Unconditional bases in tensor products of Hilbert spaces

Abstract

We prove that a tensor norm alpha (defined on tensor products of Hilbert spaces) is the Hilbert-Schmidt norm if and only if l(2) circle times(...)circle times l(2), endowed with the norm alpha, has an unconditional basis. This extends a classical result of Kwapien and Pelczynski. The symmetric version of that statement follows, and this extends a recent result of Defant, Diaz, Garcia and Maestre.