On L∞-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.