Abstract
Given a purely non-atomic, finite measure space \((\Omega ,\Sigma ,\nu )\), it is proved that for every closed, infinite-dimensional subspace V of \(L_p(\nu )\) (\(1\le p<\infty \)) there exists a decomposition \(L_p(\nu )=X_1\oplus X_2\), such that both subspaces \(X_1\) and \(X_2\) are isomorphic to \(L_p(\nu )\) and both \(V\cap X_1\) and \(V\cap X_2\) are infinite-dimensional. Some consequences concerning dense, non-closed range operators on \(L_1\) are derived.
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1 Introduction
To generalize a theorem of Dixmier concerning operator ranges into Hilbert spaces [14] to \(L_p[0,1]\) (\(1<p<\infty \)), Cross, Ostrovskii, and Shevchik introduced the following notion: a Banach space X is said to have property (B) if for every closed, infinite-dimensional subspace V of X, there exists a projection \(P\in \mathcal {L}(X)\), such that both intersections \(V\cap P(X)\) and \(V\cap (I_X-P)(X)\) are infinite-dimensional [10][Sect. 5]).
On one hand, the aforementioned authors observed in [10] that any space containing an isomorphic copy of the hereditarily indecomposable space \(X_{\text {GM}}\) of Gowers and Maurey [18] does not have property (B). In particular, since \(X_{\text {GM}}\) is separable, neither C[0, 1] nor \(L_{\infty }[0,1]\) have property (B). On the other hand, they identified two natural classes of Banach spaces for which property (B) holds.
-
(i)
Banach spaces with an unconditional basis;
-
(ii)
subprojective Banach spaces.
Let us recall that a Banach space X is said to be subprojective if every closed, infinite-dimensional subspace Y of X contains a closed, infinite-dimensional subspace Z complemented in X. For more information about subprojective spaces, see [16] and [28].
It is well known that the spaces \(L_p[0,1]\) (henceforth, \(L_p\) for short) have an unconditional basis if and only if \(1<p<\infty \) (see [1][Theorem 6.1.6] and [29]). Moreover, it is also known that \(L_p\) is subprojective if and only if \(2\le p<\infty \) (see [30] and [1][Corollary 6.4.9]). An example of a subprojective space with basis but without unconditional basis is the quasi-reflexive James space \(\mathcal {J}\) [1][Corollary 3.4.7].
It is straightforward from the comments above that \(L_p\) has property (B) if \(1<p<\infty \), and does not have it if \(p=\infty \). Since \(L_1\) does not belong to neither of types (i) nor (ii), Cross and his coauthors ask if \(L_1\) has property (B) [10][Problem 6.3]).
It is well known that for any decomposition \(L_1=X_1\oplus X_2\), at least one of the factors \(X_i\) is isomorphic to \(L_1\) [15] and the other factor is a \(\mathcal {L}_1\)-space [8], but it is unknown if it is isomorphic to a space other than \(L_1\), \(\ell _1\) or a finite-dimensional space. These facts led Odell et al. to introduce the strong property (B) and to ask if \(L_1\) has this property [27][Problem 4], where this property is defined as follows:
Definition 1.1
A Banach space X has the strong property (B) if for every closed, infinite-dimensional subspace V of X, there exists a projection \(P\in \mathcal {L}(X)\), such that both spaces P(X) and \((I-P)(X)\) are isomorphic to X and both intersections \(V\cap P(X)\) and \(V\cap (I-P)(X)\) are infinite-dimensional.
Note that \(\mathcal {J}\) does not have the strong property (B), because \(\dim {\mathcal {J}^{**}/\mathcal {J}}=1\), and therefore, \(\mathcal {J}\) cannot be isomorphic to \(\mathcal {J}\times \mathcal {J}\).
The central result in this paper (Theorem 3.3) positively answers the question posed by Odell and his coauthors. Even more, it is proved that for any finite measure space \((\Omega ,\Sigma ,\nu )\) with no atom, \(L_p(\Omega ,\Sigma ,\nu )\) has the strong property (B) for all \(1\le p<\infty \) (Theorem 4.1 and Proposition 4.3). Moreover, as a consequence of Theorem 3.3, it is proved in Corollary 4.2 that the aforementioned generalization of Dixmier’s theorem can be extended to \(L_1\).
The results presented in this article are based on three results: the discovery of Aldous that every infinite-dimensional, reflexive subspace E of \(L_1\) contains a subspace isomorphic to \(\ell _p\) for some \(1<p<\infty \) [2], a theorem by Dacunha–Castelle and Krivine that establishes that if that subspace E contains an isomorph of \(\ell _p\) then E contains \(\ell _p\) almost isometrically [11], and a change of density found by Berkes and Rosenthal [5].
The notation to be used throughout this paper is introduced next. Given a topological space \((S,\tau )\), an ultrafilter \(\mathfrak {U}\) on a set of indices I and \(x_0\in S\), the convergence of \((x_i)_{i\in I}\subset S\) to \(x_0\) following \(\mathfrak {U}\) is denoted and for short, by . The notation if there is no confusion) or \(\lim _{j}x_j=x_0\) is reserved for the usual convergence of a sequence \((x_j)_{j=1}^{\infty }\) to \(x_0\).
Given a pair of Banach spaces \((X,\Vert \cdot \Vert _X)\) and \((Y,\Vert \cdot \Vert _Y)\), the kernel and the range of an operator \(T:X\longrightarrow Y\) are denoted N(T) and R(T), respectively. Operators are linear and bounded maps; X and Y are isomorphic if there is a bijective operator \(T:X\rightarrow Y\), in which case, it is denoted \(X\simeq Y\). Given \(1\le p <\infty \), \(X\oplus _p Y\) denotes the Banach space \(X\times Y\) endowed with the norm \(\Vert (x,y)\Vert _p:=(\Vert x\Vert _X^p+\Vert y\Vert _Y^p)^{1/p}\). A sequence \((x_n)_{n=1}^{\infty }\) contained in X is said to be equivalent to a basic sequence \((y_n)_{n=1}^{\infty }\subset Y\) if the linear operator from \(\mathop {\overline{\mathrm{span}}}{\{y_n\}}_{n=1}^{\infty }\) into \(\mathop {\overline{\mathrm{span}}}{\{x_n\}}_{n=1}^{\infty }\) that maps \(y_n\) to \(x_n\) is a bijective isomorphism, in which case \((x_n)_{n=1}^{\infty }\) is also basic. For more information about Banach spaces, see [1, 25] and [26].
Given a collection \(\mathcal {A}\) of subsets of a set \(\Omega \), \(\sigma (\mathcal {A})\) denotes the \(\sigma \)-algebra generated by \(\mathcal {A}\). Given \(A\subset \Omega \), the indicator function associated with A is denoted \(\chi _A\).
