  Let $A$ be a $C^*$-algebra and $E\colon A \to A$ a conditional expectation.
The Kadison-Schwarz inequality for completely positive maps, $$E(x)^*E(x) \leq
E(x^* x),$$ implies that
  $$
  \|E(x)\|^2 \leq \|E(x^* x)\|.$$
  In this note we show that $E$ is a homomorphism if and only if
  $$\|E(x)\|^2 = \|E(x^*x)\|,$$ for every $x$ in $A$.
  We also prove that a homomorphic conditional expectation on a commutative
$C^*$-algebra $C_0(X)$ is given by composition with a continuous retraction of
$X$.
  One may therefore consider homomorphic conditional expectations as
noncommutative retractions.
