  Let $\mathcal{A}$ be a unital complex semisimple Banach algebra, and
$M_{\mathcal{A}}$ denote its maximal ideal space. For a matrix $M\in
{\mathcal{A}}^{n\times n}$, $\widehat{M}$ denotes the matrix obtained by taking
entry-wise Gelfand transforms. For a matrix $M\in {\mathbb{C}}^{n\times n}$,
$\sigma(M)\subset \mathbb{C}$ denotes the set of eigenvalues of $M$. It is
shown that if $A\in {\mathcal{A}}^{n\times n}$ and $B\in {\mathcal{A}}^{m\times
m}$ are such that for all $\varphi \in M_{\mathcal{A}}$,
$\sigma(\widehat{A}(\varphi))\cap \sigma(\widehat{B}(\varphi))=\emptyset$, then
for all $C\in {\mathcal{A}}^{n\times m}$, the Sylvester equation $AX-XB=C$ has
a unique solution $X\in {\mathcal{A}}^{n\times m}$. As an application, Roth's
removal rule is proved in the context of matrices over a Banach algebra.
