The Entire Cyclic Cohomology of Noncommutative 2-Tori

Our aim in this paper is to compute the entire cyclic cohomology of noncommutative 2-tori. First of all, we clarify their algebraic structure of noncommutative 2-tori as a $F^$-algebra, according to the idea of Elliott-Evans. Actually, they are the inductive limit of subhomogeneous $F^$-algebras. Using such a result, we compute their entire cyclic cohomology, which is isomorphic to their periodic one as a complex vector space.

💡 Research Summary

The paper addresses the computation of the entire cyclic cohomology (also called entire cyclic cohomology) of the non‑commutative two‑torus (A_{\theta}). The authors begin by recalling that the smooth subalgebra (A_{\theta}^{\infty}) of the (C^{})-torus is a Fréchet ‑algebra (an (F^{})-algebra) and that this algebra can be described as an inductive limit of sub‑homogeneous (F^{})-algebras. This description follows the Elliott‑Evans approach originally developed for the classification of irrational rotation algebras.

For rational (\theta = p/q) the smooth torus is exactly isomorphic to a matrix algebra over the smooth functions on the ordinary 2‑torus, \