Interpolation between $L_0({\mathcal M},\tau)$ and $L_\infty({\mathcal M},\tau)$

TL;DR

This paper demonstrates that symmetrically Δ-normed operator spaces associated with semifinite von Neumann algebras serve as interpolation spaces between $L_0$ and the algebra itself, with applications to ${ m K}$-functional and monotonicity properties.

Contribution

It establishes that these operator spaces are interpolation spaces between $L_0$ and the algebra, contrasting with classical results, and explores ${ m K}$-functional and monotonicity in this context.

Findings

Symmetrically Δ-normed operator spaces are interpolation spaces between $L_0$ and the algebra.

The ${ m K}$-functional of an operator coincides with that of its singular value function.

The pair $(L_0( au), ext{algebra})$ is ${ m K}$-monotone in non-atomic finite factors.

Abstract

Let ${\mathcal M}$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $\tau$. We show that the symmetrically $\Delta$-normed operator space $E({\mathcal M},\tau)$ corresponding to an arbitrary symmetrically $\Delta$-normed function space $E(0,\infty)$ is an interpolation space between $L_0({\mathcal M},\tau)$ and ${\mathcal M}$, which is in contrast with the classical result that there exist symmetric operator spaces $E({\mathcal M},\tau)$ which are not interpolation spaces between $L_1({\mathcal M},\tau)$ and ${\mathcal M}$. Besides, we show that the ${\mathcal K}$-functional of every $X\in L_0({\mathcal M},\tau)+ {\mathcal M} $ coincides with the ${\mathcal K}$-functional of its generalized singular value function $\mu(X)$. Several applications are given, e.g., it is shown that the pair $(L_0({\mathcal M},\tau),{\mathcal M})$ is ${\mathcal K}$-monotone when ${\mathcal M}$ is a non-atomic finite factor.

Equations

$$E(\mathcal{M},\tau ):=\{X\in S(\mathcal{M},\tau ):\mu (X)\in E(0,\infty )\},\Vert X{\Vert }_{E(\mathcal{M},\tau )}:=\Vert \mu (X){\Vert }_E$$

$$A(\mathcal{M},\tau )\cap B(\mathcal{M},\tau )\subset E(\mathcal{M},\tau )\subset A(\mathcal{M},\tau )+B(\mathcal{M},\tau )$$

$$\tau\left(E^{|X|}\bigl{(}\frac{1}{n},\infty\bigr{)}\right)=o(1),\quad n\to\infty.$$

$$\mu (s;X)=\inf \{\Vert XP{\Vert }_{\infty }:P=P^{\ast }\in \mathcal{M}\text{mbox}isaprojection,\tau (1-P)\le s\}.$$

$$d_X(t)=\tau (E^X(t,\infty )),t>0,$$

$$\mu (t;X)=\inf \{s\ge 0:d_{|X|}(s)\le t\}.$$

$$\mu (t;x)=\inf \{s\ge 0:m(\{|x|\ge s\})\le t\}.$$

$$\mathcal{F}(\mathcal{M},\tau )=\{X\in \mathcal{M}:\tau (r(X))<\infty \}=\{X\in \mathcal{M}:\tau (s(X))<\infty \}.$$

$$V(\epsilon ,\delta )=\{X\in S(\mathcal{M},\tau ):\exists P\in P\left(\mathcal{M}\right)\text{mbox}suchthat\Vert X(1-P)\Vert \le \epsilon ,\tau (P)\le \delta \}.$$

$$\Vert AXB{\Vert }_E\le \Vert A{\Vert }_{\infty }\Vert B{\Vert }_{\infty }\Vert X{\Vert }_E,A,B\in \mathcal{M},X\in E(\mathcal{M},\tau ).$$

$$E(\mathcal{M},\tau ):=\{X\in S(\mathcal{M},\tau ):\mu (X)\in E(0,\infty )\},\Vert X{\Vert }_{E(\mathcal{M},\tau )}:=\Vert \mu (X){\Vert }_E$$

$$\Vert f{\Vert }_{L_0}=m(\text{mbox}supp(f)),$$

$$d_{\mu (X)}(\lambda )=\tau (E^{|X|}(\lambda ,\infty ))=t.$$

$$K_u(f)=\underset{s>0}{\inf }[su+d_{\mu (f)}(s)]=\underset{t>0}{\inf }[t+u\mu (t;f)].$$

$$K_u(X):=\inf \{\Vert G{\Vert }_0+u\Vert H{\Vert }_{\infty }:X=G+H,G\in L_0(\mathcal{M},\tau ),H\in L_{\infty }(\mathcal{M},\tau )\},u>0.$$

$$\Vert X{\Vert }_S:=K_1(X)=\inf \{\Vert G{\Vert }_0+\Vert H{\Vert }_{\infty }:X=G+H,G\in L_0(\mathcal{M},\tau ),H\in L_{\infty }(\mathcal{M},\tau )\}$$

$$K_u(X)=K_u(\mu (X)),u>0.$$

$$K_u(X)$$

$$\le \underset{s>0}{\inf }\{\Vert X{E}^{|X|}(s,\infty ){\Vert }_0+u\Vert X{E}^{|X|}(0,s]{\Vert }_{\infty }\}$$

$$\le \underset{s>0}{\inf }\{d_{|X{E}^{|X|}(s,\infty )|}(0)+us\}$$

$$=\underset{s>0}{\inf }\{d_{\mu (X{E}^{|X|}(s,\infty ))}(0)+us\}\text{mbox}(by\text{cite}[cite][\text{@@bibref}DPS,ChapterIII,Eq.(4)])$$

$$=\underset{s>0}{\inf }\{d_{\mu (X{E}^{|X|}(s,\infty ))}(s)+us\}$$

$$\le \underset{s>0}{\inf }\{d_{\mu (X)}(s)+us\}$$

$$\stackrel{\text{eqref}ineq:K}{=}{K}_u(\mu (X)).$$

$$t+u\mu (t;X)$$

$$=t+u\cdot \inf \{\Vert X-Y{\Vert }_{\infty }:Y\in L_0(\mathcal{M},\tau ),\Vert Y{\Vert }_0\le t,X-Y\in \mathcal{M}\}$$

$$\le \Vert G{\Vert }_0+u\Vert X-G{\Vert }_{\infty }.$$

$$\underset{t>0}{\inf }[t+u\mu (t;X)]\le \underset{\begin{array}{c}G\in L_0(\mathcal{M},\tau )\\ X-G\in L_{\infty }(\mathcal{M},\tau )\end{array}}{\inf }(\Vert G{\Vert }_0+u\Vert X-G{\Vert }_{\infty })=K_u(X).$$

$$K_u(X)=\underset{s>0}{\inf }[su+d_{|X|}(s)],u>0.$$

$$K_u(\mu (X))+K_u(\mu (Y))$$

$$=K_u(X)+K_u(Y)$$

$$=\inf \{\Vert X_1{\Vert }_0+u\Vert X_2{\Vert }_{\infty }:X=X_1+X_2,X_1\in L_0(\mathcal{M},\tau ),X_2\in L_{\infty }(\mathcal{M},\tau )\}$$

$$+\inf \{\Vert Y_1{\Vert }_0+u\Vert Y_2{\Vert }_{\infty }:Y=Y_1+Y_2,Y_1\in L_0(\mathcal{M},\tau ),Y_2\in L_{\infty }(\mathcal{M},\tau )\}$$

