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

Abstract

Abstract. We study and classify free actions of compact quantum groups on unital C * -algebras in terms of generalized factor systems. Moreover, we use these factor systems to show that all finite coverings of irrational rotation C * -algebras are cleft.

Noncommutative coverings of quantum tori

“…This condition was originally introduced for actions of quantum groups on C * -algebras by D. A. Ellwood [9] and is known as the Ellwood condition. For equivalent formulations of freeness we may refer the reader to [22,24].…”

“…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. 4.6]).…”

Lifting spectral triples to noncommutative principal bundles

“…The tremendous work of Hopf, Stiefel, and Withney in the 1930's demonstrated the importance of principal bundles for various applications to algebraic topology, geometry and mathematical physics. In the noncommutative setting the notion of a free action of a quantum group on a C * -algebra provides a natural framework for noncommutative principal bundles (see, e. g., [5,24,45,52] and ref. therein).…”

“…therein). Their structure theory and their relation to K-theory (see, e. g., [9,12,18,50,51,52] and refs. therein) certainly appeal to operator algebraists and functional analysts.…”

“…The purpose of this paper is to give a methodical construction of spectral triples on noncommutative principal bundles that are build upon the geometry of the "quantum base spaces" and the underlying structure groups. Since the subject of noncommutative principal bundles or, equivalently, free actions is better understood in the compact case (see, e. g., [5,12,50,52] and ref. therein), we will restrict our study to free actions of compact Lie groups on unital C * -algebras.…”