Homomorphisms of C*-Algebras and Their K-theory

Let A and B be C*-algebras and ϕ : A → B be a ∗-homomorphism. We discuss the properties of the kernel and (co-)image of the induced map K0(ϕ) : K0(A) → K0(B) on the level of K-theory. In particular, we are interested in the case that the co-image is torsion free, and show that it holds when A and B are commutative and unital, B has real rank zero, and ϕ is unital and injective. We also show that A is embeddable in B if K0(ϕ) is injective and A has stable rank one and real rank zero.