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To clarify, I mean "conditional expectation" in the sense of $C^*$-algebras (a completely positive projection of norm 1, equivalently, a completely positive linear map onto a $C^*$-subalgebra which is a bimodule map over its image). I'm wondering when this can be homomorphic. The only example I can think of is as follows: if $p$ is a central projection, then $x \mapsto px$ is a homomorphic conditional expectation. (For instance, in a $C^*$-algebra that is a direct sum, projection onto a summand is an instance of this.) Are there others?

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up vote 4 down vote accepted

Let $A=C[0,1]$, let $B$ be the subalgebra of constant functions, and let $\varphi:A\to B$ be defined by $\varphi(f)=f(0)$. Then $\varphi$ is a $*$-homomorphism (hence completely positive) of norm $1$, satisfies $\varphi(f)=f$ when $f$ is in $B$. Note that $B$ is not a summand of $A$; it is not an ideal. (And the only projections in $A$ are $0$ and $1$.)

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Thanks Jonas! One could slightly generalize this to any unital $C^*$-algebra $A$ (i.e. possibly noncommutative) with a multiplicative linear functional $\omega: A \to \mathbb{C}$, and then $a \mapsto \omega(a) \mathbb{1}$ is a homomorphic conditional expectation. –  Dave Gaebler Feb 15 '12 at 17:37
You're welcome, Dave. Good point. Come to think of it, the map $\mathbb C\times\mathbb C\to \mathbb C\cdot(1,1)$ defined by $(w,z)\mapsto (w,w)$ is a simpler example. –  Jonas Meyer Feb 15 '12 at 17:51
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