Unital embeddings of Cuntz algebras from path homomorphisms of graphs
Piotr M. Hajac, Yang Liu
TL;DR
This work addresses when unital embeddings between Cuntz algebras and their matrix algebras exist and how to realize them explicitly. By leveraging both covariant and contravariant functorialities of graph C*-algebras, the authors derive explicit polynomial formulas for embeddings $\mathcal{O}_p \to M_k(\mathcal{O}_q)$ under the K-theoretic constraint $(p-1)k=(q-1)s$, and recover Kawamura’s original construction as a covariant instance. They realize $M_k(\mathcal{O}_m)$ as graph C*-algebras of blown-up graphs and provide explicit isomorphisms, enabling a concrete graph-theoretic realization of the embeddings. The results unify K-theory obstructions with explicit combinatorial graph data, yielding practical polynomial descriptions of unital embeddings between Cuntz algebras and their matrix algebras.
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.