2006/11/30 by Alexander Alldridge, Troels Roussau Johansen · 1 citation
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebra over a field #Atiyah–Singer index theorem #Geometry #Homotopy and Cohomology in Algebraic Topology #Hopf algebra #Mathematics #Pure mathematics #Regular polygon #Symbol (formal) #math.KT #math.OA #msc:19K56 #msc:47B35
paper · pdf · doi:10.1016/j.aim.2007.11.024
published as Adv. Math. 218 (2008), no. 1, 163--201 · 46 pages, 1 figure; last version prior to publication, journal reference added
openalex publication_date 2008/01/16 · arxiv created 2009/11/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study multivariate generalisations of the classical Wiener--Hopf algebra, which is the C^*-algebra generated by the Wiener--Hopf operators, given by the convolutions restricted to convex cones. By the work of Muhly and Renault, this C^*-algebra is known to be isomorphic to the reduced C^*-algebra of a certain restricted action groupoid. In a previous paper, we have determined a composition series of this C^*-algebra, and compute the K-theory homomorphisms induced by the `symbol' maps given by the subquotients of the composition series in terms of the analytical index of a continuous family of Fredholm operators. In this paper, we obtain a topological expression for these index maps in terms of geometric-topological data naturally associated to the underlying convex cone. The resulting index formula is expressed in the framework of Kasparov's bivariant KK-theory. Our proof relies heavily on groupoid methods.