A new uniform structure for Hilbert \(C^*\)-modules

We introduce and study some new uniform structures for Hilbert \(C^*\) $C^{\ast }$ -modules over a \(C^*\) $C^{\ast }$ -algebra \(\mathcal {A}.\) $\mathcal{A}.$ In particular, we prove that in some cases they have the same totally bounded sets. To define one of them, we introduce a new class of \(\mathcal {A}\) $\mathcal{A}$ -functionals: locally adjointable functionals, which have interesting properties in this context and seem to be of independent interest. A relation between these uniform structures and the theory of \(\mathcal {A}\) $\mathcal{A}$ -compact operators is established.