Cunz and Li defined defined C*-algebras for arbitrary rings with of course some condition. One can look at their article (http://arxiv.org/abs/0905.4861). My question: is the construction functorial? If not, for what kind of morphism of C*-algebras/rings will it work?

For group case I know: one has to consider morphism between C*-algebras $A$ to $B$ as essential *-homomorphism from $A$ to $\mathfrak{M}(B)$. Where $\mathfrak{M}(X)$ is the multiplier algebra of $X$.