A isomorphism between C*-algebras

Question

Let $A$ be a C*-algebra and $J\triangleleft A$ be an ideal, then $A^{**}\cong J^{**}\oplus(A/J)^{**}$ ? Why?

Answer 1

The key concept here is that of annihilator. For a subset $X\subset A$, the annihilator of $X$ is the subset of $A^*$ given by $$ X^o=\{f\in A^*: f(x)=0\,\forall x\in X\}. $$ (note that almost all equal signs below mean isomorphism)

It is not hard to prove that $X^{oo}=X^{**}$. Now if $A=X\oplus Y$, then $A^*=X^*\oplus Y^*$. Also $X=A/Y$, $Y=A/X$. And it is also not hard to check that $(A/X)^*= X^o$. Then $$ A^*=X^*\oplus Y^*=(A/Y)^*\oplus (A/X)^*=Y^o\oplus X^o $$ Then, as $A=J\oplus (A/J)$, $$ A^{**}=J^{oo}\oplus (A/J)^{oo}=J^{**}\oplus (A/J)^{**}. $$