Hilbert modules over semicrossed products of the disk algebra
Given the disk algebra A( I !D) and an automorphism [alpha], there is associated a non-self-adjoint norm closed subalgebra doubz+x[alpha]A( I !D) of the crossed product doubzx[alpha]C( T) called the semicrossed product of A( I !D) with [alpha]. It is well known that the automorphisms of A( I !D) arise via composition with conformal bijections [phi] of I !D. These automorphisms are labeled according to the corresponding conformal maps as parabolic, hyperbolic, or elliptic and each case is studied. The contractive and completely contractive representations of doubz+x[alpha]A( I !D) on a Hilbert space H (i.e. contractive Hilbert modules) are found to be in a one-to-one correspondence with pairs of contractions S and T on H satisfying TS=S[phi](T). To this end, a noncommutative dilation result is obtained. It states that given a pair of contractions S and T on H satisfying TS=S[phi](T) there exist a pair of unitaries U and V on K supseteqH satisfying VU=U[phi](V) and dilating S and T respectively. Some concrete representations of doubz+x[alpha]A( I !D) are then found in order to compute the characters, the maximal ideal space, and the strong radical. The Shilov and orthoprojective Hilbert modules over doubz+x[alpha]A( I !D) are shown to correspond to pairs of isometries S and T satisfying TS=S[phi](T).