Tensor Gaussian measures on L^p(E)

Let (i, H, E) and (j, K, F) be abstract Wiener spaces and let α be a reasonable norm on E ⊗ F. We are interested in the following problem: is (i ⊗ j, H \ ̂bo₂ K, E \ ̂bo_αF) an abstract Wiener space ? The first thing we do is to prove that the setting of the problem is meaningfull: namely, i ⊗ j is always a continuous one to one map from H \ ̂bo₂ K into E \ ̂bo_αF. Then we exhibit an example which shows that the answer cannot be positive in full generality. Finally we prove that if F=L^p(X,X,λ) for some σ-finite measure λ ≥ 0 then (i⊗j, H⊗₂K,L^p(X,X,λ) is an abstract Wiener space. By-products are some new results on γ-radonifying operators, and new examples of Banach spaces and cross norms for which the answer is affirmative (in particular α = π the projective norm, and F=L¹(X,X,λ)).