If you take a von Neumann algebra $M$ and any its maximal abelian subalgebra (masa) $D$, then there is a norm-one projection from $M$ onto $D$ (conditional expectation). The same is true if you take the Cuntz algebra $O_2$ and its canonical masa $C(\Delta)$. Is this true in general?
Let $A$ be C*-algebra. Is there a masa $D$ of $A$ onto which there is a conditional expectation (norm-one projection from $A$ onto $D$)?