$$=\inf \{\Vert X_1{\Vert }_0+\Vert Y_1{\Vert }_0+u\Vert X_2{\Vert }_{\infty }+u\Vert Y_2{\Vert }_{\infty }:$$

$$X=X_1+X_2,Y=Y_1+Y_2,X_1,Y_1\in L_0(\mathcal{M},\tau ),X_2,Y_2\in L_{\infty }(\mathcal{M},\tau )\}$$

$$\ge \inf \{\Vert X_1+Y_1{\Vert }_0+u\Vert X_2+Y_2{\Vert }_{\infty }:$$

$$X=X_1+X_2,Y=Y_1+Y_2,X_1,Y_1\in L_0(\mathcal{M},\tau ),X_2,Y_2\in L_{\infty }(\mathcal{M},\tau )\}$$

$$\ge \inf \{\Vert Z_1{\Vert }_0+u\Vert Z_2{\Vert }_{\infty }:X+Y=Z_1+Z_2,Z_1\in L_0(\mathcal{M},\tau ),Z_2\in L_{\infty }(\mathcal{M},\tau )\}$$

Full text

1. Introduction

Recall the Calkin correspondence between symmetrically $\Delta$-normed operator spaces and symmetrically $\Delta$-normed function spaces introduced in [17]. Let $E(0,\infty)$ be an arbitrary symmetrically $\Delta$-normed function space equipped with a $\Delta$-norm $\|\cdot\|_{E}$ (see Section 2) and let $\mathcal{M}$ be an arbitrary semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau$. Then,

[TABLE]

is a symmetrically $\Delta$-normed operator space, where $S({\mathcal{M}},\tau)$ is the space of all $\tau$-measurable operators affiliated with ${\mathcal{M}}$. Moreover, if $E(0,\infty)$ is complete, then $E({\mathcal{M}},\tau)$ is also complete (see [17, Theorem 3.8], see also [22, 34] for the Banach case and the quasi-Banach case). For brevity, we omit below the term “operator” and refer just to symmetrically $\Delta$-normed spaces.

Let $A({\mathcal{M}},\tau)$ and $B({\mathcal{M}},\tau)$ be two symmetrically $\Delta$-normed spaces. A symmetrically $\Delta$-normed space $E({\mathcal{M}},\tau)$ is said to be intermediate for $A({\mathcal{M}},\tau)$ and $B({\mathcal{M}},\tau)$ if the continuous embeddings

[TABLE]

hold. Let $E({\mathcal{M}},\tau)$ be a symmetrically $\Delta$-normed space intermediate between $A({\mathcal{M}},\tau)$ and $B({\mathcal{M}},\tau)$. If every linear operator on $A({\mathcal{M}},\tau)+B({\mathcal{M}},\tau)$ which is bounded from $A({\mathcal{M}},\tau)$ to $A({\mathcal{M}},\tau)$ and $B({\mathcal{M}},\tau)$ to $B({\mathcal{M}},\tau)$ is also a bounded operator from $E({\mathcal{M}},\tau)$ to $E({\mathcal{M}},\tau)$, then $E({\mathcal{M}},\tau)$ is called an interpolation space between the spaces $A({\mathcal{M}},\tau)$ and $B({\mathcal{M}},\tau)$.

Interpolation function spaces have been widely investigated (see e.g. [1, 2, 18, 23, 31, 4, 3, 26]) since Mityagin [28] and Calderón [5] gave characterizations of the class of all interpolation spaces with respect to $(L_{1}(0,\infty),L_{\infty}(0,\infty))$ (see also [10, 9] for results in the noncommutative setting). Among several real interpolation methods, the K-method of interpolation linked to the so-called ${\mathcal{K}}$-functional is very important (we refer [27, 26, 2, 36] for applications of ${\mathcal{K}}$-functionals in different areas). Calculating the ${\mathcal{K}}$-functionals for a given couple of spaces is very important in the K-method [25]. In [18], the ${\mathcal{K}}$-functionals for the couple $L_{0}(0,\infty)$ and $L_{\infty}(0,\infty)$ are obtained. We give a description of the ${\mathcal{K}}$-functional of every element $X\in(L_{0}+L_{\infty})({\mathcal{M}},\tau)$ in terms of singular value function as well as in terms of its distribution function, showing that the ${\mathcal{K}}$-functional of $X$ coincides with the ${\mathcal{K}}$-functional of its generalized singular value function $\mu(X)$ (see Section 2), which extends [18, Proposition 3].

It is well-known (see e.g. [28, 5], see also [23, 12, 9]) that the operator space $E({\mathcal{M}},\tau)$ corresponding to a fully symmetric (see e.g Section 5) function space $E(0,\infty)$ is an interpolation space between $L_{1}({\mathcal{M}},\tau)$ and ${\mathcal{M}}$. In particular, there exist symmetric normed spaces which are not interpolation spaces between $L_{1}({\mathcal{M}},\tau)$ and $L_{\infty}({\mathcal{M}},\tau)$ (see [23, Chapter II, $\S$ 4.2 and $\S$ 5.7]). However, in this paper, it is shown that if we consider $L_{0}({\mathcal{M}},\tau)$ (the set of all $\tau$-measurable operators having finite-trace support) instead of $L_{1}({\mathcal{M}},\tau)$, then the operator space $E({\mathcal{M}},\tau)$ corresponding to an arbitrary symmetrically $\Delta$-normed function space $E(0,\infty)$ is necessarily an interpolation space between $L_{0}({\mathcal{M}},\tau)$ and ${\mathcal{M}}$, which is a noncommutative version of results in [18] (see also [2]).

As an application of the previous result, we describe the orbits and ${\mathcal{K}}$-orbits for an arbitrary $A\in S({\mathcal{M}},\tau)$ in the last section. It is shown that the unit balls of ${\mathcal{K}}$-orbits do not coincide with the unit balls of orbits in the pair $(L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$, which generalises [2, Theorem 4]. In [2], it is asserted that the commutative pair $(L_{0}(0,\infty),L_{\infty}(0,\infty))$ is not ${\mathcal{K}}$-monotone, that is, ${\mathcal{K}}$-orbits do not necessarily coincide with orbits in the pair $(L_{0}(0,\infty),L_{\infty}(0,\infty))$. However, it is known that this assertion is incorrect and $(L_{0}(0,\infty),L_{\infty}(0,\infty))$ is indeed ${\mathcal{K}}$-monotone (see e.g. Section 6). A non-commutative version of this result is established, that is, the pair $(L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$ is ${\mathcal{K}}$-monotone in the setting when ${\mathcal{M}}$ is a non-atomic finite factor. We would like to thank Professor Astashkin for providing us with the proof for the commutative pair $(L_{0}(0,\infty),L_{\infty}(0,\infty))$.

2. Preliminaries

2.1. Generalized singular value functions

In what follows, ${\mathcal{H}}$ is a Hilbert space and $B({\mathcal{H}})$ is the $*$-algebra of all bounded linear operators on ${\mathcal{H}}$, and $\mathbf{1}$ is the identity operator on ${\mathcal{H}}$. Let $\mathcal{M}$ be a von Neumann algebra on ${\mathcal{H}}$. For details on von Neumann algebra theory, the reader is referred to e.g. [7], [19, 20] or [37]. General facts concerning measurable operators may be found in [30], [32] (see also [38, Chapter IX] and the forthcoming book [12]). For convenience of the reader, some of the basic definitions are recalled.

