Unital embeddings of Cuntz algebras from path homomorphisms of graphs

TL;DR

This paper develops explicit polynomial formulas for unital embeddings of Cuntz algebras into matrix algebras over Cuntz algebras, utilizing graph C*-algebra functoriality and K-theory insights.

Contribution

It introduces a method to explicitly construct all such embeddings using graph homomorphisms and functoriality, extending previous results with concrete formulas.

Findings

Provides explicit polynomial formulas for embeddings

Utilizes graph C*-algebra functoriality

Extends Kawamura's formulas to matrix embeddings

Abstract

Cuntz algebras $\mathcal{O}_n$, $n>1$, are celebrated examples of a separable infinite simple C*-algebra with a number of fascinating properties. Their K-theory allows an embedding of $\mathcal O_m$ in $\mathcal O_n$ whenever $n-1$ divides $m-1$. In 2009, Kawamura provided a simple and explicit formula for all such embeddings. His formulas can be easily deduced by viewing Cuntz algebras as graph C*-algebras. Our main result is that, using both the covariant and contravariant functoriality of assigning graph C*-algebras to directed graphs, we can provide explicit polynomial formulas for all unital embeddings of Cuntz algebras into matrices over Cuntz algebras allowed by K-theory.