Completability and optimal factorization norms in tensor products of Banach function spaces.

Given σ -finite measure spaces (Ω 1 , Σ 1 , μ 1) and (Ω 2 , Σ 2 , μ 2) , we consider Banach spaces X 1 (μ 1) and X 2 (μ 2) , consisting of L 0 (μ 1) and L 0 (μ 2) measurable functions respectively, and study when the completion of the simple tensors in the projective tensor product X 1 (μ 1) ⊗ π X 2 (μ 2) is continuously included in the metric space of measurable functions L 0 (μ 1 ⊗ μ 2) . In particular, we prove that the elements of the completion of the projective tensor product of L p -spaces are measurable functions with respect to the product measure. Assuming certain conditions, we finally show that given a bounded linear operator T : X 1 (μ 1) ⊗ π X 2 (μ 2) → E (where E is a Banach space), a norm can be found for T to be bounded, which is 'minimal' with respect to a given property (2-rectangularity). The same technique may work for the case of n-spaces. [ABSTRACT FROM AUTHOR]