Some counterexamples in the theory of quantum isometry groups

By considering spectral triples on $S_{\mu ,c}^2$ ($c>0$) 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 $C^{\ast }$-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 $C^{\ast }$-action on the $C^{\ast }$ algebra underlying the given spectral triple, may not have a universal object.