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Obstructions for semigroups of partial isometries to be self-adjoint†

Published online by Cambridge University Press:  10 March 2016

JANEZ BERNIK  and
ALEXEY I. POPOV
JANEZ BERNIK
Affiliation:
Fakulteta za Matematiko in Fiziko, Univerza v Ljubljani, Jadranska 19, 1000 Ljubljana, Slovenia. e-mail: janez.bernik@fmf.uni-lj.si
ALEXEY I. POPOV
Affiliation:
Department of Mathematics and Computer Science, Univresity of Lethbridge, C526 University Hall, 4401 University Drive, Lethbridge, Alberta T1K 3M4, Canada. e-mail: alexey.popov@uleth.ca
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Abstract

In this paper we study the following question: given a semigroup ${\mathcal S}$ of partial isometries acting on a complex separable Hilbert space, when does the selfadjoint semigroup ${\mathcal T}$ generated by ${\mathcal S}$ again consist of partial isometries? It has been shown by Bernik, Marcoux, Popov and Radjavi that the answer is positive if the von Neumann algebra generated by the initial and final projections corresponding to the members of ${\mathcal S}$ is abelian and has finite multiplicity. In this paper we study the remaining case of when this von Neumann algebra has infinite multiplicity and show that, in a sense, the answer in this case is generically negative.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 1 , July 2016 , pp. 107 - 116
Copyright
Copyright © Cambridge Philosophical Society 2016 

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Footnotes

†

Research supported in part by ARRS (Slovenia)

References

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