UC Irvine

Let A be a C*-algebra and ε: A → A a conditional expectation. The Kadison-Schwarz inequality for completely positive maps, ε(x)*ε(x) ≤ ε(x*x), implies that ε(x)2 ≤ ε(x*x) In this note we show that ε is homomorphic (in the sense that ε(xy) = ε(x)E(y) for every x, y in A) if and only if ε(x)2 = ε(x*x), for every x in A. We also prove that a homomorphic conditional expectation on a commutative C*-algebra C0(X) is given by composition with a continuous retraction of X. One may therefore consider homomorphic conditional expectations as noncommutative retractions.