Prove that $L$ is closed under Kleene star iff $L=NL$
I am trying to solve this exercise, but it is quiet difficult.
Of course first part is very easy:
Let assume that $L=NL$. Lets consider language $A\in L$. We show that also $A^*\in L$. It is easy because we can guess partition of word $w=w_1w_2..w_k$ and launch algorithm for each part. It is possible because $L=NL$ so being in class $L$ we can use guessing.
However, when we assume that $L$ is closed under Kleene star I am hopeless. Obviously, from assumption we know that $A\in L$ then also $A^*\in L$. Of course, we know that $L\subseteq NL$. However I must now show that $NL\subseteq L$ using assumption. This is hard for me.