Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] Bruce A. Barnes, Representations of the $ l^{1}$-algebra of an inverse semigroup, Trans. Amer. Math. Soc. 218 (1976), 361-396. MR 0397310 (53 #1169)
  • [2] V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. MR 2267655 (2008g:28002)
  • [3] Joachim Cuntz, Simple $ C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), no. 2, 173-185. MR 0467330 (57 #7189)
  • [4] Joachim Cuntz and Wolfgang Krieger, A class of $ C^{\ast } $-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251-268. MR 561974 (82f:46073a), https://doi.org/10.1007/BF01390048
  • [5] Kenneth R. Davidson, $ C^*$-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012 (97i:46095)
  • [6] Kenneth R. Davidson, Elias Katsoulis, and David R. Pitts, The structure of free semigroup algebras, J. Reine Angew. Math. 533 (2001), 99-125. MR 1823866 (2002a:47107), https://doi.org/10.1515/crll.2001.028
  • [7] J. Duncan and A. L. T. Paterson, $ C^\ast $-algebras of inverse semigroups, Proc. Edinburgh Math. Soc. (2) 28 (1985), no. 1, 41-58. MR 785726 (86h:46090), https://doi.org/10.1017/S0013091500003187
  • [8] P. R. Halmos and L. J. Wallen, Powers of partial isometries, J. Math. Mech. 19 (1969/1970), 657-663. MR 0251574 (40 #4801)
  • [9] John M. Howie, Fundamentals of semigroup theory, London Mathematical Society Monographs. New Series, vol. 12, The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications. MR 1455373 (98e:20059)
  • [10] Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., Orlando, FL, 1986. Advanced theory. MR 859186 (88d:46106)
  • [11] Mario Petrich, Inverse semigroups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 752899 (85k:20001)
  • [12] Gelu Popescu, Non-commutative disc algebras and their representations, Proc. Amer. Math. Soc. 124 (1996), no. 7, 2137-2148. MR 1343719 (96k:47077), https://doi.org/10.1090/S0002-9939-96-03514-9
  • [13] Alexey I. Popov and Heydar Radjavi, Semigroups of partial isometries, Semigroup Forum 87 (2013), no. 3, 663-678. MR 3128716, https://doi.org/10.1007/s00233-013-9487-6
  • [14] G. B. Preston, Representations of inverse semi-groups, J. London Math. Soc. 29 (1954), 411-419. MR 0064038 (16,216a)
  • [15] Iain Raeburn, Graph algebras, CBMS Regional Conference Series in Mathematics, vol. 103, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2005. MR 2135030 (2005k:46141)
  • [16] Charles J. Read, A large weak operator closure for the algebra generated by two isometries, J. Operator Theory 54 (2005), no. 2, 305-316. MR 2186356 (2006g:47119)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 47D03, 47A65, 47B40, 20M20, 46L10

Retrieve articles in all journals with MSC (2010): 47D03, 47A65, 47B40, 20M20, 46L10


Additional Information

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

American Mathematical Society