What is known about ideals of bidual $\mathfrak{A^{\ast\ast}}$ of a $C^{\ast}$-algebra $\mathfrak{A}$.

Question

Let $\mathfrak{A}$ be a $C^{\ast}$-algebra. What is known about ideals of bidual $\mathfrak{A^{\ast\ast}}$ of $\mathfrak{A}$. Are the ideals of both related in some sense?

Any references/ideas?

Edit: Dear downvoters, I'm not asking a solution. I'm asking a reference from where I can read about them and also it's not a homework problem obviously.

Answer 1

There is not much relation. Even when $\mathfrak A$ is simple, $\mathfrak A^{**}$ has lots and lots of ideals. Every (equivalence class of a) representation of $\mathfrak A$ corresponds to a central projection in $\mathfrak A^{**}$, and each central projection corresponds with an ideal.

There is little to gain in trying to understand the structure of $\mathfrak A$ in terms $\mathfrak A^{**}$. The usefulness of the enveloping von Neumann algebra comes from the fact that it encodes all the representation theory of $\mathfrak A$ (universality), and also the fact that sometimes one can use von Neumann algebra techniques.