2 4 A pr 1 99 2 A note on unconditional structures in weak

We prove that if a non-atomic separable Banach lattice in a weak Hilbert space, then it is lattice isomorphic to L2(0, 1). Introduction This note is to be considered as an addendum to [3], where an extensive study of Banach lattices, which are weak Hilbert spaces, was made. We prove that if a non-atomic separable Banach lattice is a weak Hilbert space then it is lattice isomorphic to L2(0, 1). The result is a consequence of Theorem 3.11 in [3] together with an easy, short argument. We believe that it may have some impact om the study of unconditional structures in weak Hilbert spaces. For the convenience of the reader we have in section 1 given the definition of a weak Hilbert space and formulated the special case of [3], Theorem 3.11, which is needed to prove our result. 1 Notation and terminology In this note we shall use the notation and terminology commonly used in Banach space theory as it appears in [1] and [2].