The structure of higher-dimensional noncommutative tori and metric diophantine approximation.

Abstract

We use the cut-down method and metric diophantine approximation techniques in order to prove that the set of elements Θ = (e jk ) i^j<k^N with the property that the noncommutative torus Α Θ is an inductive limit of direct sums of 2 N~1 circle algebras and the symmetrized noncommutative torus Α σ Θ is an AF-algebra is the complement of a first category set in [0,1) 2" . As a consequence, almost all noncommutative tori are classified by their ordered K-theory. IntroductionLet Θ = ( /fcJi^fcgjv be an antisymmetric matrix with real entries. The noncommutative torus Α Θ is the universal C*-algebra generated by 7V unitaries U 19 ...,U N subject to the commutation relations U j U k = βχρ(2π/6^Λ) U k U J9 l ^y, k ^ N. Throughout this paper we shall simply denote Θ = (e jk ) 1Â s shown by Connes ([12], [13], see also [10] and [31]), noncommutative tori provide natural examples of noncommutative differential manifolds, with differential structure induced by the ergodic action (z l9 ..., Z N ) Uj = z j U j of J N on Α θ . This class of C*algebras arises in many interesting ways. (Diesen, Pedersen and Takesaki proved ([32]) that noncommutative JV-tori are the only C*-algebras on which J N can act ergodically. They provide a natural environment for the study of almost periodic Mathieu operators and their multidimensional analogs [2], because the integrated density of states (or equivalently the trace on the C*-algebra Α θ ) describes the spectrum of these operators. On the other hand, by a result of Poguntke ([36]), the unique simple subquotient by a primitive ideal of the f ll C*-algebra associated with a connected Lie group is either the algebra JT of compact operators or 3f® Α θ , for some simple noncommutative torus Α θ . l ) EPSRC Advanced Fellow. Brought to you by | University of Arizona Authenticated Download Date | 6/8/15 12:19 PM Combining this with Elliotfs classification result for inductive limits of circle algebras ([18]), respectively with the results of Elliott and R0rdam on the automorphisms of that class of algebras ([23]), we get the following corollaries:Corollary A. "Almost all" noncommutative tori are classified up to isomorphism by their ordered K-theory (or equivalently by the r nge of the trace on Κ 0 (Α Θ )).

<inline-formula><tex-math id="M1">$ C^* $</tex-math></inline-formula>-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms

“…In [6], G. A. Elliott ( cf. [3], [13], [21], [24], etc.) initiated to study higher-dimensional non-commutative tori.…”