annihilator of an ideal in $C^*$ algebra

Question

If $H$ is a hilbert space,K is a closed subspace of $H$,then $H=K\oplus K^\perp$.

If $A$ is a $C^*$ algebra,$I$ is a closed ideal of $A$.Does there exist a similar decomposition $A=I\oplus I^\perp$?

Answer 1

Such a decomposition basically requires that $I=Ap$ for a central projection $p$. That's the way with (sot-closed) ideals in von Neumann algebras.

It also fails in many other cases. Specifically, it will fail whenever $I$ is an essential ideal.

For instance, let $A=\ell^\infty(\mathbb N)$ and let $I=c_0$. If you try $A=I\oplus J$, take $x\in J$. Since $e_n\in c_0$ for all $n$, $x_ne_n=xe_n\in J$; but then $x_ne_n\in J\cap I=\{0\}$. Thus $x_n=0$ for all $n$ and $x=0$.

Answer 2

No. If $H$ is a separable infinite-dimensional Hilbert space, then $B(H)$ has a unique nontrivial closed ideal, namely the ideal $K(H)$ of compact operators. Hence $B(H)$ cannot decompose into a direct sum of two ideals.