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.

  • 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:


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Best Paper Award 2018

Homomorphic conditional expectations as noncommutative retractions

Robert Pluta and Bernard Russo Volume 2, Number 4 (2017)


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.