Index pairings for ${\mathbb{R}}^n$-actions and Rieffel deformations

Abstract

With an action $\alpha$ of ${\mathbb{R}}^n$ on a $C^{\ast }$-algebra $A$ and a skew-symmetric $n\times n$ matrix $\Theta$ one can consider the Rieffel deformation $A_{\Theta }$ of $A$, which is a $C^{\ast }$-algebra generated by the $\alpha$-smooth elements of $A$ with a new multiplication. The purpose of this paper is to obtain explicit formulas for $K$-theoretical quantities defined by elements of $A_{\Theta }$. We assume that there is a densely defined trace on $A$, invariant under the action. We give an explicit realization of Thom class in $KK$ in any dimension $n$, and use it in the index pairings. When $n$ is odd, for example, we give a formula for the index of operators of the form $P{\pi }^{\Theta }(u)P$, where ${\pi }^{\Theta }(u)$ is the operator of left Rieffel multiplication by an invertible element $u$ over the unitization of $A$, and $P$ is projection onto the nonnegative eigenspace of a Dirac operator constructed from the action $\alpha$. The results are new also for the undeformed case $\Theta =0$. The construction relies on two approaches to Rieffel deformations in addition to Rieffel's original one: "Kasprzak deformation" and "warped convolution". We end by outlining potential applications in mathematical physics.

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Cited by in corpus (4)

• Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
• On $q$-tensor product of Cuntz algebras
• Electromagnetism in terms of quantum measurements
• Topological insulators from the perspective of non-commutative geometry and index theory