English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $T$ be the circle group and let $LT$ be its loop group. We formulate andinvestigate several topological aspects of the $LT$-equivariant index theoryfor proper $LT$-spaces, where proper $LT$-spaces are infinite-dimensionalmanifolds equipped with "proper cocompact" $LT$-actions. Concretely, weintroduce "$\mathcal{R}KK$-theory for infinite-dimensional manifolds", and byusing it, we formulate an infinite-dimensional version of the $KK$-theoreticalPoincar\'e duality homomorphism, and an infinite-dimensional version of the$\mathcal{R}KK$-theory counterpart of the assembly map, for proper $LT$-spaces.  The left hand side of the Poincar\'e duality homomorphism is formulated bythe "$C^*$-algebra of a Hilbert manifold" introduced by Guoliang Yu. Thus, theresult of this paper suggests that this construction carries some topologicalinformation of Hilbert manifolds. In order to formulate the assembly map in aclassical way, we need crossed products, which require an invariant measure ofa group. However, there is an alternative formula to define them usinggeneralized fixed-point algebras. We will adopt it as the definition of"crossed products by $LT$".

Topological Aspects of the Equivariant Index Theory of Infinite-Dimensional LT-Manifolds