Homomorphic conditional expectations as noncommutative retractions

Abstract

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

Accepted by Advances in Operator Theory, 14 pages