Advances in Operator Theory

Homomorphic conditional expectations as noncommutative retractions

Robert Pluta and Bernard Russo

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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.

Article information

Adv. Oper. Theory, Volume 2, Number 4 (2017), 396-408.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L99: None of the above, but in this section
Secondary: 17C65: Jordan structures on Banach spaces and algebras [See also 46H70, 46L70]

conditional expectation Kadison inequality retraction triple homomorphism JC*-triple


Pluta, Robert; Russo, Bernard. Homomorphic conditional expectations as noncommutative retractions. Adv. Oper. Theory 2 (2017), no. 4, 396--408. doi:10.22034/aot.1705-1161.

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