THE INDEX OF PRODUCT SYSTEMS OF HILBERT MODULES: TWO EQUIVALENT DEFINITIONS

Biljana Vujosevic

Abstract: We prove that a conditionally completely positive definite kernel, as the generator of completely positive definite (CPD) semigroup associated with a continuous set of units for a product system over a $C^*$-algebra $\mathcal{B}$, allows a construction of a Hilbert $\mathcal{B}-\mathcal{B}$ module. That construction is used to define the index of the initial product system. It is proved that such definition is equivalent to the one previously given by Keckic and Vujosevic [\emph{On the index of product systems of Hilbert modules}, Filomat, to appear, ArXiv:1111.1935v1 [math.OA] 8 Nov 2011]. Also, it is pointed out that the new definition of the index corresponds to the one given earlier by Arveson (in the case $\mathcal{B}=\mathbb{C}$).

Keywords: product system; Hilbert module; index

Classification (MSC2000): 46L53;46L55

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