Homomorphic conditional expectations as noncommutative retractions

Abstract

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.