Semicrossed products of the disk algebra: Contractive representations and maximal ideals

Abstract

Given the disk algebra A(D) and an automorphism α, there is associated a non-self-adjoint operator algebra Z ×α A(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 ×α A(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 ×α A(D) are then computed. In the last section, we compare the strong radical to the Jacobson radical.