The functional calculus of regular operators on Hilbert C∗-modules revisited
نویسنده
چکیده
In [13], Woronowicz introduced a functional calculus for normal regular operators in a Hilbert C-module. In this paper, we have translated several concepts known in Hilbert space theory to the Hilbert C-module framework. We looked for instance into the functional calculus for strictly positive elements, the Fuglede Putnam theorem in Hilbert C-modules, commuting normal regular operators, ... It appears that if we are a little bit careful, most of the Hilbert space results have their analogues in the Hilbert C-module case. An exception to this rule is of course the polar decomposition of a regular operator. Introduction Regular operators on Hilbert C-modules were studied in [1]. A nice extensive overview concerning Hilbert C-modules is given in [7]. Another standard reference for refular operators is [13]. There is a lot of interest for regular operators (or elements affiliated with a C-algebra) in the C-algebraic quantum group scene. There are several reasons for this : • An interesting object assoiated to a locally compact group is the modular function which connects the left and right Haar measure. This modular function is a continuous group homomorphism from the group in the complex numbers. In the quantum group case, the analogue of this modular function still exists, but it is now a strictly positive element affiliated with the C-algebra. • Some important quantum groups are defined by generators and relations. In the compact case, this generators belong to the C-algebra. But in the non compact case, these generators can (and will be mostly) elements affiliated to the C-algebra (see e.g [13], [14]). Research Assistant of the National Fund for Scientific Research (Belgium)
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