isomorphism between $K_1(A)$ and $K_0(SA)$.

Question

The above theorem is from Rordam's book. I have a question: In the proof of Thorem 10.1.3,the author mentioned that we can use the identifications.How to prove the above two identifications?

Answer 1

I'll do a very simplified version of what you're asking by leaving out the matrix parts, leaving that detail to you. Moreover I'll only work with the cone case. Something similar will work with the suspension.

Also it does help if you actually write out the question you want answered.

I claim that we can identify functions in $\widetilde{CA}$ with elements $g \in C([0,1],\tilde{A})$ such that $s(g(t)) = g(0)$ for all $t \in [0,1]$. Let $$ B = \{g \in C([0,1],\tilde{A}) \mid s(g(t)) = g(0) \text{ for all } t \in [0,1]\}. $$ You can check that $B$ is a C*-algebra (just check that it is a norm-closed *-subalgebra of $C([0,1],\tilde{A})$). Define $\phi: \widetilde{CA} \to B$ by

$$ \phi(f + \alpha 1_{\widetilde{CA}})(t)= \begin{cases} \alpha 1_{\tilde{A}}, & t = 0 \\ f(t) + \alpha 1_{\tilde{A}}, & t \in (0,1] \end{cases}. $$ Then $\phi$ will end up being a well-defined *-isomorphism, the details of which are left to you. It is definitely worth checking that $\phi(f + \alpha 1_{\widetilde{CA}}):[0,1] \to \tilde{A}$ is actually a continuous function (recall that $CA = C_0((0,1],A)$) and satisfies the desired scalar condition.