The non-commutative tori provide probably the most accessible interesting examples of non-commutative differentiable manifolds. We can identify an ordinary n-torus rn with its algebra, C(rn), of… (More)

Let a compact Lie group act ergodically on a unital C∗-algebra A. We consider several ways of using this structure to define metrics on the state space of A. These ways involve length functions,… (More)

One can describe an n-dimensional noncommutative torus by means of an antisymmetric n×n matrix θ. We construct an action of the group SO(n, n|Z) on the space of n × n antisymmetric matrices and show… (More)

We propose a definition of what should be meant by a proper action of a locally compact group on a C *-algebra. We show that when the C *-algebra is commutative this definition exactly captures the… (More)

Let l be a length function on a group G, and let Ml denote the operator of pointwise multiplication by l on l(G). Following Connes, Ml can be used as a “Dirac” operator for C ∗ r (G). It defines a… (More)

In contrast to the usual Lipschitz seminorms associated to ordinary metrics on compact spaces, we show by examples that Lipschitz seminorms on possibly non-commutative compact spaces are usually not… (More)

By a quantum metric space we mean a C∗-algebra (or more generally an order-unit space) equipped with a generalization of the Lipschitz seminorm on functions which is defined by an ordinary metric. We… (More)

Let l be a length function on a group G, and let Ml denote the operator of pointwise multiplication by l on l(G). Following Connes, Ml can be used as a “Dirac” operator for C ∗ r (G). It defines a… (More)

In contrast to the usual Lipschitz seminorms associated to ordinary metrics on compact spaces, we show by examples that Lipschitz seminorms on possibly non-commutative compact spaces are usually not… (More)

On looking at the literature associated with string theory one finds statements that a sequence of matrix algebras converges to the 2-sphere (or to other spaces). There is often careful bookkeeping… (More)