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Semicrossed Products of the Disk Algebra and the Jacobson Radical

Published online by Cambridge University Press:  20 November 2018

Anchalee Khemphet  and
Justin R. Peters
Anchalee Khemphet
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa, USA e-mail: khemphet@iastate.edu
Justin R. Peters
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa, USA e-mail: peters@iastate.edu
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Abstract

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We consider semicrossed products of the disk algebra with respect to endomorphisms defined by finite Blaschke products. We characterize the Jacobson radical of these operator algebras. Furthermore, in the case that the finite Blaschke product is elliptic, we show that the semicrossed product contains no nonzero quasinilpotent elements. However, if the finite Blaschke product is hyperbolic or parabolic with positive hyperbolic step, the Jacobson radical is nonzero and a proper subset of the set of quasinilpotent elements.

Keywords

47L6547L2030J1030H50semicrossed productdisk algebraJacobson radical
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 1 , 14 March 2014 , pp. 80 - 89
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Basallote, M., Contreras, M. D., and Hernández-Mancera, C., Commuting finite Blaschke productswith no fixed points in the unit disk. J. Math. Anal. Appl. 359 (2009), 547–555. http://dx.doi.org/10.1016/j.jmaa.2009.06.010 Google Scholar
[2] Buske, D. and Peters, J., Semicrossed products of the disk algebra; contractive representations andmaximal ideals. Pacific J. Math. 185 (1998), 97–113. http://dx.doi.org/10.2140/pjm.1998.185.97 Google Scholar
[3] Carleson, L. and Gamelin, T.W., Complex Dynamics. Springer-Verlag, New York, 1993.Google Scholar
[4] Contreras, M. D., Dĭaz-Madrigal, S., and Pommerenke, Ch., Iteration in the unit disk: the paraboliczoo. In: Complex and harmonic analysis, DEStech Publ., Inc., Lancaster, PA, 2007, 63–91.Google Scholar
[5] Dales, H. G., Aiena, P., Eschmeier, J., Laursen, K., and Willis, G. A., Introduction to Banach Algebras,Operators, and Harmonic Analysis. London Math. Soc. Stud. Texts 57, Cambridge University Press, Cambridge, UK, 2003.Google Scholar
[6] Davidson, K. A. and Katsoulis, E., Dilating covariant representations of the noncommutative discalgebras. J. Funct. Anal. 259 (2010), 817–831. http://dx.doi.org/10.1016/j.jfa.2010.04.005 Google Scholar
[7] Davidson, K. A., Dilating theory, commutant lifting and semicrossed products. Documenta Math. 16 (2011), 781–868.Google Scholar
[8] Davidson, K. A., Semicrossed products of the disk algebra. Preprint, arxiv:1104.1398v1, 2011.Google Scholar
[9] Davidson, K. A., Semicrossed products of simple C*-algebras. Math. Ann. 342 (2008), 515–525. http://dx.doi.org/10.1007/s00208-008-0244-1 Google Scholar
[10] Donsig, A. P., Katavolos, A., and Manoussos, A., The Jacobson radical for analytic crossed products. J. Funct. Anal. 187 (2001), 129–145. http://dx.doi.org/10.1006/jfan.2001.3819 Google Scholar
[11] Hamilton, D. H., Absolutely continuous conjugacies of Blaschke products. Adv. Math. 121 (1996), 1–20. http://dx.doi.org/10.1006/aima.1996.0044 Google Scholar
[12] Hoffman, K., Banach Spaces of Analytic Functions. Prentice-Hall, Englewood Cliffs, NJ, 1962.Google Scholar
[13] Hoover, T., Peters, J., and Wogen, W., Spectral properties of semicrossed products. Houston J. Math. 19 (1993), 649–660.Google Scholar
[14] Kakariadis, E. T. A., Semicrossed products of C*-algebras and their C*-envelopes. Preprint, arxiv:1102.2252v2, 2011. Google Scholar
[15] Kakariadis, E. and Katsoulis, E., Semicrossed products of operator algebras and their C*-envelopes. Preprint, arxiv:1008.2374v1, 2010. Google Scholar
[16] Pommerenke, C., Boundary Behaviour of Conformal Maps. Springer-Verlag, Berlin Heidelberg, 1992.Google Scholar
[17] Shapiro, J. H., Composition Operators and Classical Function Theory. Springer-Verlag, New York, 1993.Google Scholar
[18] Walters, P., An Introduction to Ergodic Theory. Springer-Verlag, New York, 1982.Google Scholar