Crossed Product C*-Algebras Associated wth Homeomorphisms of the Unit Circle

Given a homeomorphism ϕ of the unit circle T, there is associated the C*-crossed product algebra of C(T) by the Z-action induced by ϕ. We analyze the C*-crossed product algebras arising from those homeomorphisms that possess both cyclic and non-cyclic points, or are order-reversing. It is shown that upto compact perturbations, infinite-dimensional irreducible *-representations of algebras in this class can be expressed in terms of certain finite-dimensional irreducible *-representations. This relationship is used to recover the topological features of the underlying homeomorphisms upto inversion, and to construct a complete *-isomorphism invariant for these algebras. It is shown that two algebras with underlying homeomorphisms ϕ and ψ are *-isomorphic if and only if ϕ is conjugate to ψ or ψ ⁻¹.