On -envelopes of Banach spaces

Let A denote any of the following classes of L ∞ -spaces: C ( K ) -spaces, Lindenstrauss spaces, λ -separably injective spaces, universally λ -separably injective spaces, λ -Lindenstrauss–Pełczyński spaces or L ∞ , λ -spaces. We show that every Banach space X can be isometrically embedded into a space A ( X ) ∈ A so that every operator X → A with A ∈ A can be extended to an operator A ( X ) → A with the same norm.