The Podleś sphere as a spectral metric space

We study the spectral metric aspects of the standard Podleś sphere, which is a homogeneous space for quantum SU(2). The point of departure is the real equivariant spectral triple investigated by Dąbrowski and Sitarz. The Dirac operator of this spectral triple interprets the standard Podleś sphere as a 0-dimensional space and is therefore not isospectral to the Dirac operator on the 2-sphere. We show that the seminorm coming from commutators with this Dirac operator provides the Podleś sphere with the structure of a compact quantum metric space in the sense of Rieffel.