Let $A,B$ be $C^*$ algebras and $\phi, \psi : A\to B$ be *-homomorphisms. We say that they are mutually orthogonal if for all $x,y\in A$ we have $\phi(x) \psi(y)=0$.
It can be shown then that $\phi +\psi$ is also a $*$-homomorphism and if $A, B$ are unital then $K_0(\phi+ \psi)=K_0(\phi)+K_0(\psi)$. The proof using the standard picture of $K_0$ in the unital case and can be found in the book "An introduction to K-theory of C* algebras", page 44.
My question is if this result holds also in the non-unital case? If not, I would like to get a counterexample. Thank you!