Let $B\subset A$ be an inclusion of $C^*$ - algebras. I am having confusions on the existance of a conditional expectation $E: A \to B$. I could see that in general an inclusion need not have any conditional expectation. I couldnt get an example towards this.

1.But if we consider the inclusion of tracial von Neumann algebras, there exists a unique trace preserving faithful normal conditional expectation.

2.Is the same true for inclusion of finite von Neumann algebras ? I knw there exists a unique centre valued trace but can we find a scalar valued trace? Because the construction of the conditional expectation for a tracial von Neumann algebras are via the GNS construction. Is the same true for finite von Neumann algebras also?

3.For a $C^*$ algebra $A$, with $B$ is a Cartan subalgebra, then also there exists a faithful conditional expectation.

Is there any unified theorem which tells the existence of a conditional expectation(faithful) for a $C^*$ - algebra inclusion?