Part I, Free actions of compact Abelian groups on C⁎-algebras

Abstract

We study free actions of compact groups on unital C * -algebras. In particular, we provide a complete classification theory of these actions for compact Abelian groups and explain its relation to the classification of classical principal bundles.

Part II, Free actions of compact groups on C*-algebras

“…Free actions have attained special interest in the literature, see, e. g., [13,16,15,4]. For the definition given here and the relation to noncommutative principal bundles, we refer to [18,Section 3]. It should be noted that A 2 (π)A 2 (π) * is always a * -ideal in C(π).…”

Noncommutative coverings of quantum tori

“…Then A 2 θ is simple and, according to [12], the Picard group Pic(A 2 θ ) equals its subgroup Out(A 2 θ ) of outer automorphisms of A 2 θ . That is, given a covering (A, G, α) of A 2 θ with compact Abelian group G, every isotypic component A(χ), χ ∈Ĝ, contains a unitary element u(χ) (see [22,Rem. 5.15] or [24,Thm.…”

“…where Ad[u(χ)](x) := u(χ)xu * (χ) for all x ∈ A. Conversely, given a compact Abelian group G and a group homomorphism ϕ :Ĝ → Out(A 2 θ ), one may ask whether there is a free C * -dynamical system (A, G, α) with fixed point algebra A 2 θ and Picard homomorphism ϕ A = ϕ. In light of [22,Thm. 5.15], such a C * -dynamical system exists if and only if a certain cohomology class associated to ϕ vanishes in H 3 (Ĝ, T).…”

“…5.15], such a C * -dynamical system exists if and only if a certain cohomology class associated to ϕ vanishes in H 3 (Ĝ, T). In this case, all C * -dynamical systems with fixed point algebra A 2 θ and Picard homomorphism ϕ are parametrized, up to isomorphism, by H 2 (Ĝ, T) (see [22,Thm. 5.8 and Cor.…”

Part III, Free Actions of Compact Quantum Groups on C^*-Algebras

“…The present paper is a sequel of [27] and [28], where we studied free actions of compact Abelian groups and so-called cleft actions, respectively. To be more precise, we achieved in [27] a complete classification of free, but not necessary ergodic actions of compact Abelian groups on unital C * -algebras. This classification extends the results of [1,19] and relies on the fact that the corresponding isotypic components are Morita self-equivalence over the fixed point algebra.…”