an $n\times n$ matrix with entries $a_{ij} = a_{ji}^*$ in a $C^*$-algebra $\mathcal{A}_0$ and $f$ is an analytic function defined on a suitable open subset of $\mathbb{C}$ containing the spectrum of $A$, viewed as an element of the $C^*$-algebra $\mathcal{M}_n(\mathcal{A}_0)$ ($=\mathcal{A}_0^{n\times n}$). The interesting case is when the constants in the bounds do not depend on $n$. Functions of more general sparse matrices over $\mathcal{A}_0$ will also be discussed.

For the case of functions of $n\times n$ quaternion matrices, we identify the set of such matrices with a (real) subalgebra of $\mathbb{C}^{2n\times 2n}$ and treat them as a special type of complex block matrices; as we will see, this will impose some restrictions on the type of functions that we are allowed to consider.

Background on $C^*$-algebras
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In this section we provide definitions and notations used throughout the remainder of the paper, and recall some of the fundamental results from the theory of $C^*$-algebras. The reader is referred to [@MC; @Rudin] for concise introductions to this theory and to [@KRI] for a more systematic treatment.

Recall that a [*Banach algebra*]{} is a complex algebra $\mathcal{A}_0$ with a norm making $\mathcal{A}_0$ into a Banach space and satisfying $$\|ab\| \le \|a\|\|b\|$$ 