Non-commutative analogues of weak compactness criteria in symmetric spaces

Abstract

The aim of this work is to study M. Nowak’s and K. M. Chong’s weak compactness criteria in non-commutative Orlicz spaces. In addition, we obtain a non-commutative analogue of Pełczyński’s property (V) in Orlicz spaces of $\tau$-measurable operators. We also obtain Kolmogorov’s compactness criterion in terms of conditional expectations in non-commutative symmetric spaces.

1 Introduction

The notion of equi-integrability is closely related to weak compactness in function spaces, which could be directly generalized to any Banach lattice X of measurable functions over a finite measure space as follows: a set $K \subset X$ has equi-absolutely continuous norms in X if (see, for example, [4])

$$\begin{aligned} \lim _{\delta \rightarrow 0} \sup _{\nu (E)< \delta } \sup _{x \in K}\Vert x \chi _E\Vert _X=0.\end{aligned}$$

W. Orlicz himself investigated criteria of weak compactness in Orlicz spaces, precisely, he showed that in every Orlicz space $L_G$ over the interval (0, 1) with

$$\begin{aligned} \lim _{t \rightarrow \infty } \frac{G^*(2t)}{G^*(t)}=\infty , \end{aligned}$$

an analogue of Dunford–Pettis’s criterion of weak compactness holds [35, assertion 1.5] (see also [1]). In other words, every relatively weakly compact subset of $L_G$ over the interval (0, 1) has equi-absolutely continuous norms in $L_G$. Here $G^*$ denotes the complementary (or conjugate) (see [26, Chapter 1, formula (2.9)]) function to G.

In fact, the notion of equi-integrability is equivalent to De la Vallée Poussin’s criterion of weak compactness (see [37, Theorem 2, p.3])). A great number of papers has been devoted to a study of weak compactness criteria in Orlicz spaces, as to mention a few [1, 3,4,5, 9, 10, 15, 24, 28, 29, 34, 37, 38, 45, 46].

Below we also provide other results related to weak compactness that we study in this paper.

In 1962, T. Ando (see [3]) described weak compactness in Orlicz spaces using Köthe duality. The results of T. Ando were extended from the setting of finite measure spaces to the setting of $\sigma$-finite measure spaces in the work of M. Nowak in 1986, see [34].

In Sect. 3, we obtain non-commutative analogues of T. Ando and M. Novak’s results concerning weak relative compactness of a bounded subset K of a non-commutative Orlicz space (see Theorem 3.1) as well as a non-commutative analogue of K. M. Chong’s criterion (see Theorem 3.5) and its equivalence to De la Vallée Poussin’s criterion in non-commutative ${{\mathcal {L}}}_1$-spaces (see Theorem 3.3).

In Sect. 4, we extend known results concerning Pełczyński’s property (V) of Orlicz function spaces from [21, 28, 30] to the non-commutative setting. Our methods here are based on the recent study of M-ideals in [22].

Section 5 is devoted to the extension of a characterization of compact sets, originally due to Kolmogorov (see [25]) in the case of reflexive $L_p$-spaces over a bounded measurable set in ${{\mathbb {R}}}^d$, to non-commutative separable symmetric spaces. To separable Orlicz function spaces, Kolmogorov’s result was extended by Takahashi [40]. Further, an analogue of Kolmogorov’s compactness criterion for separable Orlicz spaces $L_G$ in terms of Steklov functions may be found in [26, Theorem 11.1, p. 97].

We present its non-commutative analogue in terms of conditional expectations for general separable symmetric spaces over semifinite von Neumann algebras, see Theorem 5.1.

2 Preliminaries

Recall that a subset K of a space $L_1(\nu )$ is called uniformly integrable if, for any $\varepsilon >0$, there exists $\delta >0$ such that $\sup \left\{ \int _{E}|f|{\mathrm{d}}\nu : \ f \in K\right\} < \varepsilon$ whenever $\nu (E)<\delta .$ In particular, every bounded subset of $L_2$ is uniformly integrable. Alternatively, K is bounded and uniformly integrable if and only if, for any $\varepsilon >0$, there is $N>0$ such that $\sup \left\{ \int _{|f|>c}|f|{\mathrm{d}}\nu : \ f \in K\right\} < \varepsilon$ whenever $c \ge N$ (see [1, p.2]).

2.1 Singular value functions

Let (I, m) denote the measure space, where $I = (0,\infty )$ (or (0, 1)), equipped with the Lebesgue measure m. Let L(I, m) be the space of all measurable real-valued functions on I equipped with Lebesgue measure m. Define S(I, m) to be the subset of L(I, m), which consists of all functions f such that $m(\{t : |f(t)| > s\}) < \infty$ for some $s > 0.$ Note that if $I=(0,1)$, then $S(I,m)=L(I,m).$

For $f\in S(I,m)$, we denote by $\mu (f)$ the decreasing rearrangement of the function |f|. That is,

$$\begin{aligned}\mu (t,f)=\inf \{s\ge 0:\ m(\{|f|>s\})\le t\},\quad t>0.\end{aligned}$$

We say that f is submajorized by g in the sense of Hardy–Littlewood–Pólya (written $f\prec \prec g$) if

$$\begin{aligned}\int _0^t\mu (s,f){\mathrm{d}}s\le \int _0^t\mu (s,g){\mathrm{d}}s,\quad t\ge 0.\end{aligned}$$

In addition, we say that f is majorized by g on I in the sense of Hardy–Littlewood–Pólya (written $f\prec g$) if in addition to $f\prec \prec g$, we have

$$\begin{aligned}\int _{I}\mu (s,f){\mathrm{d}}s=\int _{I}\mu (s,g){\mathrm{d}}s.\end{aligned}$$

Let ${\mathcal {M}}$ be a von Neumann algebra on a Hilbert space ${\mathcal {H}}$ equipped with a semifinite faithful normal trace $\tau$. For a given von Neumann algebra ${\mathcal {M}}$ we set ${\mathcal{M}}_+ := \{A\in {\mathcal {M}}: \ A \ge 0\},$ which is called the positive part of ${\mathcal {M}}.$ Let $\text {Proj}({\mathcal {M}})$ denote the lattice of all projections in ${\mathcal {M}}.$

A linear closed and densely defined operator A affiliated with ${\mathcal {M}}$ is called $\tau$-measurable if $\tau (E_{(s,\infty )}(|A|))<\infty$ for sufficiently large $s\ge 0$, where $E_{(s, \infty )}(|A|)$ is the spectral projection of |A| corresponding to the interval $(s, \infty ).$ For a $\tau$-measurable operator A and $s\ge 0$, $\tau (E_{(s,\infty )}(|A|))<\infty$ is called the distribution function of |A| and denoted by (see [18, Definition 1.3])

$$\begin{aligned}\lambda _{s} (A) = \tau (E_{(s, \infty )}(|A|)),\quad s\ge 0.\end{aligned}$$

