Homomorphic conditional expectations as noncommutative retractions

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Let A be a C*-algebra and E A → A a conditional expectation. The Kadison-Schwarz inequality for completely positive maps, E(x)*E(x) ≤ E(x* x), implies that E(x)² ≤ 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 C₀(X) is given by composition with a continuous retraction of X. One may therefore consider homomorphic conditional expectations as noncommutative retractions.