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This is an exercise from Brown-Ozawa book:

2.1.8. If $\theta: A\to B$ is nuclear and $C\subseteq B$ is a $C^*$-subalgebra with the properties that:

(a). $\theta(A)\subseteq C$ and

(b). There exist a sequence of c.c.p. maps $\varphi_n: B\to C$ such that $\varphi_n|_{C} \to id_C $ in the point-norm topology,

then $\theta:A\to C$ is also nuclear.

My first question is whether the conditions guarantee us that there exists a condinitional expectation $E: B\to C$ $?$
Is yes, then we're done. I thought some argument with $limsup$ could help.
Second, we hace $\varphi_n \circ \theta$ are nuclear and converges (pointwise in norm) to $\theta$. Is that true that a limit of nuclear maps is again nuclear? If yes, we're done.

Thanks.

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Well, my answer does not answer your questions regarding the existence of conditional expectation in this case or if the point-norm limit of nuclear maps is nuclear. These are interesting questions also for me, I hope someone else would answer them.

However, here is a solution to the question:

Recall the equivalent definition of nuclear map, that can be found here:
An exercise about the definition of nuclear maps

So, let $F\subseteq A$ be finite subset and $\epsilon>0$ be given.
As $\varphi_n|_C \to id|_C$ pointwise, and the range of $\theta$ is contained in $C$, there exists $N_0\in \Bbb{N}$ s.t. $||\varphi_{N_0}(\theta(a))-\theta(a)||<\epsilon/2$ for all $a\in F$.
Using the equivalent definition for nuclearity of $\theta$ we also have $n$ and pair of c.c.p. maps $\phi:A\to M_n$ and $\psi:M_n\to B$ s.t. $||\psi\circ \phi(a)-\theta(a)||<\epsilon/2 $ for all $a\in F$.
Now, $\phi: A\to M_n$ and $\varphi_{N_0}\circ \psi: M_n\to C$ are c.c.p. maps that satisfy for all $a\in F$:

$||\varphi_{N_0}\circ \psi \circ \phi (a)-\theta(a)||\leq ||\varphi_{N_0}(\psi(\phi(a)))-\varphi_{N_0}(\theta(a))||+||\varphi_{N_0}(\theta(a))-\theta(a)||<\epsilon$.

As required.

I hope this helps .

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    $\begingroup$ BTW I think similar arguments would show that limit of nuclear maps is nuclear. $\endgroup$ – Shirly Geffen Nov 1 '16 at 13:40

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