A linear operator $X:\mathfrak{D}\left(X\right)\rightarrow{\mathcal{H}}$, where the domain $\mathfrak{D}\left(X\right)$ of $X$ is a linear subspace of ${\mathcal{H}}$, is said to be affiliated with $\mathcal{M}$ if $YX\subseteq XY$ for all $Y\in\mathcal{M}^{\prime}$, where $\mathcal{M}^{\prime}$ is the commutant of $\mathcal{M}$. A linear operator $X:\mathfrak{D}\left(X\right)\rightarrow{\mathcal{H}}$ is termed measurable with respect to $\mathcal{M}$ if $X$ is closed, densely defined, affiliated with $\mathcal{M}$ and there exists a sequence $\left\{P_{n}\right\}_{n=1}^{\infty}$ in the logic of all projections of $\mathcal{M}$, ${\mathcal{P}}\left(\mathcal{M}\right)$, such that $P_{n}\uparrow\mathbf{1}$, $P_{n}({\mathcal{H}})\subseteq\mathfrak{D}\left(X\right)$ and $\mathbf{1}-P_{n}$ is a finite projection (with respect to $\mathcal{M}$) for all $n$. It should be noted that the condition $P_{n}\left({\mathcal{H}}\right)\subseteq\mathfrak{D}\left(X\right)$ implies that $XP_{n}\in\mathcal{M}$. The collection of all measurable operators with respect to $\mathcal{M}$ is denoted by $S\left(\mathcal{M}\right)$, which is a unital $\ast$-algebra with respect to strong sums and products (denoted simply by $X+Y$ and $XY$ for all $X,Y\in S\left(\mathcal{M}\right)$).

Let $X$ be a self-adjoint operator affiliated with $\mathcal{M}$. We denote its spectral measure by $\{E^{X}\}$. It is well known that if $X$ is a closed operator affiliated with $\mathcal{M}$ with the polar decomposition $X=U|X|$, then $U\in\mathcal{M}$ and $E\in\mathcal{M}$ for all projections $E\in\{E^{|X|}\}$. Moreover, $X\in S(\mathcal{M})$ if and only if $X$ is closed, densely defined, affiliated with $\mathcal{M}$ and $E^{|X|}(\lambda,\infty)$ is a finite projection for some $\lambda>0$. It follows immediately that in the case when $\mathcal{M}$ is a von Neumann algebra of type $III$ or a type $I$ factor, we have $S(\mathcal{M})=\mathcal{M}$. For type $II$ von Neumann algebras, this is no longer true. From now on, let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau$.

An operator $X\in S\left(\mathcal{M}\right)$ is called $\tau$-measurable if there exists a sequence $\left\{P_{n}\right\}_{n=1}^{\infty}$ in $P\left(\mathcal{M}\right)$ such that $P_{n}\uparrow\mathbf{1},$ $P_{n}\left({\mathcal{H}}\right)\subseteq\mathfrak{D}\left(X\right)$ and $\tau(\mathbf{1}-P_{n})<\infty$ for all $n$. The collection of all $\tau$-measurable operators is a unital $\ast$-subalgebra of $S\left(\mathcal{M}\right)$ denoted by $S\left(\mathcal{M},\tau\right)$. It is well known that a linear operator $X$ belongs to $S\left(\mathcal{M},\tau\right)$ if and only if $X\in S(\mathcal{M})$ and there exists $\lambda>0$ such that $\tau(E^{|X|}(\lambda,\infty))<\infty$. Alternatively, an unbounded operator $X$ affiliated with $\mathcal{M}$ is $\tau$-measurable (see [14]) if and only if

[TABLE]

Definition 2.1**.**

Let a semifinite von Neumann algebra $\mathcal{M}$ be equipped with a faithful normal semi-finite trace $\tau$ and let $X\in S(\mathcal{M},\tau)$. The generalized singular value function $\mu(X):t\rightarrow\mu(t;X)$ of the operator $X$ is defined by setting

[TABLE]

An equivalent definition in terms of the distribution function of the operator $X$ is the following. For every self-adjoint operator $X\in S(\mathcal{M},\tau),$ setting

[TABLE]

we have (see e.g. [14])

[TABLE]

Consider the algebra $\mathcal{M}=L^{\infty}(0,\infty)$ of all Lebesgue measurable essentially bounded functions on $(0,\infty)$. Algebra $\mathcal{M}$ can be seen as an abelian von Neumann algebra acting via multiplication on the Hilbert space $\mathcal{H}=L^{2}(0,\infty)$, with the trace given by integration with respect to Lebesgue measure $m.$ It is easy to see that the algebra of all $\tau$-measurable operators affiliated with $\mathcal{M}$ can be identified with the subalgebra $S(0,\infty)$ of the algebra of Lebesgue measurable functions which consists of all functions $x$ such that $m(\{|x|>s\})$ is finite for some $s>0$. It should also be pointed out that the generalized singular value function $\mu(x)$ is precisely the decreasing rearrangement $\mu(x)$ of the function $x$ (see e.g. [23]) defined by

[TABLE]

The two-sided ideal ${\mathcal{F}}({\mathcal{M}},\tau)$ in ${\mathcal{M}}$ consisting of all elements of $\tau$-finite rank is defined by setting

[TABLE]

For convenience of the reader we also recall the definition of the measure topology $t_{\tau}$ on the algebra $S({\mathcal{M}},\tau)$. For every $\varepsilon,\delta>0,$ we define the set

[TABLE]

The topology generated by the sets $V(\varepsilon,\delta)$, $\varepsilon,\delta>0,$ is called the measure topology $t_{\tau}$ on $S({\mathcal{M}},\tau)$ [12, 14, 30]. It is well known that the algebra $S({\mathcal{M}},\tau)$ equipped with the measure topology is a complete metrizable topological algebra [30] (see also [29]). A sequence $\{X_{n}\}_{n=1}^{\infty}\subset S({\mathcal{M}},\tau)$ converges to zero with respect to measure topology $t_{\tau}$ if and only if \tau\big{(}E^{|X_{n}|}(\varepsilon,\infty)\big{)}\to 0 as $n\to\infty$ for all $\varepsilon>0$ [12, 11].

2.2. $\Delta$-normed spaces

For convenience of the reader, we recall the definition of $\Delta$-norm. Let $\Omega$ be a linear space over the field $\mathbb{C}$. A function $\|\cdot\|$ from $\Omega$ to $\mathbb{R}$ is a $\Delta$-norm, if for all $x,y\in\Omega$ the following properties hold:

• (1)

$\|x\|\geqslant 0$, $\|x\|=0\Leftrightarrow x=0$;

• (2)

$\|\alpha x\|\leqslant\|x\|$ for all $|\alpha|\leq 1$;

• (3)

$\lim_{\alpha\rightarrow 0}\|\alpha x\|=0$;

• (4)

$\|x+y\|\leq C_{\Omega}\cdot(\|x\|+\|y\|)$ for a constant $C_{\Omega}\geq 1$ independent of $x,y$.

