La structure des sous-espaces de treillis. (The structure of lattice subspaces) (Q2760595)

In this memoir, the new notion of lattice subspace (sous-espace de treillis), defined by G. Pisier, is investigated.NEWLINENEWLINENEWLINEA Banach space \(E\) is equipped with a lattice subspace structure when a Banach lattice \(X\) and an isometric embedding \(J: E\to X\) are associated to \(E\). The fundamental fact for the study of this structure is the following result of G. Pisier: NEWLINENEWLINENEWLINEFor every isometric embedding \(J: E\to X\) of \(E\) into a Banach lattice \(X\), there exists a norm \(\alpha\) on \(E\otimes c_0\), the algrebraic tensor product, such that: NEWLINE\[NEWLINE\alpha\big[(\text{id}_E\otimes T)(x)\big]\leq \|T\|\alpha(x)\tag{P}NEWLINE\]NEWLINE for every \(T\in {\mathcal L}(c_0)\), and every \(x\in E\otimes c_0\); and, conversely, such a norm \(\alpha\) gives an isometric embedding of \(E\) into a Banach lattice (actually, it is the ``conversely'' way which needs work). NEWLINENEWLINENEWLINEThis (previously unpublished) result is proved in Section 2. Every norm with property (P) is a cross-norm on \(E\otimes c_0\). The injective tensor norm on \(E\otimes c_0\) is the smallest norm with \((P)\) (and is associated to the isometric embedding \(J: E\to {\mathcal C}(B_{E^\ast},w^\ast)\), \(J(x)(x^\ast)=\langle x^\ast,x\rangle\)), giving the lattice subspace \(\min(E)\), and the projective tensor product on \(E\otimes c_0\) is the greatest norm giving a lattice subspace structure, \(\max(E)\), on \(E\).NEWLINENEWLINENEWLINEThis notion of lattice subspace is the commutative version of that of operator space, which is the coupling of a Banach space \(E\) with a norm \(\beta\), having some properties, on \(E\otimes {\mathcal K}(\ell_2)\).NEWLINENEWLINENEWLINELet us now describe the content of this memoir. It is divided in seven sections.NEWLINENEWLINENEWLINEIn Section 1, the author recalls some facts about Banach lattices, and proves that the adjoint of every regular operator \(T\) between two Banach lattices is also regular, and \(\|T^\ast\|_r=\|T\|_r\). For this, he introduces the notion of \(\ell_1\)-regular operator.NEWLINENEWLINENEWLINEIn Section 2, he states and proves Pisier's theorem quoted above.NEWLINENEWLINENEWLINEIn Section 3, the author defines lattice subspaces. The main tool for the study of this structure is the notion of regular operator, which generalizes the classical one between Banach lattices: if \((E,\alpha)\) (\(\alpha\) norm on \(E\otimes c_0\) with property \((P)\)) is a lattice subspace, one denotes by \(E(c_0)\) the completion of \(E\otimes c_0\) with respect to \(\alpha\); an operator \(T: E\to F\) is said to be regular if \(T\otimes \text{id}_{c_0}: E(c_0)\to F(c_0)\) is bounded, and then one sets \(\|T\|_r=\|T\otimes \text{id}_{c_0}\|\). When every operator \(T: E\to E\) is regular, \(E\) is said to be homogeneous, and \(\delta(E)=\sup\{\|T\|_r\); \(T: E\to E\), \(\|T\|=1\}\) is called the homogeneous constant of \(E\). When \(T\) is bijective and \(T\) and \(T^{-1}\) are regular, \(T\) is said to be a regular isomorphism, and a regular isometry if \(\|T\|_r=f \|T\|_r=\|T^{-1}\|_r=1\). Gaussian subspaces of \(L^2({\mathbf P})\) are homogeneous with constant \(1\), and it is recalled that \textit{P. Rauch} [Stud. Math. 100, No. 3, 251-282 (1991; Zbl 0759.46022)] proved that the space generated by the Rademacher functions is not homogeneous. The