Given a C*-algebra $\mathcal{A}$.
(It may or may not contain identity!)
Consider a positive linear functional: $$\omega:\mathcal{A}\to\mathbb{C}:\quad A\geq0\implies \omega(A)\geq0$$
Construct its semi-inner product space: $$\mathcal{X}_\omega:=\mathcal{A}:\quad\langle X,Y\rangle_\omega:=\omega(Y^*X)$$
Then $$\|AX\|_\omega^2=\omega(X^*A^*AX) \leq\|A^*A\|\omega(X^*X)=\|A\|^2\|X\|_\omega^2$$
Regard its null space: $$\mathcal{N}_\omega:=\{N\in\mathcal{X}_\omega:\|X\|_\omega=0\}$$
By the above it is a left ideal: $$0\leq\|AN\|_\omega\leq\|A\|\|N\|_\omega=0$$
Quotient out the null space: $$\mathcal{X}_\omega/\mathcal{N}_\omega:=\{[X]_\omega:X\in\mathcal{X}_\omega\}$$
But why does involution lift: $$(X+\mathcal{N})^*=X^*+\mathcal{N}^*\nsubseteq X^*+\mathcal{N}$$ Do I maybe miss a crucial argument??
That would give the setting for Tomita's Modular Theory:
An ideal is linearly closed. A key step in the GNS construction is that if $x,y\in\mathcal N$, then by Cauchy-Schwarz $$\tag{1} \omega((x+y)^*(x+y))=2\text{Re}\,\omega(y^*x)\leq|\omega(y^*x)|\leq\omega(x^*x)^{1/2}\omega(y^*y)^{1/2}=0. $$
So $$ \mathcal N +\mathcal N\subset\mathcal N. $$ The inclusion for the $*$ does not hold. But we don't need it: what we need if that if $z-x\in\mathcal N$ and $w-y\in\mathcal N$, then $$ \omega(w^*z)=\omega(y^*x). $$ This follows from $(1)$: $$ \omega(w^*z)=\omega(y^*z)=\omega(y^*x), $$ since $\omega((w-y)^*x)=0$, and $\omega(y^*(z-x))=0$.