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Decomposability of multiparameter car flows

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  31 March 2023

Anbu Arjunan
Anbu Arjunan*
Affiliation:
Indian Statistical Institute, Delhi Centre, Delhi 110016, India (aanbu100@gmail.com)
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Abstract

Let P be a closed convex cone in $\mathbb{R}^d$ which is assumed to be spanning $\mathbb{R}^d$ and contains no line. In this article, we consider a family of CAR flows over P and study the decomposability of the associated product systems. We establish a necessary and sufficient condition for CAR flow to be decomposable. As a consequence, we show that there are uncountable many CAR flows which are cocycle conjugate to the corresponding CCR flows.

Keywords

E0-semigroupsCCR flowsCAR flowsadditive decomposable vectorsquantum stochastic equation

MSC classification

Primary: 46L55: Noncommutative dynamical systems
Secondary: 46L99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 1 , February 2023 , pp. 1 - 22
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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References

Anbu, A., Remarks on CCR and CAR flows over closed convex cones, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 24 (2021), 1–18.Google Scholar
Anbu, A. and Sundar, S., CCR flows associated to closed convex cones, Münster J. Math. 13 (2020), 115–143.Google Scholar
Anbu, A., Srinivasan, R. and Sundar, S., E-semigroups over closed convex cones, J. Operator Theory 84 (2020), 289–322.Google Scholar
Arveson, W., Noncommutative dynamics and E-semigroups (Springer, 2003).10.1007/978-0-387-21524-2CrossRefGoogle Scholar
Bhat, B. V. R., Lindsay, J. M. and Mukherjee, M., Additive units of product systems, Trans. Amer. Math. Soc. 370 (2018), 2605–2637.10.1090/tran/7092CrossRefGoogle Scholar
Margetts, O. T. and Srinivasan, R., Invariants for E 0-semigroups on II 1-factors, Comm. Math. Phys. 323 (2013), 1155–1184.10.1007/s00220-013-1790-2CrossRefGoogle Scholar
Murugan, S. P. and Sundar, S., $E_{0}^{P}$-semigroups and product systems, arXiv:1706.03928, 2017.Google Scholar
Powers, R. T., An index theory for semigroups of $*$-endomorphisms of B(H) and type II 1 factors, Canad. J. Math. 40 (1988), 86–114.10.4153/CJM-1988-004-3CrossRefGoogle Scholar
Powers, R. T. and Robinson, D. W., An index for continuous semigroups of $*$-endomorphisms of ${\mathfrak B}({\mathfrak H})$, J. Funct. Anal. 84 (1989), 85–96.10.1016/0022-1236(89)90111-0CrossRefGoogle Scholar
Srinivasan, R., CCR and CAR flows over convex cones, arXiv:1908.00188, 2019.Google Scholar
Stoll, M., Introduction to real analysis (Addison-Wesley Longman, 2001).Google Scholar
Sundar, S., Arveson’s characterisation of CCR flows: the multiparameter case, J. Funct. Anal. 280 (2021), 1–44.10.1016/j.jfa.2020.108802CrossRefGoogle Scholar