The Strong Property (B) for $L_p$ Spaces

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.

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

$$\begin{aligned} \mu _{\mathfrak {U}}(A) := \inf \{ \mu _{\mathfrak {U}}(C):A\subset C,\ C\in \Sigma _{\mathfrak {U}}\} = \sup \{ \mu _{\mathfrak {U}}(C):C\subset A,\ C\in \Sigma _{\mathfrak {U}}\}. \end{aligned}$$

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 $A\in \Sigma _{\mathfrak {U}}$

$$\begin{aligned} \nu _{\mathbf {f}}(A):=\lim _{i\rightarrow \mathfrak {U}}\int _{A_i}f_i\,\,\mathrm{d}{\mu },\quad A=(A_i)^{\mathfrak {U}}\in \Sigma _{\mathfrak {U}} \end{aligned}$$

and for each $A\in \sigma (\Sigma _{\mathfrak {U}})$

$$\begin{aligned} \nu _{\mathbf {f}}(A):= \inf \{ \nu _{\mathbf {f}}(C):A\subset C,\ C\in \Sigma _{\mathfrak {U}}\} =\sup \{ \nu _{\mathbf {f}}(C):C\subset A,\ C\in \Sigma _{\mathfrak {U}}\}. \end{aligned}$$

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

$$\begin{aligned} w_{\mathbf {f}}(A)=\int _{A}g_{\mathbf {f}}\,\,\mathrm{d}{\mu _{\mathfrak {U}}},\quad A\in \sigma (\Sigma _{\mathfrak {U}}). \end{aligned}$$

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

$$\begin{aligned} L_1(\mu )_{\mathfrak {U}}=L_1(\mu _{\mathfrak {U}})\oplus _1 L_1\bigl (\mu (\mathfrak {U}))\bigr ) \end{aligned}$$

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

(2.1)

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:

$$\begin{aligned} \mathop {\mathbb {E}}(e^{itu\mathbf {v}}|\mathcal {B})=\mathop {\mathbb {E}}(e^{-|t|^pu^p}|\mathcal {B})= e^{-|t|^pu^p}, \ \text {all} \quad t\in \mathbb {R}. \end{aligned}$$

Hence, Lemma 2.1 gives for all $t\in \mathbb {R}$, and therefore, from (2.1)

(2.2)

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

(2.3)

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

$$\begin{aligned} J_k^n:=\{j\in \mathbb {N}:e^{it_nx_j}\in \mathcal {V}_k^n\}\in \mathfrak {U}. \end{aligned}$$

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

$$\begin{aligned} \emptyset \not = A:= \left[ \bigcap _{m=1}^{k} J_{k+1-m}^m\right] \cap \{p\in \mathbb {N}:p>j_{k-1}\}\in \mathfrak {U}, \end{aligned}$$

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 $z_k:=x_{j_k}$

$$\begin{aligned} e^{it_nz_k}\in \mathcal {V}_{k+1-n}^{n}, \text { for all}~k\ge n, \text {all}~n\in \mathbb {N}. \end{aligned}$$

(2.4)

Then, it follows from (2.3) and (2.4) that:

(2.5)

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

$$\begin{aligned} \Vert e^{-|\tau |^pu^p}-e^{-|\tau _n|^pu^p}\Vert _{L_1}<\varepsilon /4 \ \text {for all} \quad n\ge m_0. \end{aligned}$$

(2.6)

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

$$\begin{aligned}&\int _{\mathbb {I}}|e^{i\tau _nz_j(\omega )}-e^{i\tau z_j(\omega )}|\,\mathrm{d}{\mu (\omega )}= \int _{\mathbb {I}}|e^{i(\tau _n-\tau )z_j(\omega )}-1|\,\mathrm{d}{\mu (\omega )}= \\&\int _{\{|z_j(\omega )|> K\}}|e^{i(\tau _n-\tau )z_j(\omega )}-1|\,\mathrm{d}{\mu (\omega )}+ \\&\int _{\{|z_j(\omega )|\le K\}}|e^{i(\tau _n-\tau )z_j(\omega )}-1|\,\mathrm{d}{\mu (\omega )} \le \frac{\varepsilon }{4} + |\tau _n-\tau |K. \end{aligned}$$

