On conditional expectation of tracial C star algebra

Question

Let $A$ be a C*-algebra with a trace $\tau$. $B$ be a C*-subalgebra of $A$. Do always exist trace preserving conditional expectation exist like in von Neumann algebra?

Answer 1

No, not even removing the need for the conditional expectation to be $\tau$-preserving. If $A$ is nuclear, any sub-$C^\ast$-algebra $B$ in the range of a conditional expectation will also be nuclear. But there are non-nuclear $C^\ast$-algebras that embed into nuclear ones.