I read a paper in which the author uses the propetry that if $A$ is a submodule of a

Hilbert $C^\ast$ module ($C$ is a $C^*$ algebra) such that

$A^\bot=0$ then $A$ is dense. I don't know how to prove it since in a gneral Hilbert module over a $C^*$ algebra $(A^\bot)^\bot$ is greater than $Closure(A)$ and the standard Hilbert space proof does not work.

The paper in question is: UNBOUNDED OPERATORS ON HILBERT C∗-MODULES OVER C∗-ALGEBRAS OF COMPACT OPERATORS by Guljas you can find on the web.

onlyconsiders modules over the algebra of compact operators, and such things are rather special. See the 2nd paragraph of the paper: "These modules are characterized by the property that each closed submodule is orthogonally complemented or orthogonally closed" $\endgroup$