Separable elastic Banach spaces are universal

By Dale E. Alspach, Bünyamin Sarı

A Banach area X is elastic if there's a consistent ok in order that each time a Banach house Y embeds into X, then there's an embedding of Y into X with consistent okay . We turn out that C[0,1] embeds into separable limitless dimensional elastic Banach areas, and accordingly they're common for all separable Banach areas. This confirms a conjecture of Johnson and Odell. The facts makes use of incremental embeddings into X of C(K) areas for countable compact ok of accelerating complexity. to accomplish this we advance a generalization of Bourgain's foundation index that applies to unconditional sums of Banach areas and turn out a strengthening of the vulnerable injectivity estate of those C(K) that's discovered on certain reproducible bases.

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Remember that any C(K) house with countable compact okay is isomorphic to a couple C(α) area the place the latter denotes the distance C[1, α] of continuing capabilities on a successor ordinal α + 1 < ω1 built with the order topology [6]. For a given compact metric area ok, the corresponding α is decided as follows. enable okay (1) = {k : ∃kn ∈ okay, n = 1, 2, three, . . . , kn = km , m = n, kn → ok} be the set of restrict issues of ok. positioned ok (α+1) = (K (α) )(1) , and ok (β) = α<β ok (α) if β is a restrict ordinal. allow o(K) be the smallest ordinal such that ok (o(K)) has finite cardinality or, if ok (α) is often infinite, enable o(K) = ω1 .

Allow (xn )n∈M (2) , M (2) ⊂ M , be the subsequence of components with help strictly inside the help of xn2 . If p2 = p(q(1), 1) then online game 1 is begun with (xn )n∈M (1) and (yq(1,k) )k∈N and the integer l(2) = l(1, 1) ≥ n1 . the second one participant in online game 1 chooses n2 ∈ M (2) with n2 > l(1, 1) through the tactic from the inductive speculation for α = β(i1 ) and β = γ(1). continuing during this style at every one flip okay both pk = q(0, j) for a few j and the second one participant choices nk = m(ij ) > l(k) > nk−1 with β(ij ) ≥ γ(j) or pk = q(j, r) for a few, j, r, and the r flip of the sport j is performed with l(k) = l(j, r) > nk−1 to select nk ∈ M (j).

Schechtman To the reminiscence of Edward Odell MSC: fundamental 46B03 secondary 46B25 key terms: Bourgain’s index vulnerable injective Reproducible foundation areas of continuing services on ordinals a b s t r a c t A Banach area X is elastic if there's a consistent ok in order that at any time when a Banach area Y embeds into X, then there's an embedding of Y into X with consistent ok. We turn out that C[0, 1] embeds into separable infinite dimensional elastic Banach areas, and accordingly they're common for all separable Banach areas.

Bourgain, On separable Banach areas, common for all separable reflexive areas, Proc. Amer. Math. Soc. seventy nine (1980) 241–246. [4] W. B. Johnson, E. Odell, The diameter of the isomorphism category of a Banach area, Ann. of Math. (2) 162 (2005) 423–437. [5] J. Lindenstrauss, A. Pełczyński, Contributions to the idea of the classical Banach areas, J. Funct. Anal. eight (1971) 225–249. [6] S. Mazurkiewicz, W. Sierpiński, Contribution à los angeles topologie des ensembles dénombrables, Fund. Math. 1 (1920) 17–27. [7] E. Odell, Ordinal indices in Banach areas, Extracta Math.

I hence for all N ≥ 1 there are branches of a n=N Yn c0 -tree of size i in V . seeing that X is K-elastic, for every i ∈ N there's an isomorphism Si from V i into X such that v i ≤ Si v ≤ okay v i for all v ∈ V i . hence X has the tree index (with constants ok) at the least ω. We now expand this a bit via staring at that we will be able to embed massive components of the areas V i at the same time into X as disjointly supported blocks of (wj ). enable δ > zero and allow (yi,n,k )∞ k=1 be a two-player complementably sequentially reproducible foundation of the subspace of V i spanned by means of (T yn,k )k∈Kn (in norm · i ).

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