Computing $K$-theory elements in a $C^*$ algebra $A$

Question

Let $A$ be a unital $C^*$ algebra. Let $p,q$ be projections in $M_n(A)$. Then $[p]-[q]$ defines an element in $K_0(A)$.

Now consider the matrices, the projections,

$$ \left[ \begin{pmatrix} 1-p & 0 \\ 0 & q \end{pmatrix} \right] - \left[\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \right] $$ does this define the same $K_0(A)$ element as $[p]-[q]$?

Answer 1

I will assume you mean $$\left[\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \right]-\left[ \begin{pmatrix} 1-p & 0 \\ 0 & q \end{pmatrix} \right] , $$ since your first matrix is not a projection. Recall that $$ \left[\begin{pmatrix} p & 0 \\ 0 & 0 \end{pmatrix} \right]=\left[\begin{pmatrix} 0 & 0 \\ 0 & p \end{pmatrix} \right]=[p]. $$ Recall also that if $r,s$ are projections with $rs=0$, then $[r]+[s]=[r+s]$. Then $$ \left[\begin{pmatrix} 1-p & 0 \\ 0 & q \end{pmatrix} \right] =\left[\begin{pmatrix} 1-p & 0 \\ 0 & 0 \end{pmatrix} \right]+\left[\begin{pmatrix} 0 & 0 \\ 0 & q \end{pmatrix} \right]=[1-p]+[q]. $$ So \begin{align} \left[\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \right]-\left[ \begin{pmatrix} 1-p & 0 \\ 0 & q \end{pmatrix} \right] &=\left[\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \right]-\left[ \begin{pmatrix} 1-p & 0 \\ 0 & 0 \end{pmatrix} \right] -\left[ \begin{pmatrix} 0 & 0 \\ 0 & q \end{pmatrix} \right] \\ \ \\ &=\left[\begin{pmatrix} p+(1-p) & 0 \\ 0 & 0 \end{pmatrix} \right]-\left[ \begin{pmatrix} 1-p & 0 \\ 0 & 0 \end{pmatrix} \right] -\left[ \begin{pmatrix} 0 & 0 \\ 0 & q \end{pmatrix} \right] \\ \ \\ &=[p]+[1-p]-[1-p]-[q]\\ \ \\ &=[p]-[q]. \end{align}