Let $V$ be a (possibly infinite-dimensional) vector space over a division ring $d$, and consider the projective special linear group $\mathbf{PSL}(V)$. We suppose that if the dimension of $V$ would be $2$, then $d$ is not isomorphic to $\mathbb{F}_2$ or $\mathbb{F}_3$.

My questions are:

• Is it known that $\mathbf{PSL}(V)$ is simple (so even in infinite dimension) ? (In finite dimension, the answer is positive, I have read.) If I am not mistaken, it is simple if and only if it is perfect (due to Iwasawa's criterion).

• What is the automorphism group of $\mathbf{PSL}(V)$ ? Is it isomorphic to the projective semilinear $\mathbf{P\Gamma L}(V)=\mathbf{PGL}(V)\rtimes\mathrm{Aut}(d)$ ?

I would greatly appreciate, besides an answer, good references !