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A universal reflexive space for the class of uniformly convex Banach spaces. (English) Zbl 1108.46007
In [Proc. Am. Math. Soc. 79, 241–246 (1980; Zbl 0438.46005)], J. Bourgain proved that there is no universal space in the class of separable reflexive spaces: every separable Banach space which contains an isomorphic copy of every separable reflexive Banach space contains actually an isomorphic copy of every separable Banach space. He asked if there exists a separable reflexive Banach space which contains an isomorphic copy of every separable superreflexive Banach space (or, equivalently, of every separable uniformly convex Banach space). In the paper under review, the authors answer this question affirmatively.
Their proof uses a result of S. Prus [Bull. Pol. Acad. Sci., Math. 31, 281–288 (1983; Zbl 0547.46007)], which is a partial answer to Bourgain’s question: there exists a reflexive Banach space $$X_P$$ with a basis such that every separable superreflexive space with the approximation property is isomorphic to a complemented subspace of $$X_P$$. The crucial role in his proof is played by the notions of $$p$$-lower and $$q$$-upper estimates for disjoint blocks of a given finite-dimensional decomposition ($$FDD$$ with $$(p,q)$$-estimates); in particular, this space $$X_P$$ is universal for the class of Banach spaces having $$FDD$$ with $$(p,q)$$-estimates.
The authors of the present paper introduce the notion of $$(p,q)$$-tree estimates, $$1\leq q \leq p \leq \infty$$, and show that for a reflexive Banach space $$X$$, it is equivalent to say that: (a) $$X$$ has $$(p,q)$$-tree estimates, or that: (b) $$X$$ is isomorphic to a subspace of a reflexive space having an $$FDD$$ with $$(p,q)$$-estimates. That answers Bourgain’s question, since it follows from V. I. Gurarij and N. I. Gurarij [Math. USSR, Izv. 5 (1971), 220–225 (1972; Zbl 0248.46017); Izv. Akad. Nauk SSSR, Ser. Mat. 35, 210–215 (1971; Zbl 0214.12702)] (see also [R. C. James, Pac. J. Math. 41, 409–419 (1972; Zbl 0235.46031)]) that every uniformly convex space has $$(p,q)$$-tree estimates. This proof uses blocking techniques and M. Zippin’s result that every reflexive space can be embedded into a reflexive space with a basis [Trans. Am. Math. Soc. 310, No. 1, 371–379 (1988; Zbl 0706.46015)]. Actually, to show that (a) $$\Leftrightarrow$$ (b), the authors go through a third property: (c) $$X$$ is isomorphic to a quotient of a reflexive space having an FDD with $$(p,q)$$-estimates.
Reviewer: Daniel Li (Lens)

##### MSC:
 46B03 Isomorphic theory (including renorming) of Banach spaces 46B20 Geometry and structure of normed linear spaces
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##### References:
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