In this paper, a new definition of numerical radius for adjointable operators in Hilbert -module space will be introduced. We also give a new proof of numerical radius inequalities for Hilbert space operators.

We construct an example of a Hilbert $$C^*$$ -module which shows that Troitsky’s theorem on the geometric essence $$ {\mathcal A} -compact operators between -modules cannot be extended to modules are not countably generated case (even in stronger uniform structure, is also introduced). In addition, constructed module admits no frames.