We denote the set of all $\tau$-measurable operators by $S({\mathcal {M}},\tau ).$ For every $A\in S({\mathcal {M}},\tau ),$ we define its singular value function $\mu (A)$ by setting

$$\begin{aligned}\mu (t,A)=\inf \{\Vert A(1-P)\Vert _{{\mathcal {M}}}:P\in \text {Proj}({\mathcal {M}}),\ \tau (P)\le t\}, \quad t>0.\end{aligned}$$

For more details on generalized singular value functions, we refer the reader to [18, 32]. If $A,B\in S({\mathcal {M}},\tau ),$ then we say that A is submajorized by B (in the sense of Hardy–Littlewood–Pólya), denoted by $A\prec \prec B$, if

$$\begin{aligned}\int _0^t\mu (s,A){\mathrm{d}}s\le \int _0^t\mu (s,B){\mathrm{d}}s,\quad t\ge 0.\end{aligned}$$

For $1\le p<\infty ,$ we set

$$\begin{aligned}{\mathcal {L}}_p({\mathcal {M}})=\{A\in S({\mathcal {M}},\tau ):\ \Vert A\Vert _{{\mathcal {L}}_p({\mathcal {M}})} =\left\| \mu (A)\right\| _p<\infty \},\ \left\| \mu (A)\right\| _p^p : =\int _0^\infty \mu (s,A)^p {\mathrm{d}}s.\end{aligned}$$

Such Banach spaces $({\mathcal {L}}_p({\mathcal {M}}),\left\| \cdot \right\| _{{\mathcal {L}}_p({\mathcal {M}})})$ ($1\le p<\infty$) are examples of symmetric spaces of operators (see Sect 2.3). We denote by $S_0({\mathcal {M}},\tau )$ the subspace of $S({\mathcal {M}},\tau )$, which consists of all elements in $S({\mathcal {M}},\tau )$, whose singular value functions vanish at infinity. For more information on these spaces see [32, Chapter 2, p.60], and the handbook [36].

2.2 Orlicz spaces

Definition 2.1

A continuous and convex function $G: [0, \infty ) \rightarrow [0, \infty )$ is called an N-function if

• 1. $G(0)=0$,

• 2. $G(\lambda )>0$ for $\lambda >0$,

• 3. $\frac{G(\lambda )}{\lambda } \rightarrow 0$ as $\lambda \rightarrow 0$,

• 4. $G(\lambda ) \rightarrow \infty$ as $\lambda \rightarrow \infty$.

Definition 2.2

A function $G: [0, \infty ) \rightarrow [0, \infty ]$ is said to be an Orlicz function if (see [24, p.258])

• 1. $G(0)=0$,

• 2. G is not identically equal to zero,

• 3. G is convex,

• 4. G is continuous at zero.

It follows from the definitions that every N-function is also an Orlicz function. The converse, however, does not hold. For example, the function $G(t)=t$ is an Orlicz function but not an N-function. In what follows, unless otherwise specified, we always denote by G an N-function. For such a function we shall consider an (extended) real valued functional $\mathbf {G} (f)$ (also called the modular defined by an N-function G) defined, on the class of all measurable functions f on I, by

$$\begin{aligned}\mathbf {G} (f)=\int _{I} G(|f(t)|){\mathrm{d}}t.\end{aligned}$$

The set

$$\begin{aligned}L_{G}=\{f \in S(I,m): \quad \Vert f\Vert _{L_G}<\infty \},\end{aligned}$$

where

$$\begin{aligned}\Vert f\Vert _{L_G}=\inf \left\{ c>0: \quad \int _{I} G\left( \frac{|f|}{c}\right) {\mathrm{d}}m \le 1\right\} \end{aligned}$$

is called an Orlicz space defined by the Orlicz function G (equipped with Orlicz norm).

We will denote by $G^*$ the function complementary (or conjugate) to G in the sense of Young, defined by (see [26, Chapter 1, p.11])

$$\begin{aligned}G^*(t)=\sup \{s|t| - G(s): \,\,\ s \ge 0\}.\end{aligned}$$

We notice that $G^*$ is again an N-function (see [24, p.258]).

For any $A \in S({\mathcal {M}}, \tau )$, by means of functional calculus applied to the spectral decomposition of |A|, we have

$$\begin{aligned} \tau \left( G(|A|) \right) = \int _{0}^{\tau (\mathbf {1})} \lambda _s (|A|) dG(s) = \int _{0}^{\tau (\mathbf {1})} G(\mu (s,A)){\mathrm{d}}s. \end{aligned}$$

(2.1)

For an Orlicz function G, the non-commutative Orlicz space ${\mathcal {L}}_G({\mathcal {M}},\tau )$ (or simply ${\mathcal {L}}_G({\mathcal {M}})$) is defined as the space of all $\tau$-measurable operators A affiliated with ${\mathcal {M}}$ such that

$$\begin{aligned}\tau \left( G\left( \frac{|A|}{c} \right) \right) < \infty \end{aligned}$$

for some $c>0$. The space $L_G({\mathcal {M}})$, equipped with the norm

$$\begin{aligned}\Vert A\Vert _{{\mathcal {L}}_G({\mathcal {M}})} = \inf \left\{ c>0: \tau \left( G\left( \frac{|A|}{c} \right) \right) \le 1 \right\} ,\end{aligned}$$

is a Banach space. Observe that if $\tau (\mathbf{1})=\infty$, then $\mathbf{1}\not \in {\mathcal {L}}_G({\mathcal {M}})$. Otherwise, $\tau (G((\frac{\mathbf{1}}{c})))=\infty$ for any c. Hence, for any N-function G, $L_G(0,\infty )$ is a subspace of $S_0(0,\infty )$ and hence, ${\mathcal {L}}_G({\mathcal {M}},\tau )\subset S_0({\mathcal {M}},\tau )$. Note if $G(t)= t^p$ with $1 \le p < \infty$ then ${\mathcal {L}}_G({\mathcal {M}})= {\mathcal {L}}_p({\mathcal {M}})$. For more details on non-commutative Orlicz spaces see, for example, [6, 7].

2.3 Symmetric Banach Function and Operator Spaces

For the general theory of symmetric Banach function spaces, we refer the reader to [8, 27, 31].

Definition 2.3

Let ${\mathcal {E}}$ be a linear subset in $S({\mathcal {M}},\tau )$ equipped with a complete norm $\Vert \cdot \Vert _{{\mathcal {E}}}.$ We say that ${\mathcal {E}}$ is a non-commutative symmetric space (or symmetric operator space) (on ${\mathcal {M}}$, or in $S({\mathcal {M}},\tau )$) if for every $A\in {\mathcal {E}}$ and for every $B\in S({\mathcal {M}},\tau )$ with $\mu (B)\le \mu (A),$ we have $B\in {\mathcal {E}}$ and $\Vert B\Vert _{{\mathcal {E}}}\le \Vert A\Vert _{{\mathcal {E}}}$.

