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Complexication of the injective tensor product
In this section we show that the main result of the previous section has an analogue when the projective norm is replaced with the injective norm, and the Bocnak complexication procedure is replaced with the Taylor complexication procedure that was discussed in chapter 1.
Let X; Y be real Banach spaces. Since XC YC is isomorphic, as a complex vector space, to (X Y)C; the normed vector space ((XC; ) ((YC; ); _) is isomorphic, as a vector space, to ((X Y; _); ): (Here (XC; ) means the vector space XC with the above-mentioned Taylor complexication norm, we mention again that it is a complete normed space under the complexication norm; ( ;_) means endowed with the injective tensor norm, it isIn this section we show that the main result of the previous section has an analogue when the projective norm is replaced with the injective norm, and the Bocnak complexication procedure is replaced with the Taylor complexication procedure that was discussed in chapter 1. Let X; Y be real Banach spaces. Since XC YC is isomorphic, as a complex vector space, to (X Y )C; the normed vector space ((XC; ) ((YC; ); _) is isomorphic, as a vector space, to ((X Y; _); ): (Here (XC; ) means the vector space XC with the above-mentioned Taylor complexication norm, we mention again that it is a complete normed space under the complexication norm; ( ;_) means endowed with the injective tensor norm, it is not necessarily complete; and so on.) The next result will show that they are also isomorphic, in fact isometrically isomorphic, as normed vector spaces.
An injective relationship between the real and the complex tensor norm equivalence classes
How do the real tensor norms t in with the complex tensor norms? Does every real tensor norm have a complex counterpart? Or are there \fewer » complex norms, perhaps because the conditions in the complex-scalar world are more stringent? All the examples of real tensor norms that one thinks of, such as the projective tensor norm, or the hilbertian tensor norm, have complex analogues. This chapter shows that that is to be expected, at least on the level of topological equivalence classes of (nitely generated) tensor norms.
The motivation for this chapter is the few paragraphs of the Resume [3, p 19] that A. Grothendieck uses to discuss the relation between real and complex tensor norms.
He starts by saying that the relation on the isometric level is not as simple as it appears, because the integral norm of even the identity operator on a nite-dimensional Banach space depends on whether one uses the real or complex scalars. Grothendieck then asserts that the situation is much clearer when one considers the topological equivalence classes of tensor norms. (Remember that a \tensor norm » such as for example the injective tensor norm, is actually a family of norms in the conventional sense, the family being indexed by the collection of tensor products of Banach spaces. Two tensor norms are equivalent when they are equivalent on every tensor product of Banach spaces. By an \innite Lp?sum » argument, it turns out that this happens if and only if there are equivalence constants that are independent of the Banach spaces that appear in the tensor product to which the norms are applied.) Then, to continue paraphrasing Grothendieck, there will be
an injective correspondence, between the family of real {, and the family of complex tensor norm equivalence classes. This correspondence can be constructed using classes of -integral operators.
Grothendieck doesn’t prove this statement, and the reader should keep in mind that there are very few of the statements that appear in the Resume are given explicit proofs. (See, however the book [2].) In this chapter we construct such a correspondence. On the one hand our construction uses results of complexication [4], which appeared decades after the Resume.
On the other hand, we use nite-dimensional normed operator ideals, which can be shown to be in one-to-one correspondence with Grothendieck’s – integral operators. Thus, the construction as presented in this chapter is probably the one that Grothendieck had in mind.
The plan, broadly speaking, is this (denitions will be given in the main text): We start with a real tensor norm : We can associate to it in a canonical way a unique real nite-dimensional operator ideal. Then we show that one can complexify any real nite-dimensional operator ideal. Using the one-to-one correspondence between nite-dimensional operator ideals and tensor norms, this leads to a complex tensor norm. Using a similar route via nite-dimensional operator ideals, one can not only complexify a real tensor norm, but also dene the real version of a complex tensor norm. We then show that complexifying a real tensor norm and then taking the real part gives a real tensor norm that is always equivalent to the original real tensor norm. Then, it is not dicult to deduce that complexifying a topological equivalence class gives a well-dened complex topological equivalence class, in an injective way.
Part I: Isometric results
1 Preliminaries on complexication
1.1 Complexication procedures
1.2 More about the Taylor complexication procedure
2 Complexication of the injective and projective tensor prod-ucts of Banach spaces
2.1 The algebraic tensor product and complexication
2.2 Complexication of the projective tensor product
2.3 Complexication of the injective tensor product
Part II: Isomorphic results
3 An injective relationship between the real and the complex tensor norm equivalence classes
3.1 Finite dimensional operator ideal norms and tensor norms
3.2 Complexication of real nite dimensional operator ideals
3.3 The real version of a complex nite-dimensional operator ideal norm
3.4 Preservation of topological equivalence
