---
abstract:
- 'We extend previous results on the exponential off-diagonal decay of the entries of analytic functions of banded and sparse matrices to the case where the matrix entries are elements of a $C^*$-algebra.'
- 'We extend previous results on the exponential off-diagonal decay of the entries of analytic functions of banded and sparse matrices to the case where the matrix entries are elements of a $C^*$-algebra.'
address:
- 'Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA. '
- 'Equipe Calcul Formel, DMI-XLIM UMR 7252 Université de Limoges - CNRS, 123 avenue Albert Thomas, 87060 Limoges Cedex, France.'
author:
- Michele Benzi
- Paola Boito
title: 'Decay properties for functions of matrices over $C^*$-algebras'
---

matrix functions, $C^*$-algebras, exponential decay, sparse matrices, graphs, functional calculus

Introduction
============

Decay properties of inverses, exponentials and other functions of band or sparse matrices over $\mathbb{R}$ or $\mathbb{C}$ have been investigated by several authors in recent years [@BG99; @BR07; @DMS84; @highambook; @iserles; @jaffard; @KSW; @mastronardi]. Such properties play an important role in various applications including electronic structure computations in quantum chemistry [@BBR13; @BM12], quantum information theory [@CE06; @CEPD06; @ECP10; @SCW06], high-dimensional statistics [@Aune], random matrix theory [@Molinari] and numerical 