Hilbert modules over a class of semicrossed products
HTML articles powered by AMS MathViewer
- by Dale R. Buske PDF
- Proc. Amer. Math. Soc. 129 (2001), 1721-1726 Request permission
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.References
- William B. Arveson, Subalgebras of $C^{\ast }$-algebras, Acta Math. 123 (1969), 141–224. MR 253059, DOI 10.1007/BF02392388
- Dale R. Buske, Hilbert Modules over Semicrossed Products of the Disk Algebra, Dissertation, Iowa State University, Ames, 1997.
- Dale R. Buske and Justin R. Peters, Semicrossed Products of the Disk Algebra: Contractive Representations and Maximal Ideals, Pacific J. of Math., Vol. iii, No. i, (1998), 55-71.
- Ronald G. Douglas and Vern I. Paulsen, Hilbert modules over function algebras, Pitman Research Notes in Mathematics Series, vol. 217, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR 1028546
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
- Tom Hoover, Justin Peters, and Warren Wogen, Spectral properties of semicrossed products, Houston J. Math. 19 (1993), no. 4, 649–660. MR 1251615
- Paul S. Muhly and Baruch Solel, Hilbert modules over operator algebras, Mem. Amer. Math. Soc. 117 (1995), no. 559, viii+53. MR 1271693, DOI 10.1090/memo/0559
- Eric Nordgren, Peter Rosenthal, and F. S. Wintrobe, Invertible composition operators on $H^p$, J. Funct. Anal. 73 (1987), no. 2, 324–344. MR 899654, DOI 10.1016/0022-1236(87)90071-1
- Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
- S. C. Power, Completely contractive representations for some doubly generated antisymmetric operator algebras, Proc. Amer. Math. Soc. 126 (1998), no. 8, 2355–2359. MR 1451827, DOI 10.1090/S0002-9939-98-04358-5
- I. Suciu, On the semi-groups of isometries, Studia Math. 30 (1968), 101–110. MR 229093, DOI 10.4064/sm-30-1-101-110
Additional Information
- Dale R. Buske
- Affiliation: Department of Mathematics, St. Cloud State University, St. Cloud, Minnesota 56301
- Email: dbuske@stcloudstate.edu
- 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
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1814102