Let $A$ be a $C^*$-algebra. I want to prove that $$ \phi : C_0(X) \otimes_\alpha A \to C_0(X,A): f \otimes a \mapsto a \cdot f $$ is an isomorphism, where $\alpha$ is a norm making the tensor product into a $C^*$-algebra. Note that $C_0(X)$ is commutative hence nuclear. Since $\phi(C_0(X) \otimes A)$ is dense (with $\otimes$ the algebraic tensor product) one is done, if the restriction of $\phi $ to $C_0(X) \otimes A$ is an isometry. Then $\phi$ is clearly surjective and also injective. (On the algebraic tensorproduct $\phi$ is injective but I think this does not necessarily extend to the completion.)
So my question would be what one can take for $\alpha$ to prove in an efficient way that $\phi$ is an isometry.