analysis [@BC13; @ye13], to name a few.

Answering a question posed by P.-L. Giscard and coworkers [@giscard], we consider generalizations of existing decay estimates to functions of matrices with entries in more general algebraic structures than the familiar fields $\mathbb{R}$ or $\mathbb{C}$. In particular, we propose extensions to functions of matrices with entries from the following algebras:

1.  Commutative algebras of complex-valued continuous functions;

2.  Non-commutative algebras of bounded operators on a complex Hilbert space;

3.  The real division algebra $\mathbb{H}$ of quaternions.

The theory of complex $C^*$-algebras provides the natural abstract setting for the desired extensions [@KRI; @MC; @Rudin]. Matrices over such algebras arise naturally in various application areas, including parametrized linear systems and eigenproblems [@CGI10; @Tobler], differential equations [@CT06], generalized moment problems [@Osipov], control theory [@Bunce; @Curtain1; @Curtain2], and quantum physics [@Berezansky71; @Berezansky98; @Doran]. The study of matrices over $C^*$-algebras is also of independent mathematical interest; see, e.g., [@Grove; @Kadison].

Using the holomorphic functional calculus, we establish exponential off-diagonal decay results for analytic functions of banded $n\times n$ Hermitian matrices over $C^*$-algebras, both commutative and non-commutative. Our decay estimates are expressed in the form of computable bounds on the norms of the entries of $f(A)$ where $A=[a_{ij}]$ is 