Noncommutative Geometry of Quantized Coverings

TL;DR

This paper develops a unified algebraic framework for various noncommutative coverings, generalizing classical topological and geometric concepts to noncommutative spaces, with applications to operator spaces.

Contribution

It introduces a comprehensive set of axioms that unify different noncommutative covering theories and extends classical geometric notions to the noncommutative setting.

Findings

Unified algebraic framework for noncommutative coverings

Generalizations of universal covering space and fundamental group

Applications to operator spaces and noncommutative geometry

Abstract

There are theories of coverings of $C^*$-algebras which can be included into a following list: coverings of commutative $C^*$-algebras, coverings of $C^*$-algebras of groupoids and foliations, coverings of noncommutative tori, the double covering of the quantum group $SO_q(3)$. This work is devoted to a single general theory which includes all theories of this list, i.e. we develop a system of axioms which can be applied for every element of the list. Otherwise since topological coverings are related to the set of geometric constructions one can obtain noncommutative generalizations of these constructions. Here the generalizations of the universal covering space, fundamental group, Hurewicz homomorphism, covering of the Riemannian manifold, flat connection are explained. The theory gives pure algebraic proof well known results of the topology and the differential geometry. Besides there are applications of the theory to (unbounded) operator spaces and this theme is also discussed here.