The couple $(\Omega,\left\|\cdot\right\|)$ is called a $\Delta$-normed space. We note that the definition of a $\Delta$-norm given above is the same with that given in [21]. It is well-known that every $\Delta$-normed space $(\Omega,\left\|\cdot\right\|)$ is metrizable and conversely every metrizable space can be equipped with a $\Delta$-norm [21]. Note that properties $(2)$ and $(4)$ of a $\Delta$-norm imply that for any $\alpha\in\mathbb{C}$, there exists a constant $M$ such that $\|\alpha x\|\leq M\|x\|,\,x\in\Omega$, in particular, if $\|x_{n}\|\to 0,\{x_{n}\}_{n=1}^{\infty}\subset\Omega$, then $\|\alpha x_{n}\|\to 0$.

Let $E(0,\infty)$ be a space of real-valued Lebesgue measurable functions on $(0,\infty)$ (with identification $m$-a.e.), equipped with a $\Delta$-norm $\left\|\cdot\right\|_{E}$. The space $E(0,\infty)$ is said to be absolutely solid if $x\in E(0,\infty)$ and $|y|\leq|x|$, $y\in S(0,\infty)$ implies that $y\in E(0,\infty)$ and $\|y\|_{E}\leq\|x\|_{E}.$ An absolutely solid space $E(0,\infty)\subseteq S(0,\infty)$ is said to be symmetric if for every $x\in E(0,\infty)$ and every $y\in S(0,\infty)$, the assumption $\mu(y)=\mu(x)$ implies that $y\in E(0,\infty)$ and $\|y\|_{E}=\|x\|_{E}$ (see e.g. [23]).

We now come to the definition of the main object of this paper.

Definition 2.2**.**

Let a semifinite von Neumann algebra $\mathcal{M}$ be equipped with a faithful normal semi-finite trace $\tau$. Let $\mathcal{E}$ be a linear subset in $S({\mathcal{M},\tau})$ equipped with a $\Delta$-norm $\left\|\cdot\right\|_{\mathcal{E}}$. We say that $\mathcal{E}$ is a symmetrically $\Delta$-normed operator space if $X\in\mathcal{E}$ and every $Y\in S({\mathcal{M},\tau})$ the assumption $\mu(Y)\leq\mu(X)$ implies that $Y\in\mathcal{E}$ and $\|Y\|_{\mathcal{E}}\leq\|X\|_{\mathcal{E}}$.

One should note that a symmetrically $\Delta$-normed space $E({\mathcal{M}},\tau)$ does not necessarily satisfy

[TABLE]

It is clear that in the special case, when ${\mathcal{M}}=L_{\infty}(0,1)$, or ${\mathcal{M}}=L_{\infty}(0,\infty)$, or ${\mathcal{M}}=l_{\infty}$, the definition of symmetrically $\Delta$-normed operator spaces coincides with definition of the symmetric function (or sequence) spaces. In the case, when $\mathcal{M}=B(H)$ and $\tau$ is a standard trace ${\rm Tr}$, we shall call a symmetrically $\Delta$-normed operator space introduced in Definition 2.2 a symmetrically $\Delta$-normed operator ideal (for the symmetrically normed ideals we refer to [15, 16, 33]).

As mentioned before, the operator space $E({\mathcal{M}},\tau)$ defined by

[TABLE]

is a complete symmetrically $\Delta$-normed operator space whenever the symmetrically $\Delta$-normed function space $E(0,\infty)$ equipped with a $\Delta$-norm $\|\cdot\|_{E}$ is complete [17].

3. $(L_{0}+L_{\infty})({\mathcal{M}},\tau)=S({\mathcal{M}},\tau)$

By $L_{0}(0,\infty)$ we denote the space of all measurable functions on $(0,\infty)$ whose support has finite measure. This space is endowed with the group-norm [18]

[TABLE]

where ${\mbox{supp}}(f)=\{t\in(0,\infty):f(t)\neq 0\}$. It is clearly that the corresponding operator space $L_{0}({\mathcal{M}},\tau)$ is the subspace of $S({\mathcal{M}},\tau)$ which consists of all operators $X$ such that $\tau(s(X))<\infty$. It is easy to see that $(L_{0}+L_{\infty})({\mathcal{M}},\tau)$ coincides with $S({\mathcal{M}},\tau)$. For the sake of completeness, we present a brief proof below.

Proposition 3.1**.**

$(L_{0}+L_{\infty})({\mathcal{M}},\tau)=S({\mathcal{M}},\tau)$.

Proof.

Since $(L_{0}+L_{\infty})({\mathcal{M}},\tau)\subset S({\mathcal{M}},\tau)$, it suffices to prove $S({\mathcal{M}},\tau)\subset(L_{0}+L_{\infty})({\mathcal{M}},\tau)$. For any operator $X\in S({\mathcal{M}},\tau)$, there exists $\lambda>0$ such that $t:=\tau(E^{|X|}(\lambda,\infty))<\infty$. By [12, Chapter III, Eq. (4)], we have

[TABLE]

This together with [18, Proposition 3] implies that $\mu(X)\in(L_{0}+L_{\infty})(0,\infty)$. Thus, $X\in(L_{0}+L_{\infty})({\mathcal{M}},\tau)$. ∎

In [18, Proposition 3], the description of the ${\mathcal{K}}$-functional (which is a $\Delta$-norm on $S(0,\infty)$) $K_{u}(f)=\inf\{\|g\|_{0}+u\|h\|_{\infty}:f=g+h,g\in L_{0}(0,\infty),h\in L_{\infty}(0,\infty)\}$, $u>0$, of any $f\in S(0,\infty)$ is given in terms of its distribution function and its singular value function. That is, for every $f\in S(0,\infty)$, we have

[TABLE]

Similarly, for every $X\in S({\mathcal{M}},\tau)$, the ${\mathcal{K}}$-functional is defined by

[TABLE]

In particular, we define a $\Delta$-norm (see e.g. Remark 3.4) on $S({\mathcal{M}},\tau)$ by

[TABLE]

for any $X\in S({\mathcal{M}},\tau)$. The following result complements an earlier result from [9] for the pair $(L_{1}({\mathcal{M}},\tau),L_{\infty}({\mathcal{M}},\tau))$.

Proposition 3.2**.**

For every $X\in S({\mathcal{M}},\tau)$, we have

[TABLE]

In particular, $\|X\|_{S}=\inf_{s>0}[s+d_{\mu(X)}(s)]=\inf_{t>0}[t+\mu(t;X)]$. Moreover, $S({\mathcal{M}},\tau)$ is a complete $\Delta$-normed space with respect to the $\Delta$-norm $K_{u}$ for every $u>0$.

Proof.

