A Generalization of the Cuntz Algebras and Model Actions

A Cuntz algebra OH is associated functorially with an infinite-dimensional Hilbert space H. It is a simple C*-algebra distinct from the algebra O∞ introduced by Cuntz. Every locally compact group G acts in a canonical way on OH, H = L2(G), as a Galois-closed group of automorphisms. The fixed-point subalgebra OG together with the restriction to OG of the canonical endomorphism of OH provides an abstract group dual which determines the group. If, furthermore, G is amenable, OG and OH are isomorphic, a result which is in fact valid for finite groups, too. We also consider a generalization involving a Hopf C*-algebra or, more precisely, a regular multiplicative unitary.