Can $C^*$ algebra $A$ be decomposed?

Question

If $A$ is a $C^*$ algebra ,$B$ is a finite dimensional $C^*$ subalgebra of $A$.Does there exists a $*$ subalgebra of $C$ such that $A=B \oplus C$?

Answer 1

No. In such an algebra $A$, both $B$ and $C$ would be nontrivial ideals. So if you take any simple C$^*$-algebra that has finite-dimensional subalgebras, your decomposition does not exist.

Such an example could be $A=$UHF$(2^\infty)$, which has lots of projections. Take a nontrivial projection $p\in A$, and put $B=\mathbb C\, p$. Then $B$ is one-dimensional but not an ideal, and thus the decomposition does not exit.

Any other simple real rank zero example would do.