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Autumn 2017 Homomorphic conditional expectations as noncommutative retractions
Robert Pluta, Bernard Russo
Adv. Oper. Theory 2(4): 396-408 (Autumn 2017). DOI: 10.22034/aot.1705-1161

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.

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Robert Pluta. Bernard Russo. "Homomorphic conditional expectations as noncommutative retractions." Adv. Oper. Theory 2 (4) 396 - 408, Autumn 2017. https://doi.org/10.22034/aot.1705-1161

Information

Received: 12 May 2017; Accepted: 6 June 2017; Published: Autumn 2017
First available in Project Euclid: 4 December 2017

zbMATH: 1385.46055
MathSciNet: MR3730035
Digital Object Identifier: 10.22034/aot.1705-1161

Subjects:
Primary: 46L99
Secondary: 17C65

Rights: Copyright © 2017 Tusi Mathematical Research Group

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Vol.2 • No. 4 • Autumn 2017
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