REALIZATION OF MINIMAL C*-DYNAMICAL SYSTEMS IN TERMS OF CUNTZ–PIMSNER ALGEBRAS

Abstract

In the present article, we provide several constructions of C*-dynamical systems [Formula: see text] with a compact group [Formula: see text] in terms of Cuntz–Pimsner algebras. These systems have a minimal relative commutant of the fixed-point algebra [Formula: see text] in [Formula: see text], i.e. [Formula: see text], where [Formula: see text] is the center of [Formula: see text], which is assumed to be non-trivial. In addition, we show in our models that the group action [Formula: see text] has full spectrum, i.e. any unitary irreducible representation of [Formula: see text] is carried by a [Formula: see text]-invariant Hilbert space within [Formula: see text]. First, we give several constructions of minimal C*-dynamical systems in terms of a single Cuntz–Pimsner algebra [Formula: see text] associated to a suitable [Formula: see text]-bimodule ℌ. These examples are labelled by the action of a discrete Abelian group ℭ (which we call the chain group) on [Formula: see text] and by the choice of a suitable class of finite dimensional representations of [Formula: see text]. Second, we present a more elaborate contruction, where now the C*-algebra [Formula: see text] is generated by a family of Cuntz–Pimsner algebras. Here, the product of the elements in different algebras is twisted by the chain group action. We specify the various constructions of C*-dynamical systems for the group [Formula: see text], N ≥ 2.