# $K$ Theory of Operators II, Higson's notes

What I want to understand is how the map defined in proof of Proposition 3.17 is an inverse of the defined map.

Proposition 3.17: For any (ungraded) $$C^*$$-algebra $$A$$, the map $$\Phi:K_0(A) \rightarrow [\mathcal{S}, A \otimes \mathcal{K} ]$$ defined above* is an isomoprhism. Where $$\mathcal{S}=C_0(\Bbb R)$$, $$\mathcal{K}$$ compact operators on the graded hilbert space $$H=H_0\oplus H_1$$.

1. The map (*) is given as mapping an two idempotent matrix $$p,q$$ into $$f \mapsto \begin{pmatrix} p f(0) & 0 \\ 0 & q f(0) \end{pmatrix}$$ where we regard $$A \otimes \mathcal{K} \cong M_2(A \otimes \mathcal{K})$$. More details of this map is here.

2. The details of inverse map can be found in here.

I could not really chase out how this maps are inverse - in particular, the map in 2. defines an element in $$K_0(A \otimes \mathcal{K})$$, and here we use Morita equivalence to obtain an element, $$K_0(A) \cong K_0(A \otimes \mathcal{K})$$

I would be grateful if someone can spell out more details. I believe, it would be easier for one to first read the original proof.