Wen Hua QIAN, Don HADWIN
Suppose A is a unital C*-algebra and r >1. In this paper, we define a unital C*-algebra Ccb*(A, r) and a completely bounded unital homomorphism αr: A → Ccb* (A, r) with the property that Ccb*(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: Ccb* (A, r) → B such that φ = π ○ αr. We prove that, if A is generated by a normal set {tλ: λ ∈ Λ}, then Ccb* (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 Ccb*(A, r) we obtain that, if B is a unital C*-algebra that can be embedded into A, then Ccb*(B, r) can be naturally embedded into Ccb*(A, r). We give characterizations of Ccb*(A, r) for some special situations and we conclude that Ccb*(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 Ccb*(A, r). We also define and study some analogous of Ccb* (A, r).