Firstly, for every $X\in S({\mathcal{M}},\tau)$, we have

[TABLE]

Conversely, for every $G\in L_{0}({\mathcal{M}},\tau)$ with $t:=\|G\|_{0}$ and $X-G\in L_{\infty}({\mathcal{M}},\tau)$, by [12, Chapter III, Proposition 2.20], we have

[TABLE]

Hence, we obtain

[TABLE]

The fact that $S(0,\infty)$ is complete with respect with $K_{u}$ together with [17, Theorem 3.8] implies the completeness of $S({\mathcal{M}},\tau)$ with respect with $K_{u}$. ∎

Remark 3.3**.**

Recall that $d_{\mu(X)}(s)=d_{|X|}(s)$ (see e.g. [12, Chapter III, Eq. (4)]). For every $X\in S({\mathcal{M}},\tau)$, the ${\mathcal{K}}$-functional can be also defined by the formula

[TABLE]

Remark 3.4**.**

It is still unknown whether the Calkin correspondence preserves the constant $C_{E}$ for an arbitrary symmetrically $\Delta$-normed function space $E(0,\infty)$ (see [17, 34]). However, it is well-known that $(S(0,\infty),K_{u}(\cdot))$, $u>0$, is an $F$-space (i.e., a complete $\Delta$-normed space with $C_{E}=1$) and Proposition 3.2 implies that for every $u>0$, $K_{u}(\mu(\cdot))$ is not only a $\Delta$-norm but also an $F$-norm on $S({\mathcal{M}},\tau)$. Indeed, for every $X,Y\in S({\mathcal{M}},\tau)$, by Proposition 3.2, we have

[TABLE]

where we used the fact that $\|X\|_{0}+\|Y\|_{0}=\tau({\mbox{supp}}(X))+\tau({\mbox{supp}}(Y))\geq\tau({\mbox{supp}}(X+Y))=\|X+Y\|_{0}$ for every $X,Y\in S({\mathcal{M}},\tau)$.

4. An embedding theorem

It is well-known that for every symmetrically normed function space $E(0,\infty)$, the corresponding operator space $E({\mathcal{M}},\tau)$ is symmetrically normed [22] and is an intermediate space for the noncommutative pair $(L_{1}({\mathcal{M}},\tau),{\mathcal{M}})$ [11, 12]. In this section, we prove an analogue for the $\Delta$-normed case, that is, every operator space $E({\mathcal{M}},\tau)$ corresponding to a $\Delta$-normed function space $E(0,\infty)$ is an intermediate space for the noncommutative pair $(L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$.

Before we proceed to the proof of the embedding theorem, we show that the topology given by $\|\cdot\|_{S}$ is equivalent with the measure topology.

Proposition 4.1**.**

Let $\{X_{n}\}$ be a sequence in $S({\mathcal{M}},\tau)$. Then, $\|X_{n}\|_{S}\rightarrow 0$ if and only if $X_{n}\rightarrow_{t_{\tau}}0$.

Proof.

By [17, Lemma 2.4], it suffices to show that $\|X_{n}\|_{S}\rightarrow 0$ whenever $X_{n}\rightarrow_{t_{\tau}}0$. By [12, Chapter II, Proposition 5.7], we have $\tau(E^{|X_{n}|}(\varepsilon,\infty))\rightarrow_{n}0$ for every $\varepsilon>0$. By Proposition 3.2, we have $\|X_{n}\|_{S}\leq\varepsilon+\tau(E^{|X_{n}|}(\varepsilon,\infty))$, which completes the proof. ∎

Notice that the two-sided ideal ${\mathcal{F}}(\tau)$ in ${\mathcal{M}}$ coincides with $(L_{0}\cap L_{\infty})({\mathcal{M}},\tau)$. For every $X\in{\mathcal{F}}(\tau)$, we define the group-norm $\|X\|_{\mathcal{F}}$ by

[TABLE]

The following embedding theorem is the main result of this section, which extends [18, Theorem 1] to the non-commutative case.

Theorem 4.2**.**

If $E(0,\infty)$ is a nontrivial symmetrically $\Delta$-normed function space, then

[TABLE]

Moreover, the embeddings are continuous. That is, $E({\mathcal{M}},\tau)$ is an intermediate space between $L_{0}({\mathcal{M}},\tau)$ and ${\mathcal{M}}$.

Proof.

Since $E(0,\infty)$ is not empty, there is a non-zero element $x_{0}\in E(0,\infty)$. Then, there is a scalar $t>0$ such that $\mu(t;x_{0})>0$. It is clear that $\mu(t;x_{0})\chi_{(0,t]}\leq\mu(x_{0})$, which implies that $\chi_{(0,t]}\in E(0,\infty)$. Since $m(\chi_{(0,t]})=m(\chi_{(t,2t]})=\cdots=m(\chi_{((n-1)t,nt]})$ and $E(0,\infty)$ is a linear space, it follows that $\chi_{(0,nt]}\in E(0,\infty)$. Hence, ${\mathcal{F}}(0,\infty)\subset E(0,\infty)$.

Let $\{X_{n}\}_{n}\subset{\mathcal{F}}(\tau)$ be a sequence such that $\|X_{n}\|_{\mathcal{F}}\rightarrow 0$. For every $0<\varepsilon<1$, we can find an $N$ such that for every $n\geq N$, we have $\|X_{n}\|_{{\mathcal{F}}}\leq\varepsilon$, that is,

[TABLE]

Hence, $\mu(X_{n})\leq\varepsilon\chi_{(0,\varepsilon]}\leq\varepsilon\chi_{(0,1]}$ and therefore $\|X_{n}\|_{E}\leq\|\varepsilon\chi_{(0,1]}\|_{E}$. By the continuity of $\Delta$-norm $\|\cdot\|_{E}$, we obtain that $\|X_{n}\|_{E}\rightarrow_{n}0$.

Lemma 2.4 in [17] together with Proposition 4.1 implies that $E({\mathcal{M}},\tau)$ is continuously embedded into $S({\mathcal{M}},\tau)$. ∎

The set of all self-adjoint elements in $E({\mathcal{M}},\tau)$ is denoted by $E_{h}({\mathcal{M}},\tau)$. Then, [12, Chapter II, Proposition 6.1] together with Proposition 4.1 and Theorem 4.2 implies the following results immediately.

Corollary 4.3**.**

Let $E(0,\infty)$ is a symmetrically $\Delta$-normed function space. The following statements hold.

1. (1)

The positive cone $E({\mathcal{M}},\tau)^{+}$ is closed in $E({\mathcal{M}},\tau)$ with respect to $\|\cdot\|_{E}$. 2. (2)

If $\{X_{n}\}_{n=1}^{\infty}$ is a sequence in $E({\mathcal{M}},\tau)$ and $X,Y\in E_{h}({\mathcal{M}},\tau)$ are such that $\|X_{n}-X\|_{E}\rightarrow_{n}0$ and $X_{n}\leq Y$ for all $n$, then $X\leq Y$. 3. (3)

If $\{X_{n}\}_{n=1}^{\infty}$ is an increasing sequence in $E_{h}({\mathcal{M}},\tau)$ and $X\in E_{h}({\mathcal{M}},\tau)$ with $\|X_{n}-X\|_{E}\rightarrow_{n}0$, then $X_{n}\uparrow_{n}X$.

5. Interpolation in the pair $(L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$

Introduce the dilation operator $\sigma_{s}$ on $S(0,\infty)$, $s>0$, by setting

[TABLE]

It is well-known that $\mu(X+Y)\leq\sigma_{2}(\mu(X)+\mu(Y))$, $X,Y\in S({\mathcal{M}},\tau)$ [24]. We note also that $\sigma_{2^{k}}x\in E(0,\infty)$ with

[TABLE]

for all $x\in E(0,\infty)$ and $k\in\mathbb{N}$ (see e.g. [23], see also [18]).

