Unital embeddings of C*-algebras that one can see

Cuntz algebras $O_n$, $n>1$, are celebrated examples of a separable infinite simple C*-algebra with a number of fascinating properties. Their K-theory allows an embedding of $O_m$ in $O_n$ whenever $n-1$ divides $m-1$. In 2009, Kawamura provided a simple and explicit formula for all such embeddings. It turns out that his formulas can be easily deduced by viewing Cuntz algebras as graph C*-algebras, known as operator algebras that one can see. Better still, playing the game of graphs and using both the covariant and contravariant functoriality of assigning graph C*-algebras to directed graphs, we can show how to embed Cuntz algebras into matrices over Cuntz algebras via straightforward polynomial formulas. Based on joint work with Yang Liu.