Given the disk algebra 𝒜(𝔻)
and an automorphism α, there is associated a non-self-adjoint operator algebra
ℤ+×α𝒜(𝔻) called the semicrossed product of 𝒜(𝔻) with α. We consider those
algebras where the automorphism arises via composition with parabolic, hyperbolic,
and elliptic conformal maps φ of 𝔻 onto itself. To characterize the contractive
representations of ℤ+×α𝒜(𝔻), a noncommutative dilation result is obtained. The
result states that given a pair of contractions S,T on some Hilbert space ℋ
which satisfy TS = Sφ(T), there exist unitaries U,V on some Hilbert space
𝒦⊃ℋ which dilate S and T respectively and satisfy V U = Uφ(V ). It is then
shown that there is a one-to-one correspondence between the contractive
(and completely contractive) representations of ℤ+×α𝒜(𝔻) on a Hilbert
space ℋ and pairs of contractions S and T on ℋ satisfying TS = Sφ(T).
The characters, maximal ideals, and strong radical of ℤ+×α𝒜(𝔻) are then
computed. In the last section, we compare the strong radical to the Jacobson
radical.
