2002/01/22 by Victor Nistor, Nistor, Victor, Evgenij Troitsky +1
Mathematics · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #math.KT #math.OA
paper · pdf · doi:10.48550/arxiv.math/0201207
28 pages, LaTeX
openalex publication_date 2002/01/22 · arxiv created 2002/02/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \GR → B be a bundle of compact Lie groups acting on a fiber bundle Y → B. In this paper we introduce and study gauge-equivariant K-theory groups K_\GRi(Y). These groups satisfy the usual properties of the equivariant K-theory groups, but also some new phenomena arise due to the topological non-triviality of the bundle \GR → B. As an application, we define a gauge-equivariant index for a family of elliptic operators (Pb)b ∈ B invariant with respect to the action of \GR → B, which, in this approach, is an element of K_\GR0(B). We then give another definition of the gauge-equivariant index as an element of K0(C^*(\GR)), the K-theory group of the Banach algebra C^*(\GR). We prove that K0(C^*(\GR)) ≃ K0_\GR(\GR) and that the two definitions of the gauge-equivariant index are equivalent. The algebra C^*(\GR) is the algebra of continuous sections of a certain field of C^*-algebras with non-trivial Dixmier-Douady invariant. The gauge-equivariant K-theory groups are thus examples of twisted K-theory groups, which have recently proved themselves useful in the study of Ramond-Ramond fields.