Let \((\Omega ,\Sigma , \nu )\) be a measure space; a subset \(A\in \Sigma \) is said to be an atom if \(\nu (A)>0\) and \(\nu (A\cap B)\) is equal to \(\nu (A)\) or 0 for all \(B\in \Sigma \). A well-known theorem of Sierpinski states that for a measure space \((\Omega ,\Sigma ,\nu )\) with no atom, given \(A\in \Sigma \) and given \(0\le \lambda \le \nu (A)\), A contains a subset \(B\in \Sigma \), such that \(\lambda =\nu (B)\). For \(A\in \Sigma \), \(\Sigma (A):=\{B\cap A:B\in \Sigma \}\) is a \(\sigma \)-algebra and \((A,\Sigma (A),\nu \!\mid {\!}_{\Sigma (A)}\,)\) is a measure space; we will write \(\nu \) rather than \(\nu \!\mid {\!}_{\Sigma (A)}\,\) if there is no possible confusion. A density function is a measurable function \(\varphi :\Omega \longrightarrow [0,\infty )\), such that \(\int _{\Omega }\varphi \,\mathrm{d}{\nu }=1\); if \(\varphi (t)>0\) for all \(t\in \Omega \), \(\varphi \) is said to be a strictly positive density function. If \(\nu (\Omega )=1\), \((\Omega ,\Sigma ,\nu )\) is called a probability space and every \(\nu \)-measurable function \(f:\Omega \longrightarrow \mathbb {R}\) is called a random variable. If \(\nu (\Omega )<\infty \), then \((\Omega ,\Sigma ,\nu )\) is called a finite measure space. Given \(1\le p<\infty \), the Banach space of all (classes of) real-valued (alt., complex-valued) functions f for which \(|f|^p\) is integrable with respect of \(\nu \) is denoted by \(L_p(\Omega ,\Sigma ,\nu )\) (alt., \(L_p(\Omega ,\Sigma ,\nu ,\mathbb {C})\)), and in the case \(p=1\), its weak topology is denoted as w. The unit interval [0, 1] is denoted by \(\mathbb {I}\), the \(\sigma \)-algebra of borelian subsets of \(\mathbb {I}\) is denoted by \(\mathcal {B}\), and the Lebesgue measure on \(\mathcal {B}\) is denoted by \(\mu \). Sometimes, if the context is clear, the shorter notation \(L_p(\nu )\) will be used instead of \(L_p(X,\Sigma ,\nu )\). The shorter form \(L_p\) will be used to denote \(L_p(\mathbb {I},\mathcal {B},\mu )\).
Given a random variable f, \(\sigma (f)\) denotes the sub-\(\Sigma \)-algebra generated by f, that is, the sub-\(\Sigma \)-algebra generated by all the subsets \(\{s\le f(t)<r\}\) \(\{r,s\}\subset \mathbb {R}\). Independent random variables are measurable functions that are independent in the probabilistic sense. Given a random variable f and a sub-\(\sigma \)-algebra \(\mathcal {A}\) of \(\Sigma \), f and \(\mathcal {A}\) are said to be independent if \(\sigma (f)\) and \(\mathcal {A}\) are independent; and if g is another random variable, f and g are independent if so are \(\sigma (f)\) and \(\sigma (g)\).
Given a probability space \((\Omega ,\Sigma ,\nu )\), the conditional expectation of \(f\in L_1(\Omega ,\Sigma ,\nu )\) relative to a sub-\(\sigma \)-algebra \(\mathcal {S}\) of \(\Sigma \)—denoted \(\mathop {\mathbb {E}}(f|\mathcal {S})\)—is the only function in \(L_1(\Omega ,\mathcal {S},\nu \!\mid {\!}_{\mathcal {S}}\,)\) that satisfies \(\int _{C}f\,\mathrm{d}{\nu }=\int _{C}\mathop {\mathbb {E}}(f|\mathcal {S})\,\mathrm{d}{\nu }\) for all \(C\in \mathcal {S}\); the conditional expectation \(\mathop {\mathbb {E}}(\cdot |\mathcal {S})\) is a norm one operator from \(L_1(\Omega ,\Sigma ,\nu )\) onto \(L_1(X, \mathcal {S}, \nu \!\mid {\!}_{\mathcal {S}}\,)\); exactly, \(\mathop {\mathbb {E}}(f|\mathcal {S})\) is the Radon–Nikodym derivative of f with respect to \(\nu \!\mid {\!}_{\mathcal {S}}\,\); f is said to be of zero mean if \(\mathop {\mathbb {E}}(f):=\mathop {\mathbb {E}}(f|\{\emptyset ,\Omega \})=0\), that is, if \(\int _{\Omega }f\,\mathrm{d}{\nu }=0\). Section V in [13] and [4] covers all necessary facts about conditional expectations needed here.
2 Ultrapowers of \(L_1\) Spaces
This section begins with a short summary on ultrapowers of \(L_1\) spaces, whose study was initiated by Dacunha–Castelle and Krivine [11]. For proofs and details, see [21] and [17].
Let I be a set of indices. Through this paper, every ultrafilter \(\mathfrak {U}\) is a countably uncomplete ultrafilter on I, that is, an ultrafilter for which there exists a countable partition \(\{I_n\}_{n=1}^{\infty }\) of I disjoint with \(\mathfrak {U}\).
Let \(\Omega \) be any non-empty set and let \(\mathcal {B}(I,\Omega )\) denote the set of all families \((t_i)_{i\in I}\subset \Omega \). Let \(\sim \) be the equivalence relation in \(\mathcal {B}(I,\Omega )\) given by \((t_i)_{i\in I}\sim (s_i)_{i\in I}\) if \(\{i\in I:t_i=s_i\}\in \mathfrak {U}\). The theoretic ultrapower of \(\Omega \) following \(\mathfrak {U}\) is the quotient \(\Omega ^{\mathfrak {U}}:=\mathcal {B}(I,\Omega )/\sim \). The element of \(\Omega ^{\mathfrak {U}}\) with representative \((t_i)_{i\in I}\) will be denoted \((t_i)^{\mathfrak {U}}\). The theoretic ultraproduct of a family \(\{A_i\}_{i\in I}\) where \(A_i\subset \Omega \) is the subset \((A_i)^{\mathfrak {U}}:=\{(t_i)^{\mathfrak {U}}:t_i\in A_i,\ i\in I\}\). In accordance with this definition, \((A_i)^{\mathfrak {U}}\cup (B_i)^{\mathfrak {U}}=(A_i\cup B_i)^{\mathfrak {U}}\) and \((A_i)^{\mathfrak {U}}\cap (B_i)^{\mathfrak {U}}=(A_i\cap B_i)^{\mathfrak {U}}\).
