$A$ 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?



Does a closed right ideal of a C$^*$-algebra have a C$^*$-algebra?
It need not exist. Take $A = B(l^2)$ (or the compacts if you want $A$ to be separable) and let $J$ be the set of operators of the form $$u \mapsto \langle u,v\rangle e\_1$$ for $v \in l^2$, where $e\_1$ is the first standard basis vector. It is a right ideal because for any $T \in B(l^2)$ we have $\langle Tu,v\rangle e\_1 = \langle u,T^\*v\rangle e\_1$, and it is closed because $T \in J$ if and only if $\langle Tu,v\rangle = 0$ for all $u,v \in l^2$ with $v \perp e\_1$.


The adjoint of $J$ is the set of operators of the form $u \mapsto \langle u,e\_1\rangle v$ for $v \in l^2$, and the intersection $J \cap J^\*$ is one-dimensional, so we are done by Yemon's comment.