A symmetric function space is the term reserved for a symmetric operator space when ${\mathcal {M}}=L_{\infty }(I,m),$ where $I=(0,\infty )$ (or $I=(0,1)$).

Recall the construction of a non-commutative symmetric (operator) space ${\mathcal {E}}({\mathcal {M}},\tau ).$ Let E be a symmetric Banach function space on $(0,\infty )$. Set

$$\begin{aligned} {\mathcal {E}}({\mathcal {M}},\tau )=\{A\in S({\mathcal {M}},\tau ):\ \mu (A)\in E\}.\end{aligned}$$

We equip ${\mathcal {E}}({\mathcal {M}},\tau )$ with a natural norm

$$\begin{aligned} \Vert A\Vert _{{\mathcal {E}}({\mathcal {M}},\tau )}:=\Vert \mu (A)\Vert _E,\quad A\in {\mathcal {E}}({\mathcal {M}},\tau ). \end{aligned}$$

For brevity, we shall frequently omit $\tau$ in the notation above and simply write $\Vert A\Vert _{{\mathcal {E}}({\mathcal {M}})}$. The space $({\mathcal {E}}({\mathcal {M}}), \Vert \cdot \Vert _{{\mathcal {E}}({\mathcal {M}})})$ is a Banach space with the norm $\Vert \cdot \Vert _{{\mathcal {E}}({\mathcal {M}})}$ and is called the non-commutative symmetric (operator) space associated with $({\mathcal {M}},\tau )$ corresponding to $(E,\Vert \cdot \Vert _{E})$ (see [23]). An extensive discussion of various properties of such spaces can be found in [23, 32]. Furthermore, the following fundamental theorem was proved in [23] (see also [32, Question 2.5.5, p. 58]).

Theorem 2.4

Let $(E,\Vert \cdot \Vert _{E})$ be a symmetric Banach function space on $(0,\infty )$ and let ${{\mathcal {M}}}$ be a semifinite von Neumann algebra. Set

$$\begin{aligned}{\mathcal {E}}({\mathcal {M}})=\{A\in S({\mathcal {M}},\tau ):\ \mu (A)\in E\}, \quad \Vert A\Vert _{{\mathcal {E}}({\mathcal {M}})}:=\Vert \mu (A)\Vert _E.\end{aligned}$$

So defined $({\mathcal {E}}({\mathcal {M}}),\Vert \cdot \Vert _{{\mathcal {E}}({\mathcal {M}})})$ is a non-commutative symmetric space.

The main result of [23] (see also [32, Section 3]) shows that the correspondence

$$\begin{aligned}({E}, \Vert \cdot \Vert _{{E}}) \longleftrightarrow ({\mathcal {E}}({\mathcal {M}}), \Vert \cdot \Vert _{{\mathcal {E}}({\mathcal {M}})})\end{aligned}$$

is a one-to-one correspondence between the set of all symmetric operator spaces in $S({{\mathcal {M}}, \tau })$ and the set of all symmetric function spaces in S(I, m); whenever $({{\mathcal {M}}, \tau })$ does not contain any minimal projections or is atomic and all minimal projections have equal trace. Of course, depending on $({{\mathcal {M}}, \tau })$ the symmetric function space $E\subset S(I,m)$ is considered either on (0, 1),  or on $(0,\infty )$.

3 Weak compactness criteria in Orlicz spaces of $\tau$-measurable operators

In this section, we obtain non-commutative analogues of Ando–Nowak’s and Chong’s theorems.

Let ${\mathcal {L}}_G({\mathcal {M}})$ be the non-commutative Orlicz space. Recall that the Köthe dual (or associate) space, denoted by ${\mathcal {L}}_G^{\times }({\mathcal {M}})$, is defined by setting (see [14, Definition 5.1], [13, 17])

$$\begin{aligned}{\mathcal {L}}_G^{\times }({\mathcal {M}}):=\{A\in S({\mathcal {M}}): \ AB \in {\mathcal {L}}_1({\mathcal {M}}) \ \forall \ B\in {\mathcal {L}}_G({\mathcal {M}})\}\end{aligned}$$

with the norm defined by setting

$$\begin{aligned}\Vert A\Vert _{{\mathcal {L}}_G^{\times }({\mathcal {M}})}:=\sup \{|\tau (AB)|: \, B\in {\mathcal {L}}_G({\mathcal {M}}), \, \Vert B\Vert _{{\mathcal {L}}_G({\mathcal {M}})}\le 1\}.\end{aligned}$$

The following criterion extends results obtained in [3, Theorem 1] and [34, Theorem 1.1].

Theorem 3.1

Let ${\mathcal {M}}$ be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau$. Let $L_G$ be an Orlicz space of functions on $(0, \tau (\mathbf {1}))$ and ${\mathcal {L}}_G({\mathcal {M}})$ be the corresponding non-commutative Orlicz space such that ${\mathcal {L}}_G({\mathcal {M}})\subset S_0({\mathcal {M}},\tau )$. A bounded subset ${\mathcal {K}}$ of ${\mathcal {L}}_{G}^{\times }({\mathcal {M}})$ is relatively $\sigma ({\mathcal {L}}^{\times }_{G}({\mathcal {M}}),{\mathcal {L}}_{G}({\mathcal {M}}))$-compact if and only if the following condition holds

$$\begin{aligned} \sup _{A\in {\mathcal {K}}} \frac{\tau (G(\lambda \cdot |A|))}{\lambda } \rightarrow 0\quad \text {as}\ \lambda \downarrow 0. \end{aligned}$$

(3.1)

Proof

Let ${\mathcal {K}}$ be a bounded subset of ${\mathcal {L}}_{G}^{\times }({\mathcal {M}})$. Assume that ${\mathcal {K}}$ is relatively $\sigma ({\mathcal {L}}^{\times }_{G}({\mathcal {M}}),{\mathcal {L}}_{G}({\mathcal {M}}))$-compact. Then by [15, Theorem 5.4 $(i)\Longrightarrow (ii)$] $\mu ({\mathcal {K}}):=\{\mu (A): A\in {\mathcal {K}} \}$ is relatively $\sigma (L^{\times }_{G},L_{G})$-compact. Hence $\mu ({\mathcal {K}})$ satisfies the conditions of the Nowak’s theorem (see [34, Theorem 1.1]), which in turn implies

$$\begin{aligned} \sup _{A\in {\mathcal {K}}} \frac{\mathbf {G}(\lambda \mu (A))}{\lambda } \rightarrow 0 \quad \lambda \downarrow 0. \end{aligned}$$

(3.2)

Appealing to formula (2.1), we obtain (3.1).

