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Additive units of product systems

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<mark>Journal publication date</mark>2018
<mark>Journal</mark>Transactions of the American Mathematical Society
Volume370
Number of pages33
Pages (from-to)2605-2637
StatePublished
Early online date1/12/17
Original languageEnglish

Abstract

We introduce the notion of additive units, or ‘addits’, of a pointed Arveson system.
By the latter 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 Cantor–Bendixson derivative
in the context of the random closed set approach to product systems due to Tsirelson and
Liebscher.