A short exact sequence of $C^*$ algebras

Question

Ex 4.10.15 Let $J$ be an ideal in $C^*$ algebra $A$.

1. Let $C:=C(A,A/J)= \{(a,f) : f:[0,1] \rightarrow A/J, f(0)=0, f(1)=\pi(a)\}$ be mapping cone of quotient map.

2. Let $Z:=C(C,A) = \{ ((a,f),g) \, : \, g:[0,1] \rightarrow A, g(0)=0, g(1)=a \}$ be mapping cone of projection map.

3. Let $SA:= \{ g \, : \, g:[0,1] \rightarrow A, f(0)=0, f(1)=0 \}$ be suspenseion of $A$.

Then there is an exact sequence of $C^*$ algebras, $$ 0 \rightarrow S(A) \rightarrow Z \rightarrow C \rightarrow 0 \quad (1) $$

(First part is original question and resolved by Eric Wofsey)

Below is an optional second part.

Hence, show that we have exactness at the middle of $$K_0(SA)\rightarrow K_0(SA/J) \rightarrow K_0(C)$$

We know that $K_0$ is half exact. Even if we show that $K_0(Z)\cong K_0(SA/J)$, it is not clear to me if it commutes with the diagram of maps at $(1)$.

Answer 1

The map $S(A)\to Z$ should instead be $g\mapsto((0,0),g)$, and the map $Z\to C$ is just the projection $((a,f),g)\mapsto (a,f)$. This makes exactness at $Z$ obvious: if $((a,f),g)\in Z$ maps to $0$ in $C$, then $(a,f)=(0,0)$. That means $g(1)=a=0$ so $g\in SA$, and $((a,f),g)=((0,0),g)$ is the image of $g$ under the map $S(A)\to Z$.