Banach algebras, Samelson products, and the Wang differential

Abstract

Assume that given a principal G bundle ζ : P → Sk (with k ≥ 2) and a Banach algebra B upon which G acts continuously. Let [Formula: see text] denote the associated bundle and let [Formula: see text] 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 [Formula: see text] We show that if B is a C*-algebra then the differential is given explicitly in terms of an enhanced Samelson product 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 [Formula: see text] constructed from the Hopf bundle ζ : S7 → S4 and by the conjugation action of S3 on M2 = M2(ℂ). We compare and contrast the information obtained from the homotopy groups π*( U ◦Aζ⊗M2), the rational homotopy groups π*( U ◦Aζ⊗M2) ⊗ ℚ and the topological K-theory groups K*(Aζ⊗M2), where U ◦B is the connected component of the unitary group of the C*-algebra B.