Conversely, suppose that (3.1) holds and hence by (2.1) we obtain (3.2). By [34, Theorem 1.1] the set $\mu ({\mathcal {K}})$ is relatively $\sigma (L^{\times }_{G},L_{G})$-compact. Applying [15, Theorem 5.4 $(ii)\Longrightarrow (i)$], we obtain that ${\mathcal {K}}$ is relatively $\sigma ({\mathcal {L}}^{\times }_{G}({\mathcal {M}}),{\mathcal {L}}_{G}({\mathcal {M}}))$-compact. $\square$

Remark 3.2

When $\tau (\mathbf{1})=\infty$, we note that ${\mathcal {L}}_G^\times ({\mathcal {M}})$ cannot be identified with ${\mathcal {L}}_1({\mathcal {M}})$ since ${\mathcal {L}}_\infty (M)\not \subset S_0({\mathcal {M}},\tau )$, and ${\mathcal {L}}_G({\mathcal {M}})\subset S_0({\mathcal {M}},\tau )$ in Theorem 3.1. Recall that ${\mathcal {A}} \subset {\mathcal {L}}_1({\mathcal {M}})$ is said to be of uniformly absolutely continuous norms in ${\mathcal {L}}_1({\mathcal {M}})$ if ${\mathcal {A}}$ is a bounded set and

$$\begin{aligned} \sup _{x\in {\mathcal{A}}}\Vert e_\alpha xe_\alpha \Vert _{{{\mathcal {L}}_1({\mathcal {M}}) }}\rightarrow _\alpha 0 \end{aligned}$$

for every downwards directed system $e_\alpha \downarrow _\alpha 0$ of projections in ${\mathcal {M}}$. If ${\mathcal {A}} \subset {\mathcal {L}}_1({\mathcal {M}})$ is bounded, then ${\mathcal {A}}$ is relatively weakly compact if and only if ${\mathcal {A}}$ is of uniformly absolutely continuous norms (see e.g. [16, Theorem 4.4]).

The following theorem is the non-commutative version of [33, Theorem 3.5]. It extends [45, Theorem 3.10] (see also [46, Lemma 3.4]).

Theorem 3.3

Let ${\mathcal {M}}$ be a non-atomic finite von Neumann algebra and ${\mathcal {L}}_1({\mathcal {M}})$ be the corresponding non-commutative space of $\tau$-measurable operators. Let ${\mathcal {K}}$ be a bounded subset of ${\mathcal {L}}_1({\mathcal {M}})$. Then the following conditions are equivalent

• 1. there exists an Orlicz function G with $\frac{G(t)}{t} \rightarrow \infty$ as $t \rightarrow \infty$ so that

  $$\begin{aligned}\sup _{A\in {\mathcal {K}}} \tau \left( G(|A|) \right) <\infty ;\end{aligned}$$

• 2. there exists a positive operator $B\in {\mathcal {L}}_1({\mathcal {M}})$ such that $\mu (|A|) \prec \prec \mu (B)$ for all $A \in {\mathcal {K}}.$

Proof

$1. \Longrightarrow 2.$ The condition (1) implies that ${\mathcal {K}}$ is a bounded subset of ${\mathcal {L}}_G({\mathcal {M}})$. Without loss of generality, we may assume that ${\mathcal {K}}$ is the unit ball in ${\mathcal {L}}_G({\mathcal {M}})$.

Since $A \in {\mathcal {K}}$, we have $\mu (A) \in \mu ({\mathcal {K}}) \subset L_1(0,\tau (\mathbf {1})),$ where $\mu ({\mathcal {K}}) := \{ \mu (A): A \in {\mathcal {K}} \}.$ Proceeding as in the commutative case (see [33, Theorem 3.5]), we obtain a positive function $g \in L_1(0,\tau (\mathbf {1}))$ such that $\mu (|A|) \prec \prec \mu (g)$ for all $A \in {\mathcal {K}}.$ Since $\mu (g) \in L_1(0,\tau (\mathbf {1}))$, it follows from [32, Theorem 2.5.3 (b), p. 57] that there exists a positive operator $B \in {\mathcal {L}}_1({\mathcal {M}})$ such that $\mu (g)=\mu (B).$ The fact that B may be chosen positive is explained in [11, Proposition 3.0.3], for which we refer the reader for additional references and comments. Hence $\mu (|A|) \prec \prec \mu (B)$ for all $A \in {\mathcal {K}}.$

$2.\Longrightarrow 1.$ Suppose there is a positive operator $B \in {\mathcal {L}}_1({\mathcal {M}})$ such that $\mu (|A|) \prec \prec \mu (B)$ for all $A\in {\mathcal {K}},$ i.e. $\int _{0}^{t} \mu (s,A) {\mathrm{d}}s \le \int _{0}^{t} \mu (s,B) {\mathrm{d}}s < \infty$ for all $t\in (0, {\tau (\mathbf {1})})$. Then, by [33, Theorem 3.5] $((\text {b}) \Longrightarrow (\text {a}))$, we have

$$\begin{aligned}\sup _{A\in {\mathcal {K}}} \left\{ \int _{0}^{\tau (\mathbf {1})} G(\mu (s, A)){\mathrm{d}}s \right\} < \infty . \end{aligned}$$

By (2.1) it is equivalent to

$$\begin{aligned}\sup _{A\in {\mathcal {K}}} \tau (G(|A|)) < \infty .\end{aligned}$$

$\square$

The following result shows that the statement of Theorem 3.3 holds with an additional condition when ${\mathcal {M}}$ is a non-atomic von Neumann algebra such that $\tau (\mathbf {1})=\infty .$

Theorem 3.4

Let ${\mathcal {M}}$ be a non-atomic von Neumann algebra equipped with a faithful normal trace $\tau$ such that $\tau (\mathbf {1})=\infty .$ Let ${\mathcal {K}}$ be a bounded subset of ${\mathcal {L}}_1({\mathcal {M}})$. If

$$\begin{aligned}\sup _{A \in {\mathcal {K}}} \int _{N}^{\infty }\mu (s,A){\mathrm{d}}s\rightarrow 0, \quad N \rightarrow \infty , \end{aligned}$$

then the following conditions are equivalent

• 1. there exists an Orlicz function G with $\frac{G(t)}{t} \rightarrow \infty$ as $t \rightarrow \infty$ so that

  $$\begin{aligned} \sup _{A\in {\mathcal {K}}} \tau \left( G(|A|) \right) <\infty ; \end{aligned}$$

• 2. there exists a positive operator $B\in {\mathcal {L}}_1({\mathcal {M}})$ such that $\mu (|A|) \prec \prec \mu (B)$ for all $A \in {\mathcal {K}}.$

Proof

By [33, Remark 3.8], the argument for the implication $2. \Longrightarrow 1.$ follows the same line as in the proof of Theorem 3.3 and is therefore omitted.

