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

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 ∥[ M ℓ , ·]∥ on C c (Γ, σ ) is a Leibniz Lip-norm on the twisted reduced group C *-algebra C r *(Γ, σ ) for the pointwise multiplication operator M ℓ on ℓ 2 (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.