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 αr: A → C*cb(A, r) with the property that C*cb(A, r) = C*(αr(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 ◦ αr. We prove that, if A is generated by a normal set {tλ: λ ∈ Λ}, then C*cb(A, r) is generated by the set {αr(tλ): 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).