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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 m′ on the C∗ algebra
generated by U, and if m, m′ are 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 m′ occurring as any given pair
of nonzero relatively prime numbers.
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