Let $B$ be a $C^*$ algebra, and $A(B)$ denote the algebra of bounded continuous functions from $[1,\infty)$ to $B$. Let $I(B)$ be the ideal of functions which vanish at infinity. Let $Q(B)$ be the quotient $C^*$ algebra.
We have an exact sequence $$0 \rightarrow I(B) \rightarrow A(B) \rightarrow Q(B) \rightarrow 0$$
Claim: $$K_0(A(B)) \cong K_0(Q(B))$$
I know: from algebraic $K$ theory of rings we have a middle exact sequence, and from homotopy invaraince, that $K_0(I(B)) = 0$.
How does one deduce surjectivity of $K_0(A(B)) \rightarrow K_0(Q(B))$?
I suppose this follows from six-term exact sequence of $C^*$ algebras. I wonder if there is an elementary way to see this.
Reference: Page 47 of Higson's notes.