Let us show that $1. \Longrightarrow 2.$ Suppose there exists an Orlicz function G so that

$$\begin{aligned} \sup _{A\in {\mathcal {K}}} \tau \left( G(|A|) \right) <\infty .\end{aligned}$$

In other words, ${\mathcal {K}} \subset {\mathcal {L}}_G({\mathcal {M}})$. Without loss of generality, we may assume that ${\mathcal {K}}$ is the unit ball in ${\mathcal {L}}_G({\mathcal {M}})$. For all $A \in {\mathcal {K}},$ we have that $\mu (A) \in \mu ({\mathcal {K}}) \subset L_1(0,\infty ),$ where $\mu ({\mathcal {K}}) := \{ \mu (A): A \in {\mathcal {K}} \}.$ Since

$$\begin{aligned} \sup _{A \in {\mathcal {K}}} \int _{N}^{\infty }\mu (s,A){\mathrm{d}}s\rightarrow 0, \quad N \rightarrow \infty , \end{aligned}$$

by [33, Remark 3.8] there exists a positive function $g \in L_1(0,\infty )$ such that $\mu (|A|) \prec \prec \mu (g)$ for all $A \in {\mathcal {K}}.$ Since $\mu (g) \in L_1(0,\infty ),$ it follows from [32, Theorem 2.5.3 (a), p. 57] and [11, Proposition 3.0.3] that there exists a positive operator $B \in {\mathcal {L}}_1({\mathcal {M}})$ such that $\mu (g)=\mu (B).$ Hence $\mu (|A|) \prec \prec \mu (B)$ for all $A \in {\mathcal {K}},$ thereby completing the proof. $\square$

The following theorem is the non-commutative analogue of the Chong’s criterion [9, Lemma 4.1].

Theorem 3.5

Let ${\mathcal {M}}$ be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau .$ Let ${\mathcal {K}} \subset {\mathcal {L}}_1({\mathcal {M}})$ be any family of operators. Then ${\mathcal {K}}$ satisfies

• 1. $\sup \{\tau (|A|): A \in {\mathcal {K}} \} < \infty ;$

• 2. $\sup \left\{ \int _{0}^{\infty } \left( \mu (s, |A|)-u\right) ^+ {\mathrm{d}}s: \,\ A \in {\mathcal {K}} \right\} \rightarrow 0 \ \text {as}\ u \rightarrow \infty ;$

if and only if there exists a positive operator $B \in {\mathcal {L}}_1({\mathcal {M}})$ such that $\mu (|A|) \prec \prec \mu (B)$ for all $A \in {\mathcal {K}}$.

Proof

Let 1. and 2. hold. By [32, Theorem 2.6.3. p. 61, formula (2.4)], we have

$$\begin{aligned}\sup \{\tau (|A|): A \in {\mathcal {K}} \}=\sup \left\{ \int _{0}^{\infty }\mu (s,|A|){\mathrm{d}}s: A\in {\mathcal {K}}\right\} <\infty .\end{aligned}$$

Then, there exists a positive function $f \in L_1(I),$ where $I=(0,\tau (1))$, such that $\mu (|A|) \prec \prec f$ (see [9, Lemma 4.1]). Since $\mu (f)\in L_1(I),$ it follows from [32, Theorem 2.5.3, p. 57] (see also [11, Proposition 3.0.3]) that there exists a positive operator $B \in {\mathcal {L}}_1({\mathcal {M}})$ such that $\mu (f)=\mu (B)$, and hence $\mu (|A|) \prec \prec \mu (B)$ for all $A \in {\mathcal {K}}$.

Conversely, let B be a positive operator in ${\mathcal {L}}_1({\mathcal {M}})$ such that $\mu (|A|) \prec \prec \mu (B)$ for all $A \in {\mathcal {K}}$ or, equivalently

$$\begin{aligned} \int _{0}^{t} \mu (s, |A|){\mathrm{d}}s \le \int _{0}^{t} \mu (s, B){\mathrm{d}}s, \,\ t \in [0, {\infty }),\quad \forall \ A \in {\mathcal {K}}.\end{aligned}$$

Since $B \in {\mathcal {L}}_1({\mathcal {M}})$, it follows immediately that

$$\begin{aligned}\sup _{A \in {\mathcal {K}}}\int _{0}^{\infty }\mu (s, |A|)ds < \infty, \end{aligned}$$

which implies that the condition $1.$ of the theorem holds.

By [39, Theorem 4], for every $A \in {\mathcal {K}}$ and all $u\ge 0$, we have

$$\begin{aligned}{\begin{matrix}\int _{0}^{\infty } (\mu (s, |A|)-u)^+ {\mathrm{d}}s &{}\le \int _{\mu (A)>u}(\mu (s, |A|)-u){\mathrm{d}}s\le \int _{\mu (B)>u}(\mu (s, B)-u){\mathrm{d}}s\\ &{}\le \int _{\mu (B)>u}\mu (s, B){\mathrm{d}}s. \end{matrix}} \end{aligned}$$

Since $\int _{\mu (B)>u}\mu (s, B){\mathrm{d}}s\rightarrow 0$ as $u \rightarrow \infty$, we conclude that

$$\begin{aligned}\sup \left\{ \int _0^{\infty } \left( \mu (s, |A|)-u\right) ^+ {\mathrm{d}}s: \,\ A \in {\mathcal {K}} \right\} \rightarrow 0\quad \text {as}\ u \rightarrow \infty ,\end{aligned}$$

so the condition 2. of the theorem holds. $\square$

4 Pełczyński’s property (V) of Orlicz spaces

In this section, we consider the Pełczyński’s property (V) of non-commutative Orlicz spaces. For the treatment of the property $(V^*)$ in symmetric non-commutative (operator) spaces, we refer the reader to [12].

Definition 4.1

A Banach space X is said to have the Pełczyński’s property (V) if every subset ${\mathcal {F}}$ of $X^*$ is relatively weakly compact whenever it has the following property

$$\begin{aligned}\lim _{n\rightarrow \infty }\sup _{x^*\in {\mathcal {F}}} |x^* (x_n)|=0\end{aligned}$$

for every weakly unconditionally Cauchy sequence $\{x_n\}_{n\ge 1}$ in X (i.e. such that $\sum _{n\ge 1}|x^*(x_n)|<\infty$ for any $x^* \in X^*$). Equivalently, X has the Pełczyński’s property (V) if and only if for every Banach space Z and for every non-weakly compact operator $T:X\rightarrow Z$, there exists a subspace $X_0$, isomorphic to $c_0$, such that T is an isomorphism between $X_0$ and $T(X_0)$.

