Higher Dimensional Noncommutative Tori

N. Christopher Phillips, University of Oregon

Abstract. A unital C*-algebra can be thought of as a noncommutative version of a compact space (more accurately, as a noncommutative version of the algebra of continuous functions on a compact space). Noncommutative C*-algebras can sometimes be used as substitutes for compact spaces in situations in which no reasonable space exists. One example is the orbit space of a minimal homeomorphism, such as the rotation by an irrational multiple of 2? on the circle. The orbit space itself is an uncountable set with the indiscrete topology. For irrational rotations, the C*-algebras, called irrational rotation algebras, preserve information about the rotation angle, and this information is visible in their algebraic topology.

An irrational rotation algebra can also be obtained as the C*-algebra generated by two unitaries which commute up to a scalar (exp (ir) for rotation by r). It is a simple C*-algebra. Thought of this way, it is a noncommutative deformation of (the continuous functions on) the 2-torus. A higher dimensional noncommutative torus is a C*-algebra obtained as an analogous deformation of a higher dimensional torus. Most of these algebras are simple. Our main result is a structure theorem for the simple case, generalizing the Elliott-Evans theorem for irrational rotation algebras, which allows one to determine exactly when two of them are isomorphic. Surprisingly, actions of finite groups are used in the proof.