Reducible operators in non-$\Gamma$ type ${\rm II}_1$ factors

TL;DR

This paper investigates whether reducible operators are dense in non-$ ext{II}_1$ factors, showing they are nowhere dense in non-$ ext{Gamma}$ factors, contrasting with the classical case on Hilbert spaces.

Contribution

It introduces a new characterization of Property $ ext{Gamma}$ and a spectral gap property, demonstrating the non-density of reducible operators in non-$ ext{Gamma}$ type $ ext{II}_1$ factors.

Findings

Reducible operators form a nowhere dense set in non-$ ext{Gamma}$ factors of type $ ext{II}_1$

The paper develops a new characterization of Property $ ext{Gamma}$ for type $ ext{II}_1$ factors

Spectral gap properties are established for operators in non-$ ext{Gamma}$ factors

Abstract

A famous question of Halmos asks whether every operator on a separable infinite-dimensional Hilbert space is a norm limit of reducible operators. In [30], Voiculescu gave this problem an affirmative answer by his remarkable non-commutative Weyl-von Neumann theorem. We investigate the existence or non-existence of an analogue of Voiculescu's result in factors of type ${\rm II}_1$. In the paper, we prove that, in the operator norm topology, the set of reducible operators is ${\it nowhere}$ dense in a non-$\Gamma$ factor $\mathcal M$ of type ${\rm II}_1$, where separable and non-separable cases of $\mathcal M$ are both considered. Main tools developed in the paper are a new characterization of Murray and von Neumann's Property $\Gamma$ for a factor of type ${\rm II}_1$ and a spectral gap property for a single operator in a non-$\Gamma$ factor of type ${\rm II}_1$.

Equations

$$W^{\ast }(pxp{)}^{\prime }\cap (p\mathcal{M}p{)}^{\omega }\ne \mathbb{C}p.$$

$$\Vert [x_1,e]{\Vert }_2+\Vert [x_2,e]{\Vert }_2\ge \alpha \Vert e{\Vert }_2\Vert I-e{\Vert }_2,\text{ for every projection }e\in \mathcal{M}.$$

$$\Vert [x_1,p]{\Vert }_2+\Vert [x_2,p]{\Vert }_2\ge \alpha \Vert p{\Vert }_2,$$

$$\Vert [x_1,p]\Vert +\Vert [x_2,p]\Vert \ge \frac{\alpha }{\sqrt{2}}.$$

$$l^{\infty }(\mathcal{M})=\{(a_n{)}_n:\forall n\in \mathbb{N},a_n\in \mathcal{M}\text{ and }\underset{n\in \mathbb{N}}{\sup }\Vert a_n\Vert <\infty \},$$

$${\mathcal{I}}_{\omega }(\mathcal{M})=\{(a_n{)}_n\in l^{\infty }(\mathcal{M}):\underset{n\to \omega }{\lim }\Vert a_n{\Vert }_2=0\}.$$

$${\mathcal{M}}^{\omega }=l^{\infty }(\mathcal{M})/{\mathcal{I}}_{\omega }(\mathcal{M}).$$

$$(\mathcal{A}\otimes {\mathcal{M}}_k(\mathbb{C}){)}^{\prime }\cap (\mathcal{P}\otimes {\mathcal{M}}_k(\mathbb{C}){)}^{\omega }=({\mathcal{A}}^{\prime }\cap {\mathcal{P}}^{\omega })\otimes 1_k,$$

$${\mathrm{dist}}_{\Vert \cdot {\Vert }_2}(x,\mathcal{S})=\inf \{\Vert x-y{\Vert }_2:y\in \mathcal{S}\}.$$

$$W^{\ast }(x_1,\dots ,x_n{)}^{\prime }\cap {\mathcal{M}}^{\omega }\ne \mathbb{C}I$$

$$\underset{n\to \infty }{\lim }{\tau }_{\mathcal{N}}(u_n{b}_1{u}_n^{\ast }{b}_2)={\tau }_{\mathcal{N}}(b_1){\tau }_{\mathcal{N}}(b_2),\forall b_1,b_2\in \mathcal{N}.$$

$$|{\tau }_{\mathcal{N}}(u_{k_n}{p}_n^{(m)}{u}_{k_n}^{\ast }{p}_n^{(m)})-({\tau }_{\mathcal{N}}(p_n^{(m)}){)}^2|=|{\tau }_{\mathcal{N}}(u_{k_n}{p}_n^{(m)}{u}_{k_n}^{\ast }{p}_n^{(m)})-{\lambda }^2|\le 1/n,\forall 1\le m\le n.$$

$$|\Vert [u_{k_n},p_n^{(m)}]{\Vert }_{2,\mathcal{N}}^2-(2\lambda -2{\lambda }^2)|=|2{\tau }_{\mathcal{N}}(p_n^{(m)})-2{\tau }_{\mathcal{N}}(u_{k_n}{p}_n^{(m)}{u}_{k_n}^{\ast }{p}_n^{(m)})-(2\lambda -2{\lambda }^2)|\le 2/n.$$

$$\Vert [V,P_m]{\Vert }_{2,\mathcal{M}}^2=\underset{n\to \omega }{\lim }\Vert [u_{k_n},p_n^{(m)}]{\Vert }_{2,\mathcal{N}}^2=2\lambda -2{\lambda }^2>0,\forall m\ge 1.$$

$$\{p_1,\dots ,p_n,x_1,\dots ,x_n\}\subseteq {\mathrm{C}}^{\ast }(x)\subseteq W^{\ast }(x).$$

$$f_i(t)=\{\begin{array}{ll}1,& t\in {\sigma }_{p_i\mathcal{M}{p}_i}(x_i);\\ 0,& t\in \sigma (x)\backslash {\sigma }_{p_i\mathcal{M}{p}_i}(x_i).\end{array}$$

$$\mathrm{supp}(x,\{p_i{\}}_{i=1}^k)=\bigvee \{p_i:p_{i}x\ne 0\text{ or }x{p}_i\ne 0,1\le i\le k\}.$$

$$W^{\ast }(x,\mathcal{A}{)}^{\prime }\cap {\mathcal{M}}^{\omega }\text{ is diffuse}.$$

$$\sum _{i=1}^m{f}_{ii}x{f}_{ii}=y=\sum _{j=1}^m{f}_{jj}y{f}_{jj}=\sum _{s,t=1}^k{e}_{ss}(\sum _{i=1}^m{f}_{ii}y{f}_{ii})e_{tt}=\sum _{s,t=1}^k\sum _{i=1}^m{e}_{ss}{f}_{ii}y{e}_{tt}{f}_{ii}.$$

$$\frac{|\{(e_{ss}{f}_{ii},e_{tt}{f}_{jj}):e_{ss}{f}_{ii}y{e}_{tt}{f}_{jj}\ne 0\text{mbox}for1\le s,t\le k\text{mbox}and1\le i,j\le m\}|}{(km{)}^2}\le \frac{k^{2}m}{k^2{m}^2}=\frac{1}{m}.$$

$$(km)\cdot\tau\Big{(}\operatorname{supp}(q,\{e_{ss}f_{ii}\}_{1\leq s\leq k;1\leq i\leq m})\Big{)}\leq(km)\cdot\frac{1}{2k}=\frac{m}{2}\leq\ |\{f_{ii}e_{11}:2\leq i\leq m\}|.$$

$$q=q(\sum _{i=2}^m{f}_{ii}{e}_{11})=(\sum _{i=2}^m{f}_{ii}{e}_{11})q.$$

$$a=e_{11}+2e_{22}+3e_{33}$$

$$b=$$

$$y^{\ast }{e}_{21}$$

$$e_{31}$$

$$W^{\ast }(y,z_1,z_2,{\mathcal{N}}_0)\subseteq W^{\ast }(a+ib).$$

$$q=\bigvee \{e:e\text{ is a projection in }{\mathcal{M}}_2\text{ such that }e{C}_p=0\}.$$

$$\delta =\Vert (q_n{)}_{\omega }-E_{\mathcal{M}}^{{\mathcal{M}}^{\omega }}((q_n{)}_{\omega }){\Vert }_{2,{\tau }_{\omega }}\le \Vert (q_n{)}_{\omega }-y{\Vert }_{2,{\tau }_{\omega }}=\underset{n\to \omega }{\lim }\Vert q_n-y{\Vert }_2,\forall y\in \mathcal{M}.$$

$$\Vert q_{k_n}-y_n^{(m)}{\Vert }_2\ge \delta /2\text{mbox}and\Vert [q_{k_n},a_m]{\Vert }_2\le 1/n,\forall 1\le m\le n.$$

Full text

In the paper we prove that, in the operator norm topology, the set of reducible operators is nowhere dense in a non-$\Gamma$ factor $\mathcal{M}$ of type ${\rm II}_{1}$, where separable and non-separable cases of $\mathcal{M}$ are both considered. Main tools developed in the paper are a new characterization of Murray and von Neumann’s Property $\Gamma$ for a factor of type II1 and a spectral gap property for a single operator in a non-$\Gamma$ factor of type ${\rm II}_{1}$.

