Suppose one has a densely defined symmetric operator $T:\mathcal{M}\rightarrow\mathcal{M}$, where $\mathcal{M}$ is a Hilbert $A$-module for a $C^*$-algebra $A$. Suppose that $T$ is non-negative, so that for all $x\in\mathcal{M}$, $$\langle x,Tx\rangle_{\mathcal{M}}\geq 0.$$
When $A=\mathbb{C}$, $T$ has a Friedrichs extension to a self-adjoint operator.
Question: Has an analogous result been proved for $A$ a general $C^*$-algebra?