The property (V) of Orlicz function spaces has been considered in [21, 28, 30]. We characterize below the Orlicz functions such that the corresponding non-commutative Orlicz spaces have the property (V).

A subspace X of a Banach space Y is called an M-ideal of Y if there is an L-projection P on $Y^*$ whose kernel is $X^\bot$, the annihilator of X; that is, we have

$$\begin{aligned}\Vert y^*\Vert = \Vert P y^*\Vert +\Vert y^* - P y^*\Vert , \quad y^*\in Y^*.\end{aligned}$$

In particular, when $Y=X^{**}$, X is called an M-embedded space (see e.g. [21, Chapter III, Definition 1.1]). The theory of M-embedded spaces has been developed intrinsically since it was introduced by Alfsen and Effros [2] in 1972. Apart from the intrinsic mathematical beauty of the theory in its own right, the interest to the theory of M-embedded spaces has been maintained by its numerous applications in diverse areas of mathematics such as $C^*$-algebras, ordered Banach spaces and $L^1$-preduals (see e.g. [21]). Examples of M-embedded spaces are given by special examples of Orlicz sequence and function spaces, by the predual space of Lorentz function space ${L_{p,1}(0,\infty )}$, $1<p<\infty$, and by the set $K({\mathcal {H}})$ of all compact operators on a Hilbert space ${\mathcal {H}}$, see e.g. [21, Chapter III, Example 1.4], [43] and [28]. Furthermore, Werner [43, Proposition 4.1] proved that, under some mild additional conditions imposed on a symmetric sequence space E, the property of being an M-embedded space carries over to its non-commutative counterpart ${{\mathcal {E}}}$, the symmetric ideal of bounded operators on a separable Hilbert space associated with E. This result was recently extended to the setting of arbitrary semifinite von Neumann algebra [22, Theorem 3.3].

Theorem 4.2

Assume that $E(0,\infty )$ is a fully symmetric function space having order continuous norm (i.e. separable), which fails to be a superset of $C_0(0,\infty )$, where $C_0(0,\infty )$ is the space of all bounded vanishing functions. If $E(0,\infty )$ is an M-embedded space, then ${\mathcal {E}}({\mathcal {M}})$ is an M-embedded space, too.

In fact, an M-embeddedness is a stronger property than the Pełczyński’s property (V).

Theorem 4.3

(See e.g. [28, Theorem 1] and [19, 20]) Every Banach space, which is an M-embedded space, has the property (V).

It has been proved in [22] that the separable part ${\mathcal {L}}_{q,\infty }^0({\mathcal {M}})$ of ${\mathcal {L}}_{q,\infty }({\mathcal {M}})$ is M-embedded when $1<q<\infty$. Hence, it has the property (V).

We define $H_G(0,\infty )$ by setting [21, p. 103]

$$\begin{aligned}H_G (0,\infty ) =\left\{ \int_0^\infty G\left( \frac{|f(s)|}{\rho }\right) {\mathrm{d}}s <\infty \text{ for } \text{ all } \rho >0 \right\} . \end{aligned}$$

Let $G^*$ be the complementary (in the sense of Young) function to G. We say that $G^*$ satisfies the $\Delta _2$-condition if

$$\begin{aligned}\limsup _{t\rightarrow 0} \frac{G^*(2t)}{G^*(t)}<\infty \quad \text {and} \quad \limsup _{t\rightarrow \infty } \frac{G^*(2t)}{G^*(t)}<\infty .\end{aligned}$$

Theorem 4.4

Let ${\mathcal {M}}$ be a semifinite von Neumann algebra equipped with a semifinite faithful normal trace $\tau$. Let $G: [0, \infty )\rightarrow [0, \infty )$ be a continuous convex function such that $G(0)=0$ and $G(t)>0$, $t>0$. If $G^*$ satisfies the $\Delta _2$-condition but G fails it, then ${\mathcal {H}}_G({\mathcal {M}})$ has the Property (V).

Proof

When $G^*$ satisfies the $\Delta _2$-condition while G fails it, $H_G (0,\infty )$ is an M-embedded space [21, p. 105, Example 1.4] (see also [43]). Moreover, since the decreasing rearrangement of any element $L_G(0,\infty )$ vanishes at infinity and

$$\begin{aligned}H_G(0,\infty )^{\times \times } = L_G(0,\infty )\end{aligned}$$

[21, p. 103], it follows that $H_G(0,\infty ) \not \supset C_0(0,\infty )$ [22, Proposition 2.4]. Hence, by Theorem 4.2, ${\mathcal {H}}_G({\mathcal {M}})$ is an M-embedded space and therefore, by Theorem 4.3 it has the property (V). $\square$

Remark 4.5

The proof in [28, Theorem 2] works only for finite measure spaces and could not be adjusted for infinite measure spaces. On the other hand, the result of Theorem 4.4, holds for an arbitrary semifinite von Neumann algebra.

5 Non-commutative analogue of Kolmogorov’s compactness criterion in terms of conditional expectations

This section is devoted to the extension of a well-known Kolmogorov’s criterion of compactness (see [25], see also [26, Theorem 11.1, p. 97]). This criterion found non-trivial generalizations in several directions, see, for example [40, 26, Theorem 11.1, p. 97]. In addition, in [16, Section 5] relatively compact sets in symmetrically normed spaces characterized completely in terms of sets of uniformly absolutely continuous norms, that is, $\sup _{x\in {\mathcal {A}}} \Vert e_{n}xe_{n}\Vert _{{\mathcal {E}}}\rightarrow _{n} 0$ for all mutually disjoint sequences $\{e_n\}_{n\ge 1}$ of projections in ${\mathcal {M}}.$ In particular, if ${\mathcal {M}}$ is atomic von Neumann algebra such that $\tau (\mathbf {1})<\infty$ and ${\mathcal {E}}({\mathcal {M}})$ is a symmetrically normed space with the order continuous norm, then a bounded set ${\mathcal {A}}\subset {\mathcal {E}}({\mathcal {M}})$ is relatively compact if and only if ${\mathcal {A}}$ is of uniformly (equi-) absolutely continuous norms (see [16, Corollary 5.3]).