1. Introduction

Let $\mathcal{H}$ be a complex Hilbert space. Denote by $\mathcal{B}(\mathcal{H})$ the set of all bounded linear operators on $\mathcal{H}$. A von Neumann algebra is a self-adjoint subalgebra of $\mathcal{B}(\mathcal{H})$ that is closed in the weak operator topology. A factor is a von Neumann algebra whose center consists of scalar multiples of the identity. Factors are further classified by Murray and von Neumann into type In, I∞, II1, II∞ and III factors (see [15]). By definition, $\mathcal{B}(\mathcal{H})$ is a type I factor.

When $\mathcal{H}$ is separable, Halmos [12] proved that the set of irreducible operators in $\mathcal{B}(\mathcal{H})$ is a dense $G_{\delta}$ subset of $\mathcal{B}(\mathcal{H})$ in the operator norm topology. Recall that an operator $a\in\mathcal{B}(\mathcal{H})$ is reducible if $a$ has nontrivial reducing closed subspaces. And $a\in\mathcal{B}(\mathcal{H})$ is irreducible if $a$ has no nontrivial reducing closed subspaces, i.e. if $p$ is a projection in $\mathcal{B}(\mathcal{H})$ such that $pa=ap$, then $p=0$ or $p=I$.

Similarly, an element $a$ in a factor $\mathcal{N}$ is reducible if there is a nontrivial projection $p$ in $\mathcal{N}$ such that $ap=pa$. Furthermore, an element $a$ in $\mathcal{N}$ is irreducible if $a$ is not reducible in $\mathcal{N}$. Note that a single generator of a factor with separable predual is an irreducible operator. Thus, in a factor with separable predual, there always exist irreducible operators (see [22, 31]). Recently, the authors in [8] proved that in each factor $\mathcal{N}$ with separable predual, the set of irreducible operators in $\mathcal{N}$ is operator norm dense and $G_{\delta}$.

On the other hand, the eighth problem raised by Halmos in his ten problems in Hilbert space in [13] is stated as follows.

Problem 8**.**

Is every operator (on a separable Hilbert space) the norm limit of reducible ones?

On a finite-dimensional Hilbert space, the answer to the problem is negative, since the set of reducible operators is closed and nowhere dense in the operator norm. On a separable, infinite-dimensional Hilbert space, the problem was answered affirmatively by Voiculescu as an application of his non-commutative Weyl-von Neumann theorem in [30].

Inspired by some recent research on irreducible operators in factors [8] and normal operators in semi-finite factors [11, 16, 17], we investigate Problem 8 in the setting of type ${\rm II}_{1}$ factors.

Let $\mathcal{M}$ be a factor of type ${\rm II}_{1}$ with trace $\tau$. Through out the paper, we denote by $\|\cdot\|$ the operator norm on $\mathcal{M}$, and by $\|\cdot\|_{2}$ the $2$-norm on $\mathcal{M}$, i.e., $\|x\|_{2}=\sqrt{\tau(x^{*}x)}$ for all $x\in\mathcal{M}$. For elements $x$ and $y$ in $\mathcal{M}$, we denote by $[x,y]=xy-yx$ the commutator of $x$ and $y$.

We will frequently mention Murray and von Neumann’s Property $\Gamma$ for type ${\rm II}_{1}$ factors (see [20]). Among different invariants applied in the classification of ${\rm II}_{1}$ factors with separable predual, Property $\Gamma$ is the first invariant used by Murray and von Neumann in [20] to show the existence of non hyperfinite type ${\rm II}_{1}$ factors and it plays a critical role in Connes’ celebrated paper [3]. Recall that $\mathcal{M}$ has Property $\Gamma$ if and only if for any $x_{1},\ldots,x_{n}$ in $\mathcal{M}$ and any $\epsilon>0$, there exists a unitary element $u$ in $\mathcal{M}$ with $\tau(u)=0$ such that $\|[x_{i},u]\|_{2}\leq\epsilon$ for all $1\leq i\leq n$. Notice that the definition of property $\Gamma$ for a type ${\rm II}_{1}$ factor $\mathcal{M}$ doesn’t require $\mathcal{M}$ having separable predual. For simplicity, if a type ${\rm II}_{1}$ factor $\mathcal{M}$ doesn’t have Property $\Gamma$, we say that $\mathcal{M}$ is non-$\Gamma$ (When $\mathcal{M}$ has separable predual, $\mathcal{M}$ is non-$\Gamma$ if and only if $\mathcal{M}$ is full [2]).

The purpose of the paper is the following theorem.

THEOREM 6.6. *Let $\mathcal{M}$ be a non-$\Gamma$ type ${\rm II}_{1}$ factor. Then, in the operator norm topology, the set of reducible operators in $\mathcal{M}$ is nowhere dense and not closed in $\mathcal{M}$. *

In order to prove Theorem 6.6, we take four main steps.

*Step One: * Inspired by Dixmier’s ideas in [4], we develop another characterization of Property $\Gamma$ for type ${\rm II}_{1}$ factors.

PROPOSITION 3.8. Let $\mathcal{M}$ be a type ${\rm II}_{1}$ factor with trace $\tau$. Then the following statements are equivalent:

1. (i)

$\mathcal{M}$* has Property $\Gamma$ of Murray and von Neumann.* 2. (ii)

For every $x$ in $\mathcal{M}$, $W^{*}(x)^{\prime}\cap\mathcal{M}^{\omega}$ is diffuse. 3. (iii)

For every $x$ in $\mathcal{M}$ and every nonzero projection $p$ in $\mathcal{M}$,

[TABLE] 4. (iv)

For every $x$ in $\mathcal{M}$, $W^{*}(x)^{\prime}\cap\mathcal{M}^{\omega}\neq\mathbb{C}I$.

Here $W^{*}(x)$ is the von Neumann subalgebra generated by $x$ in $\mathcal{M}$ and $\omega$ is a free ultrafilter on the set $\mathbb{N}$ of natural numbers.

*Step Two: * By Proposition 3.8 and a lemma by Connes in [3], we obtain a single operator with spectral gap in a non-$\Gamma$ type ${\rm II}_{1}$ factor, which provide another important technique in the proof of Theorem 6.6.

THEOREM 4.9. Let $\mathcal{M}$ be a non-$\Gamma$ type ${\rm II}_{1}$ factor with trace $\tau$. Then there exist two self-adjoint elements $x_{1}$, $x_{2}$ in $\mathcal{M}$ and a positive number $\alpha>0$ such that

[TABLE]

*Step Three: * A key observation, connecting spectral gap property of an operator with operator norm closure of reducible operators, is the following lemma.

LEMMA 5.8

• Let $x_{1}$ and $x_{2}$ be self-adjoint elements in $\mathcal{M}$. If there exist a positive number $\alpha>0$ and a projection $p\in\mathcal{M}$ with $\tau(p)>0$, satisfying*

[TABLE]

then

[TABLE]

Now the existence of elements, which are not contained in the operator norm closure of reducible operators, in a non-$\Gamma$ type II1 factor is a combination of Theorem 4.9 with Lemma 5.8.

THEOREM 5.11 *Let $\mathcal{M}$ be a non-$\Gamma$ type ${\rm II}_{1}$ factor. Then $\operatorname{Red}(\mathcal{M})^{\|\cdot\|}\neq\mathcal{M}$, where $\operatorname{Red}(\mathcal{M})^{\|\cdot\|}$ is the operator norm closure of reducible operators in $\mathcal{M}$. *

*Step Four: * Finally, based on Theorem 5.11, Theorem 6.6 is proved.

