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Cuntz Algebra States Defined by Implementers of Endomorphisms of the CAR Algebra

Published online by Cambridge University Press:  20 November 2018

Michael J. Gabriel
Michael J. Gabriel*
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, Ontario, K1N 6N5, email: mamjg@swansea.ac.uk
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Abstract

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We investigate representations of the Cuntz algebra ${{\mathcal{O}}_{2}}$ on antisymmetric Fock space ${{F}_{a}}({{\mathcal{K}}_{1}})$ defined by isometric implementers of certain quasi-free endomorphisms of the CAR algebra in pure quasi-free states $\varphi {{P}_{1}}$. We pay special attention to the vector states on ${{\mathcal{O}}_{2}}$ corresponding to these representations and the Fock vacuum, for which we obtain explicit formulae. Restricting these states to the gauge-invariant subalgebra ${{\mathcal{F}}_{2}}$, we find that for natural choices of implementers, they are again pure quasi-free and are, in fact, essentially the states $\varphi {{P}_{1}}$. We proceed to consider the case for an arbitrary pair of implementers, and deduce that these Cuntz algebra representations are irreducible, as are their restrictions to ${{\mathcal{F}}_{2}}$.

The endomorphisms of $B\left( {{F}_{a}}({{\mathcal{K}}_{1}}) \right)$ associated with these representations of ${{\mathcal{O}}_{2}}$ are also considered.

Keywords

46L0546L30
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 4 , 01 August 2002 , pp. 694 - 708
Copyright
Copyright © Canadian Mathematical Society 2002

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