Proving B* = B on a given set
(Criterion of no-op Kleene star) Let $V$ be a language such that $\epsilon\in V$. Then $V=V^*$ if and only if $V=VV$, where $V^*$ is the Kleene star.

Proof: "$\Longrightarrow$":  $V=V\epsilon\subseteq VV$ while $VV\subseteq V^*=V$.

"$\Longleftarrow$". $V\subseteq V^*$ by definition. Let $V_i$ as defined in Wikipedia. Since $V_0\subseteq V$ and assuming $V_n\subseteq V$, $V_{n+1}=V_nV\subseteq VV=V$, we know $V_i\subseteq V$ for all $i\ge0$ by induction. Hence $$V^*=\bigcup_{i=0}^\infty V_{i+1}\subseteq \bigcup_{i=0}^\infty V=V\,.$$



Exercise 1. Show that $\epsilon\in B$ and $B=BB$.

Exercise 2. Let $V$ be a non-empty language such that $V=VV$. Show that $\epsilon\in V$. (Hence for a non-empty language $V$, $V=V^*$ if and only if $V=VV$.)

Exercise 3. Show that $V^*=(V^*)^*$.