We mention that there exist examples of nonseparable type II1 factors with Property $\Gamma$ in which all operators are reducible (see Example 5.3). In Proposition 5.2 we show that, if a type II1 factor has separable predual, then the set of reducible operators is not operator norm closed. It remains an open question whether, in a type II1 factor with separable predual and with Property $\Gamma$, the set of reducible operators is operator norm dense.

This paper is organized as follows. In Section 2, we recall ultra-power algebras, central sequence algebras, and Property $\Gamma$ for type ${\rm II}_{1}$ factors. Some useful techniques are also prepared. In Section 3, in the view of a single operator, we prove a characterization of Property $\Gamma$ for type ${\rm II}_{1}$ factors in Proposition 3.8. As an application of Proposition 3.8, we provide an answer to Sherman’s question in Problem 2.11 of [26], where we show equivalent characterizations of McDuff factors. In Section 4, the existence of a single operator with spectral gap in a non-$\Gamma$ type ${\rm II}_{1}$ factor is shown in Theorem 4.9. In Section 5, we prove in Theorem 5.11 that reducible operators are not dense in non-$\Gamma$ type ${\rm II}_{1}$ factors. In Section 6, we further prove in Theorem 6.6 that reducible operators are nowhere dense in non-$\Gamma$ type ${\rm II}_{1}$ factors with the techniques developed in the preceding sections.

2. Preliminaries and Notation

As one goal, we develop a new characterization of Property $\Gamma$ for type ${\rm II}_{1}$ factors in the following sections. For this purpose, we first recall ultra-power algebras and central sequence algebras related to a type ${\rm II}_{1}$ factor.

Let $\mathcal{M}$ be a type ${\rm II}_{1}$ factor with trace $\tau$. Let $\mathbb{N}$ be the set of all the natural numbers and $\omega\in\beta(\mathbb{N})\setminus\mathbb{N}$ a fixed free ultrafilter over $\mathbb{N}$. Let

[TABLE]

and

[TABLE]

Then $\mathcal{I}_{\omega}(\mathcal{M})$ is a two sided ideal of $l^{\infty}(\mathcal{M})$ and the ultra-power of $\mathcal{M}$ along $\omega$, denoted by $\mathcal{M}^{\omega}$, is defined to be

[TABLE]

If no confusion arises, an element in $\mathcal{M}^{\omega}$ will be denoted by $(a_{n})_{\omega}$. By [32] or [24], $\mathcal{M}^{\omega}$ is a type ${\rm II}_{1}$ factor with a natural trace $\tau_{\omega}$ (also see Theorem A.3.5 in [27]). If $\mathcal{P}$ is a von Neumann subalgebra of $\mathcal{M}$, then we view $\mathcal{P}^{\omega}\subseteq\mathcal{M}^{\omega}$ (see the discussion after Definition A.4.1. in [27]). Moreover there is a natural embedding from $\mathcal{M}$ into $\mathcal{M}^{\omega}$ by sending each $a\in\mathcal{M}$ to a constant sequence $(a,a,a,\ldots)_{\omega}$ in $\mathcal{M}^{\omega}$. Thus we view $\mathcal{M}\subseteq\mathcal{M}^{\omega}$.

In the following lemma, item (i) is Lemma A.5.5 of [27]. Item (ii) follows from (i).

Lemma 2.1** (Lemma A.5.5 of [27]).**

Suppose that $\mathcal{P}$ is a type ${\rm II}_{1}$ factor with trace $\tau$ and $\mathcal{M}_{k}(\mathbb{C})$ is a matrix algebra of size $k\in\mathbb{N}$. Then the following statements are true:

1. (i)

$\displaystyle(\mathcal{P}\otimes\mathcal{M}_{k}(\mathbb{C}))^{\omega}=\mathcal{P}^{\omega}\otimes\mathcal{M}_{k}(\mathbb{C})$; 2. (ii)

If $\mathcal{A}$ is a von Neumann subalgebra of $\mathcal{P}$, then

[TABLE]

where $1_{k}$ is the identity of $\mathcal{M}_{k}(\mathbb{C})$.

Recall that a type ${\rm II}_{1}$ factor $\mathcal{M}$ has Property $\Gamma$ of Murray and von Neumann [20] if and only if *for any family $x_{1},\ldots,x_{n}$ of finitely many elements in $\mathcal{M}$ and any $\epsilon>0$, there exists a unitary element $u$ in $\mathcal{M}$ with $\tau(u)=0$ such that $\|[x_{i},u]\|_{2}\leq\epsilon$ for all $1\leq i\leq n$. *

Definition 2.2**.**

For $x\in\mathcal{M}$ and $\mathcal{S}\subseteq\mathcal{M}$, we define a distance function as follows:

[TABLE]

Denote by $W^{*}(\mathcal{S})$ the von Neumann subalgebra generated by $\mathcal{S}$ in $\mathcal{M}$ and by ${\rm C}^{*}(\mathcal{S})$ the unital ${\rm C}^{*}$-subalgebra generated by $\mathcal{S}$ in $\mathcal{M}$. The relative commutant of $W^{*}(\mathcal{S})$ in $\mathcal{M}$ is denoted by $W^{*}(\mathcal{S})^{\prime}\cap\mathcal{M}$.

Remark 2.3**.**

It is obvious that if a type ${\rm II}_{1}$ factor $\mathcal{M}$ has Property $\Gamma$, then

[TABLE]

for any finitely many elements $x_{1},\ldots,x_{n}$ in $\mathcal{M}$. If $\mathcal{M}$ has separable predual, then $\mathcal{M}$ has Property $\Gamma$ if and only if $\mathcal{M}^{\prime}\cap\mathcal{M}^{\omega}\neq\mathbb{C}I$ (see [4]). It is worthwhile noting that there exist examples of (nonseparable) type ${\rm II}_{1}$ factors $\mathcal{M}$ with Property $\Gamma$ such that $\mathcal{M}^{\prime}\cap\mathcal{M}^{\omega}=\mathbb{C}I$ (see Proposition $3.7$ in [9]).

Examples of (nonseparable) type II1 factors $\mathcal{M}$ with Property $\Gamma$ and with $\mathcal{M}^{\prime}\cap\mathcal{M}^{\omega}=\mathbb{C}I$ can also be found in the following proposition, which is a consequence of Popa’s result in [23].

Proposition 2.4**.**

Let $\mathcal{N}$ be a type ${\rm II}_{1}$ factor with separable predual and with Property $\Gamma$. Let $\omega$ be a free ultrafilter on $\mathbb{N}$ and $\mathcal{M}=\mathcal{N}^{\omega}$ an ultra-power of $\mathcal{N}$ along $\omega$. Then $\mathcal{M}$ is a type ${\rm II}_{1}$ factor with Property $\Gamma$ and with $\mathcal{M}^{\prime}\cap\mathcal{M}^{\omega}=\mathbb{C}I$.

Proof.

It is known that $\mathcal{M}=\mathcal{N}^{\omega}$ is a type II1 factor (see Theorem A.3.5 in [27]). It is straight forward to verify that $\mathcal{M}$ has Property $\Gamma$. We need only to show that $\mathcal{M}^{\prime}\cap\mathcal{M}^{\omega}=\mathbb{C}I$.

The traces on $\mathcal{N}$, $\mathcal{M}$, and $\mathcal{M}^{\omega}$ will be denoted by $\tau_{\mathcal{N}}$, $\tau_{\mathcal{M}}$, and $\tau_{\mathcal{M}^{\omega}}$ respectively. The 2-norms induced by the corresponding traces on $\mathcal{N}$, $\mathcal{M}$ and $\mathcal{M}^{\omega}$ will be denoted by $\|\cdot\|_{2,\mathcal{N}}$, $\|\cdot\|_{2,\mathcal{M}}$ and $\|\cdot\|_{2,\mathcal{M}^{\omega}}$ respectively. Elements in $\mathcal{N}$, $\mathcal{M}$ and $\mathcal{M}^{\omega}$ will be denoted by $x$, $X$ or $(x_{n})_{\omega}$, and $(X_{m})_{\omega}$ respectively if there is no confusion.