Given a Banach space X, its ultrapower following \(\mathfrak {U}\) is the quotient Banach space \(X_{\mathfrak {U}}:=\ell _{\infty }(I,X)/{N}\) where N is the subspace of all families \((x_i)_{i\in I}\), such that \(\lim _{i\rightarrow \mathfrak {U}}x_i=0\). The element of \(X_{\mathfrak {U}}\) with representative \((x_i)_{i\in I}\) is denoted \([x_i]_{\mathfrak {U}}\) or \([x_i]\) for short and its norm can be calculated as \(\Vert [x_i]\Vert =\lim _{i\rightarrow \mathfrak {U}}\Vert x_i\Vert \). The space X is canonically embedded into \(X_{\mathfrak {U}}\) by means of the isometry \(J:X\longrightarrow X_{\mathfrak {U}}\) that maps each x to the constant class [x]. For simplicity, J is dropped in the notation, so Jx will be denoted x if the context is clear.
Let \((\Omega ,\Sigma ,\mu )\) be a probability space. The set \(\Sigma _{\mathfrak {U}}:=\{(A_i)^{\mathfrak {U}}:A_i\in \Sigma \}\) is a Boolean algebra. The smallest \(\sigma \)-algebra in \(\Omega ^{\mathfrak {U}}\) containing \(\Sigma _{\mathfrak {U}}\) will be denoted \(\sigma (\Sigma _{\mathfrak {U}})\). The measure \(\mu \) induces a probability measure \(\mu _{\mathfrak {U}}\) on \(\sigma (\Sigma _{\mathfrak {U}})\) defined on each element \((A_i)^{\mathfrak {U}}\in \Sigma _{\mathfrak {U}}\) as \(\mu _{\mathfrak {U}}\bigl ((A_i)^{\mathfrak {U}}\bigr ):=\lim _{i\rightarrow \mathfrak {U}}\mu (A_i)\) and extended to each \(A\in \sigma (\Sigma _{\mathfrak {U}})\) as
Each \(\mathbf {f}=[f_i]\in L_1(\mu _{\mathfrak {U}})\) induces a signed measure \(\nu _{\mathbf {f}}\) on \(\sigma (\Sigma _{\mathfrak {U}})\) as follows: given \(A\in \Sigma _{\mathfrak {U}}\)
and for each \(A\in \sigma (\Sigma _{\mathfrak {U}})\)
The space \(L_1(\mu _{\mathfrak {U}})\) is canonically embedded into \(L_1(\mu )_{\mathfrak {U}}\) by means of the isometry \(L:L_1(\mu _{\mathfrak {U}})\longrightarrow L_1(\mu )_{\mathfrak {U}}\) that maps each \(\chi _{(A_i)^{\mathfrak {U}}}\) to \([\chi _{A_i}]\). Unless it be necessary, the notation L will be omitted, so, for \(\mathbf {f}\in L_1(\mu _{\mathfrak {U}})\), \(L\mathbf {f}\) will be also denoted \(\mathbf {f}\). In turn, \(L_1(\mu )\) embeds into \(L_1(\mu _{\mathfrak {U}})\) via the isometry \(D:L_1(\mu )\longrightarrow L_1(\mu _{\mathfrak {U}})\) that maps \(\chi _A\) to \(\chi _{(A^\mathfrak {U})}\) for all \(A\in \Sigma \). For simplicity, the notation D will be also omitted, and consequently, the element Df that maps each \((t_i)^{\mathfrak {U}}\in \Omega ^{\mathfrak {U}}\) to \(\bigl ( f(t_i)\bigr )^{\mathfrak {U}}\) will be denoted f.
Given \(\mathbf {f}=[f_i]\in \bigl (L_1(\mu )\bigr )_{\mathfrak {U}}\), consider the Hahn decomposition \(\nu _{\mathbf {f}}=w_{\mathbf {f}}+m_{\mathbf {f}}\) where \(w_{\mathbf {f}}\) and \(m_{\mathbf {f}}\) are the absolutely continuous part and the singular part of \(\nu _{\mathbf {f}}\) with respect to \(\mu _{\mathfrak {U}}\). Let \(g_{\mathbf {f}}\) be the Radon–Nikodym derivative of \(w_{\mathbf {f}}\) with respect to \(\mu _{\mathfrak {U}}\), so that
The operator \(Q:L_1(\mu )_{\mathfrak {U}}\longrightarrow L_1(\mu )_{\mathfrak {U}}\) that maps each \(\mathbf {f}\) to \(g_{\mathbf {f}}\) satisfies QL is the identity on \(L_1(\mu _{\mathfrak {U}})\). Hence, LQ is a norm one projection from \(L_1(\mu )_{\mathfrak {U}}\) on \(L_1(\mu _{\mathfrak {U}})\), which determines a decomposition
for certain measure \(\mu (\mathfrak {U})\). Given \(\mathbf {f}\in L_1(\mu )_{\mathfrak {U}}\), the two following statements hold [17]:
-
(a)
\(\mathbf {f}\in L_1(\mu _{\mathfrak {U}})\) if and only if \(\mathbf {f}\) has a relatively weakly compact representative \((f_i)_{i\in I}\);
-
(b)
\(\mathbf {f}\in L_1\bigl (\mu (\mathfrak {U})\bigr )\) if and only if \(\mathbf {f}\) has a representative \((f_i)_{i\in I}\) for which \(\lim _{i\rightarrow \mathfrak {U}}\mu (\{|f_i|>0\})=0\).
An ultrafilter \(\mathfrak {U}\) on \(\mathbb {N}\) is said to be a p-point if every bounded sequence \((a_n)_{n=1}^{\infty }\) of real numbers contains a convergent subsequence \((a_{k_n})_{n=1}^{\infty }\), such that \(\{k_n:n\in \mathbb {N}\}\in \mathfrak {U}\) [7] [Definition 4.6 and Theorem 4.7]; an ultrafilter \(\mathfrak {U}\) on \(\mathbb {N}\) is said to be rare if for every countable partition \(\{F_n\}_{n=1}^{\infty }\) of \(\mathbb {N}\) into finite sets, there exists \(A\in \mathfrak {U}\), such that for every positive integer n, \(A\cap F_n\) has at most one element [9]. In the case when \(\mathfrak {U}\) is a rare p-point over \(\mathbb {N}\), statements (a) and (b) admit the following equivalent forms [17]:
-
(a’)
\(\mathbf {f}\in L_1(\mu _{\mathfrak {U}})\) if and only if \(\mathbf {f}\) has a representative weakly convergent following \(\mathfrak {U}\);
-
(b’)
\(\mathbf {f}\in L_1\bigl (\mu (\mathfrak {U})\bigr )\) if and only if \(\mathbf {f}\) has a representative \((f_n)_{n\in \mathbb {N}}\), such that \(f_n\cdot f_m=0\) \(\mu \)-a.e. for all n and all \(m\not =n\).
