theta-deformations as compact quantum metric spaces

Abstract

Let M be a compact spin manifold with a smooth action of the n-torus. Connes and Landi constructed theta-deformations M_{theta} of M, parameterized by n by n real skew-symmetric matrices theta. The M_{theta}'s together with the canonical Dirac operator (D, H) on M are an isospectral deformation of M. The Dirac operator D defines a Lipschitz seminorm on C(M_{theta}), which defines a metric on the state space of C(M_{theta}). We show that when M is connected, this metric induces the weak-* topology. This means that M_{theta} is a compact quantum metric space in the sense of Rieffel.