Universal C*-algebras defined by completely bounded unital homomorphisms

Suppose A is a unital C^*-algebra and r >1. In this paper, we define a unital C^*-algebra C_cb^*(A, r) and a completely bounded unital homomorphism α_r: A → C_cb^* (A, r) with the property that C_cb^*(A, r) = C^*(α_r(A)) and, for every unital C^*-algebra B and every unital completely bounded homomorphism φ: A → B, there is a (unique) unital *-homomorphism π: C_cb^* (A, r) → B such that φ = π ○ α_r. We prove that, if A is generated by a normal set {t_λ: λ ∈ Λ}, then C_cb^* (A, r) is generated by the set {α_r(t_λ): λ ∈ Λ}. By proving an equation of the norms of elements in a dense subset of C_cb^*(A, r) we obtain that, if B is a unital C^*-algebra that can be embedded into A, then C_cb^*(B, r) can be naturally embedded into C_cb^*(A, r). We give characterizations of C_cb^*(A, r) for some special situations and we conclude that C_cb^*(A, r) will be "nice" when dim(A) ≤ 2 and "quite complicated" when dim(A) ≥ 3. We give a characterization of the relation between K-groups of A and K-groups of C_cb^*(A, r). We also define and study some analogous of C_cb^* (A, r).