Suppose that $\mathcal{M}^{\prime}\cap\mathcal{M}^{\omega}\neq\mathbb{C}I$ and $(P_{m})_{\omega}$ is a nontrivial projection in $\mathcal{M}^{\prime}\cap\mathcal{M}^{\omega}$. Let $\lambda=\tau_{\mathcal{M}^{\omega}}((P_{m})_{\omega})$. Then $0<\lambda<1$. By Theorem A.5.3 in [27], we assume that each $P_{m}$ is a projection in $\mathcal{M}$ with $\tau_{\mathcal{M}}(P_{m})=\lambda$ for $m\geq 1$. As $P_{m}=(p_{n}^{(m)})_{\omega}$ is in $\mathcal{M}=\mathcal{N}^{\omega}$, we further assume that each $p_{n}^{(m)}$ is a projection in $\mathcal{N}$ with $\tau_{\mathcal{N}}(p_{n}^{(m)})=\lambda$ for $n,m\geq 1$.

By Corollary on page 187 in [23], there exists a family $\{u_{n}\}_{n=1}^{\infty}$ of unitary elements in $\mathcal{N}$ such that $\tau_{\mathcal{N}}(u_{n})=0$ and

[TABLE]

For each $n\geq 1$, by Equation (2.1), we let $k_{n}$ be a positive integer such that

[TABLE]

So, for $1\leq m\leq n$,

[TABLE]

Let $V=(u_{k_{n}})_{\omega}$ be a unitary element in $\mathcal{M}=\mathcal{N}^{\omega}$. Then

[TABLE]

This contradicts with the assumption that $(P_{m})_{\omega}$ is in $\mathcal{M}^{\prime}\cap\mathcal{M}^{\omega}$. Hence $\mathcal{M}^{\prime}\cap\mathcal{M}^{\omega}=\mathbb{C}I$. ∎

The next lemma is well-known. We include its proof for the purpose of completeness.

Lemma 2.5**.**

Let $\mathcal{M}$ be a type ${\rm II}_{1}$ factor with trace $\tau$. Suppose that $p$ is a nonzero projection in $\mathcal{M}$. Then $\mathcal{M}$ has Property $\Gamma$ if and only if $p\mathcal{M}p$ has Property $\Gamma$.

Proof.

When $\mathcal{M}$ has separable predual, the result can be found in Proposition 1.11 of [21]. Now we assume that $\mathcal{M}$ has nonseparable predual.

(i). Suppose that $\mathcal{M}$ has Property $\Gamma$. Let $x_{1},\ldots,x_{n}$ be in $p\mathcal{M}p$ and $\epsilon>0$. By Proposition 7.1 in [1], there exists a subfactor $\mathcal{M}_{1}$, with separable predual and with Property $\Gamma$, such that $\{p,x_{1},\ldots,x_{n},I\}\subseteq\mathcal{M}_{1}\subseteq\mathcal{M}$. Then $p\mathcal{M}_{1}p$ has Property $\Gamma$ by Lemma 2.5 of [3]. So there exists a unitary $u$ in $p\mathcal{M}_{1}p\subseteq p\mathcal{M}p$, with $\tau(u)=0$, such that $\|[x_{i},u]\|_{2}\leq\epsilon$ for $1\leq i\leq n$. By definition, $p\mathcal{M}p$ has Property $\Gamma$.

(ii). Assume that $p\mathcal{M}p$ has Property $\Gamma$. Let $n\in\mathbb{N}$ and $q$ be a subprojection of $p$ such that $\tau(q)=1/n$. By part (i), $q\mathcal{M}q$ has Property $\Gamma$. Notice that $\mathcal{M}$ is $\ast$-isomorphic to the von Neumann algebra tensor product $q\mathcal{M}q\otimes\mathcal{M}_{n}(\mathbb{C})$, which is denoted by $\mathcal{M}\cong q\mathcal{M}q\otimes\mathcal{M}_{n}(\mathbb{C})$. From Theorem 13.4.5 of [27], it follows that $\mathcal{M}$ has Property $\Gamma$. ∎

A quick consequence of spectral theory is needed in the paper and its proof is sketched.

Lemma 2.6**.**

Let $\{p_{i}\}_{i=1}^{n}$ be a family of mutually orthogonal projections in $\mathcal{M}$ such that $p_{1}+\cdots+p_{n}=I$. Suppose that $\{x_{i}\}_{i=1}^{n}$ is a family of elements in $\mathcal{M}$ satisfying

1. (i)

$x_{i}$* is in $p_{i}\mathcal{M}p_{i}$ for each $1\leq i\leq n$,* 2. (ii)

as an operator in $p_{i}\mathcal{M}p_{i}$, $x_{i}$ is self-adjoint and invertible for each $1\leq i\leq n$, and 3. (iii)

$\sigma_{p_{i}\mathcal{M}p_{i}}(x_{i})\cap\sigma_{p_{j}\mathcal{M}p_{j}}(x_{j})=\emptyset$, $\forall\ 1\leq i\neq j\leq n$, i.e. the spectra of $x_{i}$ and $x_{j}$ are pairwise disjoint for $i\neq j$.

If $x=x_{1}+x_{2}+\cdots+x_{n},$ then

[TABLE]

Proof.

Since $x_{i}$ is self-adjoint for $1\leq i\leq n$, it follows that $x$ is self-adjoint. Note that the spectra of $x_{i}$ and $x_{j}$ are pairwise disjoint for $1\leq i\neq j\leq n$. Define continuous functions $f_{i}(t)$ on $\sigma(x)$ as follows:

[TABLE]

That $x_{i}$ is invertible in $p_{i}\mathcal{M}p_{i}$ entails $p_{i}=f_{i}(x)\in{\rm C}^{\ast}(x)$ for $1\leq i\leq n$. Moreover, $x_{i}=p_{i}xp_{i}$ belongs to ${\rm C}^{\ast}(x)$ for $1\leq i\leq n$. This completes the proof. ∎

3. A characterization of Property $\Gamma$ for type ${\rm II}_{1}$  factors

Let $\mathcal{M}$ be a type ${\rm II}_{1}$ factor with trace $\tau$. It is an open question whether a type ${\rm II}_{1}$ factor with separable predual is generated by a single operator. When $\mathcal{M}$ is singly generated, the following Theorem 3.1 is a direct consequence of the definition of Property $\Gamma$. The main goal of this section is to provide an equivalent characterization of Property $\Gamma$ for type ${\rm II}_{1}$ factors without the assumption on cardinality of generators.

Theorem 3.1**.**

Let $\mathcal{M}$ be a type ${\rm II}_{1}$ factor with trace $\tau$. Then $\mathcal{M}$ has Property $\Gamma$ if and only if, for any element $x$ in $\mathcal{M}$ and any $\epsilon>0$, there exists a unitary element $u$ in $\mathcal{M}$ such that $\tau(u)=0$ and $\|[x,u]\|_{2}\leq\epsilon$.

The proof of Theorem 3.1 is postponed after a few technical lemmas. We start with a definition of support of an operator with respect to a family of mutually orthogonal projections.

Definition 3.2**.**

Let $x\in\mathcal{M}$ and $\{p_{i}\}_{i=1}^{k}\subseteq\mathcal{M}$ be a family of mutually orthogonal equivalent projections with $p_{1}+\cdots+p_{k}=I$. Define

[TABLE]

Lemma 3.3**.**

Let $k\in\mathbb{N}$. Suppose that $\mathcal{A}$ is a type ${\rm I}_{k}$ subfactor of $\mathcal{M}$ and $\{e_{ij}\}_{i,j=1}^{k}$ is a system of matrix units of $\mathcal{A}$. Let $q_{1}$ be a projection in $\mathcal{M}$.

If $\{i_{1},\ldots,i_{l}\}\subseteq\{1,\ldots,k\}$ satisfies the inequality $l\geq k\cdot\tau(\operatorname{supp}(q_{1},\{e_{ii}\}_{i=1}^{k}))$, then there exists a projection $q\in\mathcal{M}$ such that

1. (i)

$\operatorname{supp}(q,\{e_{ii}\}_{i=1}^{k})\leq e_{i_{1}i_{1}}+\cdots+e_{i_{l}i_{l}}$* and* 2. (ii)

$W^{*}(q_{1},\mathcal{A})=W^{*}(q,\mathcal{A})$.

Proof.

