The Index of Product Systems of Hilbert Modules: Two Equivalent Definitions

Biljana Vujošević

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 $\mathcal{B}=\mathbb{C}$).