We consider a convex optimization problem on linearly constrained cones in Euclidean Jordan algebras. The problem is solved using a damped Newton algorithm. Quadratic convergence to the global minimum is shown using an explicit step-size selection. Moreover, we prove that the algorithm is a smooth discretization of a Newton flow with Lipschitz continuous derivative.

Let α and β be two Furstenberg transformations on 2-torus associated with irrational numbers θ1, θ2, integers d1, d2 and Lipschitz functions f1 and f2. We show that α and β are approximately conjugate in a measure theoretical sense if (and only if) θ1 ± θ2 = 0 in R/Z. Closely related to the classification of simple amenable C∗-algebras, we show that α and β are approximately K-conjugate if (and...

In an earlier paper of mine relating vector bundles and Gromov–Hausdorff distance for ordinary compact metric spaces, it was crucial that the Lipschitz seminorms from the metrics satisfy a strong Leibniz property. In the present paper, for the now noncommutative situation of matrix algebras converging to the sphere (or to other spaces) for quantum Gromov–Hausdorff distance, we show how to const...

Introductory and historical remarks Clifford (1878) introduced his ‘geometric algebras’ as a generalization of Grassmann algebras, complex numbers and quaternions. Lipschitz (1886) was the first to define groups constructed from ‘Clifford numbers’ and use them to represent rotations in a Euclidean space. É. Cartan discovered representations of the Lie algebras son(C) and son(R), n > 2, that do ...