality in Kirchberg algebras between K-theory groups and strong extension groups. In the paper, we will find a certain class of unital simple Exel--Laca algebras for which the reciprocal duals are simple Cuntz--Krieger algebras in terms of the underlying infinite matrices. In our procedure to obtain simple Cuntz--Krieger algebras from Exel--Laca algebras, we compute the strong extension groups for Exel--Laca algebras belonging to the class.