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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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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 -