  In this paper we consider a fractional analogue of the Monge-Amp\`ere
operator. Our operator is a concave envelope of fractional linear operators of
the form $ \inf_{A\in \mathcal{A}}L_Au, $ where the set of operators
corresponds to all affine transformations of determinant one of a given
multiple of the fractional Laplacian. We set up a relatively simple framework
of global solutions prescribing data at infinity and global barriers. In our
key estimate, we show that the operator remains strictly elliptic, which allows
to apply known regularity results for uniformly elliptic operators and deduce
that solutions are classical.
