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Yemon Choi
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Just so that this question can be marked as answered: Bill Johnson states in comments that those in the know call these Banach spaces "script $L_p$-spaces", and I trust his awareness/judgement of the norms in the community.

I was hoping that in the years since the foundational work, other terminology might have emerged; but so it goes.

Post Made Community Wiki by Yemon Choi