Given a smooth manifold $X$; by definition the topological Brauer group $B(X)$ is the group of classes of Azumaya algebras over the sheaf $\mathcal{O}_X$ of $\mathbb{C}$-valued functions on $X$.

If we replace the sheaf $\mathcal{O}_X$ by the sheaf of $C^{\infty}$-functions; will the resulting Brauer group be the same?