I am trying to understand the proof of the GNS construction. I don't understand the follwoing step: Let $\phi:A\to\mathbb{C}$ be a state, i.e. a positive ($\phi(a^*a)\geq 0$), linear functional. We consider the following set: $N_\phi=\{a\in A|\phi(a^*a)=0\}$. Then I want to show that this is a closed leftideal. Let $a\in N, b\in A$ then $\phi((ba)^*ba)=0$ is to show for left ideal.
I did read that we have $a^*a\leq ||a||^21$, but why is this true? I have the definition of the order $a\leq b$ if and only if $b-a$ is a positive element, i.e. of the form $b-a=c^*c$ for some $c\in A$.
Now if the above holds we have $a^*b^*ba\leq a^*||b||^2a=||b||^2a^*a$, since $b\mapsto a^*ba$ preserves positivity.
But then why does $\phi$ preserve $\leq$, i.e. why does $\phi(a^*b^*ba)\leq \phi(a^*||b||^2a)$ hold?
I would like to avoid the consequence $a\leq b \Rightarrow ||a||\leq||b||$, since I want to show this, using the GNS construction. However if someone can tell me how to prove this implication without GNS then I could use it.