Does a closed right ideal of a C$^*$-algebra have a C$^*$-algebra? $A$ is an infinite dimensional C$^*$-algebra and $J\subset A$ is a closed right ideal. $A$ and $J$ are infinite dimensional(as a vector space). I want to find an infinite dimensional C$^*$-algebra subset of $J$. How can I find it?
I know an infinite dimensional C$^*$-algebra has an infinite dimensional commutative C$^*$-subalgebra. So if $A_1$ is infinite dimensional commutative C$^*$-subalgebra of $A$, Is the set $A_1\cap J$ an infinite dimensional C$^*$-algebra? If no, so what can I do?

You can't do that in general. Many C$^*$-algebras are simple. In such a case if $B\subset J$ is a C$^*$-algebra, then 
$$
B=B\cap B^*\subset J\cap J^*=\{0\}, 
$$
since $J\cap J^*$ is a closed ideal. 