Obviously, if \((f_i)_{i\in I}\subset L_1(\mu )\) is relatively weakly compact, then there exists the weak limit w-\(\lim _{i\rightarrow \mathfrak {U}}f_i\in L_1(\mu )\). However, there are ultrafilters \(\mathfrak {W}\) on the set of positive integers \(\mathbb {N}\) for which there exists \(\mathbf {f}\in L_1\bigl (\mu (\mathfrak {W})\bigr )\setminus \{0\}\) and each of its representatives is weakly convergent following \(\mathfrak {W}\) [17][Proposition 16]; with a similar construction, it is not difficult to find a weakly null family \((f_i)_{i\in \mathbb {N}}\subset L_1(\mu )\) following \(\mathfrak {W}\), such that \((f_i)_{i\in J}\) is unbounded for all \(J\in \mathfrak {W}\).
Moreover, the existence of rare \(\mathrm{p}\)-points relies upon set theories accepting the Continuum Hypothesis or some of its weaker forms such as Martin’s axiom (see [17][Sect. 5] for a brief discussion).
The first result translates weak convergence in \(L_1\) into ultrapower and probabilistic language. In spite of its simplicity, it is worthy of a proof.
Lemma 2.1
Let \((\Omega ,\Sigma ,\mu )\) be a probability measure space, let \(\mathfrak {U}\) be an ultrafilter on a set I, and let \((f_i)_{i\in I}\) be a relatively weakly compact subset of \(L_1(\mu )\), so that \(\mathbf {f}:=[f_i]\in L_1(\mu _{\mathfrak {U}})\). Then, if and only if \(f=\mathop {\mathbb {E}}(\mathbf {f}|\Sigma )\).
Proof
By definition of conditional expectation, \(f=\mathop {\mathbb {E}}(\mathbf {f}|\Sigma )\) if and only if \(\int _Af\,\mathrm{d}{\mu }=\int _A\mathbf {f}\,\mathrm{d}{\mu _{\mathfrak {U}}}\) for all \(A\in \Sigma \). However, \(\mathbf {f}\in L_1(\mu _{\mathfrak {U}})\); hence, \(\int _A\mathbf {f}\,\mathrm{d}{\mu _{\mathfrak {U}}}=\lim _{i\rightarrow \mathfrak {U}}\int _Af_i\,\mathrm{d}{\mu }. \) and the result follows. \(\square \)
According to Berkes and Rosenthal [5], the following result on extraction of sequences can be derived from [11], but they do not offer any proof.
Let us recall that for \(1\le p\le 2\), a p-stable random variable on a probability space \((\Omega ,\Sigma ,\nu )\) is a random variable f for which there exists a constant \(c>0\), such that \(\mathop {\mathbb {E}}(e^{itf(\omega )})=e^{-c|t|^p}\) for all \(t\in \mathbb {R}\).
Proposition 2.2
Let E be a reflexive, infinite-dimensional subspace of \(L_1\). Then, there is \(1<p\le 2\), there is a normalized sequence \((x_n)_{n=1}^{\infty }\subset E\), and there is a density function u, such that for all real number t.
Proof
Let \(\mu \) denote the Lebesgue measure on \(\mathbb {I}\), so that \(L_1=L_1(\mu )\).
The main result in [2] establishes that E contains a subspace isomorphic to \(\ell _p\) for some \(1<p<\infty \) (see [23] for an alternative proof). Thus, Théorèmes 0.1 and 0.2 in [11] provides us with an ultrafilter \(\mathfrak {U}\) on \(\mathbb {N}\), an element \(0\le u\in L_1(\mu )\) and a p-stable random variable \(\mathbf {v}\in L_1(\mu _{\mathfrak {U}})\), such that \(\mathbf {z}:=\mathbf {v}u\in E_{\mathfrak {U}}\setminus \{0\}\), \(\Vert \mathbf {z}\Vert =1\) and such that \(\sigma (\mathbf {v})\) and \(\mathcal {B}\) are independent. (The fact that u is non-negative follows from the identity \(u=\mathop {\mathbb {E}}(U^p|\mathcal {B})^{1/p}\) where \(0\le U\in L_1(\mu _{\mathfrak {U}})\setminus \{0\}\). Indeed, u and \(\mathbf {v}\) are, respectively, the functions \(\bar{U}\) and \(\bar{V}\) that occur at the end of the proof 3\(\Rightarrow \)4 of Théorème 0.2 in [11], page 348). Hence, \(0\not =u\ge 0\), and dividing by \(\int _{\mathbb {I}}u\,\mathrm{d}{\mu }\) if necessary, u can be assumed to be a density function.
Let \((x_j)_{j\in \mathbb {N}}\) and \((v_j)_{j\in \mathbb {N}}\) be respective representatives of \(\mathbf {z}\) and \(\mathbf {v}\), such that \(\{x_j\}_{j=1}^{\infty }\) is normalized and relatively weakly compact. Thus
Since \(\sigma (\mathbf {v})\) and \(\mathcal {B}\) are independent sub-\(\sigma \)-algebras of \(\sigma (\mathcal {B}_{\mathfrak {U}})\), \(\mathbf {v}\) is p-stable and u is \(\mathcal {B}\)-measurable; it follows:
Hence, Lemma 2.1 gives for all \(t\in \mathbb {R}\), and therefore, from (2.1)
Let \(\{t_n\}_{n=1}^{\infty }\) be a dense, countable subset of \(\mathbb {R}\), and for every positive integer n, let \(K_n:=\{e^{it_nx_j}:j\in \mathbb {N}\}\subset L_1(\mathbb {I},\mathcal {B},\mu ,\mathbb {C})\).
