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the tag (oa.operator-algebras) also seems suitable here

edited Nov 23, 2018 at 11:24

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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?