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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 $X$.

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If you mean the full ring C*-algebra, it is functorial with respect to ring homomorphisms preserving regular elements (and the chosen algebra on the subsets): This follows from the universal property. –  Martin Brandenburg Jan 22 '13 at 9:54
    
What is your morphism between C*-algebras? –  Sayan Jan 22 '13 at 17:49

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