On the Non-commutative Torus of Real Dimension Two

Abstract

The aim of the following notes is to recall how, according to the theory of Alain Connes, one may introduce a differentiable structure on a dense sub-algebra of the irrational rotation C*-algebra introduced in the lectures of Jean Bellissard. This sub-algebra may be viewed as a non-commutative generalisation of the classical torus of real dimension two. We compute the variation of the value at the origin of the meromorphic continuation of the zeta-function of the Laplacian on this algebra as the metric on the non-commutative torus varies within a given conformal class. This manifests the involved intervention into pseudo-differential computations of the non-commutativity, even for the simplest of non-commutative differential objects.