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

December 06, 2000

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.

open access link

Fields Inst.Commun. 30 (2001) 1-10

Keywords:

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