List $\{e_{ii}:e_{ii}q_{1}\neq 0\text{ or }q_{1}e_{ii}\neq 0\ \mbox{ for }\ 1\leq i\leq k\}$ as $\{e_{j_{1}j_{1}},\ldots,e_{j_{m}j_{m}}\}$, where $m=k\cdot\tau(\operatorname{supp}(q_{1},\{e_{ii}\}_{i=1}^{k}))$ is an integer by Definition 3.2. As $\tau(\operatorname{supp}(q_{1},\{e_{ii}\}_{i=1}^{k}))\leq l/k$, we have $m\leq l$. Thus, from the fact that $\mathcal{A}$ is a type ${\rm I}_{k}$ factor, it induces that there is a unitary element $u\in\mathcal{A}$ such that $ue_{j_{n}j_{n}}u^{*}=e_{i_{n}i_{n}}$ for $1\leq n\leq m.$ Now $q=uq_{1}u^{*}$ is a desired projection in $\mathcal{M}$. ∎

Recall that a von Neumann algebra is called diffuse if it doesn’t contain nonzero minimal projections. The following lemma is prepared for an induction argument of Claim 3.9.

Lemma 3.4**.**

Let $k\in\mathbb{N}$. Suppose that $\mathcal{A}$ is a type ${\rm I}_{k}$ subfactor of $\mathcal{M}$ and $\{e_{ij}\}_{i,j=1}^{k}$ is a system of matrix units of $\mathcal{A}$. Assume that $x$ is an element in $\mathcal{M}$ such that

[TABLE]

Then, for any $\epsilon>0$, there exist a positive integer $m$, a type ${\rm I}_{m}$ subfactor $\mathcal{N}$ of $\mathcal{A}^{\prime}\cap\mathcal{M}$, a system of matrix units $\{f_{i,j}\}_{i,j=1}^{m}$ of $\mathcal{N}$ and a projection $q$ in $\mathcal{M}$ such that

1. (i)

$q=q(\sum_{i=2}^{m}f_{i,i}e_{11})=(\sum_{i=2}^{m}f_{i,i}e_{11})q$; 2. (ii)

$\operatorname{dist}_{\|\cdot\|_{2}}(x,W^{*}(q,\mathcal{A},\mathcal{N}))\leq\epsilon$.

Proof.

Fix $\epsilon>0$. Let $m=64k^{2}$ and $\mathcal{P}=\mathcal{A}^{\prime}\cap\mathcal{M}$. As $\mathcal{A}$ is a type ${\rm I}_{k}$ subfactor of $\mathcal{M}$, we identify $\mathcal{M}$ with $\mathcal{P}\otimes\mathcal{A}$. Notice that $W^{*}(x,\mathcal{A})^{\prime}\cap\mathcal{M}^{\omega}$ is diffuse. There exists a family $\{(p^{(i)}_{n})_{\omega}:1\leq i\leq m\}$ of mutually orthogonal projections in $W^{*}(x,\mathcal{A})^{\prime}\cap\mathcal{M}^{\omega}$ with the same trace such that $\sum_{1\leq i\leq m}(p^{(i)}_{n})_{\omega}=I$.

By Lemma 2.1, $W^{*}(x,\mathcal{A})^{\prime}\cap\mathcal{M}^{\omega}\subseteq\mathcal{A}^{\prime}\cap\mathcal{M}^{\omega}=\mathcal{P}^{\omega},$ whence $\{(p^{(i)}_{n})_{\omega}:1\leq i\leq m\}\subseteq\mathcal{P}^{\omega}$. By Lemma A.5.3 in [27], we can further assume that $\{p_{n}^{(i)}\}_{1\leq i\leq m}$ is a family of mutually orthogonal equivalent projections in $\mathcal{P}$ such that $\sum_{1\leq i\leq m}p^{(i)}_{n}=I$. Thus $\{(p^{(i)}_{n})_{\omega}:1\leq i\leq m\}\subseteq W^{*}(x,\mathcal{A})^{\prime}\cap\mathcal{M}^{\omega}$ implies that there exists a family $\{p_{i}\}_{1\leq i\leq m}$ of mutually orthogonal equivalent projections in $\mathcal{P}$ such that

1. (a)

$p_{1}+p_{2}+\cdots+p_{m}=I$; 2. (b)

$\tau(p_{i})=1/m$ for each $1\leq i\leq m$; 3. (c)

$\|x-\sum_{1\leq i\leq m}p_{i}xp_{i}\|_{2}\leq\epsilon$.

Since $\mathcal{P}$ is a subfactor, there exist a type ${\rm I}_{m}$ subfactor $\mathcal{N}$ of $\mathcal{P}$ and a system of matrix units $\{f_{i,j}\}_{i,j=1}^{m}$ of $\mathcal{N}$ such that $f_{ii}=p_{i}$ for each $1\leq i\leq m$.

Define $y=\sum_{i=1}^{m}p_{i}xp_{i}=\sum_{i=1}^{m}f_{ii}xf_{ii}$. Recall that $\{e_{st}\}^{k}_{s,t=1}$ is a system of matrix units of $\mathcal{A}$. As $\mathcal{A}$ commutes with $\mathcal{N}$, it follows that $W^{*}(\mathcal{A},\mathcal{N})$ is a subfactor of type ${\rm I}_{km}$. Moreover, we have that $\{e_{st}f_{ij}\}_{1\leq s,t\leq k;1\leq i,j\leq m}$ is a system of matrix units of $W^{*}(\mathcal{A},\mathcal{N})$ and $\{e_{ss}f_{ii}\}_{1\leq s\leq k;1\leq i\leq m}$ is a family of mutually orthogonal projections in $\mathcal{M}$ such that $\tau(e_{ss}f_{ii})=\frac{1}{km}$ for each $1\leq s\leq k$ and $1\leq i\leq m$. It follows that

[TABLE]

When no confusion can arise, we write $|\SS|$ for the cardinality of a set $\SS$. Thus we obtain the following inequality:

[TABLE]

By the Cut-and-Past Theorem (Theorem 4.1 of [25]), there exists a projection $q$ in $\mathcal{M}$ such that

1. (d)

$W^{*}(y,\mathcal{A},\mathcal{N})=W^{*}(q,\mathcal{A},\mathcal{N})$; 2. (e)

\displaystyle\tau\Big{(}\operatorname{supp}(q,\{e_{ss}f_{ii}\}_{1\leq s\leq k;1\leq i\leq m})\Big{)}\leq 2\left(\frac{1}{m}\right)^{1/2}+\frac{2}{km}=2\left(\frac{1}{64k^{2}}\right)^{1/2}+\frac{2}{64k^{3}}\leq\frac{1}{2k}.

Note that, from (e), we obtain that

[TABLE]

By Lemma 3.3, we can further assume that $\operatorname{supp}(q,\{e_{ss}f_{ii}\}_{1\leq s\leq k;1\leq i\leq m})\leq\sum_{i=2}^{m}f_{ii}e_{11}$, which implies (i)

[TABLE]

Now (ii) follows from (c), the choice of $y$, and (d). ∎

An easy exercise of spectral theory is needed.

Lemma 3.5**.**

Let $\mathcal{N}_{0}\subseteq\mathcal{M}$ be a subfactor of type ${\rm I}_{3}$ and $\{e_{ij}\}_{1\leq i,j\leq 3}$ a system of matrix units of $\mathcal{N}_{0}$. Let $z_{1}$ and $z_{2}$ be self-adjoint elements in $\mathcal{N}_{0}^{\prime}\cap\mathcal{M}$ and $y$ an element in $\mathcal{N}_{0}^{\prime}\cap\mathcal{M}$. Assume that

[TABLE]

and

[TABLE]

Then

[TABLE]

Proof.

Apparently, $e_{11},e_{22},e_{33}$ are in $W^{*}(a)$. Observe that $e_{ii}be_{33}=e_{i3}$ and $e_{33}be_{ii}=e_{3i}$ for $i=1,2,3$. We have $e_{i3}e_{3j}=e_{ij}$ is in $W^{*}(a+ib)$ for $1\leq i,j\leq 3$. Thus $\mathcal{N}_{0}\subseteq W^{*}(a+ib)$. From the fact that $e_{i1}be_{1i}=z_{1}e_{ii}$ for $i=1,2,3$, it follows that $z_{1}=z_{1}e_{11}+z_{1}e_{22}+z_{1}e_{33}$ is in $W^{*}(a+ib)$. Similarly, it can be shown that $z_{2},y\in W^{*}(a+ib)$. Therefore,

[TABLE]

This ends the proof. ∎

The next lemma, used repeatedly in the proof of Proposition 3.8, might have been known, but we can’t find a reference to it.

Lemma 3.6**.**

Suppose that $\mathcal{M}_{1}$ is a finite von Neumann algebra and $\mathcal{M}_{2}\subseteq\mathcal{M}_{1}$ is a von Neumann subalgebra containing the identity $I$ of $\mathcal{M}_{1}$. If $\mathcal{M}_{2}$ is diffuse, then $\mathcal{M}_{1}$ is diffuse.

