Summary: The aim of this paper is to transfer the Gauss map, which is a Bernoulli shift for continued fractions, to the noncommutative setting. We feel that a natural place for such a map to act is on the AF algebra \(\mathfrak U\) considered separately by F. P. Boca [Can. J. Math. 60, No. 5, 975–1000 (2008; Zbl 1158.46039)] and D. Mundici [“Revisiting the Farey AF Algebra”, Milan J. Math. 79, No. 2, 643–656 (2011), doi:10.1007/s00032-011-0166-3]. The center of \(\mathfrak U\) is isomorphic to \(C[0, 1]\); therefore, we first consider the action of the Gauss map on \(C[0, 1]\) and extend then the map to \(\mathfrak U\) and show that the extension inherits many desirable properties.