Suppose $M$ is a is a von Neumann algebra factor of type II$_{\infty}$, and $N$ is a is a von Neumann algebra factor of type III. I have no idea how to prove that $K_0(M)=K_0(N)=0$.
What are the definitions of von Neumann algebra factor of type II$_{\infty}$ and von Neumann algebra factor of type III?
What happens is that the Grothendieck group of a semigroup that has an "infinity" is always trivial. This is because $\infty+d=\infty+c$ for any $c,d$, so $(\infty,\infty)\sim(c,d)$. Both type II$_\infty$ and type III factors have infinite projections, so the above applies.
When the algebra is non-separable, we can still do the above. There will be infinite projections of different cardinalities, so it is enough to choose an infinity that is greater than both $c$ and $d$.