# Homomorphic conditional expectations as noncommutative retractions

## TL;DR

This paper characterizes when a conditional expectation on a $C^*$-algebra is a homomorphism, showing it corresponds to a noncommutative retraction, and explores the structure of such expectations especially in commutative cases.

## Contribution

It establishes a criterion for homomorphic conditional expectations and links them to noncommutative retractions, extending the understanding of their structure in $C^*$-algebras.

## Key findings

- A conditional expectation is a homomorphism if and only if a specific norm equality holds.
- Homomorphic conditional expectations on commutative $C^*$-algebras correspond to composition with continuous retractions.
- Homomorphic conditional expectations can be viewed 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.