2006/11/30 by Alexander Alldridge, Troels Roussau Johansen · 1 citation
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebra over a field #Compactification (mathematics) #Homomorphism #Mathematical Analysis and Transform Methods #Mathematics #Pure mathematics #math.OA #msc:19K56 #msc:47B35
paper · pdf · doi:10.1016/j.jfa.2007.03.009
published as J. Funct. Anal. 249 (2007), no. 2, 425--453 · 34 pages, 2 figures; last version prior to publication, journal reference added
openalex publication_date 2007/05/04 · arxiv created 2009/11/05 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · 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, given by the action of Euclidean space on a certain compactification. Using groupoid methods, we construct composition series for the Wiener--Hopf C^*-algebra by a detailed study of this compactification. We compute the spectrum, and express homomorphisms in K-theory induced by the symbol maps which arise by the subquotients of the composition series in analytical terms. Namely, these symbols maps turn out to be given by an analytical family index of a continuous family of Fredholm operators. In a subsequent paper, we also obtain a topological expression of these indices.