Index pairings for $\mathbb{R}^n$-actions and Rieffel deformations
Andreas Andersson
June 16, 2014
With an action $\alpha$ of $\mathbb{R}^n$ on a $C^*$-algebra $A$ and a
skew-symmetric $n\times n$ matrix $\Theta$ one can consider the Rieffel
deformation $A_\Theta$ of $A$, which is a $C^*$-algebra generated by the
$\alpha$-smooth elements of $A$ with a new multiplication. The purpose of this
paper is to get explicit formulas for $K$-theoretical quantities defined by
elements of $A_\Theta$. We assume there is a densely defined trace on $A$,
invariant under the action. 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 elements $u\in A$,
and $P$ is projection onto the nonnegative eigenspace of a Dirac operator
constructed from the action $\alpha$. We first obtain the $K$-theoretical index
as a Kasparov product and then apply to it a trace on the crossed product
$A\rtimes_\alpha\mathbb{R}^n$ to obtain a real-valued index. The same quantity
has the meaning of a pairing between a semifinite Fredholm module and a
$K_1$-class of $A$, and as a pairing between cohomology and homology coming
from the dynamical system $(A,\alpha,\mathbb{R}^n)$. 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.
Keywords:
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