the Relationship between Projections and Unitaries

Update. We can use functional calculus to obtain unitaries or projections. ----- If $u$ is a unitary, then $\frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & u^*\\ u &1 \end{array}\right)$ is a projection; If $p$ is a projection, then $\left(\begin{array}{cc} p & 1-p\\ 1-p & p \end{array}\right)$ is a unitary.

The isomorphism between $K_0$ groups under the unital case implies that under the non-unital case

This is a supplementary explanation for lemma 6.2.10, [[1](#wegge-olsen)]. The author states that to prove the map $a\mapsto \diag(a,0)$ from $A$ to $M_n(A)$ induces an isomorphism between $K_0(A)$ and $K_0(M_n(A))$, it suffices to prove that the map induces a isomorphism between $V(A)$ and $V(M_n(A))$. It is true for the unital case in which $K_0(A)$ is just the Grothendieck group of $V(A)$. But for the non-unital case? It can be proved by the unital case!(known from Dadarlat and Chung) Consider the following diagram: $\newcommand{\C}{\mathbb{C}}$ \begin{xy} \xymatrix{ 0\ar[r]\ar[d] & A\ar[r]^i\ar[d]^\alpha &A^+\ar@/^/[r]^{\pi}\ar[d]^{\alpha^+} &\C\ar[r]\ar[d]^{\alpha_{\C}}\ar@/^/ [l]^j &0\ar[d] \\ 0\ar[r] & M_n(A)\ar[r]^{i_n} &M_n(A^+)\ar@/^/[r]^{\pi_{ n}}&M_n(\C)\ar[r]\ar@/^/[l]^{j_n} &0 } \end{xy} (1)It is commutative. For example, $\alpha_\C\circ\pi(a,\lambda)=\alpha_\C(\lambda)=\diag(\lambda,0)$ and $\pi_n\circ\alpha^+(a,\lambda)=\pi_n((a,\lambda),0)=(\lambda,0)$ for each $(a,\lambda)\in A^+$, thus $\alpha_\C\circ\pi=\pi_n\circ\alpha^+$. (2)Its rows are split exact sequences of C*-algebras. It is easy to verify that $\operatorname{im}i_n=\ker \pi_n$ and $j_n\circ\pi_n=\operatorname{id}$. Since $K_0$ is a covariant functor that preserves split exactness (corollary 8.2.2, [[1](#wegge-olsen)]), the following diagram is also communicative and the rows are split exact too. \begin{xy} \xymatrix{ 0\ar[r]\ar[d] & K_0(A)\ar[r]^{i_*}\ar[d]^{\alpha_*} & K_0(A^+)\ar[r]^{\pi_*}\ar[d]^{\alpha^+_*} &K_0(\C)\ar[r]\ar[d]^{\alpha_{\C *}} &0\ar[d] \\ 0\ar[r] & K_0(M_n(A))\ar[r]^{i_{n*}} &K_0(M_n(A^+))\ar[r]^{\pi_{ n *}}&K_0(M_n(\C))\ar[r] &0 } \end{xy} From the unital case, we know that $\alpha^+_*:K_0(A^+)\to K_0(M_n(A^+))$ and $\alpha_{\C *}:K_0(\C)\to K_0(M_n(\C))$ are isomorphisms. Proof ends by applying the five-lemma to \begin{xy} \xymatrix{ 0\ar[r]\ar[d] & 0\ar[r]\ar[d] & K_0(A)\ar[r]^{i_*}\ar[d]^{\alpha_*} & K_0(A^+)\ar[r]^{\pi_*}\ar[d]^{\alpha^+_*} &K_0(\C)\ar[d]^{\alpha_{\C *}} \\ 0\ar[r] & 0\ar[r] & K_0(M_n(A))\ar[r]^{i_{n*}} &K_0(M_n(A^+))\ar[r]^{\pi_{ n *}}&K_0(M_n(\C)) } \end{xy} $1$ Wegge-Olsen, N. E. (1993). K-theory and C*-algebras.