Proof.

Assume that $\mathcal{M}_{1}$ is not diffuse. Then there exists a nonzero minimal projection $p$ in $\mathcal{M}_{1}$. Let $C_{p}$ be the central carrier of $p$ in $\mathcal{M}_{1}$. By Proposition 6.4.3 of [15], $\mathcal{M}_{1}C_{p}$ is a factor. As $\mathcal{M}_{1}$ is finite, $\mathcal{M}_{1}C_{p}$ is a factor of type ${\rm I}_{k}$ for some positive integer $k$.

Define

[TABLE]

Then $I-q$ is a nonzero projection in $\mathcal{M}_{2}$ satisfying, for every nonzero subprojection $f$ of $I-q$ in $\mathcal{M}_{2}$, $fC_{p}\neq 0$. If $\mathcal{M}_{2}$ contains no minimal projections, then there exists a family of mutually orthogonal nonzero projections $f_{1},\ldots,f_{k+1}$ in $\mathcal{M}_{2}$ such that $I-q=f_{1}+\cdots+f_{k+1}$. The choice of $q$ ensures that $f_{1}C_{p},\ldots,f_{k+1}C_{p}$ is a family of mutually orthogonal nonzero projections in $\mathcal{M}_{1}C_{p}$, which contradicts with the fact that $\mathcal{M}_{1}C_{p}$ is a factor of type ${\rm I}_{k}$. This completes the proof. ∎

Central sequence algebras of type ${\rm II}_{1}$ factors play an important role in the paper. The following result is a slight modification of Lemma 3.5 of [7] (or Theorem A.6.5 of [27]) by removing the condition that $\mathcal{M}$ is separable. Recall that a finite von Neumann algebra with a faithful, normal, tracial state is separable if it has a separable predual, which is equivalent to it is countably generated (see [5] Exercise I.7.3 b and c).

Lemma 3.7**.**

Let $\mathcal{M}$ be a type ${\rm II}_{1}$ factor with trace $\tau$. Suppose that $\mathcal{Q}$ is a separable irreducible ${\rm II}_{1}$ subfactor of $\mathcal{M}$. If $\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}\neq\mathbb{C}I$, then $\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}$ is diffuse.

Proof.

Assume that $\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}\neq\mathbb{C}I$ and $\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}$ is not diffuse. Note that $\mathcal{M}^{\omega}$ is a type II1 factor with a natural trace $\tau_{\omega}$ and $\mathcal{M}\subseteq\mathcal{M}^{\omega}$. Let $E_{\mathcal{M}}^{\mathcal{M}^{\omega}}:\mathcal{M}^{\omega}\rightarrow\mathcal{M}$ be the trace-preserving conditional expectation from $\mathcal{M}^{\omega}$ onto $\mathcal{M}$. Suppose that $\{a_{m}\}_{m=1}^{\infty}$ is a countable family of self-adjoint generators of $\mathcal{Q}$.

Let $(q_{n})_{\omega}\neq 0,I$ be a minimal projection in $\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}$ and $z=(Y_{n})_{\omega}$ the central carrier of $(q_{n})_{\omega}$ in $\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}$. Then $z(\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega})$ is a factor of type ${\rm I}_{k}$ for some positive integer $k$ by Proposition 6.4.3 of [15]. Assume that $\tau_{\omega}((q_{n})_{\omega})=r$ with $0<r<1$, then $\tau_{\omega}(z)=kr\leq 1$. We can further assume that $q_{n}$ and $Y_{n}$ are projections in $\mathcal{M}$ with $\tau(q_{n})=r$ and $\tau(Y_{n})=kr$ for each $n\geq 1$ by Theorem A.5.3 in [27].

We claim that $\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}$ is a nonseparable subspace of $\mathcal{M}^{\omega}$ with respect to $\|\cdot\|_{2,\tau_{\omega}}$, the trace norm of $\mathcal{M}^{\omega}$. Assume, to the contrary, that $\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}$ is separable with respect to $\|\cdot\|_{2,\tau_{\omega}}$ and assume that $\{(y_{n}^{(m)})_{\omega}\}_{m=1}^{\infty}$ is a dense subset of $\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}$. As $\mathcal{Q}^{\prime}\cap\mathcal{M}=\mathbb{C}I$, $(q_{n})_{\omega}\notin\mathcal{M}$, whence $\delta=\|(q_{n})_{\omega}-E_{\mathcal{M}}^{\mathcal{M}^{\omega}}((q_{n})_{\omega})\|_{2,\tau_{\omega}}>0$. It follows that

[TABLE]

Combining it with the fact that $(q_{n})_{\omega}$ is in $\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}$, for each $n\geq 1$ we let $k_{n}\in\mathbb{N}$ be such that

[TABLE]

Therefore, $(q_{k_{n}})_{\omega}\in\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}$ and $\|(q_{k_{n}})_{\omega}-(y_{n}^{(m)})_{\omega}\|_{2,\tau_{\omega}}\geq\delta/2>0$ for $m\geq 1$. This contradicts with the assumption that $\{(y_{n}^{(m)})_{\omega}\}_{m=1}^{\infty}$ is dense in $\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}$. Hence $\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}$ is nonseparable.

Observe that $E_{\mathcal{M}}^{\mathcal{M}^{\omega}}(\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega})\subseteq\mathcal{Q}^{\prime}\cap\mathcal{M}=\mathbb{C}I$. Thus $E_{\mathcal{M}}^{\mathcal{M}^{\omega}}((q_{n})_{\omega})=\tau_{\omega}((q_{n})_{\omega})=r,$ which implies that

[TABLE]

Combining it with the fact that $(q_{n})_{\omega}$ is in $\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}$, for each $n\geq 1$ we let $j_{n}\in\mathbb{N}$ be such that

[TABLE]

Hence, $(q_{j_{n}}Y_{n})_{\omega}=(q_{j_{n}})_{\omega}z$ is a projection in $z(\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega})$ with trace $kr^{2}$. On the other hand, $z(\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega})$ is a type Ik factor with a minimal projection $(q_{n})_{\omega}$ of trace $r$. Therefore, $kr^{2}\geq r$, whence $kr=1$ and $z=I$. This means that $\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}$ is a type Ik factor, which contradicts with the fact that $\mathcal{Q}^{\prime}\cap\mathcal{M}^{\omega}$ is nonseparable. This ends the proof of the lemma. ∎

If $\mathcal{M}$ is a singly generated type ${\rm II}_{1}$ factor, then, by [4], that $\mathcal{M}$ has Property $\Gamma$ is equivalent to $W^{*}(x)^{\prime}\cap\mathcal{M}^{\omega}\neq\mathbb{C}I$ for all $x\in\mathcal{M}$. In fact a more general statement, without the assumption on cardinality of generators of $\mathcal{M}$, is still valid. We develop this in the following proposition.

Proposition 3.8**.**

Let $\mathcal{M}$ be a type ${\rm II}_{1}$ factor with trace $\tau$. Then the following statements are equivalent:

1. (i)

$\mathcal{M}$* has Property $\Gamma$.* 2. (ii)

For every $x$ in $\mathcal{M}$, $W^{*}(x)^{\prime}\cap\mathcal{M}^{\omega}$ is diffuse. 3. (iii)

For every $x$ in $\mathcal{M}$ and every nonzero projection $p$ in $\mathcal{M}$,

[TABLE] 4. (iv)

For every $x$ in $\mathcal{M}$, $W^{*}(x)^{\prime}\cap\mathcal{M}^{\omega}\neq\mathbb{C}I$.

Proof.

(i) $\Rightarrow$ (ii). Assume that (i) holds. Let $x$ be an element in $\mathcal{M}$. By Proposition 7.1 in [1], there exists a separable subfactor $\mathcal{M}_{1}$ with Property $\Gamma$ such that $x\in\mathcal{M}_{1}\subseteq\mathcal{M}$. It follows that $\mathcal{M}_{1}^{\prime}\cap\mathcal{M}_{1}^{\omega}$ is diffuse (see [4] or Lemma 3.7). Hence, by Lemma 3.6, $W^{*}(x)^{\prime}\cap\mathcal{M}^{\omega}\supseteq\mathcal{M}_{1}^{\prime}\cap\mathcal{M}_{1}^{\omega}$ is diffuse, i.e., (ii) is true.

