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.


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