Dual space of quotient of C$^*$-algebras

Question

I'm interested in understanding what the dual space of a quotient of a $C^*$-algebra $A$ looks like.

Let $A$ denote a $C^*$-algebra and $I$ a closed ideal therein. Denote the dual space of $A$ by $B$. I think one can say something like:

The dual space $B_I$ of the quotient $A/I$ is canonically isometrically isomorphic to a weak$^*$ closed subset of B.

Is this true?

Thank you very much!

Answer 1

This has nothing to do with algebras. Whnever you have a closed subspace $L$ of a normed (or, more generally, locally convex) vector space $X$, then the dual of the quotient $X/L$ (with the quotient locally convex topology) is (canonically isomorphic to) $L^\perp=\lbrace f\in X^\ast: f|_L=0\rbrace$ which is weak$^\ast$-closed in $X^*$.