Tensor products of ordered Banach spaces
by
Richar Becker
ParisVI/CNRS

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This lecture is devoted to studying a generalization of the Lapreste norms on tensor products of Banach spaces, based on generating closed convex cones of the Banach spaces concerned.

Given 1 ≤ p ≤ ∞, we put:

Mp((zi))= sup {|| < zi, z'i > ||p:z' ∈ G', ||z'|| ≤ 1}

for every Banach space G and every sequence (z_i) in G.

If 1 ≤ p, q, r ≤ ∞ satisfy 1/r+1/p'+1/q'=1, given Banach spaces E and F and two convex cones X ⊂ E and Y ⊂ F as above, we consider on E⊗F the norm:

a+pq(z)= inf {||(li)||r Mq'((xi)) Mp'((yi))}

when the infimum is taken over all possible representations z=∑ⁿ₁ l_i x_i⊗y_i, with x_i ∈ X and y_i ∈ Y.

If X=E and Y=F, the classical Lapreste norm a_pq is obtained.

We extend, as far as possible, some of the results about Lapreste norms to this new setting. Two main topics are considered concerning these norms:

1) The determination of the topological dual of (E⊗F, a⁺_pq). We pay a special attention to the case when Y=F, and to the case when E is reflexive and r=∞.

2) The equivalence between norms a⁺_pq with different pairs (p, q). For this purpose we use a parameter i(X), which extends the cotype index q_E of B. Maurey and G. Pisier to the case of general cones.

We show that, when q₁, q₂ > i(X) and p₁, p₂ > i(Y), the norms a⁺_{p1 q1} and a⁺_{p2 q2} are equivalent on E⊗F.

Moreover, the index i(X) and its conjugate I(X) are involved in the study of convex cones, contained in a Banach space, with no dependence on tensor products.

Let X be a separable Banach space

The hardy spaces H_p(C) has the following property ... for 1 ≤ p < ∞.

Fermat was right

Is the universe finite?

References

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R. Becker, R. Convex cones, tensor products, and operators, Rev. Roumaine Math. Pures Appl. 52, (2007), 509-527.

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R. Becker, Ordered Banach spaces Hermann, 68, Paris, 2008.

Date received: January 5, 2010