I have a continuous curve $A:\mathbf{R}_+\rightarrow \text{M}_N(\mathbf{R})$ such that $[A(t),A(s)] \operatorname*{\longrightarrow}_{t,s\rightarrow +\infty} 0$, where $[A(t),A(s)] = A(t)A(s)-A(s)A(t)$.
Question : does there exists a continuous curve $B:\mathbf{R}_+\rightarrow\text{M}_N(\mathbf{R})$ such that
- for all $t,s\geq 0$, $B(t)B(s) = B(s)B(t)$ ;
- $(A(t)-B(t)) \operatorname*{\longrightarrow}_{t\rightarrow +\infty} 0$ ?
If the map $A$ is furthermore assumed bounded, it is rather elementary (thanks Olivier Benoist !) to build the map $B$ (in that case the set of possible limit points of $A$ contains only matrices which commute pairwise, so it suffices to take the orthogonal projection onto the corresponding spanned vector space).
I wonder if this statement is true, if $A$ is not assumed bounded. If not, I would be glad to learn a counterexample !
I found two posts on MO related to this question, this one and this one. I don't see directly how to use the answers to the first one, while the second one seems to restrict to the bounded case.
PS: I feel that this question could be related to algebraic geometry as it is linked to the distance between the curve $A$ and the (variety) of pairwise commuting matrices, that's why I added the tag but I am not so sure about it !
