Concerning Theorem 3.3.3 Murphy's C*-algebras and operator theory ( characterization of positive functionals)

Question

i'm working through Murphy's book on C*-algebras and don't get why the proof of theorem 3.3.3 should be as exhaustive as it is.

Briefly the point is : If $\varphi$ is a continuous (bounded) linear functional on a C*-algebra $A$ and $(u_\lambda)_\lambda$ an approximate unit s.t. $||\varphi||=\lim_\lambda\varphi(u_\lambda)$, then $\varphi$ is positive.

Throughout the proof one may assume $\varphi$ to be normalized!

Now, Murphy begins (in awfully boring way) to show that $\varphi$ maps hermitian elements of $A$ to real numbers and proceeds as follows:

Now suppose that $a$ is positive and $||a||\leq 1$. Then $u_\lambda - a$ is hermitian and $||u_\lambda -a||\leq 1$, so $\varphi(u_\lambda -a) \leq 1$. But then $1 - \varphi(a) = \lim_\lambda\varphi(u_\lambda-a)\leq 1$ and so $\varphi(a)\geq 0$

So, could you just point me to where we use that $\varphi$ maps self-adjoints on self-adjoints?

Thanks!

Answer 1

On the codomain, $\mathbb C$, the selfadjoints are the reals. So $\varphi$ maps selfadjoints to reals. This allows one to say that $\varphi(u_\lambda-a)\leq1$, as opposed to $|\varphi(u_\lambda-a)|$, which would not yield the desired inequality.