Do Kleene star and complement commute? I am having hard time solving the following problem.


  Are there any languages for which
  $$
\overline{L^*} = (\overline{L})^*
$$


Assuming $\emptyset^* = \emptyset$, if I consider $\Sigma = \{a\}$ and L = $\Sigma^*$, I get that $L^* = L$ and that $\overline{L^*} = \emptyset$. For the right side I get $\overline{L} = \emptyset$ and $(\overline{L})^* = \emptyset$. Thus, both sides are equal. 

Is it true that $\emptyset^* = \emptyset$?
Hint: The star of a language always contains the empty string. The complement of a language containing the empty string never does. With that in mind, look at the left and the right hand sides of your proposed equality.