As each \(K_n\) is countable and relatively weakly compact in \(L_1(\mathbb {I},\mathcal {B},\mu ,\mathbb {C})\), the weak topology is metrizable on every \(\overline{\!K_n}^{w}\) (see the proof of Eberlein–Smulian Theorem in [12]). Moreover, by virtue of (2.2), \(e^{-|t_n|^pu^p}\) is a weak limit point of \(K_n\) for every natural number n. Hence, for each \(n\in \mathbb {N}\), there exists a local basis of decreasing weak neighborhoods \(\{\mathcal {V}_i^n\}_{i=1}^{\infty }\) of \(e^{-|t_n|^pu^p}\). This means that given a sequence \((y_i)_{i=1}^{\infty }\subset K_n\), if for every \(k\in \mathbb {N}\), there exists a natural number \(i_0\), such that \(y_i\in \mathcal {V}_k^n\) for all \(i\ge i_0\), then
Next, we extract a subsequence \((z_j)_{j=1}^{\infty }\) of \((x_j)_{j=1}^{\infty }\), such that for all \(n\in \mathbb {N}\) as follows:
Given any pair (k, n) of natural numbers, formula (2.2) yields
Fix any \(j_1\in J_1^1\) and assume that the natural numbers \(j_1<j_2<\ldots <j_{k-1}\) satisfying \(j_l\in \bigcap _{m=1}^{l} J_{l+1-m}^m\) for all \(1\le l\le k-1\) have been already chosen. Then, as
we can pick \(j_k\in A\). Continuing this process recursively, as \(\{\mathcal {V}_{i}^{n}\}_{i=1}^{\infty }\) is decreasing, we get an increasing sequence \((j_k)_{k=1}^{\infty }\) of natural numbers, such that for \(z_k:=x_{j_k}\)
Then, it follows from (2.3) and (2.4) that:
Next, we will prove that (2.5) also holds for all \(t\in \mathbb {R}\). Fix a real number \(\tau \) and a balanced, weak neighborhood \(\mathcal {W}\subset L_1(\mathbb {I},\mathcal {B},\mu ,\mathbb {C})\) of 0. Choose \(\varepsilon >0\) and another balanced, weak neighborhood \(\mathcal {V}\subset L_1(\mathbb {I},\mathcal {B},\mu ,\mathbb {C})\) of 0 such that \(B(0;\varepsilon )+\mathcal {V}\subset \mathcal {W}\), where \(B(0;\varepsilon )\) is the closed ball of \(L_1(\mathbb {I},\mathcal {B},\mu ,\mathbb {C})\) centered at 0 of radius \(\varepsilon \). Pick a sequence \((\tau _n)_n\) in \(\{t_n\}_{n=1}^{\infty }\), such that . The Dominated Convergence Theorem gives a positive integer \(m_0\), such that
Next, as \(\{z_j\}_{j=1}^{\infty }\) is equiintegrable, there is \(K>0\) such that \(\mu (\{\omega :|z_j(\omega )|>K\})<\varepsilon /8\) for all \(j\in \mathbb {N}\). Moreover, for every \(x\in \mathbb {R}\), \( |e^{ix}-1|=\sqrt{2}\sqrt{1-\cos {x}} \le |x|. \) Thus
Therefore, there is \(m\ge m_0\), such that
Next, (2.5) yields \(j_0\), such that
Thus, for every \(j\ge j_0\), consecutive applications of (2.7), (2.8) and (2.6) give
and by virtue of (2.3), the proof is done. \(\square \)
Given a sequence of random variables \(\bigl (x_n(\omega )\bigr )_n\) on a probability space \((\Omega ,\Sigma ,\nu )\), assume that for each real number t, there exists the limit of the sequence \((e^{itx_n})_n\) in the weak topology of \(L_1(\nu )\), and denote this limit by \(h(t,\omega )\): the function \(h(t,\omega )\) is called the limit conditional characteristic function of \((x_n)_n\) (page 498 in[5]).
Remark
The extraction of the subsequence \((z_j)_j\) in the proof of Proposition 2.2 becomes simpler if the ultrafilter \(\mathfrak {U}\) were a p-point. Indeed, the metrizability conditions of each \((K_n,w)\) enable to take a subsequence \((x_{k_j})_j\) of \((x_j)_j\), such that and if \(\mathfrak {U}\) is a p-point, it can be proved this subsequence satisfies \(J_n:=\{k_j:j\in \mathbb {N}\}\in \mathfrak {U}\). Thus, relabeling the subsequence \((x_{k_j})_j\) taken from each \(K_n\) as \((x^n_j)_j\), we may assume that \((x^n_j)_j\supset (x^{n+1}_j)_j\) for all n. Then, a diagonalization argument shows that \((z_j)_j:=(x_j^j)_j\) satisfies for all n as required. However, it is necessary to point out that the ultrafilter \(\mathfrak {U}\) in the proof of Proposition 2.2 is the one provided in the proof of 3\(\Rightarrow \)4 of Théorème 0.2 in [11] and that ultrafilter is a product of two ultrafilters, so it is not a p-point. These facts and a comparison of statements (a), (b) to (a’) and (b’) clearly suggest an explicit proof for Proposition 2.2 was needed.
3 Decompositions of \(L_1\)
Let us begin this section recalling that if \((\Omega ,\sigma ,\nu )\) is an atomless finite measure space, such that \(L_1(\nu )\) is separable, then \(L_1(\nu )\simeq L_1\) (see §41 in[20] or I.B.1 in [31]).
Theorem 3.1
Let \((f_n)_{n=1}^{\infty }\setminus \{0\}\) be a bounded sequence of independent random variables with zero mean defined on an atomless probability space \((\Omega ,\Sigma ,\nu )\) for which \(L_1(\Omega ,\Sigma ,\nu )\) is separable. Then, there exists a norm one projection \(Q:L_1(\Omega ,\Sigma ,\nu )\longrightarrow L_1(\Omega ,\Sigma ,\nu )\) satisfying the following statements:
-
(i)
\(R(Q)\simeq L_1\) and \((f_{2n-1})_{n=1}^{\infty }\subset R(Q)\);
-
(ii)
\(N(Q)\simeq L_1\) and \((f_{2n})_{n=1}^{\infty }\subset N(Q)\).
Proof
Let \(\mathcal {S}_n:=\sigma (f_n)\) for all n. Let \(\Sigma _e\) and \(\Sigma _o\), respectively, denote the sub-\(\sigma \)-algebras of \(\Sigma \) generated by \(\cup _{n=1}^{\infty }\mathcal {S}_{2n}\) and by \(\cup _{n=1}^{\infty }\mathcal {S}_{2n-1}\).
Denote \(L_1(\Sigma _e):=L_1(\Omega ,\Sigma _e,\nu \!\mid {\!}_{\Sigma _e}\,)\) and \(L_1(\Sigma _o):=L_1(\Omega ,\Sigma _o,\nu \!\mid {\!}_{\Sigma _o}\,)\).
Since the random variables \(f_n\) are independent, neither \((\Omega ,\Sigma _o,\mu \!\mid {\!}_{\Sigma _o}\,)\) nor \((\Omega ,\Sigma _e,\mu \!\mid {\!}_{\Sigma _e}\,)\) has an atom. This fact and the separability of \(L_1(\nu )\) yields \( L_1\simeq L_1(\Omega ,\Sigma ,\nu )\simeq L_1(\Sigma _o)\simeq L_1(\Sigma _e). \) In addition, as \(\Sigma _e\) and \(\Sigma _o\) are independent, then (see [4] or (2.18) and (2.21) in [22])
Let Q be the norm one projection on \(L_1(\Omega ,\Sigma ,\nu )\) given by the conditional expectation \(Q(f):=\mathop {\mathbb {E}}(f|\Sigma _o)\).
It is straightforward from (3.2) that
In particular, \((f_{2n-1})_{n=1}^{\infty }\subset R(Q)\) and (i) is proved.
To prove (ii), note that (3.1) implies that all \(\Sigma _e\)-measurable, integrable functions with zero mean belong to N(Q). In particular, \((f_{2n})_{n=1}^{\infty }\subset N(Q)\), so the proof will be completed as soon as it is demonstrated that \(N(Q)\simeq L_1\).
