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An application of the DR-duality theory for compact groups to endomorphism categories of C*-algebras with nontrivial center

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Lledo,  Fernando
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Citation

Baumgärtel, H., & Lledo, F. (2001). An application of the DR-duality theory for compact groups to endomorphism categories of C*-algebras with nontrivial center. In R. Longo (Ed.), Mathematical physics in mathematics and physics: quantum and operator algebraic aspects (pp. 1-10). Providence, RI: American Mathematical Society.


Cite as: https://hdl.handle.net/11858/00-001M-0000-0013-55CC-F
Abstract
In Rev. Math. Phys. 4 (1997) 785 we study Hilbert-C* systems {F,G} where the fixed point algebra A has nontrivial center Z and where A'cap F=Z is satisfied. The corresponding category of all canonical endomorphisms of A contains characteristic mutually isomorphic subcategories of the Doplicher/Roberts-type which are connected with the choice of distinguished G-invariant algebraic Hilbert spaces within the corresponding G-invariant Hilbert Z-modules. We present in this paper the solution of the corresponding inverse problem. More precisely, assuming that the given endomorphism category T of a C*-algebra A with center Z contains a certain subcategory of the DR-type, a Hilbert extension {F,G} of A is constructed such that T is isomorphic to the category of all canonical endomorphisms of A w.r.t. {F,G} and A'cap F=Z. Furthermore, there is a natural equivalence relation between admissible subcategories and it is shown that two admissible subcategories yield A-module isomorphic Hilbert extensions iff they are equivalent. The essential step of the solution is the application of the standard DR-theory to the assigned subcategory.