Let ${\mathcal {M}}$ be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau .$ An increasing sequence $({\mathcal {M}}_n)_{n\ge 0}$ of von Neumann subalgebras of ${\mathcal {M}}$ such that the union $\bigcup _{n\ge 0}\mathcal M_n$ is weak$^{*}$ dense in ${\mathcal {M}}$ is called a filtration of ${\mathcal {M}}$. Assume that for every $n\ge 0,$ the restriction $\tau |_{{\mathcal {M}}_n}$ is semifinite. Then there exists a map ${\mathbb {E}}_n:{\mathcal {M}} \rightarrow {\mathcal {M}}_n$ satisfying the following properties:

• 1. ${\mathbb {E}}_n$ is a normal contractive positive projection from ${\mathcal {M}}$ onto ${\mathcal {M}}_n;$

• 2. ${\mathbb {E}}_n(axb)=a{\mathbb {E}}_n(x)b$ for any $x\in {\mathcal {M}}$ and $a,b\in {\mathcal {M}}_n;$

• 3. $\tau \circ {\mathbb {E}}_n=\tau .$

The map ${\mathbb {E}}_n$ satisfying above conditions is called the conditional expectation with respect to ${\mathcal {M}}_n.$ For more information we refer to [42, 44]. Since each ${\mathbb {E}}_{n}$ preserves the trace, it may be extended to a contractive projection from ${\mathcal {L}}_p({\mathcal {M}})$ onto ${\mathcal {L}}_p({\mathcal {M}}_n)$ for all $1\le p\le \infty .$

In this section, we characterize relatively compact sets in separable symmetric spaces of $\tau$-measurable operators associated with a hyperfinite von Neumann algebra from the perspective of conditional expectations. In particular, we obtain relative compactness criterion in separable non-commutative Orlicz spaces. Observe that when ${\mathcal {M}}$ acts on a separable Hilbert space, the space ${\mathcal {E}}({\mathcal {M}})$ is separable provided that the space E(I) is separable (see [12, Corollary 6.10] and detailed discussion in [11, p.54]).

Theorem 5.1

Let $({\mathcal {M}},\tau )$ be a hyperfinite non-commutative probability space and let $({\mathcal {M}}_n)_{n\ge 0}$ be a filtration, which consists of finite dimensional von Neumann algebras. Let ${\mathcal {E}}({\mathcal {M}})$ be the corresponding separable symmetric Banach space of $\tau$-measurable operators. If ${\mathcal {F}}\subset {\mathcal {E}}({\mathcal {M}})$ is a bounded set, then the following conditions are equivalent

• 1. ${\mathcal {F}}$ is relatively compact;

• 2.
  $$\begin{aligned}\sup _{x\in {\mathcal {F}}}\Vert x-{\mathbb {E}}_nx\Vert _\mathcal {E({\mathcal {M}})}\rightarrow 0,\quad n\rightarrow \infty .\end{aligned}$$

Lemma 5.2

Let ${\mathcal {E}}({\mathcal {M}})$ be a separable symmetric Banach function space. Let $({\mathcal {M}},\tau )$ be a non-commutative probability space and let $({\mathcal {M}}_n)_{n\ge 0}$ be a filtration. For every $y\in {\mathcal {E}}({\mathcal {M}}),$ we have

$$\begin{aligned}\Vert y-{\mathbb {E}}_ny\Vert _\mathcal {E({\mathcal {M}})}\rightarrow 0,\quad n\rightarrow \infty .\end{aligned}$$

Proof

Let $\psi$ be the fundamental function of ${\mathcal {E}}({\mathcal {M}}).$ If $\Vert z\Vert _{{\mathcal {L}}_\infty ({\mathcal {M}})}=\alpha$ and $\Vert z\Vert _{{\mathcal {L}}_1({\mathcal {M}})}=\beta ,$ then $z\prec \alpha \chi _{(0,\frac{\beta }{\alpha })}.$ Thus,

$$\begin{aligned}\Vert z\Vert _\mathcal {E({\mathcal {M}})}\le \Vert \alpha \chi _{(0,\frac{\beta }{\alpha })}\Vert _\mathcal {E({\mathcal {M}})}= \alpha \psi \left( \frac{\beta }{\alpha }\right) =\Vert z\Vert _{{\mathcal {L}}_{\infty }({\mathcal {M}})} \psi \left( \frac{\Vert z\Vert _{{\mathcal {L}}_1({\mathcal {M}})}}{\Vert z\Vert _{{\mathcal {L}}_\infty ({\mathcal {M}})}}\right) .\end{aligned}$$

Take $u\in {\mathcal {L}}_{\infty }({\mathcal {M}}).$ Let $z_n=u-{\mathbb {E}}_nu$ and note that $\Vert z_n\Vert _{{\mathcal {L}}_{\infty }({\mathcal {M}})}\le 2\Vert u\Vert _{{\mathcal {L}}_{\infty }({\mathcal {M}})}.$ Thus,

$$\begin{aligned}\Vert z_n\Vert _\mathcal {E({\mathcal {M}})}\le \Vert z_n\Vert _{{\mathcal {L}}_{\infty }({\mathcal {M}})}\psi \left( \frac{\Vert z_n\Vert _{{{\mathcal {L}}_1}({\mathcal {M}})}}{\Vert z_n\Vert _{{\mathcal {L}}_{\infty }({\mathcal {M}})}}\right) \le 2\Vert u\Vert _{{\mathcal {L}}_{\infty }({\mathcal {M}})}\psi \left( \frac{\Vert z_n\Vert _{{\mathcal {L}}_{1}({\mathcal {M}})}}{2\Vert u\Vert _{{\mathcal {L}}_{\infty }({\mathcal {M}})}}\right) .\end{aligned}$$

The latter inequality follows from the fact that the mapping $t\rightarrow \frac{\psi (t)}{t}$ is decreasing. Since $\Vert z_n\Vert _{{\mathcal {L}}_1({\mathcal {M}})}=\Vert u-{\mathbb {E}}_nu\Vert _{{\mathcal {L}}_1({\mathcal {M}})}\rightarrow 0$ (see Theorem 2 in [41]) it follows that $\Vert z_n\Vert _{{\mathcal {E}}({\mathcal {M}})}\rightarrow 0.$ In other words,

$$\begin{aligned}\Vert u-{\mathbb {E}}_nu\Vert _{{\mathcal {E}}({\mathcal {M}})}\rightarrow 0, \quad n \rightarrow \infty \ u\in {\mathcal {L}}_{\infty }({\mathcal {M}}).\end{aligned}$$

Now, let $y\in {\mathcal {E}}({\mathcal {M}})$ and fix $\epsilon >0.$ By the separability of ${\mathcal {E}}({\mathcal {M}}),$ one can find $u\in {\mathcal {L}}_{\infty }({\mathcal {M}})$ such that $\Vert y-u\Vert _{{\mathcal {E}}({\mathcal {M}})}\le \epsilon .$ By triangle inequality, we have

$$\begin{aligned}\Vert y-{\mathbb {E}}_ny\Vert _{{\mathcal {E}}({\mathcal {M}})}\le & {} \Vert u-{\mathbb {E}}_nu\Vert _{{\mathcal {E}}({\mathcal {M}})}+\Vert u-y\Vert _{{\mathcal {E}}({\mathcal {M}})}+\Vert {\mathbb {E}}_ny-{\mathbb {E}}_nu\Vert _{{\mathcal {E}}({\mathcal {M}})}\\&\le \Vert u-{\mathbb {E}}_nu\Vert _{{\mathcal {E}}({\mathcal {M}})}+2\Vert u-y\Vert _{{\mathcal {E}}({\mathcal {M}})}.\end{aligned}$$

