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Endomorphisms of \(C^*\)-algebras, cross products and duality for compact groups. (English) Zbl 0702.46044

Summary: We define a new kind of cross product of a \(C^*\)-algebra by semigroups of endomorphisms with properties related to permutation symmetry and the existence of conjugates. A compact group is then defined intrinsically by its dual action on the cross product. This construction yields a characterization of abstract compact group duals which is out of reach of the Tannaka-Krejn duality theory and is independent of that theory. It thereby solves the problem of proving the existence of a compact global gauge group in particle physics given only the local observables.

MSC:

46L55 Noncommutative dynamical systems
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