# Endomorphisms of $C^\ast$-algebras, cross products, and duality for compact groups

### Abstract

We define a new kind of cross product of a $C^\ast$-algebra by semigroups of endomorhpisms 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-Krein 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.

Sergio Doplicher

John E. Roberts