Homomorphic conditional expectations as noncommutative retractions

Abstract

Let $A$ be a $C^*$-algebra and $\mathcal{E}: A \to A$ a conditional expectation. The Kadison-Schwarz inequality for completely positive maps, $$\mathcal{E}(x)^* \mathcal{E}(x) \leq \mathcal{E}(x^* x),$$ implies that $$\Vert \mathcal{E}(x)\Vert ^2 \leq \Vert \mathcal{E}(x^* x)\Vert.$$ In this note we show that $\mathcal{E}$ is homomorphic (in the sense that $\mathcal{E}(xy) = \mathcal{E}(x)\mathcal{E}(y)$ for every $x, y$ in $A$) if and only if $$\Vert \mathcal{E}(x)\Vert^2 = \Vert \mathcal{E}(x^*x)\Vert,$$ 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.