Vol. 7, No. 1, 2022

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An infinite-dimensional index theorem and the Higson–Kasparov–Trout algebra

Doman Takata

Vol. 7 (2022), No. 1, 1–76

We study index theory for some special infinite-dimensional manifolds equipped with a “proper cocompact” action of the loop group LT of the circle T, from the viewpoint of noncommutative geometry. We introduce LT-equivariant KK-theory and construct three KK-elements: the index element, the Clifford symbol element and the Dirac element. These elements satisfy a certain equality, which should be called the (KK-theoretical) index theorem, or the KK-theoretical Poincaré duality for infinite-dimensional manifolds. We also discuss the assembly maps.

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index theory, noncommutative geometry, equivariant $KK\mkern-2mu$-theory, loop groups, infinite-dimensional Heisenberg group, $C^*$-algebra of Hilbert space
Mathematical Subject Classification 2010
Primary: 19K56, 22E67, 46T05
Secondary: 19K35, 26E15, 58B34
Received: 7 January 2019
Revised: 26 December 2021
Accepted: 12 January 2022
Published: 20 June 2022
Doman Takata
Faculty of Education
Niigata University