When is c 0 (τ ) complemented in tensor products of l p (I) and X?

International audienceLet X be a Banach space, I an infinite set, τ an infinite cardinal and p ∈ [1, ∞). In contrast to a classical c 0 result due independently to Cembranos and Freniche, we prove that if the cofinality of τ is greater than the cardinality of I, then the injective tensor product p(I) ⊗ ε X contains a complemented copy of c 0 (τ) if and only if X does. This result is optimal for every regular cardinal τ. On the other hand, we provide a generalization of a c 0 result of Oya by proving that if τ is an infinite cardinal, then the projective tensor product p(I) ⊗ π X contains a complemented copy of c 0 (τ) if and only if X does. These results are obtained via useful descriptions of tensor products as convenient generalized sequence spaces