Local index formula for the quantum double suspension

TL;DR

This paper advances the understanding of the local index formula in noncommutative geometry by explicitly computing it for quantum double suspensions of specific noncommutative spaces, aiding in interpreting multilinear functionals.

Contribution

It provides the second explicit computation of the local index formula for quantum double suspensions, expanding the set of concrete examples in noncommutative geometry.

Findings

Computed local index formula for quantum double suspension of C(S^2)

Computed local index formula for quantum double suspension of noncommutative 2-torus

Enhanced understanding of multilinear functionals in index formulas

Abstract

Our understanding of local index formula in noncommutative geometry is stalled for a while because we do not have more than one explicit computation, namely that of Connes for quantum SU(2) and do not understand the meaning of the various multilinear functionals involved in the formula. In such a situation further progress in understanding necessitates more explicit computations and here we execute the second explicit computation for the quantum double suspension, a construction inspired by the Toeplitz extension. More specifically we compute local index formula for the quantum double suspensions of $C(S^2)$ and the noncommutative $2$-torus.