By the preceding paragraph, we have

$$\begin{aligned}\limsup _{n\rightarrow \infty }\Vert y-{\mathbb {E}}_ny\Vert _{{\mathcal {E}}({\mathcal {M}})}\le 2\Vert u-y\Vert _{{\mathcal {E}}({\mathcal {M}})}\le 2\epsilon .\end{aligned}$$

Since $\epsilon >0$ is arbitrary, the assertion follows. $\square$

Proof of Theorem 5.1

$1. \Longrightarrow 2.$ ${\mathcal {F}}$ is relatively compact and, therefore, is totally bounded. Fix $\epsilon >0$ and choose natural number $m=m(\epsilon )$ and points $(y_k)_{1\le k\le m}$ in ${\mathcal {E}}({\mathcal {M}})$ such that

$$\begin{aligned}\min _{1\le k\le m}\Vert x-y_k\Vert _{{\mathcal {E}}({\mathcal {M}})}\le \epsilon ,\quad \text {for all}\ x\in {\mathcal {F}}.\end{aligned}$$

By triangle inequality, we have

$$\begin{aligned}{\begin{matrix}\Vert x-{\mathbb {E}}_nx\Vert _{{\mathcal {E}}({\mathcal {M}})}&{}\le \Vert y_k-{\mathbb {E}}_ny_k\Vert _{{\mathcal {E}}({\mathcal {M}})}+\Vert x-y_k\Vert _{{\mathcal {E}}({\mathcal {M}})}+\Vert {\mathbb {E}}_nx-{\mathbb {E}}_ny_k\Vert _{{\mathcal {E}}({\mathcal {M}})}\\ &{}\le \Vert y_k-{\mathbb {E}}_ny_k\Vert _{{\mathcal {E}}({\mathcal {M}})}+2\Vert x-y_k\Vert _{{\mathcal {E}}({\mathcal {M}})} \quad \text {for all} \ 1\le k\le m. \end{matrix}}\end{aligned}$$

Hence,

$$\begin{aligned}{\begin{matrix}\Vert x-{\mathbb {E}}_nx\Vert _{{\mathcal {E}}({\mathcal {M}})}&{}\le \min _{1\le k\le m}\big (\Vert y_k-{\mathbb {E}}_ny_k\Vert _{{\mathcal {E}}({\mathcal {M}})}+2\Vert x-y_k\Vert _{{\mathcal {E}}({\mathcal {M}})}\big )\\ &{}\le \max _{1\le k\le m}\Vert y_k-{\mathbb {E}}_ny_k\Vert _{{\mathcal {E}}({\mathcal {M}})}+2\min _{1\le k\le m}\Vert x-y_k\Vert _{{\mathcal {E}}({\mathcal {M}})}\\ &{}\le \max _{1\le k\le m}\Vert y_k-{\mathbb {E}}_ny_k\Vert _{{\mathcal {E}}({\mathcal {M}})}+2\epsilon \le \sum _{1\le k\le m}\Vert y_k-{\mathbb {E}}_ny_k\Vert _{{\mathcal {E}}({\mathcal {M}})}+2\epsilon . \end{matrix}}\end{aligned}$$

Thus,

$$\begin{aligned}\limsup _{n\rightarrow \infty }\sup _{x\in X}\Vert x-{\mathbb {E}}_nx\Vert _{{\mathcal {E}}({\mathcal {M}})}\le \sum _{1\le k\le m}\limsup _{n\rightarrow \infty }\Vert y_k-{\mathbb {E}}_ny_k\Vert _{{\mathcal {E}}({\mathcal {M}})}+2\epsilon .\end{aligned}$$

By Lemma 5.2, we have

$$\begin{aligned}\limsup _{n\rightarrow \infty }\sup _{x\in {\mathcal {F}}}\Vert x-{\mathbb {E}}_nx\Vert _{{\mathcal {E}}({\mathcal {M}})}\le 2\epsilon .\end{aligned}$$

Since $\epsilon >0$ is arbitrary, the assertion follows.

$2. \Longrightarrow 1.$ Fix $\epsilon >0.$ Choose $n=n(\epsilon )$ such that

$$\begin{aligned}\Vert x-{\mathbb {E}}_nx\Vert _{{\mathcal {E}}({\mathcal {M}})}\le \epsilon ,\quad x\in {\mathcal {F}}.\end{aligned}$$

The set

$$\begin{aligned}\{{\mathbb {E}}_nx,\quad x\in {\mathcal {F}}\}\end{aligned}$$

is bounded and finite dimensional since it is a subset of ${\mathcal {M}}_n,$ which is assumed to be finite dimensional. Thus, it is relatively compact and, therefore, totally bounded. Hence, there exist a natural number $m=m(\epsilon )$ and points $(y_k)_{1\le k\le m}$ such that

$$\begin{aligned}\min _{1\le k\le m}\Vert {\mathbb {E}}_nx-y_k\Vert _{{\mathcal {E}}({\mathcal {M}})}\le \epsilon ,\quad x\in {\mathcal {F}}.\end{aligned}$$

By triangle inequality, we have

$$\begin{aligned}\min _{1\le k\le m}\Vert x-y_k\Vert _{{\mathcal {E}}({\mathcal {M}})}\le 2\epsilon ,\quad x\in {\mathcal {F}}.\end{aligned}$$

Since $\epsilon >0$ is arbitrary, it follows that ${\mathcal {F}}$ is totally bounded. Therefore, ${\mathcal {F}}$ is relatively compact. $\square$

Remark 5.3

Note that the proof of necessity $(1. \Longrightarrow 2.)$ works without the assumption that filtration is finite dimensional. The latter condition is only used in the proof of sufficiency.

The following corollary is a direct consequence of Theorem 5.1.

Corollary 5.4

Let ${\mathcal {L}}_G({\mathcal {M}})$ be a separable Orlicz space. Let $({\mathcal {M}},\tau )$ be a hyperfinite non-commutative probability space and let $({\mathcal {M}}_n)_{n\ge 0}$ be a filtration, which consists of finite dimensional von Neumann algebras. Let ${\mathcal {F}}\subset {\mathcal {L}}_G({\mathcal {M}})$ be a bounded set. Then the following conditions are equivalent

• 1. ${\mathcal {F}}$ is relatively compact;

• 2.
  $$\begin{aligned}\sup _{x\in {\mathcal {F}}} \inf \left\{ c>0: \tau \left( G\left( \frac{|x-{\mathbb {E}}_nx|}{c} \right) \right) \le 1\right\} \rightarrow 0,\quad n\rightarrow \infty .\end{aligned}$$