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Additive units of product systems

Authors: B. V. Rajarama Bhat, J. Martin Lindsay and Mithun Mukherjee
Journal: Trans. Amer. Math. Soc. 370 (2018), 2605-2637
MSC (2010): Primary 46L55; Secondary 46C05, 46L53
Published electronically: December 1, 2017
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Abstract: We introduce the notion of additive units, or ``addits'', of a pointed Arveson system and demonstrate their usefulness through several applications. By a pointed Arveson system we mean a spatial Arveson system with a fixed normalised reference unit. We show that the addits form a Hilbert space whose codimension-one subspace of ``roots'' is isomorphic to the index space of the Arveson system and that the addits generate the type I part of the Arveson system. Consequently the isomorphism class of the Hilbert space of addits is independent of the reference unit. The addits of a pointed inclusion system are shown to be in natural correspondence with the addits of the generated pointed product system. The theory of amalgamated products is developed using addits and roots, and an explicit formula for the amalgamation of pointed Arveson systems is given, providing a new proof of its independence of the particular reference units. (This independence justifies the terminology ``spatial product'' of spatial Arveson systems.) Finally a cluster construction for inclusion subsystems of an Arveson system is introduced, and we demonstrate its correspondence with the action of the Cantor-Bendixson derivative in the context of the random closed set approach to product systems due to Tsirelson and Liebscher.

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Additional Information

B. V. Rajarama Bhat
Affiliation: Stat-Math Unit, Indian Statistical Institute, R.V. College Post, Bangalore-560059, India

J. Martin Lindsay
Affiliation: Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, United Kingdom

Mithun Mukherjee
Affiliation: School of Mathematics, IISER Thiruvananthapuram, CET Campus, Kerala - 695016, India

Keywords: Arveson systems, inclusion systems, quantum dynamics, completely positive semigroups, Cantor--Bendixson derivative, cluster construction
Received by editor(s): November 4, 2014
Received by editor(s) in revised form: July 29, 2016
Published electronically: December 1, 2017
Additional Notes: This work was supported by the UK-India Education and Research Initiative (UKIERI), under the research collaboration grant Quantum Probability, Noncommutative Geometry & Quantum Information.
The third-named author was also suported by DST-Inspire Fellowship IFA-13 MA 20.
Article copyright: © Copyright 2017 American Mathematical Society

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