8 research outputs found
Non-Hausdorff Symmetries of C*-algebras
Symmetry groups or groupoids of C*-algebras associated to non-Hausdorff
spaces are often non-Hausdorff as well. We describe such symmetries using
crossed modules of groupoids. We define actions of crossed modules on
C*-algebras and crossed products for such actions, and justify these
definitions with some basic general results and examples.Comment: very minor changes. To appear in Math. An
Inverse semigroup actions as groupoid actions
To an inverse semigroup, we associate an \'etale groupoid such that its
actions on topological spaces are equivalent to actions of the inverse
semigroup. Both the object and the arrow space of this groupoid are
non-Hausdorff. We show that this construction provides an adjoint functor to
the functor that maps a groupoid to its inverse semigroup of bisections, where
we turn \'etale groupoids into a category using algebraic morphisms. We also
discuss how to recover a groupoid from this inverse semigroup.Comment: Corrected a typo in Lemma 2.14 in the published versio
Radicals of some semigroup algebras
In this paper we seek to determine the Jacobson radical of certain algebras based on semigroups, and in particular on the semigroups (βS,□), where S is a cancellative, countable, abelian semigroup and βS is its Stone–Čech semigroup compactification. In particular, we wish to determine the radical of ℓ  1(βℕ)
Braided Categories of Endomorphisms as Invariants for Local Quantum Field Theories
We want to establish the “braided action” (defined in the paper) of the DHR category on a universal environment algebra as a complete invariant for completely rational chiral conformal quantum field theories. The environment algebra can either be a single local algebra, or the quasilocal algebra, both of which are model-independent up to isomorphism. The DHR category as an abstract structure is captured by finitely many data (superselection sectors, fusion, and braiding), whereas its braided action encodes the full dynamical information that distinguishes models with isomorphic DHR categories. We show some geometric properties of the “duality pairing” between local algebras and the DHR category that are valid in general (completely rational) chiral CFTs. Under some additional assumptions whose status remains to be settled, the braided action of its DHR category completely classifies a (prime) CFT. The approach does not refer to the vacuum representation, or the knowledge of the vacuum state