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Abstract:
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In this doctoral dissertation we de ne the index of product systems of
Hilbert B B modules over a unital C -algebra B. In detail, 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 Hilbert B B module after
identifying similar units by the suitable equivalence relation and we use that
construction to de ne the index of a given product system. We prove that such
de ned index is a covariant functor from the category of continuous product
systems to the category of two-sided BB modules. We prove that the index
is subadditive with respect to the outer tensor product of product systems
and we, also, prove additional properties of the index of product system that
can be embedded into a spatial one (a product system that contains a central
unital unit). We prove that such de ned index is a generalization of earlier
de ned indices by Arveson (in the case B = C) and Skeide (in the case of
spatial product systems). We, also, de ne the index of product systems in a
di erent way and prove that the new de nition is equivalent to the previous
one. Actually, it corresponds to Arveson's original de nition of the index. |