(ii) $\Rightarrow$ (i). Assume that (ii) holds. Suppose that $x_{1},\ldots,x_{n}$ are elements in $\mathcal{M}$ and $\epsilon>0$ is a positive number. Let $\mathcal{N}_{0}$ be a type ${\rm I}_{3}$ subfactor of $\mathcal{M}$ and $\mathcal{P}=\mathcal{N}_{0}^{\prime}\cap\mathcal{M}$ so that $\mathcal{M}\cong\mathcal{P}\otimes\mathcal{N}_{0}$. Assume that $\{e_{ij}\}_{i,j=1}^{3}$ is a system of matrix units of $\mathcal{N}_{0}$. Then there is a family of elements $\{y_{ij}^{(r)}\}_{1\leq i,j\leq 3;1\leq r\leq n}$ in $\mathcal{P}$ such that

[TABLE]

List elements in $\{y_{ij}^{(r)}\}_{1\leq i,j\leq 3;1\leq r\leq n}$ as $\{y_{r}\}_{1\leq r\leq 9n}$. We will prove the following claim first to complete the proof of the implication (ii) $\Rightarrow$ (i).

Claim 3.9**.**

There exist

1. (a)

a family of positive integers $\{m_{r}\}_{r=1}^{9n}$; 2. (b)

a family of commuting subfactors

[TABLE] 3. (c)

a family of systems of matrix units

[TABLE] 4. (d)

a family of projections $\{q_{r}\}_{r=1}^{9n}$ in $\mathcal{M}$

such that

1. (1)

$q_{r}=q_{r}\left(\sum_{i=2}^{m_{r}}f_{i,i}^{(r)}\right)f_{1,1}^{(r-1)}\cdots f_{1,1}^{(1)}e_{11}=\left(\sum_{i=2}^{m_{r}}f_{i,i}^{(r)}\right)f_{1,1}^{(r-1)}\cdots f_{1,1}^{(1)}e_{11}q_{r},\ \forall 1\leq r\leq 9n$; 2. (2)

$\operatorname{dist}_{\|\cdot\|_{2}}(y_{r},W^{*}(q_{r},\mathcal{N}_{0},\mathcal{N}_{1},\ldots,\mathcal{N}_{r}))\leq\epsilon/2,\ \forall\ 1\leq r\leq 9n$.

Proof of the Claim.

We prove the claim by induction.

*Step One: * When $r=1$, let

[TABLE]

and

[TABLE]

Then Lemma 3.5 implies that $W^{*}(y_{1},\mathcal{N}_{0})\subseteq W^{*}(a,b_{1})$. By the Assumption (ii) and Lemma 3.6,

[TABLE]

Applying Lemma 3.4 for $\mathcal{A}=\mathcal{N}_{0}$ and $x=y_{1}$, there exist a positive integer $m_{1}$, a type ${\rm I}_{m_{1}}$ subfactor $\mathcal{N}_{1}$ of $\mathcal{N}_{0}^{\prime}\cap\mathcal{M}$, a system of matrix units $\{f_{i,j}^{(1)}\}_{i,j=1}^{m_{1}}$ of $\mathcal{N}_{1}$ and a projection $q_{1}$ in $\mathcal{M}$ such that

1. (1)

$q_{1}=q_{1}(\sum_{i=2}^{m_{1}}f_{i,i}^{(1)})e_{11}=(\sum_{i=2}^{m_{1}}f_{i,i}^{(1)})e_{11}q_{1}$; 2. (2)

$\operatorname{dist}_{\|\cdot\|_{2}}(y_{1},W^{*}(q_{1},\mathcal{N}_{0},\mathcal{N}_{1}))\leq\epsilon/2$.

Step Two: Assume that the claim is true for $r=k$, where $1\leq k<9n$, i.e., the desired $\{m_{r}\}_{r=1}^{k}$, $\{\mathcal{N}_{r}\}_{r=1}^{k}$, $\{\{f_{i,j}^{(r)}\}_{i,j=1}^{m_{r}}\}_{r=1}^{k}$ and $\{q_{r}\}_{r=1}^{k}$ have been obtained. Notice that $\{\mathcal{N}_{r}\}_{r=1}^{k}$ is a family of commuting subfactors in $\mathcal{N}_{0}^{\prime}\cap\mathcal{M}$. So $W^{*}(\mathcal{N}_{1},\ldots,\mathcal{N}_{k})$ is a subfactor of type I${}_{m_{1}\cdots m_{k}}$, which has two self-adjoint generators $z_{1},z_{2}$. Moreover, $W^{*}(\mathcal{N}_{0},\mathcal{N}_{1},\ldots,\mathcal{N}_{k})$ is a subfactor of type I${}_{3m_{1}\cdots m_{k}}$ with a system of matrix units

[TABLE]

Define

[TABLE]

Recall that $a=e_{11}+2e_{22}+3e_{33}$. Then Lemma 3.5 implies that

[TABLE]

By Assumption (ii) and Lemma 3.6,

[TABLE]

By writing $\mathcal{A}=W^{*}(\mathcal{N}_{0},\mathcal{N}_{1},\ldots,\mathcal{N}_{k})$ and $x=y_{k+1}$ in Lemma 3.4, there exist a positive integer $m_{k+1}$, a type I${}_{m_{k+1}}$ subfactor $\mathcal{N}_{k+1}$ of $W^{*}(\mathcal{N}_{0},\mathcal{N}_{1},\ldots,\mathcal{N}_{k})^{\prime}\cap\mathcal{M}$, a system of matrix units $\{f_{i,j}^{(k+1)}\}_{i,j=1}^{m_{k+1}}$ of $\mathcal{N}_{k+1}$ and a projection $q_{k+1}$ in $\mathcal{M}$ such that

1. (1)

$q_{k+1}=q_{k+1}\left(\sum_{i=2}^{m_{k+1}}f_{ii}^{(k+1)}\right)f_{11}^{(k)}\cdots f_{11}^{(1)}e_{11}=\left(\sum_{i=2}^{m_{k+1}}f_{ii}^{(k+1)}\right)f_{11}^{(k)}\cdots f_{11}^{(1)}e_{11}q_{k+1}$; 2. (2)

$\operatorname{dist}_{\|\cdot\|_{2}}(y_{k+1},W^{*}(q_{k+1},\mathcal{N}_{0},\mathcal{N}_{1},\ldots,\mathcal{N}_{k+1}))\leq\epsilon/2$.

Thus the claim is true for $r=k+1$. By the principle of mathematical induction, Claim 3.9 is true. This finishes the proof of the claim. ∎

(End of the proof of Proposition 3.8:) Now we have obtained $\{m_{r}\}_{r=1}^{9n}$, $\{\mathcal{N}_{r}\}_{r=1}^{9n}$, $\{\{f_{i,j}^{(r)}\}_{i,j=1}^{m_{r}}\}_{r=1}^{9n}$ and $\{q_{r}\}_{r=1}^{9n}$ with the properties as listed in Claim 3.9. Observe that $W^{*}(\mathcal{N}_{1},\ldots,\mathcal{N}_{9n})$ is a subfactor of type I${}_{m_{1}\cdots m_{9n}}$. Assume that $\tilde{z}_{1},\tilde{z}_{2}$ are two self-adjoint generators of $W^{*}(\mathcal{N}_{1},\mathcal{N}_{2},\ldots,\mathcal{N}_{9n})$. The Conclusion (1) of Claim 3.9 entails that $q_{1},\ldots,q_{9n}$ are mutually orthogonal sub-projections of $e_{11}.$ The spectral theorem for a self-adjoint operator implies that

[TABLE]

Recall that $a=e_{11}+2e_{22}+3e_{33}$. Now we let

[TABLE]

A similar proof to Lemma 3.5 yields that

[TABLE]

By Assumption and Lemma 3.6, the inclusion

[TABLE]

implies that $W^{*}(q_{1},\ldots,q_{9n},\mathcal{N}_{0},\mathcal{N}_{1},\ldots,\mathcal{N}_{9n})^{\prime}\cap\mathcal{M}^{\omega}$ is diffuse. By Conclusion (2) of Claim 3.9, there is an element $w_{k}$ in $W^{*}(q_{1},\ldots,q_{9n},\mathcal{N}_{0},\mathcal{N}_{1},\ldots,\mathcal{N}_{9n})$ such that $\|y_{k}-w_{k}\|_{2}\leq\epsilon$ for each $1\leq k\leq 9n$. As $W^{*}(q_{1},\ldots,q_{9n},\mathcal{N}_{0},\mathcal{N}_{1},\ldots,\mathcal{N}_{9n})^{\prime}\cap\mathcal{M}^{\omega}$, a subset of $(\mathcal{N}_{0}^{\prime}\cap\mathcal{M})^{\omega}$, is diffuse, there exists a unitary element $u$ in $\mathcal{N}_{0}^{\prime}\cap\mathcal{M}$ such that $\tau(u)=0$ and $\|[w_{k},u]\|_{2}\leq\epsilon$ for $1\leq k\leq 9n$. Since $\{y_{ij}^{(r)}\}_{1\leq i,j\leq 3;1\leq r\leq n}$ was listed as $\{y_{r}\}_{1\leq r\leq 9n}$, we can rename $\{w_{r}\}_{1\leq r\leq 9n}$ as $\{w_{ij}^{(r)}\}_{1\leq i,j\leq 3;1\leq r\leq n}$ correspondingly with

[TABLE]

Therefore, for each $1\leq r\leq n$, it follows that

[TABLE]

By the definition of Property $\Gamma$, (i) of Proposition 3.8 is proved.

