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 (Commun Math Phys 240(3):447-456, 2000), we show that in general the quantum group of volume and orientation preserving isometries (in the sense of Bhowmick and Goswami in J Funct Anal 257:2530-2572, 2009) 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.