Gelfand-Naimark-Segal construction confused

Question

I think I understand this some what, but still kinda confused. Let $A$ be a $^*$-algebra and $\varphi$ be a functional on $A$. My professor defined positive functional as $\varphi(x^*x) \geq 0$. From this functional we can define a function $$\langle x, y \rangle_\varphi = \varphi(y^*x)$$I know that I can show that this is a semi-inner product. Then I can define $\mathcal{N}_\varphi := \{x \in A: \langle x, x \rangle_\varphi = 0\}$. I can show that this is a left ideal for our algebra $A$. Then, they induce an inner product on $A/\mathcal{N}_\varphi$. Would this inner product be $\langle x + \mathcal{N}_\varphi, y + \mathcal{N}_\varphi \rangle := \langle x, y \rangle_\varphi$? Also, I never understood what they mean by the completion? I obviously know that a metric space that is complete means that every cauchy sequence converges. I know that if $A/\mathcal{N}_\varphi$ is complete under our innerproduct norm, then it's a Hilbert space. However, I know that not all the time $A/\mathcal{N}_\varphi$ may not be complete. Sorry if this is obvious since I feel that I understand most of this.

Answer 1

Everything you say is fine. And yes, the inner product in the quotient is defined by taking the inner product of two representatives.

A metric space will usually not be complete, but it can be completed.

Since the metric you are using is induced by the inner product, it is easy to show that the inner product is bi-continuous and hence it extends to the completion. Now you have a complete inner product space, hence a Hilbert space.