Geodesic
The Index of Product Systems of Hilbert Modules: Two Equivalent Definitions
Publications de l'Institut Mathématique, _N_S_97 (2015) no. 111, p. 49 .

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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 Ke\v cki\'c and Vujo\v sevi\'c [\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 $\mathcal{B}=\mathbb{C}$).
DOI : 10.2298/PIM141114001V
Classification : 46L53, 46L55
Keywords: product system, Hilbert module, index 
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     title = {The {Index} of {Product} {Systems} of {Hilbert} {Modules:} {Two} {Equivalent} {Definitions}},
     journal = {Publications de l'Institut Math\'ematique},
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Biljana Vujošević. The Index of Product Systems of Hilbert Modules: Two Equivalent Definitions. Publications de l'Institut Mathématique, _N_S_97 (2015) no. 111, p. 49 . doi : 10.2298/PIM141114001V. http://geodesic.mathdoc.fr/articles/10.2298/PIM141114001V/

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