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 $\mathfrak A$ considered separately by F. Boca and D. Mundici. The center of $\mathfrak A$ is isomorphic to $C[0,1]$, so we first consider the action of the Gauss map on $C[0,1]$ and then extend the map to $\mathfrak A$ and show that the extension inherits many desirable properties.