Therefore, there is $m\ge m_0$, such that

$$\begin{aligned} \int _{\mathbb {I}}|e^{i\tau _nz_j(\omega )}-e^{i\tau z_j(\omega )}|\,\mathrm{d}{\mu (\omega )}\le \varepsilon /2, \ \text {for all} \quad n\ge m~ \text {and all}~ j\in \mathbb {N}. \end{aligned}$$

(2.7)

Next, (2.5) yields $j_0$, such that

$$\begin{aligned} e^{i\tau _mz_j}\in e^{-|\tau _m|^pu^p}+\mathcal {V}, \text {for all} \quad j\ge j_0. \end{aligned}$$

(2.8)

Thus, for every $j\ge j_0$, consecutive applications of (2.7), (2.8) and (2.6) give

$$\begin{aligned} \begin{aligned} e^{i\tau z_j}\in e^{i\tau _m z_j} +B(0;\varepsilon /2)&\subset e^{-|\tau _m|^pu^p} + \mathcal {V}+ B(0;\varepsilon /2) \subset \\&e^{-|\tau |^pu^p} + \mathcal {V}+ B(0;\varepsilon ) \subset e^{-|\tau |^pu^p} +\mathcal {W}, \end{aligned} \end{aligned}$$

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])

$$\begin{aligned}&\mathop {\mathbb {E}}(f_{2n}|\Sigma _o)=\mathop {\mathbb {E}}(f_{2n}) \end{aligned}$$

(3.1)

$$\begin{aligned}&\mathop {\mathbb {E}}(f_{2n-1}|\Sigma _o)=f_{2n-1}, \ n\in \mathbb {N}. \end{aligned}$$

(3.2)

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

$$\begin{aligned} L_1\simeq L_1(\Sigma _o)=R(Q)\subset L_1(\Omega ,\Sigma ,\nu ). \end{aligned}$$

(3.3)

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

$$\begin{aligned}&L_1\simeq R(P)=L_1(\Sigma _e)\subset L_1(\Omega ,\Sigma ,\nu ) \end{aligned}$$

(3.4)

$$\begin{aligned}&P(f)=\mathop {\mathbb {E}}(f)\cdot \chi _{\Omega } \quad \text { for all} f\in L_1(\Sigma _o). \end{aligned}$$

(3.5)

The combination of (3.3), (3.1), (3.4), and (3.5) readily yields

$$\begin{aligned} PQ(f)=QP(f) =\mathop {\mathbb {E}}(f),\quad f\in L_1(\Omega ,\Sigma ,\nu ), \end{aligned}$$

(3.6)

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

$$\begin{aligned} L_1\simeq R(P) = N(Q\!\mid {\!}_{R(P)}\,)\oplus R(Q\!\mid {\!}_{R(P)}\,) =\bigl [N(Q)\cap R(P)\bigr ]\oplus R(QP). \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} N(Q)=N(P\!\mid {\!}_{N(Q)}\,)\oplus R(P\!\mid {\!}_{N(Q)}\,) \\ =\bigl [N(Q)\cap N(P)\bigr ]\oplus \bigl [ R(P)\cap&N(Q)\bigr ]\simeq \bigl [N(Q)\cap N(P)\bigr ]\oplus L_1. \end{aligned} \end{aligned}$$

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

$$\begin{aligned} A_n:=|f_n|^{-1}\bigl ((\varepsilon /2,\infty )\bigr ). \end{aligned}$$

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

$$\begin{aligned} \nu (\limsup _{n}{A_{j_n}})= \lim _{n}\nu \left( \bigcup _{k=n}^{\infty }{A_{j_n}}\right) \ge \delta . \end{aligned}$$

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

$$\begin{aligned} \sum _{n=1}^{\infty }\left| \frac{f_{k_n}}{\phi }-g_n\right| (\omega )<\infty \qquad \text {for}~\mu -\text {a.e.}~\omega \in \Omega . \end{aligned}$$

