For 0 < s < 1, let phi_s(z)=sz+(1-s). We investigate the unital C*-algebra
generated by the semigroup {C_{phi_s} : 0 < s < 1} of composition operators
acting on the Hardy space of the unit disk. We determine the joint approximate
point spectrum of a related collection of operators and show that the quotient
of the C*-algebra by its commutator ideal is isomorphic to the direct sum of
the complex numbers and the algebra of almost periodic functions on the real
line. In addition, we show that the C*-algebra is irreducible.