## Covariant strong Morita theory of star product algebras.(English)Zbl 1195.81083

Caenepeel, Stefaan (ed.) et al., Noncommutative structures in mathematics and physics. Proceedings of the congress, Brussels, Belgium, July 22–26, 2008. Contactforum. Brussel: Koninklijke Vlaamse Academie van Belgie voor Wetenschappen en Kunsten (ISBN 978-90-6569-061-6/pbk). 245-258 (2010).
Summary: We recall some recent progress in understanding the representation theory of *-algebras over rings $$C=R(i)$$ where $$R$$ is ordered and $$i^2=-1$$. The representation spaces are modules over auxiliary *-algebras with inner products taking values in this auxiliary *-algebra. The ring ordering allows to implement positivity requirements for the inner products. Then the representations are required to be compatible with the inner product. Moreover, one can add the notion of symmetry in form of Hopf algebra actions. For all these notions of representations there is a well-established Morita theory which we review. The core of each version of Morita theory is the corresponding Picard groupoid for which we give tools to compute and determine both the orbits and the isotropy groups.
For the entire collection see [Zbl 1192.00072].

### MSC:

 81S10 Geometry and quantization, symplectic methods 53D55 Deformation quantization, star products 46L60 Applications of selfadjoint operator algebras to physics 46L65 Quantizations, deformations for selfadjoint operator algebras

### Keywords:

representation theory of *-algebras over rings
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