  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
direct sums of finite dimensional normed spaces that satisfy this criterion.
This yields many new examples of uncountably categorical Banach spaces, in the
model theory of Banach space structures.
