The goal of my question is to understand a bijection between $K_0(A)$ to $[C_0(\Bbb R), M_2(M_\infty(A))]_*$

$$[C_0(\Bbb R), M_2(M_\infty(A))]_*$$ is the homotopy class of graded $*$-homomorphisms. $C_0(\Bbb R)$ is graded by even and odd functions. $M_2(M_\infty(A))$ by diagonal (even) and off diagonal elements (odd). This extends uniquely to a graded $*$-homomorphism of their unitzation, $$[C(\Bbb R)_+, M_2(M_\infty(A)_+)]$$ where we may regarde $C_0(\Bbb R)_+ $ as $C(S^1)$ by the Cayley trasform $x \mapsto (x+i)(x-i)^{-1}$. The grading on $C(S^1)$ to make everything is given by $f(z)\mapsto f(\bar{z})$ as explained by s.harp.

A $*$-homomoprhism is determined by the image $j:S^1 \hookrightarrow \Bbb C$. Denote this image by $u$. As shown in this post. $$\alpha(u)=u^* \quad (I) $$

Note that there is also a grading on $\varepsilon \in M_2(M_\infty(A)_+)$ by $$ \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$ and that $\alpha(x)=\varepsilon x \varepsilon$ for all $x$.

There isa bijection between all elements $u$ that satisfy (I) and the projections, given by $$ u \mapsto \frac{1}{2}(1+u\varepsilon)$$

MAP I : $ \Phi: K_0(A) \rightarrow [C_0(\Bbb R), M_2(M_\infty(A))_+]$.

Let $[p]-[q] \in K_0(A)$, then we send $$f \mapsto \begin{pmatrix} pf(0) & 0 \\ 0 & qf(0) \end{pmatrix} $$

MAP II: $ \Psi: [C(S^1), M_2(M_\infty(A))_+] \rightarrow K_0(A)$.

Given a unitary $u \in M_2(M_\infty(A))_+$ satisfying $(I)$ assign it the element

$$[p_1] - [p_\phi] \in K_0(A)$$

I would like to understand how the maps defined are bijections. I am more concerned with the direction with how showing $\Phi \circ \Psi$ is the identity.

This is a translation of Proposition 3.17, pg43 of Higson's notes. The post should be self contained.


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