Let $X$ be a von Neumann algebra. Let $f$ be a faithful semi-finite normal weight on $X$.
$$N_f:=\{x \in X : f(x^*x) < +\infty \}$$
$$n_f:=\{x \in X : f(x^*x)=0 \}$$
Most of operator algebras books start GNS construction by defining an inner product on the quotient space $N_f/n_f$ by:
$$\langle\alpha(x),\alpha(y)\rangle := f(y^*x)$$
Where $\alpha$ is the quotient map from $N_f$ to $N_f/n_f$.
Except in E. C. Lance. Ergodic Theorems for Convex Sets and Operator Algebras. Page 205. He defined $N_0= N_f \cap N_f^* $ then an inner product $$\langle x,y \rangle := f(y^*x)$$
How could this be done ?
Is this $N_f/n_f=N_0$ true ?
Is there any reference explaining GNS construction for weight on von Neumann hot for state of positive linear functional defined on $C^*$- algebras ?
The word you need to pay attention to is "faithful". If $f$ is a faithful semi-finite weight on $X$ then by definition $f(x)=0$ implies $x=0$, for $x\in X^+$. Thus $n_f$ is $\{0\}$. You only need to consider $n_f$ when $f$ is not faithful (which is more comment when considering e.g. states rather than weights).
That Lance considers $N_f \cap N_f^*$ and not just $N_f$ is because he wishes to appeal to the theory of (left) Hilbert algebras (i.e. Tomita-Takesaki Theory) and $N_0$ is a full left Hilbert algebra (or rather the image of $N_0$ in $H$ is).