Does a closed right ideal of a C$^*$-algebra have a C$^*$-algebra?

Question

$A$ 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?

Answer 1

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.