Vol. 111, No. 2, 1984

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Representations and automorphisms of the irrational rotation algebra

Berndt Brenken

Vol. 111 (1984), No. 2, 257–282

Given an irrational number α, Aα is the unique C-algebra generated by two unitary operators, U and V , satisfying the twisted commutation relation UV = exp(2πiα)V U. We investigate separable representations of Aα 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-invariant Borel measure on the circle (w.r.t. rotation by the angle 2πα) 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 mon the C algebra generated by U, and if m, mare relatively prime, the representation is irreducible. By use of an action of SL(2,Z) as *-automorphisms of Aα, that we construct, we arrive at a separating family of pure states of Aα whose corresponding irreducible representations provide explicit examples with m and moccurring as any given pair of nonzero relatively prime numbers.

Mathematical Subject Classification 2000
Primary: 46L55
Secondary: 22D30
Received: 9 July 1982
Revised: 30 November 1982
Published: 1 April 1984
Berndt Brenken