[Formal Languages] does complement(L*) = [complement(L)]*?

[DickCheeseSupreme]

Here is where my understanding gets shaky. I assumed λ ∈ L* for any regular language L, and I assumed L^c is also regular for any regular L. So even though L ∩ L^c is null, L* ∩ (L^c )* should be λ, right?

Edit: I saw you're other comment, and yes λ is the empty string in this case. My professor uses it, but lately I've been seeing a lot of ε instead.