(3.7)

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

$$\begin{aligned} \sum _{n=1}^{\infty }|f_{m_n}(t)|<\infty \qquad \text {for}~\mu -\text {a.e.}~t\in \mathbb {I}\setminus \Omega . \end{aligned}$$

(3.8)

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

$$\begin{aligned}&\sum _{n=1}^{\infty }\int _{\Omega }{\left| \frac{x_n}{\phi }-y_n\right| \phi \,\mathrm{d}{\mu }<\infty } \end{aligned}$$

(3.9)

$$\begin{aligned}&\sum _{n=1}^{\infty }\int _{\mathbb {I}\setminus \Omega }|x_n|\,\mathrm{d}{\mu }<\infty . \end{aligned}$$

(3.10)

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

$$\begin{aligned}&\int _{\Omega }{\left| \frac{x_n}{\phi }-y_n\right| \phi \,\mathrm{d}{\mu }}<\varepsilon _n, \qquad n\in \mathbb {N} \end{aligned}$$

(3.11)

$$\begin{aligned}&\int _{\mathbb {I}\setminus \Omega }|x_n|\,\mathrm{d}{\mu }<\varepsilon _n,\quad n\in \mathbb {N}. \end{aligned}$$

(3.12)

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

$$\begin{aligned} J:L_1(\mathbb {I})\longrightarrow L_1(\Omega ,\mathcal {B}(\Omega ),\nu )\oplus _1 L_1(\mathbb {I}\setminus \Omega ,\mathcal {B}(\mathbb {I}\setminus \Omega ),\mu ) \end{aligned}$$

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

$$\begin{aligned} z_n:=\left( {y_n}-\frac{x_n}{\phi }\!\mid {\!}_{\Omega }\,\,,\, -x_n\!\mid {\!}_{\mathbb {I}\setminus \Omega }\,\right) \in L_1(\Omega ,\nu )\oplus _1 L_1(\mathbb {I}\setminus \Omega ,\mu ). \end{aligned}$$

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

$$\begin{aligned}&X_1:= (J+K)^{-1}\bigl (Z_1\oplus _1 L_1(\mathbb {I}\setminus \Omega ,\mu )\bigr ), \\&X_2:=(J+K)^{-1}(Z_2) \end{aligned}$$

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 $y_n:=\sum _{i=k_n}^{k_{n+1}-1}\lambda _ih_i$

$$\begin{aligned} \Vert x_n'-y_n\Vert <\varepsilon _n \qquad \text {for all}~ n. \end{aligned}$$

(4.1)

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

$$\begin{aligned}&Z_1:= \mathop {\overline{\mathrm{span}}}\{h_i:k_{2n-1}\le i\le k_{2n}-1,\ n\in \mathbb {N}\} \\&Z_2:= \mathop {\overline{\mathrm{span}}}\{h_i:k_{2n}\le i \le k_{2n+1}-1,\ n\in \mathbb {N}\}\\&Y_1:=\mathop {\overline{\mathrm{span}}}\{y_{2n-1}:n\in \mathbb {N}\}\subset Z_1 \\&Y_2:=\mathop {\overline{\mathrm{span}}}\{y_{2n}:n\in \mathbb {N}\}\subset Z_2\\&V_1:=\mathop {\overline{\mathrm{span}}}\{x'_{2n-1}:n\in \mathbb {N}\} \\&V_2:=\mathop {\overline{\mathrm{span}}}\{x'_{2n}:n\in \mathbb {N}\}. \end{aligned}$$

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

$$\begin{aligned} E\subset L_p(\Omega ,\Sigma ,\nu )= A_1\oplus A_2\oplus (B\oplus N). \end{aligned}$$

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

$$\begin{aligned} E\subset A_1\oplus A_2\oplus D_1\oplus D_2=(A_1\oplus D_1)\oplus (A_2\oplus D_2)=L_p(\Omega ,\Sigma ,\nu ). \end{aligned}$$

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). $\square$