Homomorphic conditional expectations as noncommutative
retractions
Let A be a C*-algebra and E a conditional expectation on A.
In this note we show that E is a homomorphism
if and only if equality holds for norms in the Kadison-Schwarz
inequality.
Jordan versions of this result are given.
We also prove that a homomorphic conditional expectation on a
commutative
C*-algebra is given by composition with a continuous retraction of its
maximal ideal space. One may therefore consider homomorphic conditional
expectations
as noncommutative retractions.
Robert Pluta < plutar@tcd.ie >
Bernard Russo < brusso@uci.edu >