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We give a criterion ensuring that the elementary class of a modular Banach space \(E\) (that is, the class of Banach spaces, some ultrapower of which is linearly isometric to an ultrapower of \(E\)) consists of all direct sums \(E\oplus_m H\), where \(H\) is an arbitrary Hilbert space and \(\oplus_m\) denotes the modular direct sum. Also, we give several families of examples in the class of Nakano...
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