Extension groups for the $C^*$-algebras associated with $\lambda$-graph systems

TL;DR

This paper computes strong extension groups for $C^*$-algebras derived from $mbda$-graph systems, expanding understanding of their algebraic structure and relation to weak extension groups.

Contribution

It introduces the computation of strong extension groups for $C^*$-algebras associated with $mbda$-graph systems and explores their relation to weak extension groups.

Findings

Computed strong extension groups for the $C^*$-algebras of $mbda$-graph systems.

Analyzed the relationship between strong and weak extension groups.

Extended the algebraic understanding of $C^*$-algebras associated with symbolic dynamical systems.

Abstract

A $\lambda$-graph system is a labeled Bratteli diagram with certain additional structure, which presents a subshift. The class of the $C^*$-algebras $\mathcal{O}_{\frak L}$ associated with the $\lambda$-graph systems is a generalized class of the class of Cuntz--Krieger algebras. In this paper, we will compute the strong extension groups $\operatorname{Ext}_{\operatorname{s}}(\mathcal{O}_{\frak L})$ for the $C^*$-algebras associated with $\lambda$-graph systems ${\frak L}$ and study their relation with the weak extension group $\operatorname{Ext}_{\operatorname{w}}(\mathcal{O}_{\frak L})$.