author then shows that for every isometric embedding \(J: E\to X\) of \(E\) into a Banach lattice \(X\), the norm on \(E\otimes \ell_p\), as a subspace of \(X(\ell_p)\), does not depend on the isometry \(J\), nor of \(X\). That allows him to define the spaces \(E(\ell_p)\), and then \(p\)-convexity and \(q\)-concavity for lattice subspaces. NEWLINENEWLINENEWLINEIn Section 4, the author studies homogeneous Banach lattices (every Banach lattice has an obvious canonical lattice subspace structure); he computes \(\delta(\ell_p^n)\), for \(n=2^m\), and shows that \(\ell_p\) and \(L^p\), \(1<p<\infty\), are not homogeneous. Actually, he proves (Théorème 4.1): let \(X\) and \(Y\) be Banach lattices such that every operator \(T: X\to Y\) is regular; thenNEWLINENEWLINENEWLINE1) \(X\) is an \(AL\)-space if \(p(Y)\equiv \sup\{p\geq 1\); \(Y\) \(p\)-convex\(\}<+\infty\), andNEWLINENEWLINENEWLINE2) \(Y\) is an \(AM\)-space if \(q(X)\equiv \sup\{q\geq 1\); \(Y\) \(p\)-concave\(\}>1\). NEWLINENEWLINENEWLINEThis is a generalization of a result of \textit{D. I. Cartwright} and \textit{H. P. Lotz} [Math. Z. 142, 97-103 (1975; Zbl 0293.46011)], and the proof uses \textit{J. L. Krivine}'s theorem on finite representability, in the lattice meaning, of \(\ell_{p(X)}\) and \(\ell_{q(X)}\) into every Banach lattice \(X\) [Ann. Math., II. Ser. 104, 1-29 (1976; Zbl 0329.46008)]. It follows that the only homogeneous Banach lattices are \(AL\)-spaces and \(AM\)-spaces.NEWLINENEWLINENEWLINESection 5 is devoted to the homogeneous subspaces of \(L^p\), \(1\leq p<+\infty\). It is proved (Théorème 5.1) that, if \(X\) is a \(q\)-concave Banach lattice, with \(q<+\infty\), and \(E\subseteq X\) is isomorphic to a Hilbert space and is a homogeneous lattice subspace, for the lattice subspace structure induced by \(X\), then \(E\) is regularly isomorphic to a Gaussian space. This needs a generalization of a result of P. Rauch (op. cit.). It follows, using a generalization of Kadeč-Pełczyńki's result on the subspaces of \(L^p\), that every homogeneous subspace \(E\) of \(L^p\), \(2\leq p<+\infty\), is regularly isomorphic to a Gaussian space.NEWLINENEWLINENEWLINEIn Section 6, the author gives a version of Dvoretzky theorem for lattice subspaces, whose proof relies deeply on Pisier's one for operator spaces [\textit{G. Pisier}, Houston J. Math. 22, No. 2, 399-416 (1996; Zbl 0860.47029)]. He shows then that every finite dimensional space can regularly almost isometrically be embedded into \(\ell_\infty^n(\ell_1^n)\) for some integer \(n\). It is recalled that, in the non commutative situation, there are finite dimensional operator spaces which are not completely isomorphic to a subspace of any \(M_n\) [\textit{G. Pisier}, Astérisque 232, 159-186 (1995; Zbl 0844.46031)].NEWLINENEWLINENEWLINEFinally, in Section 7, the author studies the regular Banach-Mazur distance NEWLINE\[NEWLINEd_r(E,F)=\inf\{\|T\|_r\|T^{-1}\|_r;\;T\: E\to F\}NEWLINE\]NEWLINE for finite-dimensional Banach spaces \(E,F\). He shows that, if \(\dim E=\dim F=n\), then, in several situations, \(d_r(E,F)\leq n\). He left as an open problem whether it is always true. He shows also that \(\max(\ell_2)\), which is a \(2\)-concave lattice subspace, is not regularly isomorphic to a Gaussian space, and so cannot be embedded into a \(q\)-concave Banach lattice, for any \(q<+\infty\).NEWLINENEWLINENEWLINEIt is also a pleasure to say that this memoir is very well written.