Hilbert modules over a class of semicrossed products

Abstract:

Given the disk algebra $\mathcal {A}(\mathbb {D})$ and an automorphism $\alpha$, there is associated a non-self-adjoint operator algebra $\mathbb {Z}^+\times _{\alpha }\mathcal {A}(\mathbb D)$ called the semicrossed product of $\mathcal {A}(\mathbb {D})$ with $\alpha$. Buske and Peters showed that there is a one-to-one correspondence between the contractive Hilbert modules $\mathcal {H}$ over $\mathbb {Z}^+\times _{\alpha }\mathcal {A}(\mathbb D)$ and pairs of contractions $S$ and $T$ on $\mathcal {H}$ satisfying $TS=S\alpha (T)$. In this paper, we show that the orthogonally projective and Shilov Hilbert modules $\mathcal {H}$ over $\mathbb {Z}^+\times _{\alpha }\mathcal {A}(\mathbb D)$ correspond to pairs of isometries on $\mathcal {H}$ satisfying $TS=S\alpha (T)$. The problem of commutant lifting for $\mathbb {Z}^+\times _{\alpha }\mathcal {A}(\mathbb D)$ is left open, but some related results are presented.