2012/09/19 by Claude L. Schochet, Schochet, Claude L.
Mathematics · #46L80 #46L85 #46M20 #55Q52 #55R20 #55T25 #Algebraic Topology (math.AT) #FOS: Mathematics #Operator Algebras (math.OA) #math.AT #math.OA #msc:46L80 #msc:46L85 #msc:46M20 #msc:55Q52 #msc:55R20 #msc:55T25
paper · pdf · doi:10.48550/arxiv.1209.4131
A few typos corrected. Final version, to appear in Journal of Topology and Analysis
arxiv created 2014/02/23 · arxiv updated 2014/02/25
Supppose given a principal G bundle ζ: P → Sk (with k ≥ 2) and a Banach algebra B upon which G acts continuously. Let ζ⊗ B : P ×G B \longrightarrow Sk denote the associated bundle and let Aζ⊗ B = Γ(Sk, P ×G B) denote the associated Banach algebra of sections. Then π_*\GL Aζ⊗ B is determined by a mostly degenerate spectral sequence and by a Wang differential dk : π_*(\GL B) \longrightarrow π*+k-1 (\GL B) . We show that if B is a C^*-algebra then the differential is given explicitly in terms of an \esp with the clutching map of the principal bundle. Analogous results hold after localization and in the setting of topological K-theory. We illustrate our technique with a close analysis of the invariants associated to the C^*-algebra of sections of the bundle ζ⊗ M2 : S7 ×S3 M2 → S4 constructed from the Hopf bundle ζ: S7 → S4 and by the conjugation action of S3 on M2 = M2(\CC). We compare and contrast the information obtained from the homotopy groups π_*(Aζ⊗ M2), the rational homotopy groups π_*(Aζ⊗ M2)⊗\QQ and the topological K-theory groups K_*(Aζ⊗ M2).