A Noncommutative Gauss map

TL;DR

This paper extends the classical Gauss map, which relates to continued fractions, into a noncommutative algebraic setting, specifically on AF algebras, preserving key properties of the original map.

Contribution

The paper introduces a noncommutative version of the Gauss map on AF algebras, bridging classical dynamical systems with noncommutative operator algebra theory.

Findings

Successfully extended the Gauss map to AF algebra $rak{A}$

Preserved many properties of the classical Gauss map in the noncommutative setting

Connected the map's action on $C[0,1]$ to its extension on $rak{A}$

Abstract

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 $\ga$ 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.