Consider the norm one projection P defined on \(L_1(\Omega ,\Sigma ,\nu )\) and given by \(P(f):=\mathop {\mathbb {E}}(f|\Sigma _e)\). The same arguments given for Q show
The combination of (3.3), (3.1), (3.4), and (3.5) readily yields
and subsequently, \(P\bigl (R(Q)\bigr )\subset R(Q)\) and \(P\left( N(Q)\right) \subset N(Q)\). This proves that \(P\!\mid {\!}_{R(Q)}\,\) and \(P\!\mid {\!}_{N(Q)}\,\) are projections. Hence
However, (3.6) yields \(\dim {R(QP)}=1\), so that \(N(Q)\cap R(P)\) is a hyperplane of \(L_1(\Omega ,\Sigma ,\nu )\). Thus, as \(L_1\) is primary [15], then \(L_1\simeq N(Q)\cap R(P)\), and
Therefore, N(Q) is a complemented subspace of \(L_1(\Omega ,\Sigma ,\nu )\) containing a complemented subspace isomorphic to \(L_1\), and by means of Pelczyński decomposition method (Theorem 2.2.3 in[1]), \(N(Q)\simeq L_1\). The proof is done.
\(\square \)
Lemma 3.2
Given a probability space \((\Omega ,\Sigma ,\nu )\), a real number \(\varepsilon >0\) and an equi-integrable sequence \((f_n)_{n=1}^{\infty }\) in \(L_1(\Omega ,\Sigma ,\nu )\), such that \(\sum _{n=1}^{\infty }|f_n(t)|<\infty \) for \(\nu \)-almost every t, there is a subsequence \((f_{k_n})_{n=1}^{\infty }\) of \((f_n)_{n=1}^{\infty }\), such that \(\Vert f_{k_n}\Vert <\varepsilon \) for all n.
Proof
If the result fails, then there exists \(\varepsilon >0\), such that \(\Vert f_n\Vert _1\ge \varepsilon \) for all but for finitely many n. Without loss of generality, we may assume that \(\Vert f_n\Vert _1\ge \varepsilon \) for all n. Let
Since \(\Vert \chi _{A_n^c}f_n\Vert _1\le \Vert \chi _{A_n^c}f_n\Vert _{\infty }\le \varepsilon /2\), it follows \(\Vert \chi _{A_n}f_n\Vert _1\ge \varepsilon /2\), where \(A_n^c:=\Omega \setminus A_n\). But \((\chi _{A_n}f_n)_{n=1}^{\infty }\) is equi-itegrable, so there is \(\delta >0\), such that \(\nu (A_n)>\delta \) for infinitely many \(A_n\). Pass to a subsequence \((A_{j_n})_{n=1}^{\infty }\), so that \(\nu (A_{j_n})>\delta \) for all n. Thus
Hence, for all \(t\in \limsup _{n}{A_{j_n}}\), we get \(\sum _{n=1}^{\infty }|f_{j_n}(t)|\ge \infty \cdot \varepsilon /2=\infty \), a contradiction. \(\square \)
Theorem 3.3
Let V be an infinite-dimensional, reflexive subspace of \(L_1\). Then, \(L_1\) has two subspaces X and Y satisfying the following properties:
-
(i)
\(L_1=X\oplus Y\);
-
(ii)
\(X\simeq Y\simeq L_1\);
-
(ii)
\(X\cap V\) and \(Y\cap V\) are infinite-dimensional.
Proof
By Proposition 2.2, V contains a normalized sequence \((f_n)_{n=1}^{\infty }\) for which there exists a density function \(\phi \) and a real number \(1<p\le 2\), such that for all \(t\in \mathbb {R}\). Let \(H(t,\omega ):=e^{-|t|^p\phi (\omega )^p}\) and \(\Omega :=\{\omega :\phi (\omega )\not =0\}\).
Since V is reflexive and \((f_n)_{n=1}^{\infty }\) is norm bounded, it follows that \((f_n)_{n=1}^{\infty }\) is also bounded in probability, and passing to a subsequence if necessary, \((f_n)_{n=1}^{\infty }\) is determining (see the first Definition and Proposition 2.2, both in Sect. 2 of [5]). Thus, as \(\mu (\Omega )\phi \) is a strictly positive density on the probability space \((\Omega ,\mathcal {B}(\Omega ),\,\mathrm{d}{P})\), where \(\,\mathrm{d}{P}:=\mu /\mu (\Omega )\), Lemma 3.3 in [5] yields \((\frac{f_n\!\mid {\!}_{\,\Omega }\,}{\phi })_n\) is a determining sequence on the probability space \((\Omega , \mathcal {B}(\Omega ),\nu )\), where \(\nu =\phi \,\mathrm{d}{\mu }\), whose limit conditional characteristic function is \(h(t,\omega )=h(t):=e^{-|t|^p/\mu (\Omega )^p}\).
As h(t) is the characteristic function of a p-stable random variable, Theorem 3.1 in [5] (see its proof and also the comments at the beginning of page 500 in [5]) shows \((f_n)_n\) contains a subsequence \((f_{k_n})_n\) for which there exists a normalized sequence \((g_n)_{n=1}^{\infty }\) of independent, p-stable random variables defined on \((\Omega , \mathcal {B}(\Omega ),\nu )\) whose common characteristic function is \(e^{-|t|^p/\mu (\Omega )^p}\), and moreover
The sequence \((g_n)_n\) is basic; indeed, it is equivalent to the unit vector basis of \(\ell _p\) and the value of its basis constant is 1 (page 182 [26]).
