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On selfadjoint extensions of semigroups of partial isometries


Authors: Janez Bernik, Laurent W. Marcoux, Alexey I. Popov and Heydar Radjavi
Journal: Trans. Amer. Math. Soc. 368 (2016), 7681-7702
MSC (2010): Primary 47D03; Secondary 47A65, 47B40, 20M20, 46L10
DOI: https://doi.org/10.1090/tran/6619
Published electronically: February 25, 2016
MathSciNet review: 3546779
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Abstract: Let $ \mathcal {S}$ be a semigroup of partial isometries acting on a complex, infinite-dimensional, separable Hilbert space. In this paper we seek criteria which will guarantee that the selfadjoint semigroup $ \mathcal {T}$ generated by $ \mathcal {S}$ consists of partial isometries as well. Amongst other things, we show that this is the case when the set $ \mathcal {Q}(\mathcal {S})$ of final projections of elements of $ \mathcal {S}$ generates an abelian von Neumann algebra of uniform finite multiplicity.


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Janez Bernik
Affiliation: Fakulteta za Matematiko in Fiziko, Univerza v Ljubljani, Jadranska 19, 1000 Ljubljana, Slovenia
Email: janez.bernik@fmf.uni-lj.si

Laurent W. Marcoux
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: LWMarcoux@uwaterloo.ca

Alexey I. Popov
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Address at time of publication: Department of Mathematics & Computer Science, University of Lethbridge, Lethbridge, Alberta T1K 3M4, Canada
Email: alexey.popov@uleth.ca

Heydar Radjavi
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: hradjavi@uwaterloo.ca

DOI: https://doi.org/10.1090/tran/6619
Keywords: Partial isometry, semigroup, selfadjoint, abelian von Neumann algebra, multiplicity
Received by editor(s): September 13, 2013
Received by editor(s) in revised form: October 19, 2014
Published electronically: February 25, 2016
Additional Notes: The research of the first author was supported in part by ARRS (Slovenia)
The research of the second, third, and fourth authors was supported in part by NSERC (Canada)
Article copyright: © Copyright 2016 American Mathematical Society