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Wold decomposition on odometer semigroups

Part of: General theory of linear operators Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  13 July 2021

Boyu Li Open the ORCID record for Boyu Li [Opens in a new window]
Boyu Li*
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., Canada [email protected]
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Abstract

We establish a Wold-type decomposition for isometric and isometric Nica-covariant representations of the odometer semigroup. These generalize the Wold-type decomposition for commuting pairs of isometries due to Popovici and for pairs of doubly commuting isometries due to Słociński.

Keywords

Wold decompositionodometer semigroup

MSC classification

Primary: 47A13: Several-variable operator theory (spectral, Fredholm, etc.)
Secondary: 47A45: Canonical models for contractions and nonselfadjoint operators 47D03: Groups and semigroups of linear operators
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 3 , June 2022 , pp. 738 - 755
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

an Huef, A., Raeburn, I. and Tolich, I.. HNN extensions of quasi-lattice ordered groups and their operator algebras. Doc. Math. 23 (2018), 327–351.Google Scholar
Bratteli, O. and Jorgensen, P. E. T.. Iterated function systems and permutation representations of the Cuntz algebra. Mem. Amer. Math. Soc. 139 (1999), :x+89.Google Scholar
Brownlowe, N., Ramagge, J., Robertson, D. and Whittaker, M. F.. Zappa-Szép products of semigroups and their $C^{\ast }$-algebras. J. Funct. Anal. 266 (2014), 3937–3967.10.1016/j.jfa.2013.12.025CrossRefGoogle Scholar
Coburn, L. A.. The $C^{\ast}$-algebra generated by an isometry. Bull. Amer. Math. Soc. 73 (1967), 722–726.10.1090/S0002-9904-1967-11845-7CrossRefGoogle Scholar
Davidson, K. R., Dor-On, A. and Li, B.. Structure of free semigroupoid algebras. J. Funct. Anal. 277 (2019), 3283–3350.10.1016/j.jfa.2019.06.002CrossRefGoogle Scholar
Davidson, K. R., Katsoulis, E. and Pitts, D. R.. The structure of free semigroup algebras. J. Reine Angew. Math. 533 (2001), 99–125.Google Scholar
Davidson, K. R. and Pitts, D. R.. Invariant subspaces and hyper-reflexivity for free semigroup algebras. Proc. London Math. Soc. (3) 78 (1999), 401–430.10.1112/S002461159900180XCrossRefGoogle Scholar
Davidson, K. R., Power, S. C. and Yang, D.. Atomic representations of rank 2 graph algebras. J. Funct. Anal. 255 (2008), 819–853.10.1016/j.jfa.2008.05.008CrossRefGoogle Scholar
Nica, A.. $C^{\ast}$-algebras generated by isometries and Wiener-Hopf operators. J. Operator Theory 27 (1992), 17–52.Google Scholar
Popescu, G.. Isometric dilations for infinite sequences of noncommuting operators. Trans. Amer. Math. Soc. 316 (1989), 523–536.10.1090/S0002-9947-1989-0972704-3CrossRefGoogle Scholar
Popescu, G.. Noncommutative Wold decompositions for semigroups of isometries. Indiana Univ. Math. J. 47 (1998), 277–296.10.1512/iumj.1998.47.1509CrossRefGoogle Scholar
Popovici, D.. AWold-type decomposition for commuting isometric pairs. Proc. Amer. Math. Soc. 132 (2004), 2303–2314.10.1090/S0002-9939-04-07331-9CrossRefGoogle Scholar
Sarkar, J.. Wold decomposition for doubly commuting isometries. Linear Algebra Appl. 445 (2014), 289–301.10.1016/j.laa.2013.12.011CrossRefGoogle Scholar
Skalski, A. and Zacharias, J.. Wold decomposition for representations of product systems of $C^{*}$-correspondences. Int. J. Math. 19 (2008), 455–479.10.1142/S0129167X08004765CrossRefGoogle Scholar
Słociński, M.. On the Wold-type decomposition of a pair of commuting isometries. Ann. Polon. Math. 37 (1980), 255–262.10.4064/ap-37-3-255-262CrossRefGoogle Scholar
Spielberg, J.. $C^{\ast }$-algebras for categories of paths associated to the Baumslag-Solitar groups. J. Lond. Math. Soc. (2) 86 (2012), 728–754.10.1112/jlms/jds025CrossRefGoogle Scholar
Suciu, I.. On the semi-groups of isometries. Studia Math. 30 (1968), 101–110.10.4064/sm-30-1-101-110CrossRefGoogle Scholar
Sz.-Nagy, B., Foias, C., Bercovici, H. and Kérchy, L.. Harmonic analysis of operators on Hilbert space (New York, second edition: Universitext. Springer, 2010).10.1007/978-1-4419-6094-8CrossRefGoogle Scholar