# Proof for a classifying space for $K$ Theory.

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.