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Additive units of product systems

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Additive units of product systems. / Bhat, B.V.R.; Lindsay, J. Martin; Mukherjee, Mithun.
In: Transactions of the American Mathematical Society, Vol. 370, 01.04.2018, p. 2605-2637.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

Harvard

Bhat, BVR, Lindsay, JM & Mukherjee, M 2018, 'Additive units of product systems', Transactions of the American Mathematical Society, vol. 370, pp. 2605-2637. https://doi.org/10.1090/tran/7092

APA

Bhat, B. V. R., Lindsay, J. M., & Mukherjee, M. (2018). Additive units of product systems. Transactions of the American Mathematical Society, 370, 2605-2637. https://doi.org/10.1090/tran/7092

Vancouver

Bhat BVR, Lindsay JM, Mukherjee M. Additive units of product systems. Transactions of the American Mathematical Society. 2018 Apr 1;370:2605-2637. Epub 2017 Dec 1. doi: 10.1090/tran/7092

Author

Bhat, B.V.R. ; Lindsay, J. Martin ; Mukherjee, Mithun. / Additive units of product systems. In: Transactions of the American Mathematical Society. 2018 ; Vol. 370. pp. 2605-2637.

Bibtex

@article{bf7f91d1ce744561aac9a2414f33b55e,
title = "Additive units of product systems",
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.",
author = "B.V.R. Bhat and Lindsay, {J. Martin} and Mithun Mukherjee",
year = "2018",
month = apr,
day = "1",
doi = "10.1090/tran/7092",
language = "English",
volume = "370",
pages = "2605--2637",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",

}

RIS

TY - JOUR

T1 - Additive units of product systems

AU - Bhat, B.V.R.

AU - Lindsay, J. Martin

AU - Mukherjee, Mithun

PY - 2018/4/1

Y1 - 2018/4/1

N2 - 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.

AB - 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.

U2 - 10.1090/tran/7092

DO - 10.1090/tran/7092

M3 - Journal article

VL - 370

SP - 2605

EP - 2637

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

ER -