Additionally, as for all \(t\in \mathbb {R}\), where w denotes here the weak topology of \(L_1(\Omega ^c,\mathcal {B}(\Omega ^c),\,\mathrm{d}{\text {m}})\), \(\Omega ^c:=\mathbb {I}\setminus \Omega \) and \(\,\mathrm{d}{\text {m}}:=\,\mathrm{d}{\mu }/\mu (\Omega ^c)\), then pointwise in the probability space \((\Omega ^c,\mathcal {B}(\Omega ^c),\,\mathrm{d}{\text {m}})\) for all \(t\in \mathbb {R}\). This implies that \((f_n\!\mid {\!}_{\mathbb {I}\setminus \Omega }\,)_n\) converges in probability to 0 (Theorem 25.3 in [6]). From here, it is very easy to pick a subsequence \((f_{m_n})_n\) of \((f_{k_n})_n\), such that
Next, as \((f_n)_{n=1}^{\infty }\) is equi-integrable, an application of Lemma 3.2 on (3.7) and on (3.8) yields a pair of subsequences \((x_n)_{n=1}^{\infty }\subset (f_{m_n})_{n=1}^{\infty }\) and \((y_n)_{n=1}^{\infty }\subset (g_n)_{n=1}^{\infty }\), such that
Fix a sequence \((\varepsilon _n)_{n=1}^{\infty }\) of positive numbers, such that \(\sum _{n=1}^{\infty }\varepsilon _n<1/4\). Passing to subsequences if necessary, we can assume by virtue of (3.9) and (3.10) that
The inequality (3.11) and the principle of small perturbations (Proposition 1.a.9 in [25]) prove that \((\frac{x_n}{\phi }\!\!\!\mid {\!}_{\!\!\Omega }\,)_n\) is a basic sequence equivalent to \((y_n)_n\) as sequences in \(L_1(\Omega ,\mathcal {B}(\Omega ),\nu )\). However, the operator \(G:L_1(\Omega ,\mathcal {B}(\Omega ),\mu )\longrightarrow L_1(\Omega ,\mathcal {B}(\Omega ),\nu )\) that maps each f to \(f/\phi \) is a bijective isometry, so \((x_n\chi _{\Omega })_n\) is a basic sequence in \(L_1(\mathbb {I},\mathcal {B},\mu )\). Therefore, a second application of the principle of small perturbations in combination with (3.12) yields a positive integer \(n_0\), such that \((x_n)_n=(x_n\chi _{\Omega }+x_n\chi _{\mathbb {I}\setminus \Omega })_{n=n_0}^{\infty }\) is basic in \(L_1(\mathbb {I},\mathcal {B},\mu )\). Let \((a^*_n)_{n=n_0}^{\infty }\) be the sequence of coordinate functionals on \(\mathrm{span}\,{\{x_n\}_{n=n_0}^{\infty } }\) associated with \((x_n)_{n=n_0}^{\infty }\) and take a Hahn–Banach extension \(x^*_n\in L_1(\mathbb {I})^*\) of each \(a_n^*\), so that \((x_n^*)_{n=n_0}^{\infty }\) is bounded.
As G is a surjective isometry, so is the operator
that maps each f to \(\bigl (\frac{f}{\phi }\!\mid {\!}_{\Omega }\,, f\!\mid {\!}_{\mathbb {I}\setminus \Omega }\,\bigr )\). For every \(n\in \mathbb {N}\), let
It follows from (3.11) and (3.12) that \(\Vert z_n\Vert <2\varepsilon _n\) for all n.
Fix \(0<\varepsilon <1\). Thus, as J is a bijective isometry, \(J+K\) is a surjective isomorphism for every operator K from \(L_1(\mathbb {I})\) into \(L_1(\Omega ,\nu )\oplus _1 L_1(\mathbb {I}\setminus \Omega ,\mu )\) with \(\Vert K\Vert <\varepsilon \). For \(M=\sup {\{\Vert x^*_n\Vert \}_{n=1}^{\infty }}\), we can take a positive integer \(n_1\ge n_0\) large enough, so that \(\sum _{n=n_1}^{\infty }2\varepsilon _n<\varepsilon /M\), and therefore, the operator \(K:L_1(\mathbb {I})\longrightarrow L_1(\Omega ,\nu )\oplus _1 L_1(\mathbb {I}\setminus \Omega ,\mu )\) that maps each x to \(\sum _{n=n_1}^{\infty }\langle x_n^*,x\rangle z_n\) is well defined and \(\Vert K\Vert <\varepsilon \). Hence, \(J+K\) is a bijective isomorphism and \((J+K)(x_n)=(y_n,0)\) for all \(n\ge n_1\).
Next, Theorem 3.1 provides a decomposition \(L_1(\Omega ,\nu )=Z_1\oplus Z_2\), such that \(Z_1\simeq Z_2\simeq L_1(\mathbb {I})\) and each \(Z_i\) contains infinitely many elements \(y_n\). Therefore, the subspaces
are both isomorphic to \(L_1(\mathbb {I})\) and each \(X_i\) contains infinitely many elements \(x_n\). Hence, \(\dim {X_i\cap V}=\infty \) for all \(1\le i\le 2\), and the proof is done. \(\square \)
4 The Strong Property (B) on \(L_p\) Spaces
We first prove that each \(L_p(\mathbb {I})\), \(1\le p<\infty \), has the strong property (B).
Theorem 4.1
Let \(1\le p<\infty \) and let V be an infinite-dimensional subspace of \(L_p\). Then, there exists a decomposition \(L_p=X_1\oplus X_2\), such that both subspaces \(X_1\) and \(X_2\) are isomorphic to \(L_p\) and both intersections \(V\cap X_1\) and \(V\cap X_2\) are infinite-dimensional.
Proof
The proof is organized into the cases (a) \(p=1\); (b) \(p=2\); (c) \(p\in (1,2)\cup (2,\infty )\).
(a) It is well known that an infinite-dimensional subspace of \(L_1\) is either reflexive or contains a subspace isomorphic to \(\ell _1\) and complemented in \(L_1\) [1] [Proposition 5.6.2]. In the first case, when V is reflexive, the result follows from Theorem 3.3. In the second case, V has a pair of subspaces E and Y, such that \(E\simeq \ell _1\) and \(L_1=E\oplus Y\). Since \(L_1\) is primary (Corollary 5.5 in [15]), Y is isomorphic to \(L_1\). Hence, E and Y can be decomposed as \(E=E_1\oplus E_2\) and \(Y=Y_1\oplus Y_2\) where \(\ell _1\simeq E_1\simeq E_2\) and \(L_1\simeq Y_1\simeq Y_2\). Then, \(X_i:=E_i\oplus Y_i\) for \(i=1,2\) are the required spaces in the statement.
It is straightforward that \(L_1=X_1\oplus X_2\), \(L_1\simeq L_1\oplus \ell _1\simeq X_i\) and \(E_i\) is an infinite-dimensional subspace of \(E\cap X_i\) for \(i=1,2\).
(b) The proof for this case is immediate, because \(L_2\) is a Hilbert space, and therefore, each of its infinite-dimensional subspaces is complemented and isomorphic to \(L_2\).
(c) Take a normalized basic sequence \((x_n)_{n=1}^{\infty }\) in V and a sequence \((\varepsilon _n)_{n=1}^{\infty }\) of positive real numbers, such that \(\sum _{n=1}^{\infty }\varepsilon _n <1/4\). As \((x_n)_{n=1}^{\infty }\) is weakly null, it contains a subsequence \((x_n')_{n=1}^{\infty }\) which can be approximated by a block basic sequence of the Haar basis \((h_i)_{i=0}^{\infty }\) in \(L_p\), that is, there exist real numbers \(\lambda _i\) and an increasing sequence \((k_n)_{n=1}^{\infty }\) of non-negative integers with \(k_1=0\), such that for \(y_n:=\sum _{i=k_n}^{k_{n+1}-1}\lambda _ih_i\)
Let \((y_n^*)_{n=1}^{\infty }\subset L_p^*\) be a sequence of Hahn-Banach extensions of the coordinate functionals \((a_n^*)_{n=1}^{\infty }\subset \mathop {\overline{\mathrm{span}}}{\{y_n\}_{n=1}^{\infty }}\) associated with the basic sequence \((y_n)_{n=1}^{\infty }\).
