A Category of Spectral Triples and Discrete Groups with Length Function

Quantum Isometry Group for Spectral Triples with Real Structure

Given a spectral triple of compact type with a real structure in the sense of [Dabrowski L., J. Geom. Phys. 56 (2006), 86-107] (which is a modification of Connes' original definition to accommodate examples coming from quantum group theory) and references therein, we prove that there is always a universal object in the category of compact quantum group acting by orientation preserving isometries (in the sense of [Bhowmick J., Goswami D., J. Funct. Anal. 257 (2009), 2530-2572]) and also preserving the real structure of the spectral triple. This gives a natural definition of quantum isometry gro…

Nonunital spectral triples and metric completeness in unbounded KK-theory

By considering the general properties of approximate units in differentiable algebras, we are able to present a unified approach to characterising completeness of spectral metric spaces, existence of connections on modules, and the lifting of Kasparov products to the unbounded category. In particular, by strengthening Kasparov's technical theorem, we show that given any two composable KK-classes, we can find unbounded representatives whose product can be constructed to yield an unbounded representative of the Kasparov product.

Some counterexamples in the theory of quantum isometry groups

By considering spectral triples on $S^{2}_{μ, c}$ ($c>0$) constructed by Chakraborty and Pal (\cite{chak_pal}), we show that in general the quantum group of volume and orientation preserving isometries (in the sense of \cite{goswami2}) for a spectral triple of compact type may not have a $C^*$-action, and moreover, it can fail to be a matrix quantum group. It is also proved that the category with objects consisting of those volume and orientation preserving quantum isometries which induce $C^*$-action on the $C^*$ algebra underlying the given spectral triple, may not have a universal object.

Roe bimodules as morphisms of discrete metric spaces

For two discrete metric spaces, $X$ and $Y$ we consider metrics on $X\sqcup Y$ compatible with the metrics on $X$ and $Y$. As morphisms from $X$ to $Y$ we consider the Roe bimodules, i.e. the norm closures of bounded finite propagation operators from $l^2(X)$ to $l^2(Y)$. We study the corresponding category $\mathcal M$, which is also a 2-category. We show that almost isometries determine morphisms in $\mathcal M$. We also consider the case $Y=X$, when there is a richer algebraic structure on the set of morphisms of $\mathcal M$: it is a partially ordered semigroup with the neutral element, wi…

连通局部有限图的量子自同构群与离散群的量子化

Quantum automorphism groups of connected locally finite graphs and quantizations of discrete groups

本文为连通局部有限图构造量子自同构群，引入酉张量范畴以定义Haar泛函，并证明图的量子同构等价于其量子自同构群的幺半范畴等价。