Recall that $(L_{0}+L_{\infty})({\mathcal{M}},\tau)=S({\mathcal{M}},\tau)$ (see Proposition 3.1). Let $T:S({\mathcal{M}},\tau)\rightarrow S({\mathcal{M}},\tau)$ be a homomorphism, i.e.,

[TABLE]

for any $X,Y\in S({\mathcal{M}},\tau)$. Let $E(0,\infty)$ be a $\Delta$-normed function space. A homomorphism $T:E({\mathcal{M}},\tau)\rightarrow E({\mathcal{M}},\tau)$ is called continuous if for any given $\varepsilon>0$, there exists $\delta(\varepsilon)>0$ such that $\|X\|_{E}<\delta(\varepsilon)$ implies that $\|TX\|_{E}<\varepsilon$ [21, Chapter I, Section 4]. A homomorphism is called bounded if

[TABLE]

The homomorphism $T$ is said to be bounded on the pair $(L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$ if $T$ is a bounded mapping from $L_{0}({\mathcal{M}},\tau)$ into $L_{0}({\mathcal{M}},\tau)$ and from ${\mathcal{M}}$ into ${\mathcal{M}}$.

Theorem 5.1**.**

Let $E(0,\infty)$ be a symmetrically $\Delta$-normed function space and $T:S({\mathcal{M}},\tau)\rightarrow S({\mathcal{M}},\tau)$ be a homomorphism which is bounded on $(L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$ with

[TABLE]

[TABLE]

for some constants $M_{0},M_{1}>0$. Then, $T$ maps $E({\mathcal{M}},\tau)$ into itself and

[TABLE]

Proof.

For any $P\in{\mathcal{P}}({\mathcal{M}})$ with $\tau(\mathbf{1}-P)<t$, we have $\|TX({\bf 1}-P)\|_{0}\leq M_{0}\|X(\mathbf{1}-P)\|_{0}\leq M_{0}t$. By [24, Theorem 2.3.13], for every $t>0$, we have

[TABLE]

By Definition 2.1, we have

[TABLE]

This implies that $\sigma_{1/M_{0}}\mu(TX)\in E(0,\infty)$ and therefore, by appealing to (3), we conclude that $\mu(TX)=\sigma_{M_{0}}(\sigma_{1/M_{0}}\mu(TX))\in E(0,\infty)$. ∎

If $X,Y\in S({\mathcal{M}},\tau)$, then $X$ is said to be submajorized by $Y$, denoted by $X\prec\prec Y$, if

[TABLE]

A linear subspace $E$ of $S({\mathcal{M}},\tau)$ equipped with a complete norm $\|\cdot\|_{E}$, is called fully symmetric space (of $\tau$-measurable operators) if $X\in S({\mathcal{M}},\tau)$, $Y\in E$ and $X\prec\prec Y$ imply that $X\in E$ and $\|X\|_{E}\leq\|Y\|_{E}$ [11, 12, 24].

For a symmetric normed function space $E(0,\infty)$, by [12, Theorem 10.13] (see also [9] and [23]), $E({\mathcal{M}},\tau)$ is an interpolation space between $L_{1}({\mathcal{M}},\tau)$ and ${\mathcal{M}}$ if $E(0,\infty)$ is fully symmetric. In particular, one can find symmetric normed spaces $E({\mathcal{M}},\tau)$ which are not interpolation spaces between $L_{1}({\mathcal{M}},\tau)$ and $L_{\infty}({\mathcal{M}},\tau)$ [23, Chapter II, $\S$ 5.7]. However, for an arbitrary symmetrically $\Delta$-normed function space $E(0,\infty)$, $E({\mathcal{M}},\tau)$ is, in fact, an interpolation space between $L_{0}({\mathcal{M}},\tau)$ and ${\mathcal{M}}$.

Corollary 5.2**.**

Let $E(0,\infty)$ be a symmetrically $\Delta$-normed function space and $T:S({\mathcal{M}},\tau)\rightarrow S({\mathcal{M}},\tau)$ be a homomorphism which is bounded on $(L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$ with

[TABLE]

[TABLE]

for some constants $M_{0},M_{1}>0$. Then, $T$ is a bounded homomorphism from $E({\mathcal{M}},\tau)$ into itself. In particular, $\sup_{X\in S({\mathcal{M}},\tau)}\frac{\|TX\|_{S}}{\|X\|_{S}}<\infty$.

Proof.

By Theorem 5.1, for any $X\in E({\mathcal{M}},\tau)$, we have $TX\in E({\mathcal{M}},\tau)$ with

[TABLE]

Let $k\geq 0$ be an integer such that $2^{k}\geq M_{0}$. Noticing that $\sigma_{2^{k}}(\mu(X))\geq\sigma_{M_{0}}(\mu(X))$. By (3), we have

[TABLE]

for any $X\in E({\mathcal{M}},\tau)$.

Then, we get

[TABLE]

where $[M_{1}]$ is the integer part of $M_{1}$. The proof is complete. ∎

The results in this section are applied to the study of orbits and ${\mathcal{K}}$-orbits in the pair of $(L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$ in the next section.

6. Orbits and ${\mathcal{K}}$-orbits

For an element $X\in S({\mathcal{M}},\tau)$, the orbit ${\mbox{Orb}}(X;L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$ of $X$ is the set of all $Y\in S({\mathcal{M}},\tau)$ such that $Y=TX$ for some homomorphism $T$ which is bounded on the pair $(L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$. Furthermore, we define

[TABLE]

where the infimum is taken over all bounded homomorphisms $T$ such that $TX=Y$ and $\|T\|_{S({\mathcal{M}},\tau)}=\max\{\|T\|_{L_{0}\rightarrow L_{0}},\|T\|_{L_{\infty}\rightarrow L_{\infty}}\}$.

By Theorem 5.1, we have the following proposition, which is an analogue of [2, Theorem 1].

Proposition 6.1**.**

Let $X\in S({\mathcal{M}},\tau)$. Then, for every $Y\in{\mbox{Orb}}(X;L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$, we have

[TABLE]

Proof.

Notice for every $\varepsilon>0$, we can find a $T$ such that $Y=TX$ with $\|T\|_{S({\mathcal{M}},\tau)}\leq\|Y\|_{{\mbox{Orb}}}+\varepsilon$. Then, by (4), we have

[TABLE]

Hence, $\mu((\|Y\|_{{\mbox{Orb}}}+\varepsilon)t;Y)\leq(\|Y\|_{{\mbox{Orb}}}+\varepsilon)\mu(t;X)$ for every $t>0$. By the right-continuity of singular value functions, we have

[TABLE]

for every $t>0$, which completes the proof. ∎

Let $(X_{0},X_{1})$ be a pair of symmetrically $\Delta$-normed spaces. The ${\mathcal{K}}$-orbit of $A\in X_{0}+X_{1}$ is defined by the set ${\mathcal{K}}{\mathcal{O}}(A;X_{0},X_{1})$ of all $X\in X_{0}+X_{1}$ such that

[TABLE]

where ${\mathcal{K}}(t,Z;X_{0},X_{1}):=\inf\{\|Z_{0}\|_{X_{0}}+t\|Z_{1}\|_{X_{1}}:Z=Z_{0}+Z_{1},Z_{0}\in X_{0},Z_{1}\in X_{1}\}$.

