Filomat 2015 Volume 29, Issue 5, Pages: 1093-1111
https://doi.org/10.2298/FIL1505093K
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On the index of product systems of Hilbert modules
Kečkić Dragoljub J. (Faculty of Mathematics, Belgrade)
Vujošević Biljana (Faculty of Mathematics, Belgrade)
In this note we prove that the set of all uniformly continuous units on a
product system over a C* algebra B can be endowed with a structure of
left-right B-B Hilbert module after identifying similar units by the
suitable equivalence relation. We use this construction to define the index
of the initial product system, and prove that it is a generalization of
earlier defined indices by Arveson (in the case B=C) and Skeide (in the
case of spatial product system). We prove that such defined index is a
covariant functor from the category of continuous product systems to the
category of B bimodules. We also prove that the index is subadditive with
respect to the outer tensor product of product systems, and prove additional
properties of the index of product systems that can be embedded into a
spatial one.
Keywords: product system, Hilbert module, index, noncommutative dynamics, quantum probability
Projekat Ministarstva nauke Republike Srbije, br. 174034