The spectral distance in the moyal plane

We study the noncommutative geometry of the Moyal plane from a metric point of view. Starting from a non-compact spectral triple based on the Moyal deformation A of the algebra of Schwartz functions on R2, we explicitly compute Connes' spectral distance between the pure states of A corresponding to eigenfunctions of the quantum harmonic oscillator. For other pure states, we provide a lower bound to the spectral distance, and show that the latest is not always finite. As a consequence, we show that the spectral triple (Gayral et al. (2004) [17]) is not a spectral metric space in the sense of Bellissard et al. (2010) [19]. This motivates the study of truncations of the spectral triple, based on Mn(C) with arbitrary n∈N, which turn out to be compact quantum metric spaces in the sense of Rieffel. Finally the distance is explicitly computed for n=2. © 2011 Elsevier B.V.