A pair $(X_{0},X_{1})$ is called ${\mathcal{K}}$-monotone if ${\mathcal{K}}{\mathcal{O}}(A;X_{0},X_{1})={\mbox{Orb}}(A;X_{0},X_{1})$ for all $A\in X_{0}+X_{1}$. The pair $(L_{1}(0,\infty),L_{\infty}(0,\infty))$ is a classical example of a ${\mathcal{K}}$-monotone pair (see e.g. [5]). Moreover, the noncommutative pair $(L_{1}({\mathcal{M}},\tau),L_{\infty}({\mathcal{M}},\tau))$ is ${\mathcal{K}}$-monotone (see [9, Proposition 2.5, Theorem 4.7]).

It follows from the definition that the unit ball of ${\mbox{Orb}}(A;X_{0},X_{1})$ is a subset of the unit ball of ${\mathcal{K}}{\mathcal{O}}(A;X_{0},X_{1})$. Moreover, [9, Proposition 2.5 and Theorem 4.7] imply that the the closed unit ball of ${\mbox{Orb}}(A;L_{1}({\mathcal{M}},\tau),{\mathcal{M}})$ coincides with the unit ball of ${\mathcal{K}}{\mathcal{O}}(A;L_{1}({\mathcal{M}},\tau),{\mathcal{M}})$. However, it is known that the reverse inclusion may fail for certain element in the pair $L_{0}(0,\infty)+L_{\infty}(0,\infty)$ [2]. One of the main results of this section is a non-commutative version of [2, Theorem 4].

By Proposition 3.2, the ${\mathcal{K}}$-orbit ${\mathcal{K}}{\mathcal{O}}(A;L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$ of every ${\mathcal{A}}\in S({\mathcal{M}},\tau)$ is the set of all $X\in S({\mathcal{M}},\tau)$ such that

[TABLE]

Theorem 6.2**.**

If ${\mathcal{M}}$ is a non-trivial von Neumann algebra (${\mathcal{M}}\neq\mathbb{C}{\bf 1}$ and ${\mathcal{M}}\neq 0$), then there exist $A,X\in S({\mathcal{M}},\tau)$ such that

[TABLE]

whereas $\mu(t;A)<\mu(t;X)$, $t\in E$, for some measurable set $E$, $m(E)>0$. In particular, the unit ball of the ${\mathcal{K}}{\mathcal{O}}(A;L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$ does not coincide with the unit ball of ${\mbox{Orb}}(A;L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$.

Proof.

Since ${\mathcal{M}}\neq\mathbb{C}\bf 1$, there exist two $\tau$-finite projections $P_{1},P_{2}\in{\mathcal{P}}({\mathcal{M}})$ such that $P_{1}\perp P_{2}$. Let $\tau_{1}:=\tau(P_{1})>0$ and $\tau_{2}:=\tau(P_{2})>0$.

Let $k_{1},k_{2}>0$ be such that

[TABLE]

Define

[TABLE]

Then, $\mu(X)=k_{1}\chi_{(0,\tau_{1}+\tau_{2})}$. By (2), we have

[TABLE]

Then, we have

[TABLE]

Define

[TABLE]

Then, $\mu(A)=k_{1}\chi_{(0,\tau_{1})}+k_{2}\chi_{[\tau_{1},\tau_{1}+\tau_{2})}$. By (2), we have

[TABLE]

However, (8) implies that there is no such a $t$ such that $\tau_{1}+tk_{2}\leq\min\{tk_{1},\tau_{1}+\tau_{2}\}$. Hence, we have

[TABLE]

That is, $K_{t}(A)=K_{t}(X)$. However, it is clear that

[TABLE]

Assume that $X$ lies in the unit ball of ${\mbox{Orb}}(A;L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$. By (7)

[TABLE]

which is a contradiction with (9). Hence, $X$ lies in the unit ball of ${\mathcal{K}}{\mathcal{O}}(A;L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$ but not in the unit ball of ${\mbox{Orb}}(A;L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$. ∎

In [2], it is asserted incorrectly that the ${\mbox{Orb}}(A;L_{0}(0,\infty),L_{\infty}(0,\infty))\neq{\mathcal{K}}{\mathcal{O}}(A;L_{0}(0,\infty),L_{\infty}(0,\infty))$. The following proposition together [2, Theorem 1] explains why ${\mbox{Orb}}(A;L_{0}(0,\infty),L_{\infty}(0,\infty))={\mathcal{K}}{\mathcal{O}}(A;L_{0}(0,\infty),L_{\infty}(0,\infty))$, i.e., the pair $(L_{0}(0,\infty),L_{\infty}(0,\infty))$ is ${\mathcal{K}}$-monotone. We would like to thank Professor Astashkin for providing the proof for the special case when ${\mathcal{M}}=L_{\infty}(0,\infty)$.

Proposition 6.3**.**

Let $A,X\in S({\mathcal{M}},\tau)$. Then, the following statements are equivalent.

1. (1)

There exists $C>0$ such that $\mu(s;X)\leq C\mu(\frac{s}{C};A)$ for every $s>0$. 2. (2)

$\sup_{t>0}\frac{K_{t}(X)}{K_{t}(A)}<\infty$.

Proof.

(i) For every $A,X$ satisfying condition (1), we have

[TABLE]

which proves the validity of condition (2).

(ii) Conversely, assume that

[TABLE]

for some $C>0$. For every $Z\in S({\mathcal{M}},\tau)$, we define

[TABLE]

Clearly, we have

[TABLE]

Therefore,

[TABLE]

Let $s\in(0,\infty)$ and let $t:=\frac{s}{\mu(s;X)}$ (without loss of generality, we may assume that $\mu(s;X)>0$). Notice that $s=t\mu(s;X)\leq t\mu(s-\Delta_{1};X)$ and $s=t\mu(s;X)\leq s+\Delta_{2}$ for any $\Delta_{1},\Delta_{2}>0$ with $s-\Delta_{1}>0$. We have,

[TABLE]

Let $s_{1}:=M_{t}(A)$. Then, we have

[TABLE]

(otherwise, we have $\max\{t\mu(s_{1}-\varepsilon;A),s_{1}-\varepsilon\}<s_{1}=M_{t}(A)$ for some $\varepsilon>0$, which is a contradiction to the definition of $M_{t}(A)$.).

Since $s\stackrel{{\scriptstyle\eqref{s=}}}{{=}}M_{t}(X)\stackrel{{\scriptstyle\eqref{eq1}}}{{\leq}}2CM_{t}(A)=2Cs_{1}$, it follows that

[TABLE]

Then, we obtain

[TABLE]

that is,

[TABLE]

Since $s>0$ is arbitrary taken, it follows that

[TABLE]

for every $s>0$, which completes the proof. ∎

By the above proposition and [2, Theorem 1], we obtain the following result immediately, which implies that the commutative pair $(L_{0}(0,\infty),L_{\infty}(0,\infty))$ is indeed ${\mathcal{K}}$-monotone.

Corollary 6.4**.**

For every $a\in S(0,\infty)$, we have

[TABLE]

It is known that there cannot in general be a conditional expectation from $S({\mathcal{M}},\tau)$ onto a subalgebra of $S({\mathcal{M}},\tau)$ (see e.g. [13, Appendix B]), which is the main obstacle in extending [2, Theorem 1] to the non-commutative case. The following theorem is the last result of this section, giving a non-commutative version of [2, Theorem 1] in the setting of non-atomic finite factors by approaches which are completely different from those used in [2].

