## Advances in Operator Theory

*Advances in Operator Theory* (*AOT*) is a peer-reviewed quarterly electronic journal published by the Tusi Mathematical Research Group (TMRG). *AOT* publishes survey articles and original research papers of high standards with deep results, new ideas, profound impact, and significant implications in all areas of operator theory and all modern related topics (e.g., functional analysis).

*AOT* is indexed by the Emerging Sources Citation Index, MathSciNet, and Zentralblatt MATH. Advance publication of articles online is available.

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**ISSN:**2538-225X (electronic)**Publisher:**Tusi Mathematical Research Group**Discipline(s):**Mathematics**Full text available in Euclid:**2016--**Access:**Articles older than 5 years are open**Euclid URL:**https://projecteuclid.org/aot

### Submissions

Manuscripts must be submitted via http://aot-math.org/.

### Best Paper Award 2018

* *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.