Semigroups of Non-unital Endomorphisms of C*-Algebras and Compact Group Duality

We show that the fixed point algebra OG of the Cuntz algebra On n = 2, 3...,∞, under the canonical action of a strongly compact group G of unitaries on the generating Hubert space H is simple if the defining representation of G is a direct sum of finite dimensional representations whose determinant takes values in finite subgroups of the circle. We use this result to define the Doplicher-Roberts cross product of a unital C*-algebra A with centre Cl by a semigroup of (not necessarily unital) endomorphisms Δ generated by a countable orthogonal collection P={ρi, i ∈ N} fulfilling natural conditions. As an application we give necessary and sufficient conditions in order that (A, P) is isomorphic to (Oλ(G), Σ) with λ the regular representation of a compact and metrizable group G and Σ the set of endomorphisms induced by an irreducible decomposition of λ.