No abstract

An isometry of a unital [Formula: see text]-algebra with respect to a spectral triple is a [Formula: see text]-automorphism of the [Formula: see text]-algebra given by the conjugation by a unitary operator which commutes with the Dirac operator. We give a semidirect product topological characterization on the isometry group of a twisted reduced group [Formula: see text]-algebra of a discrete group with respect to the standard spectral triple induced by a length function on the group. We prove that this isometry group is compact in the point-norm topology, and in particular, for a finitely generated discrete group, this isometry group is a compact Lie group in the point-norm topology. We also extend this result to a unital [Formula: see text]-algebra with a filtration, and prove that its isometry group is a compact topological group in the point-norm topology.

No abstract

A compact quantum metric space is a complete order unit space endowed with a Lip-norm. We introduce a metric on the state space and give several equivalent conditions to characterize the Lipschitz morphisms and Lipschitz isomorphisms between two compact quantum metric spaces.

A compact quantum metric space is a complete order unit space endowed with a Lip-norm. We study morphisms and isomorphisms between compact quantum metric spaces. We give two sufficient conditions for a unital positive linear map of a compact quantum metric space with lower semicontinuous Lip-norm to be in fact a bi-Lipschitz map.