On modular semifinite index theory

Abstract

We propose a definition of a modular spectral triple which covers existing examples arising from KMS-states, Podles sphere and quantum SU(2). The definition also incorporates the notion of twisted commutators appearing in recent work of Connes and Moscovici. We show how a finitely summable modular spectral triple admits a twisted index pairing with unitaries satisfying a modular condition. The twist means that the dimensions of kernels and cokernels are measured with respect to two different but intimately related traces. The twisted index pairing can be expressed by pairing Chern characters in reduced versions of twisted cyclic theories. We end the paper by giving a local formula for the reduced Chern character in the case of quantum SU(2). It appears as a twisted coboundary of the Haar-state. In particular we present an explicit computation of the twisted index pairing arising from the sequence of corepresentation unitaries. As an important tool we construct a family of derived integration spaces associated with a weight and a trace on a semifinite von Neumann algebra.

36 pages

References in corpus (6)

• A twisted spectral triple for quantum SU(2)
• The Hochschild cohomology ring of the standard Podles quantum sphere
• A residue formula for the fundamental Hochschild 3-cocycle for SUq(2)
• A residue formula for the fundamental Hochschild class of the Podles sphere
• The resolvent cocycle in twisted cyclic cohomology and a local index formula for the Podles sphere
• Twisted Cyclic Cohomology and Modular Fredholm Modules

Cited by in corpus (5)

• A modular spectral triple for $\kappa$-Minkowski space
• On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for $C^{\ast }$-Dynamical Systems
• Quantum Dimension and Quantum Projective Spaces
• On the spectral and homological dimension of k-Minkowski space
• A twisted local index formula for curved noncommutative two tori