Consider a collection of $C^*$-algebras $\{A_i\}_{i \in I}$. We can form the direct sum $$\bigoplus_{i \in I}^{c_0} A_i:= \left\{(a_i)_{i \in I} \in \prod_{i\in I} A_i: \lim_{i \in I} \|a_i\| = 0\right\}$$ which is an ideal in the $C^*$-algebra $$\bigoplus_{i \in I}^{\ell^\infty} A_i:= \left\{(a_i)_{i \in I} \in \prod_{i\in I} A_i: \sup_{i \in I} \|a_i\| <\infty\right\}.$$ Given a $C^*$-algebra $A$, denote its multiplier algebra by $M(A)$. One possible realisation of the multiplier algebra is by setting $M(A):= \mathcal{L}_A(A)$, the adjointable operators when we view $A$ as a (right) Hilbert $C^*$-module over itself.
I believe I have proven that $$M\left(\bigoplus_{i \in I}^{c_0}A_i\right) \cong \bigoplus_{i \in I}^{\ell^\infty} M(A_i)\cong M\left(\bigoplus_{i \in I}^{\ell^\infty}A_i\right)$$ but I can't find a reference for this statement. So, is my assertion true?
Here is a proof sketch: We use the implementation of the multiplier $C^*$-algebra as adjointable operators. We then have natural maps $$M\left(\bigoplus_{i \in I}^{c_0}A_i\right) \to \bigoplus_{i \in I}^{\ell^\infty} M(A_i): t \mapsto (\iota_i ^* t\iota_i)_{i \in I}$$ where $\iota_i: A_i \hookrightarrow \bigoplus_{i \in I}^{c_0} A_i$ is the inclusion map and $$\bigoplus_{i \in I}^{\ell^\infty} M(A_i) \to M\left(\bigoplus_{i \in I}^{c_0}A_i\right) : (t_i)_{i \in I} \mapsto [(a_i)_i \mapsto (t_i(a_i))]$$ These are easily checked to be $*$-isomorphisms that are inverse to each other, and this establishes the isomorphism $M\left(\bigoplus_{i \in I}^{c_0}A_i\right) \cong \bigoplus_{i \in I}^{\ell^\infty} M(A_i)$. The other isomorphism is shown similarly.