Twisted bounded-dilation group C*-algebras as C*-metric algebras

Abstract

We construct a class of C*-metric algebras. We prove that for a discrete group Γ with a 2-cocycle σ, the closure of the seminorm ∥[Ml, ·]∥ on Cc(Γ, σ) is a Leibniz Lip-norm on the twisted reduced group C*-algebra Cr*(Γ, σ) for the pointwise multiplication operator Ml on l2(Γ), induced by a proper length function l on Γ with the property of bounded θ-dilation. Moreover, the compact quantum metric space structures depend only on the cohomology class of 2-cocycles in the Lipschitz isometric sense.