Additive units of product systems
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- by B. V. Rajarama Bhat, J. Martin Lindsay and Mithun Mukherjee PDF
- Trans. Amer. Math. Soc. 370 (2018), 2605-2637 Request permission
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.References
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Additional Information
- B. V. Rajarama Bhat
- Affiliation: Stat-Math Unit, Indian Statistical Institute, R.V. College Post, Bangalore-560059, India
- MR Author ID: 314081
- Email: bhat@isibang.ac.in
- J. Martin Lindsay
- Affiliation: Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, United Kingdom
- MR Author ID: 114365
- Email: j.m.lindsay@lancaster.ac.uk
- Mithun Mukherjee
- Affiliation: School of Mathematics, IISER Thiruvananthapuram, CET Campus, Kerala - 695016, India
- MR Author ID: 820990
- Email: mithunmukh@iisertvm.ac.in
- 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. - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 2605-2637
- MSC (2010): Primary 46L55; Secondary 46C05, 46L53
- DOI: https://doi.org/10.1090/tran/7092
- MathSciNet review: 3748579