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Small subspaces of \(L_p\). (English) Zbl 1234.46017

The investigation of the subspaces of the classical Banach space \(L_p=L_p[0,1]\) has been in the centre of functional analytic interest since the days of Banach. It has long been known that the subspace structure of \(L_p\) is different in the ranges \(p<2\) and \(p>2\); for example \(L_r\) is isometric to a subspace of \(L_p\) if \(1\leq p\leq r\leq 2\), but this is not so if \(2<p<\infty\) and \(r\notin\{2,p\}\). This paper makes a deep contribution to understanding the subspaces of \(L_p\) for \(p>2\). Throughout this review \(p\) will always denote a real number greater than \(2\).
By now classical results due to M. Kadets and A. Pełczyński on the one hand and W.B. Johnson and E. Odell on the other hand imply, for a subspace \(X\subset L_p\), that if \(X\) is not hilbertian, then \(X\) contains a copy of \(\ell_p\), and if \(X\) does not contain a copy of \(\ell_2\), then \(X\) embeds into \(\ell_p\). The authors point out a combination of these results as the starting point for their studies: If \(X\subset L_p\) neither embeds into \(\ell_p\) nor into \(\ell_2\), then \(X\) contains a copy of \(\ell_p\oplus \ell_2\), which isolates \(\ell_p\oplus \ell_2\) as another “small” subspace of \(L_p\).
The main result of the paper takes this idea one step further: If \(X\subset L_p\) does not embed into \(\ell_p\oplus \ell_2\), then \(X\) contains a copy of \(\ell_p(\ell_2)\). Indeed, a more precise version is proved in Theorem B: If \(X\subset L_p\) does not embed into \(\ell_p\oplus \ell_2\), then for every \(\varepsilon>0\) there is a subspace \(Y_\varepsilon\subset X\) that is \((1+\varepsilon)\)-isometric to \(\ell_p(\ell_2)\) and complemented in \(L_p\) by a projection of norm \(\leq (1+\varepsilon)\gamma_p\), where \(\gamma_p\) denotes the \(L_p\)-norm of a standard Gaussian variable, and this estimate is optimal.
The proof of this result is very involved and difficult, and it takes the better part of this well written paper to complete it. The first step is an intrinsic characterisation of those subspaces of \(L_p\) that embed into \(\ell_p\oplus \ell_2\) (Theorem A); a simplified version of this result reads as follows: \(X\subset L_p\) embeds into \(\ell_p\oplus \ell_2\) if and only if for every normalised weakly null tree in \(X\) there exists a branch \((x_i)\) such that for some \(K\geq1\) and all finite sets of scalars one has \[ \frac1K \Bigl\|\sum a_i x_i \Bigr\|_{L_p} \leq \|(a_i)\|_{\ell_p} + \Bigl\|\sum a_i x_i \Bigr\|_{L_2} \leq K \Bigl\|\sum a_i x_i \Bigr\|_{L_p}. \] The proof of this result, given in Section 3, depends on the techniques developed by Odell, Schlumprecht and their coauthors in the last decade.
In Section 4 a dichotomy of Kadets-Pełczyński type for subspaces \(X\) of \(L_p\) is proved. Its first half leads to the conclusion that \(X\) embeds into \(\ell_p\oplus \ell_2\); for this, inequalities due to Rosenthal and Burkholder play an important role. Section 5 is devoted to the proof that in the alternative case of the dichotomy \(\ell_p(\ell_2)\) \((1+\varepsilon)\)-embeds into \(X\). Section 6 details the argument that there is even a well-complemented \((1+\varepsilon)\)-copy of \(\ell_p(\ell_2)\); this proof relies on Aldous’s theory of random masures and the Krivine-Maurey theory of stable Banach spaces. A blend of both approaches that are commonly regarded as two faces of the same medal is needed here. Incidentally, the authors also record an easier proof due to G. Schechtman to obtain a \((1+\varepsilon)\)-copy of \(\ell_p(\ell_2)\) that is complemented by some norm.
The final two sections contain miscellaneous material: a new proof of the theorem due to W. B. Johnson and E. Odell [“Subspaces and quotients of \(\ell_p\oplus \ell_2\) and \(X_p\),” Acta Math.147, 117–147 (1981; Zbl 0484.46020)] saying that a subspace of \(L_p\) that is isomorphic to a quotient of \(\ell_p\oplus \ell_2\) in fact embeds into \(\ell_p\oplus \ell_2\), a discussion of the impossibility of a uniform (let alone almost isometric) embedding in the context of Theorem A, and remarks about \({\mathcal L}_p\)-subspaces of \(\ell_p\oplus \ell_2\) and Rosenthal’s space \(X_p\).

MSC:

46B25 Classical Banach spaces in the general theory
46B03 Isomorphic theory (including renorming) of Banach spaces
46B06 Asymptotic theory of Banach spaces

Citations:

Zbl 0484.46020
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References:

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