We present a classification framework for saturated Fell bundles over groups, utilizing data associated with their base group and unit fiber. This framework provides a unified perspective on the structure and properties of such bundles and yields key insights into their classification. In the case of discrete groups, we obtain a complete and transparent classification in terms of generalized factor systems, leading to a cohomological description of equivalence classes. For general locally compact groups, the situation is more delicate: While our construction extends to this setting, the classification depends on choices of topology on the underlying Banach bundle.

Let G be a group with identity element e, and suppose that S is an associative G-graded ring that is not necessarily unital. In the case where G is an ordered group, we show that a graded ideal is prime if and only if it is graded prime. Consequently, in that setting, a graded ring is prime if and only if it is graded prime. For any group G, if S is what we call ideally symmetrically G-graded, then we show that there is a bijective correspondence between the G-graded prime ideals of S and the G-prime ideals of S_e. We use this correspondence in the case where G is ordered and S is ideally symmetrically G-graded to show that S is prime if and only if S_e is G-prime. These results generalize classical theorems by Nastasescu and Van Oystaeyen to a nonunital setting. As applications, we provide a new proof of a primeness criterion for Leavitt path rings and establish conditions for primeness of symmetrically G-graded subrings of group rings over fully idempotent rings.

We present a geometrically oriented classification theory for non-Abelian extensions of groupoids generalizing the classification theory for Abelian extensions of groupoids by Westman as well as the familiar classification theory for non-Abelian extensions of groups by Schreier and Eilenberg-MacLane. As an application of our techniques we demonstrate that each extension of groupoids N→E→G gives rise to a groupoid crossed product of G by the groupoid ring of N which recovers the groupoid ring of E up to isomorphism. Furthermore, we make the somewhat surprising observation that our classification methods naturally transfer to the class of groupoid crossed products, thus providing a classification theory for this class of rings. Our study is motivated by the search for natural examples of groupoid crossed products.

The question of whether a right Hilbert bimodule admits a non-commutative frame bundle-i.e., a C*-algebraic noncommutative principal bundle with which the right Hilbert bimodule is associated via some fundamental representation-is both pivotal and difficult. In this paper, we contribute to this topic by providing an axiomatic characterization of a right Hilbert bimodule, let's say M, that ensures the existence of a unique (up to isomorphism) free C*-dynamical system (A(M), SO(n), alpha(M)) with the property that its associated noncommutative vector bundle, with respect to the standard representation of SO(n), is isomorphic to M. Our approach is inspired by potential applications in noncommutative Riemannian spin geometry.

We generalize a classical result by Passman on primeness of unital strongly group graded rings to the class of nearly epsilon-strongly group graded rings which are not necessarily unital. Using this result, we obtain (i) a characterization of prime s-unital strongly group graded rings, and, in particular, of infinite matrix rings and of group rings over s-unital rings, thereby generalizing a well-known result by Connell; (ii) characterizations of prime s-unital partial skew group rings and of prime unital partial crossed products; (iii) a generalization of the well-known characterizations of prime Leavitt path algebras, by Larki and by Abrams-Bell-Rangaswamy. © 2024 The Author(s)

Let G be a compact group, let B be a unital C*-algebra, and let (A, G, a) be a free C*-dynamical system, in the sense of Ellwood, with fixed point algebra B. We prove that (A, G, a) can be realized as the G-continuous part of the invariants of an equivariant coaction of G on a corner of B ⊕ K(H) for a certain Hilbert space H that arises from the freeness of the action.This extends a result byWassermann for free and ergodic C*-dynamical systems. As an application, we show that any faithful *-representation of B on a Hilbert space HB gives rise to a faithful covariant representation of (A, G, a) on some truncation of HB ⊕ H.

Let N and H be groups, and let G be an extension of H by N. In this article, we describe the structure of the complex group ring of G in terms of data associated with N and H. In particular, we present conditions on the building blocks N and H guaranteeing that G satisfies the zero-divisor and idempotent conjectures. Moreover, for central extensions involving amenable groups we present conditions on the building blocks guaranteeing that the Kadison–Kaplansky conjecture holds for the group C∗-algebra of G. © 2022, The Author(s).

We know from research that students learn best through active learning (e.g. Freeman et al., 2014).  In fact, students should work on their own as much as possible in order to improve understanding and to develop problem-solving skills.  As a teacher, it is therefore important to consider the "principle of minimal help" as introduced by the well-known Swiss educationalist Hans Aebli. Simply put, its aim is to provide as much help as necessary whilst as little as possible. Note that this also spares the resources of the teacher.

We present a derivation-based Atiyah sequence for noncommutative principal bundles. Along the way we treat the problem of deciding when a given ∗∗-automorphism on the quantum base space lifts to a ∗∗-automorphism on the quantum total space that commutes with the underlying structure group.

Given a free action of a compact Lie group G on a unital C*-algebra A and a spectral triple on the corresponding fixed point algebra A^G, we present a systematic and in-depth construction of a spectral triple on A that is build upon the geometry of A^G and G. We compare our construction with a selection of established examples.