For the sake of convenience, we denote $L_{\infty}(0,\tau(s(X)))$ by $M_{\mu(X)}$. If ${\mathcal{M}}$ is a non-atomic semifinite von Neumann algebra, then for every $X\in S_{0}({\mathcal{M}},\tau)$, there exists a non-atomic commutative von Neumann subalgebra ${\mathcal{M}}_{|X|}$ in $s(|X|){\mathcal{M}}s(|X|)$ and a trace-preserving $*$-isomorphism $J$ from $S({\mathcal{M}}_{|X|},\tau)$ onto the algebra $S(M_{\mu(X)},m)$ [6, 8].

Theorem 6.5**.**

If ${\mathcal{M}}$ is a non-atomic finite von Neumann factor with a faithful normal finite trace $\tau$, then for every $0\neq A\in S({\mathcal{M}},\tau)$, ${\mbox{Orb}}(A;L_{0}({\mathcal{M}},\tau),L_{\infty}({\mathcal{M}},\tau))$ is the set of all $X\in S({\mathcal{M}},\tau)$ for which there exists $C>0$ such that

[TABLE]

In particular, ${\mbox{Orb}}(A;L_{0}({\mathcal{M}},\tau),L_{\infty}({\mathcal{M}},\tau))={\mathcal{K}}{\mathcal{O}}(A;L_{0}({\mathcal{M}},\tau),L_{\infty}({\mathcal{M}},\tau))$.

Proof.

(i) It follows from Proposition 6.1 that for every $X\in{\mbox{Orb}}(A;L_{0}({\mathcal{M}},\tau),L_{\infty}({\mathcal{M}},\tau))$, there exists such a $C>0$ satisfying (14).

(ii) Conversely, assume that $A,X\in S({\mathcal{M}},\tau)$ satisfies (14).

Then, by [6, Lemma 1.3], there are isomorphisms $J_{A}$ between $S(M_{\mu(A)},m)$ and $S({\mathcal{M}}_{|A|},\tau)$ such that $J_{A}\mu(A)=|A|$ and $J_{X}$ between $S(M_{\mu(X)},m)$ and $S({\mathcal{M}}_{|X|},\tau)$ such that $J_{X}\mu(X)=|X|$.

Then, by (14), we have

[TABLE]

for every $t\in(0,\tau(s(X)))$. Without loss of generality, we can assume that $C$ is an integer which is large enough such that

[TABLE]

for every $t\in(0,\tau(s(X)))$.

Let $\varepsilon<\frac{1}{2C}$. Since $J_{A}\mu(A)=|A|$, we can define $A_{n}:=[a_{n},b_{n})$, $n\geq 0$, by

[TABLE]

For every $0\leq j\leq 2C-1$, we set

[TABLE]

By [6, Lemma 1.3], for every $[a_{nj},b_{nj})\subset[0,\tau(s(X)))$, we have

[TABLE]

Since ${\mathcal{M}}$ is a finite factor, due to (16), there exist partial isometries $U_{nj}\in{\mathcal{M}}$ such that $U_{nj}U_{nj}^{*}=E^{|A|}(n\varepsilon,(n+1)\varepsilon]=J_{A}\chi_{[a_{n},b_{n})}$ and $U_{nj}^{*}U_{nj}=J_{X}\chi_{[a_{nj},b_{nj})}$. If $[a_{nj},b_{nj})\cap[0,\tau(s(X)))=\varnothing$, we define $U_{nj}:=0$. In the case when $[a_{nj},b_{nj})\nsubseteq[0,\tau(s(X)))$ but $[a_{nj},b_{nj})\cap[0,\tau(s(X)))\neq\varnothing$, we define $U_{nj}$ as the partial isometry such that $U_{nj}U_{nj}^{*}=J_{A}\chi_{[a_{n},a_{n}-a_{nj}+\tau(s(X)))}\leq E^{|A|}(n\varepsilon,(n+1)\varepsilon]$ and $U_{nj}^{*}U_{nj}=J_{X}\chi_{[a_{nj},\tau(s(X)))}$.

Denote $U_{j}:=\sum_{n=0}^{\infty}U_{nj}$, $0\leq j\leq 2C-1$ (note that $U_{kj}^{*}U_{lj}=0$ and $U_{lj}U_{kj}^{*}=0$ whenever $k\neq l$). Note that every $U_{j}$ is a partial isometry. Let

[TABLE]

and

[TABLE]

Then, noting that $U_{n,j}^{*}E^{|A|}(n\varepsilon,(n+1)\varepsilon]U_{n,j}=U_{n,j}^{*}J_{A}\chi_{[a_{n},b_{n})}U_{n,j}=J_{X}(\chi_{[a_{nj},b_{nj})}\chi_{[0,\tau(s(X)))})$ for every $n,j$, we have

[TABLE]

Since $\varepsilon<\frac{1}{2C}$, it follows that $0\leq\mu(A)-\mu(B_{1})<\frac{1}{2C}$ and therefore, by (15), we have

[TABLE]

for every $t\in[0,\tau(s(X)))$ (note that the $\mu(t;B_{2})=\mu(\frac{t}{2C};B_{1})$ follows immediately from the definitions of $B_{1}$ and $B_{2}$).

Let $A_{\Delta}:=\int\frac{[\lambda/\varepsilon]\varepsilon}{\lambda}dE^{|A|}(\lambda)$ and $B_{\Delta}:=J_{X}\frac{\mu(X)}{\mu(B_{2})}$. Here, $[\lambda/\varepsilon]$ is the integer part of $\lambda/\varepsilon$. Note that (6) implies that $B_{\Delta}$ is a bounded operator. Clearly, we have $A_{\Delta}|A|=B_{1}$ and $|X|=J_{X}\frac{\mu(X)}{\mu(B_{2})}J_{X}\mu(B_{2})=B_{\Delta}B_{2}$ (notice that $\sum_{n,j}n\varepsilon\chi_{[a_{nj},b_{nj})}\chi_{[0,\tau(s(X)))}=\mu(B_{2})$). Let $U_{A}|A|=A$ and $U_{X}|X|=X$ be the polar decompositions. Define a homomorphism $T:S({\mathcal{M}},\tau)\rightarrow S({\mathcal{M}},\tau)$ by setting

[TABLE]

It is easy to verify that $TA=X$ and $T$ is a bounded homomorphism on the pair $(L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$ (one should note that operators of multiplication by $U_{j}$, $0\leq j\leq 2C-1$, $A_{\Delta}$ and $B_{\Delta}$ are bounded homomorphisms on the pair $(L_{0}({\mathcal{M}},\tau),{\mathcal{M}})$ by Corollary 5.2).

The last statement follows immediately from Proposition 6.3. ∎

Remark 6.6**.**

The assumption that ${\mathcal{M}}$ is a finite factor plays a crucial role in the above proof. The authors did not succeed in extending the result to the case for general semifinite von Neumann algebras.

Acknowledgements The authors would like to thank Sergei Astashkin and Dima Zanin for helpful discussions.

The first author acknowledges the support of University International Postgraduate Award (UIPA). The second author was supported by the Australian Research Council.