  We study the $C^*$-algebra $\mathbb{E}_{\mathscr{M}}$ of a smooth
4-dimensional manifold $\mathscr{M}$ introduced by G\'abor Etesi. It is proved
that the $\mathbb{E}_{\mathscr{M}}$ is a stationary AF-algebra. We calculate
the topological and smooth invariants of $\mathscr{M}$ in terms of the K-theory
of the $C^*$-algebra $\mathbb{E}_{\mathscr{M}}$. Using Gompf's Stable
Diffeomorphism Theorem, it is shown that all smoothings of $\mathscr{M}$ form a
torsion abelian group. The latter is isomorphic to the Brauer group of a number
field associated to the K-theory of $\mathbb{E}_{\mathscr{M}}$.