Since the Haar basis of \(L_p\) is monotone, so is \((y_n)_{n=1}^{\infty }\) and, therefore, as \(\Vert y_n\Vert <1\,+\,1/4\), we have \(\Vert y_n^*\Vert <2\,+\,1/2\) for all n.
Thus, the operator \(K:L_p\longrightarrow L_p \) that maps each x to \(\sum _{n=1}^{\infty }\langle y_{n}^*,x\rangle (x'_{n}-y_{n})\) is well defined and \(\Vert K\Vert \le \sum _{n=1}^{\infty }\varepsilon _n\Vert y_n^*\Vert< 1/2\,+\, 1/8<1\).
Consider the subspaces of \(L_p\) given by
Note that \((I+K)(y_n)=x_n'\). Hence, \((I+K)(Y_1)=V_1\) and \((I+K)(Y_2)=V_2\). As \(\Vert K\Vert <1\), it follows that \(I+K\) is a bijective isomorphism, and therefore, \(Y_1\simeq V_1\) and \(Y_2\simeq V_2\); analogously, for \(W_1:=(I+K)(Z_1)\) and \(W_2:=(I+K)(Z_2)\), we have \(W_i\simeq Z_i\) for \(i=1,2\). Moreover, since \((h_i)_{i=0}^{\infty }\) is unconditional, then \(L_p=W_1\oplus W_2\) and one of the following two cases happen (see [19] and also [3][Theorem 1, page 129]:)
-
(i)
both subspaces \(W_1\) and \(W_2\) are isomorphic to \(L_p\);
-
(ii)
one of the subspaces \(W_1\), \(W_2\) is isomorphic to \(L_p\) and the other is isomorphic to \(\ell _p\).
If (i) holds, since \(V_i\subset W_i\) for \(i=1,2\), the subspaces \(X_1:=W_1\) and \(X_2:=W_2\) satisfy the statement of the theorem.
If (ii) holds, assume \(W_1\simeq L_p\) and \(W_2\simeq \ell _p\). Then, a similar argument to one given in case (a) works: since \(V_2\subset W_2\simeq \ell _p\), \(W_2\) can be decomposed as \(W_2=F_1\oplus F_2\oplus G\) where \(F_1\simeq F_2\simeq \ell _p\) and \(F_1\oplus F_2\subset V_2\) [1][Proposition 2.2.1]). Decompose \(W_1\) as \(W_1=H_1\oplus H_2\) with \(H_1\simeq H_2\simeq L_p\). The subspaces \(X_1:=H_1\oplus F_1\) and \(X_2:= H_2\oplus F_2\oplus G\) are both isomorphic to \(L_p\), \(L_p=X_1\oplus X_2\) and \(F_i\subset V\cap X_i\) are infinite-dimensional for \(i=1,2\). The proof is complete.
\(\square \)
Cross and his coauthors proved the following result for \(L_p\) with \(1<p<\infty \) from the fact that these spaces have property (B). Thus, their proof also works for \(L_1\):
Corollary 4.2
Given any Banach space Y and an operator \(T\in \mathcal {L}(L_1,Y)\), such that R(T) is dense and non-closed in Y, there exists a pair of operators \(T_1\) and \(T_2\) in \(\mathcal {L}(L_1,Y)\) satisfying the following conditions:
-
(i)
\(R(T_i)\) is dense and non-closed in Y for \(i=1\) and \(i=2\);
-
(ii)
\(R(T)=R(T_1)+R(T_2)\);
-
(iii)
\(R(T_1)\cap R(T_2)=\{0\}\).
Proof
Since \(L_1\) has property (B) and is separable, Theorem 5.3 in [10] yields the desired result. \(\square \)
Theorem 4.1 can be extended to finite measure spaces with no atom.
Proposition 4.3
For every purely non-atomic, finite measure space \((\Omega ,\Sigma ,\nu )\) and every \(1\le p<\infty \), the space \(L_p(\Omega ,\Sigma ,\nu )\) has the strong property (B).
Proof
Let V be an infinite-dimensional subspace of \(L_p(\Omega ,\Sigma ,\nu )\), and let E be an infinite-dimensional, separable subspace of V. Let \(\{x_j\}_{j\in \mathbb {N}}\) be a countable dense subset of E. Let \(A_{j,a,b}:=\{\omega \in \Omega :a<x_j(\omega )<b\}\) and let \(\mathcal {M}\) be the sub-\(\sigma \)-algebra of \(\Sigma \) generated by \(\{A_{j,a,b}:j\in \mathbb {N},\ a\in \mathbb {Q},\ b\in \mathbb {Q}\}\). Let \(M:=L_p(\Omega ,\mathcal {M},\nu \!\mid {\!}_{\mathcal {M}}\,)\), so that \(M\simeq L_p(\mathbb {I})\).
As observed in Proposition III.A.2 in [31], E is a separable subspace of M, and M is a complemented subspace of \(L_p(\Omega ,\Sigma ,\nu )\) via the projection \(\mathop {\mathbb {E}}(\cdot |\mathcal {M})\). Therefore, \(E\subset M\oplus N=L_p(\Omega ,\Sigma ,\nu )\).
It is immediate from Theorem 9, page 127 in [24] that \( L_p(\mathbb {I})\times L_p(\Omega ,\Sigma ,\nu )\simeq L_p(\Omega ,\Sigma ,\nu )\) and \( L_p(\Omega ,\Sigma ,\nu )\times L_p(\Omega ,\Sigma ,\nu )\simeq L_p(\Omega ,\Sigma ,\nu )\).
Hence, by virtue of Theorem 4.1, M can be decomposed as \(M=A_1\oplus A_2\oplus B\) where \(A_1\simeq A_2\simeq B\simeq L_p(\mathbb {I})\) and the intersections \(E\cap A_1\) and \(E\cap A_2\) are infinite-dimensional.
Thus
However, \(B\oplus N\) can be decomposed as \(D_1\oplus D_2\) where \(D_1\simeq D_2\simeq L_p(\Omega ,\Sigma ,\nu )\), which leads to
Clearly, \(L_p(\Omega ,\Sigma ,\nu )\simeq A_i\oplus D_i\) and \(V\cap (A_i\oplus D_i)\) is infinite-dimensional for \(i=1,2\), which proves that \(L_p(\Omega ,\Sigma ,\nu )\) has the strong property (B). \(\square \)
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Martínez-Abejón, A. The Strong Property (B) for \(L_p\) Spaces. Mediterr. J. Math. 19, 5 (2022). https://doi.org/10.1007/s00009-021-01907-1
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DOI: https://doi.org/10.1007/s00009-021-01907-1