On C*-algebras related to constrained representations of a free group

We consider representations of the free group F₂ on two generators for which the norm of the sum of the generators and their inverses is bounded by some number μ ∈ [0, 4]. These μ-constrained representations determine a C*-algebra A_μ for each μ ∈ [0, 4]. If μ = 4, this gives the full group C*-algebra of F₂. We prove that these C*-algebras form a continuous bundle of C*-algebras over [0, 4] and evaluate their K-groups.