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

We prove that a conditionally completely positive definite k ernel, as the generator of completely positive definite (CPD) semig roup associated with a continuous set of units for a product system over a C-star-algebra B , allows a construction of a Hilbert B - B module. That construction is used to define the index of the initial product system. It is proved that suc h definition is equivalent to the one previously given by Keckic and VujoSevic [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 defi nition of the index corresponds to the one given earlier by Arveson (in the case B = C )