On frames for countably generated Hilbert 𝐶*-modules

Abstract

Let V V be a countably generated Hilbert C ∗ C^* -module over a C ∗ C^* -algebra A . A. We prove that a sequence { f i : i ∈ I } ⊆ V \{f_i:i\in I\}\subseteq V is a standard frame for V V if and only if the sum ∑ i ∈ I ⟨ x , f i ⟩ ⟨ f i , x ⟩ \sum _{i\in I}\langle x,f_i\rangle \langle f_i,x\rangle converges in norm for every x ∈ V x\in V and if there are constants C , D > 0 C,D>0 such that C ‖ x ‖ 2 ≤ ‖ ∑ i ∈ I ⟨ x , f i ⟩ ⟨ f i , x ⟩ ‖ ≤ D ‖ x ‖ 2 C\Vert x\Vert ^2\le \Vert \sum _{i\in I}\langle x,f_i\rangle \langle f_i,x\rangle \Vert \le D\Vert x\Vert ^2 for every x ∈ V . x\in V. We also prove that surjective adjointable operators preserve standard frames. A class of frames for countably generated Hilbert C ∗ C^* -modules over the C ∗ C^* -algebra of all compact operators on some Hilbert space is discussed.