I found in the book of Murphy, C\*- Algebras and Operator Theory, the Theorem 7.1.2 :

> 
> Let A be an unital C\* algebra, the semi group $V(A)$ of equivalent projections (under Murray Von
>  Neumann equivalence) in $M\_∞(A)$ is cancellative.
> 
> 
> 

For the proof he proceeds as follow : 

* take some projection $p, q, r$ such that $[p] \oplus [r] = [q] \oplus [r]$
* add to it $[I\_n - r]$ if $r\in M\_n(A)$ to obtain $[p] \oplus [I\_n] = [q] \oplus [I\_n]$

And he concludes with that, and I don't see why that should be true that one can simplify $[I\_n]$.

Furthermore, one knows that there exists C\*algebras where the $K^0$ group (the Grothendieck group of $V(A)$ ) is of Torsion, for exemple, the Cuntz algebra with $K^0$ group $\mathbb{Z} / n\mathbb{Z}$. For this algebra the morphism of semi group $V(A) \rightarrow K^0(A) $ cannot be injective and $V(A)$ cannot be cancellative, right ?

So my question is who is right ? Me or Murphy ^^ ?