Complemented copies of c_0(tau) in tensor products of L_p[0,1]

International audienceLet X be a Banach space and τ an infinite cardinal. We show that if τ has uncountable cofinality, p ∈ [1, ∞) and either the Lebesgue-Bochner space L p ([0, 1], X) or the injective tensor product L_p [0, 1] ⊗_ ε X contains a complemented copy of c_0 (τ), then so does X. We show also that if p ∈ (1, ∞) and the projective tensor product L_p [0, 1] ⊗_π X contains a complemented copy of c_0 (τ), then so does X