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arXiv · 1408.2878

$kk$-Theory for Banach Algebras II: Equivariance and Green-Julg type theorems

Walther Paravicini
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Abstract

We extend the definition of the bivariant $K$-theory $kk^{ban}$ from plain Banach algebras to Banach algebras equipped with an action of a locally compact Hausdorff group $G$. We also define a natural transformation from Lafforgue's theory $KK^{ban}_G$ into the new equivariant theory, overcoming some technical difficulties that are particular to the equivariant case. The categorical framework allows us to systematically define a descent homomorphism and to prove a Green-Julg theorem, a dual version of it and a generalised version that involves the action of a proper $G$-space. We also include a naïve Poncaré duality theorem.

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Walther Paravicini. 2014-08-12. $kk$-Theory for Banach Algebras II: Equivariance and Green-Julg type theorems. https://arxiv.org/abs/1408.2878

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