'projections' on $C^*$-algebras. Under certain assumptions, is $f(a)=a$ for all $a\in A$?

Question

Let $B$ a unital $C^*$-algebra and $A$ a $C^*$-subalgebra of $B$ containing the unit of $B$. Let $f:B\to B$ be a linear, unit-preserving, completely positive and idempotent map (i.e. $f^2=f$). ..[I have seen this as a definition of projections on a $C^*$-algebra]

Is then $f(a)=a$ for all $a\in A$?

I don't have a proof and I don't have a counterexample, I am undecided whether it is true or not (I tend to say 'not true'). I appreciate any suggestion and hints.

Answer 1

No. For instance, let $A=B=\mathbb{C}^2$ and let $f:\mathbb{C}^2\to\mathbb{C}^2$ be given by $f(x,y)=(x,x)$.

Answer 2

The answer is brutally no, in the sense that it is false for any $C^*$-algebra other than $\mathbb C$, if you choose $B$ carefully. Take $A$ to be any nontrivial unital C$^*$-algebra, $B=\mathbb C\,1$, and $g$ any state. Define $f(a)=g(a)1$.

Of course there are particular cases where the answer is yes. For instance, when $A$ is the injective envelope of $B$, as it happens when $A=B(H)$ and $B=K(H)$ (or $K(H)+\mathbb C 1$, if one requires $B$ to be unital).