Let $B$ be a $C^{\*}$-subalgebra of a $C^{\*}$-algebra $A$ with a faithful conditional expectation $P: A\rightarrow B$. [Kumjian](https://cms.math.ca/openaccess/cjm/v38/cjm1986v38.0969-1008.pdf) suggests on page 15 that since $P$ is faithful we have


$\left\Vert a\right\Vert =\text{sup}\left\{ \left\Vert P\left(c^{\*}a^{\*}ac\right)\right\Vert ^{1/2}\mid c\in A\text{, }P\left(c^{\*}c\right)\leq1\right\} $.


I assume the easiest way to see this is to check that the right side actually gives a $C^{\*}$-norm on $A$ and then make use of it's uniqueness. The faithfulness obviously gives the definiteness. What I don't see is why the right expression satisfies the triangular equation and the $\*$-condition. How can I check this?



Characterization of $C^{*}$-algebra norms via conditional expectations
I think this can be answered with GNS representation theory.
Note that all is needed is that $P$ is a faithful positive map between C\*-algebras.


Let $S(B)$ be the state space of $B$. Consider the representation
$$
\pi = \bigoplus\big\{ \pi\_{\phi\circ P} \mid \phi\in S(B) \big\}
$$
of $A$ on some (huge) Hilbert space, where each $\pi\_{\phi\circ P}$ is the GNS representation associated to the state $\phi\circ P$ on $A$.
As $P$ is faithful, it follows that so is $\pi$.
Thus $\pi$ is isometric. 
Given $a\in A$ and $\delta>0$, we may therefore find some $\phi\in S(B)$ with
$$
\|a\|^2 \leq \delta+\|\pi\_{\phi\circ P}(a)\|^2 = \delta+\sup\big\{ \|\pi\_{\phi\circ P}(a)\xi\|^2 \mid \xi\in H\_{\phi\circ P},\ \|\xi\|\leq 1 \big\}.
$$
But now each $\xi$ is approximated by an element of the form $\pi\_{\phi\circ P}(c)\xi\_{\phi\circ P}$, where $\xi\_{\phi\circ P}$ is the cyclic vector, and this element having norm at most one means $(\phi\circ P)(c^\*c)\leq 1$.
Applying this to the above supremum yields
$$
\begin{array}{ccl}
\|a\|^2 &\leq& \delta+\sup\big\{ (\phi\circ P)(c^\*a^\*ac) \mid c\in A,\ (\phi\circ P)(c^\*c)\leq 1 \big\} \\
&\leq& \delta+\sup\big\{ P(c^\*a^\*ac) \mid c\in A,\ P(c^\*c)\leq 1 \big\}.
\end{array}
$$
As $\delta>0$ was an arbitrary parameter, this shows the $\leq$-part in your equation. The inequality in the other direction is trivial as $P$ is positive.


