decomposition induced by a conditional expectation on $C^*$-algebras

Question

Let $B$ be a $C^*$-subalgebra of a $C^*$-algebra $A$ and suppose that $\theta: A\rightarrow B$ is a conditional expectation. Does there exist a $C^*$-subalgebra $C$ of $A$ such that

$A=B\oplus C$ ?

Can we take $C=\ker \theta$?

Answer 1

Can we take $C=\ker\theta$?

Only if $A=B\oplus C$. In general, that's not the case; $\ker\theta$ is not even a subalgebra in general. For instance take $A=M_2(\mathbb C)$, $B$ the diagonal matrices, and $$ \theta:\begin{bmatrix} a&b\\ c&d\end{bmatrix}\longmapsto\begin{bmatrix} a&0\\0&d\end{bmatrix}. $$

Answer 2

In general, $A$ is not $\ast$-isomorphic to $B\oplus C$ for a $C^\ast$-algebra $C$. Take for example $A=M_2(\mathbb C)$, $B=\mathbb C 1$, and $\theta\colon A\to B,\,a\mapsto \frac 1 2\mathrm{tr}(a)1$ (or any conditional expectation induced by a state). If $A\cong B\oplus C$, then $C$ must be 3-dimensional. But the only 3-dimensional $C^\ast$-algebra is $\mathbb C^3$, and $M_2(\mathbb C)\not\cong \mathbb C\oplus\mathbb C^3=\mathbb C^4$ since the left side is noncommutative, while the right side is commutative.

In fact, since it has trivial center, $M_n(\mathbb C)$ cannot be written as direct sum of any non-zero $C^\ast$-algebras $B,C$.