Kleene star operations Let $$ be any alphabet and let $_ \subseteq ^{∗}$ and  $_2 \subseteq ^{∗}$ be two non-empty languages.

a. If  $_ ^{∗} \neq ^{∗}$ than what can we say about $L_1$.

b.Let $\Lambda \in L_1$ and $\Lambda \in L_2$. Show using axioms and theorems of languages that $_ ^{∗}_2 =  ^{∗}$

For (a), $\Lambda$ should not belong to $L_1$ but I do not know how to prove that.

For (b),we have to prove that $_ ^{∗}_2 \subseteq  ^{∗}$ and $  ^{∗} \subseteq_ ^{∗}_2$ for equality to exist. We can also distinguish two cases, when $L = \Lambda $,  then $\Lambda ^{∗} \Lambda = ^{∗}$, but how can we prove that when $L \neq \Lambda $

Any idea
Hint:

You are right on (a). The proof involves key results for Kleene star as follows.
$$
\Sigma^{*}=\bigcup_{n\geqslant1}\Sigma^{n}\quad\text{and}\quad L\Sigma^{*}=\bigcup_{n\geqslant1}L\Sigma^{n}\quad\text{and}\quad \Sigma^{*}L=\bigcup_{n\geqslant1}\Sigma^{n}L
$$
Where $\Sigma^{n}$ can be considered as the set of the concatenation of i strings in $\Sigma$, and $L\subset \Sigma^{*}$.

For (a), $L_1\Sigma^{*}\subset\Sigma^{*}$ since for all $n,L_1\Sigma^{n}\subset\Sigma^{n}$. If $\Lambda\in L_1$, then $n,L_1\Sigma^{n}=\Sigma^{n}$ for all $n$. Thus $L_1\Sigma^{*}=\Sigma^{*}$.

For (b), notice that if $\Lambda\in L_1$ and $L_2$, then $L_1\Sigma^{*}=\Sigma^{*}$ and $\Sigma^{*}L_2=\Sigma^{*}$. Thus
$L_1\Sigma^{*}L_2=\Sigma^{*}L_2=\Sigma^{*}$.