Lifting Commutation Relations in Cuntz Algebras

We examine splitting of the quotient map from the full free product $A*B$, or the unital free product $A*_{\mathbb C}B$, to the (maximal) tensor product $A\otimes B$, for unital C*-algebras $A$ and $B$. Such a splitting is very rare, but we show there is one if $A$ and $B$ are both the Cuntz algebra $O_2$ or $O_\infty$, and in a few other cases. The splitting is not explicit (and in principle probably cannot be). We also describe severe $K$-theoretic obstructions to a splitting.