The quantised interval as a quantum metric space

Abstract: The study of metrics on state spaces arising from semi-norms dates back to Connes and was formalised as the notion of a compact quantum metric space by Rieffel, whose notion of quantum Gromov-Hausdorff distance on the class of compact quantum metric spaces, has established a new famework for the study of approximations of C*-algebras.

In a recent paper, Aguilar and Kaad have shown that the standard Podleś sphere, originally introduced as the homogeneous space for Woronowicz' quantum SU(2), is in fact a compact quantum metric space, and they posed the rather natural question, whether the standard Podleś sphere converges to the standard 2-sphere in the quantum analogues of the Gromov-Hausdorff distance as the deformation parameter tends to 1 . In my talk based on joint work with Jens Kaad and David Kyed, I will present some new developments to the above question. In particular we have shown that the commutative C*-subalgebras generated by the self-adjoint generator of the standard Podleś sphere, converge to the interval of length \pi as one would expect if the more general convergence result is true, and that the spaces in fact vary continuously. This provides some evidence that the convergence result for the Podles spheres may hold true as well (this is currently work in progress).

functional analysisgroup theoryoperator algebras

Audience: researchers in the topic

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