Representations and automorphisms of the irrational rotation algebra

Abstract

Given an irrational number «, Aa is the unique C*-algebra generated by two unitary operators, U and V, satisfying the twisted commutation relation UV= exp(2 πia)VU. We investigate separable representations of Aa which, when restricted to the abelian C* algebra generated by V, are of uniform multiplicity m. These representations are classified by their multiplicity, a quasi-invarian t Borel measure on the circle (w.r.t. rotation by the angle lira) and a unitary one cocycle. Separable factor representations lie in this class, the measure being ergodic in this case. A factor representation is of uniform multiplicity mf on the C* algebra generated by £/, and if m, m' are relatively prime, the representation is irreducible. By use of an action of SL(2, Z) as *-automorphisms of Aa9 that we construct, we arrive at a separating family of pure states of Aa whose corresponding irreducible representations provide explicit examples with m and m' occurring as any given pair of nonzero relatively prime numbers.