# Analogue of Friedrichs extension for Hilbert $C^*$-modules

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?