Abstract. Let V be a countably generated Hilbert C * -module over a C * -algebra A. We prove that a sequence {f i : i ∈ I} ⊆ V is a standard frame for V if and only if the sum i∈I x, f i f i , x converges in norm for every x ∈ V and if there are constants C,We also prove that surjective adjointable operators preserve standard frames. A class of frames for countably generated Hilbert C * -modules over the C * -algebra of all compact operators on some Hilbert space is discussed.

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In this paper we introduce a strong version of the Birkhoff-James orthogonality in Hilbert C * -modules. More precisely, we consider elements x and y of a Hilbert C * -module V over a C * -algebra A which satisfy x ≤ x + ya for all a ∈ A. We show that this relation can be described as the Birkhoff-James orthogonality of appropriate elements of V, and characterized in terms of states acting on the underlying C * -algebra A. Some analogous relations of this type are considered as well.

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