I am confused about the process of the construction of the Grothendieck group $K_0$ in Murphy's $C^*$-algebras and operator theory section 7.1.
Let $A$ be a $*$-algebra and $P[A]=\bigcup_{n=1}^\infty\{p\in M_n(A):\text {$p$ is a projection} \}$. By projection I mean $p=p^*=p^2$.
Define the an equivalence relation on $P[A]$ by $p \sim q \Leftrightarrow $ there is a rectangle matrix $u$ with entries in $A$ such that $p=u^*u$ and $q=uu^*$.
Two projections $p$, $q$ in $P[A]$ are said to be stably equivalent(denoted by $p\approx q$) if $1_n\oplus p{\sim}1_n\oplus q$ for some identity matrix $1_n$.($1_n\oplus p$ is the block diagonal matrix with blocks $1_n$ and $p$)
There is a well-defined operation on ${P[A]}/{\sim}$ by defining the sum of two square matrices to be $p\oplus q$(block diagonal matrix). This also induces an operation on ${P[A]}/{\approx}$.
Murphy proved that $K_0(A)^+:={P[A]}/{\approx}$ is a cancellative abelian group with zero. But I want to find an example of $A$ such that ${P[A]}/{\sim}$ is not cancellative which shows the necessity of defining stable equivalence.
Thanks for your help!