1988/04/01 by Marc A. Rieffel · 29 citations
Mathematics · #Advanced Operator Algebra Research #Holomorphic and Operator Theory #Advanced Topics in Algebra
paper · pdf · doi:10.4153/cjm-1988-012-9
The non-commutative tori provide probably the most accessible interesting examples of non-commutative differentiable manifolds. We can identify an ordinary n -torus T n with its algebra, C(T n ) , of continuous complex-valued functions under pointwise multiplication. But C(T n ) is the universal C *-algebra generated by n commuting unitary operators. By definition, [ 15, 16, 50 ], a non-commutative n -torus is the universal C *-algebra generated by n unitary operators which, while they need not commute, have as multiplicative commutators various fixed scalar multiples of the identity operator. As Connes has shown [ 8, 10 ], these algebras have a natural differentiable structure, defined by a natural ergodic action of T n as a group of automorphisms. The non-commutative tori behave in inany ways like ordinary tori. For instance, it is an almost immediate consequence of the work of Pimsner and Voiculescu [ 37 ] that the K -groups of a non-commutative torus are the same as those of an ordinary torus of the same dimension. (In particular, non-commutative tori are KK -equivalent to ordinary tori by Corollary 7.5 of [ 52 ].) Furthermore, the structure constants of non-commutative tori can be continuously deformed into those for ordinary tori. (This is exploited in [ 17 ].)