Suppose it is known that $scrM$ is an invariant subspace for $A$. What invariant subspaces for $Aotimes A$ can be obtained from this information alone?
Solution. It is $scrMotimes scrM$ that is an invariant subspace of $Aotimes A$. Indeed, if $x,yin M$, then $$ex (Aotimes A)(xotimes y)=(Ax)otimes (Ay)in Motimes M. eex$$