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Hilbert modules over a class of semicrossed products


Author: Dale R. Buske
Journal: Proc. Amer. Math. Soc. 129 (2001), 1721-1726
MSC (2000): Primary 47H20, 46M18; Secondary 47A15, 47A45, 47A56
DOI: https://doi.org/10.1090/S0002-9939-00-05691-4
Published electronically: November 2, 2000
MathSciNet review: 1814102
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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.


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Additional Information

Dale R. Buske
Affiliation: Department of Mathematics, St. Cloud State University, St. Cloud, Minnesota 56301
Email: dbuske@stcloudstate.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05691-4
Keywords: Commutant lifting, Hilbert modules, Shilov modules, orthogonally projective modules
Received by editor(s): June 23, 1998
Received by editor(s) in revised form: September 17, 1999
Published electronically: November 2, 2000
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

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