Pluta, R., Russo, B. (2017). Homomorphic conditional expectations as noncommutative retractions. Advances in Operator Theory, 2(4), 396-408. doi: 10.22034/aot.1705-1161

Robert Pluta; Bernard Russo. "Homomorphic conditional expectations as noncommutative retractions". Advances in Operator Theory, 2, 4, 2017, 396-408. doi: 10.22034/aot.1705-1161

Pluta, R., Russo, B. (2017). 'Homomorphic conditional expectations as noncommutative retractions', Advances in Operator Theory, 2(4), pp. 396-408. doi: 10.22034/aot.1705-1161

Pluta, R., Russo, B. Homomorphic conditional expectations as noncommutative retractions. Advances in Operator Theory, 2017; 2(4): 396-408. doi: 10.22034/aot.1705-1161

Homomorphic conditional expectations as noncommutative retractions

^{1}Department of Mathematics, University of California, Irvine

^{2}Department of Mathematics, University of
California, Irvine

Abstract

Let $A$ be a $C^*$-algebra and $\mathcal{E}\colon A \to A$ a conditional expectation. The Kadison-Schwarz inequality for completely positive maps, $$\mathcal{E}(x)^* \mathcal{E}(x) \leq \mathcal{E}(x^* x),$$ implies that $$\left\Vert\mathcal{E}(x)\right\Vert^2 \leq \left\Vert\mathcal{E}(x^* x)\right\Vert.$$ 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 $$\left\Vert\mathcal{E}(x)\right\Vert^2 = \left\Vert\mathcal{E}(x^*x)\right\Vert,$$ 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.