$C^{*}$ algebras positively dominated by finite dimensional algebras

Question

Assume that $A$ is a $C^{*}$ algebra and $B$ and $C$ are two sub $C^{*}$ algebras of $A$ such that:

1. $B$ is finite dimensional algebra.

2. For all positive $c\in C$, there exist a positive $b\in B$ such that $c<b$.

Can we say that $C$ is finite dimensional algebra, too?

Answer 1

If $A$ is any unital C*-algebra, the hypotheses are satisfied with $C=A$ and $B=\mathbb C1$.