(i) $\Rightarrow$ (iii). Assume that $\mathcal{M}$ has Property $\Gamma$. Let $x$ be an element in $\mathcal{M}$ and $p$ a nonzero projection in $\mathcal{M}$. By virtue of Lemma 2.5, $p\mathcal{M}p$ has Property $\Gamma$. This entails the inequality $W^{*}(pxp)^{\prime}\cap(p\mathcal{M}p)^{\omega}\neq\mathbb{C}p$.

(iii) $\Rightarrow$ (ii). Let $x$ be an element in $\mathcal{M}$. Define $\mathcal{N}=W^{*}(x)^{\prime}\cap\mathcal{M}$. If $\mathcal{N}$ contains no minimal projections, then $\mathcal{N}$ is diffuse, whence $W^{*}(x)^{\prime}\cap\mathcal{M}^{\omega}\supseteq\mathcal{N}$ is diffuse by Lemma 3.6. Otherwise, we assume that $\{p_{n}\}_{n=1}^{N}$ is a maximal family of mutually orthogonal minimal projections in $\mathcal{N}$, where $1\leq N\leq\infty$. Define $p_{0}=I-\sum_{n=1}^{N}p_{n}$. Then $p_{0}\mathcal{N}p_{0}$ is diffuse. For each $n\geq 1$, it is not hard to verify that $W^{*}(p_{n}xp_{n})$ is an irreducible subfactor of $p_{n}\mathcal{M}p_{n}$. The assumption $W^{*}(p_{n}xp_{n})^{\prime}\cap(p_{n}\mathcal{M}p_{n})^{\omega}\neq\mathbb{C}p_{n}$ and Lemma 3.7 guarantee that $W^{*}(p_{n}xp_{n})^{\prime}\cap(p_{n}\mathcal{M}p_{n})^{\omega}$ is diffuse. Obviously,

[TABLE]

This entails that $W^{*}(x)^{\prime}\cap\mathcal{M}^{\omega}$ is diffuse, by virtue of Lemma 3.6.

(iii) $\Rightarrow$ (iv). This is trivial.

(iv) $\Rightarrow$ (iii). We use a contrapositive proof here. Assume that (iii) is false. Thus there exist two self-adjoint elements $y_{1}$ and $y_{2}$ in $\mathcal{M}$ and a nonzero projection $p$ in $\mathcal{M}$ such that

[TABLE]

Denote $p$ by $p_{0}$. As $\mathcal{M}$ is a type ${\rm II}_{1}$ factor, there exists a family of mutually orthogonal subprojections $p_{1},\ldots,p_{k-1},p_{k}$ of $I-p_{0}$ such that

1. (a1)

$\tau(p_{0})=\tau(p_{1})=\cdots=\tau(p_{k-1})\geq\tau(p_{k})$ and 2. (b1)

$p_{0}+p_{1}+\cdots+p_{k}=I$.

Furthermore, there exists a family of partial isometries $v_{1},\ldots,v_{k}$ in $\mathcal{M}$ such that

[TABLE]

Without loss of generality, we assume that $\|p_{0}y_{1}p_{0}\|<1$. Define two self-adjoint elements $x_{1}$ and $x_{2}$ in $\mathcal{M}$ as follows:

[TABLE]

and

[TABLE]

By spectral theory, we obtain that $p_{0},p_{1},\ldots,p_{k},p_{0}y_{1}p_{0}$ are in $W^{*}(x_{1})$ and $p_{0}y_{2}p_{0},v_{1},\ldots,v_{k}$ are in $W^{*}(x_{1}+ix_{2})$.

We claim that $W^{*}(x_{1}+ix_{2})^{\prime}\cap\mathcal{M}^{\omega}=\mathbb{C}I$. In fact, assume that $(q_{n})_{\omega}$ is a projection in $W^{*}(x_{1},x_{2})^{\prime}\cap\mathcal{M}^{\omega}$. From the fact that $p_{0},p_{0}y_{1}p_{0}$ are in $W^{*}(x_{1})$, we conclude that

[TABLE]

whence $p_{0}(q_{n})_{\omega}p_{0}$ is a projection in $\mathcal{M}^{\omega}$. From the fact that $p_{0}y_{1}p_{0},p_{0}y_{2}p_{0}$ are in $W^{*}(x_{1},x_{2})$, it follows that $p_{0}y_{1}p_{0}$ and $p_{0}y_{2}p_{0}$ commute with $(q_{n})_{\omega}$ in $\mathcal{M}^{\omega}$. Therefore

[TABLE]

Thus $p_{0}(q_{n})_{\omega}p_{0}=0$ or $p_{0}$. We proceed the proof by considering the following two cases.

Case $1$. Assume that $p_{0}(q_{n})_{\omega}p_{0}=0$. Since $v_{1},\ldots,v_{k}$ are in $W^{*}(x_{1},x_{2})$, it follows that $(q_{n})_{\omega}v_{i}=v_{i}(q_{n})_{\omega}$. This, together with (3.1) and (3.2), implies that $0=p_{0}(q_{n})_{\omega}p_{0}v_{i}=(q_{n})_{\omega}v_{i}=v_{i}(q_{n})_{\omega}$, whence $p_{i}(q_{n})_{\omega}=v_{i}^{*}v_{i}(q_{n})_{\omega}=0$ for all $1\leq i\leq k$. So $(q_{n})_{\omega}=\sum^{k}_{i=0}p_{i}(q_{n})_{\omega}=0$.

Case $2$. Assume that $p_{0}(q_{n})_{\omega}p_{0}=p_{0}$. As $v_{1},\ldots,v_{k}\in W^{*}(x_{1},x_{2})$, we have the equality $(q_{n})_{\omega}v_{i}=v_{i}(q_{n})_{\omega}$. This, together with (3.1) and (3.2), implies that $v_{i}=p_{0}v_{i}=p_{0}(q_{n})_{\omega}p_{0}v_{i}=(q_{n})_{\omega}v_{i}=v_{i}(q_{n})_{\omega}$, whence $p_{i}=v_{i}^{*}v_{i}=v_{i}^{*}v_{i}(q_{n})_{\omega}=p_{i}(q_{n})_{\omega}$ for all $1\leq i\leq k$. It follows that $(q_{n})_{\omega}=\sum^{k}_{i=0}p_{i}(q_{n})_{\omega}=\sum^{k}_{i=0}p_{i}=I$.

In a summary, we conclude that $(q_{n})_{\omega}$ is either [math] or $I$. Thus $W^{*}(x_{1}+ix_{2})^{\prime}\cap\mathcal{M}^{\omega}=\mathbb{C}I$, whence (iv) is false. This ends the proof of the implication (iv) $\Rightarrow$ (iii). ∎

Now Theorem 3.1 follows directly from Proposition 3.8.

Proof of Theorem 3.1.

The implication “$\Rightarrow$” is obvious. For the implication “$\Leftarrow$”, the assumption implies that $W^{*}(x)^{\prime}\cap\mathcal{M}^{\omega}\neq\mathbb{C}I$, for every $x$ in $\mathcal{M}$. Now Proposition 3.8 guarantees that $\mathcal{M}$ has Property $\Gamma$. ∎

The following result, implied in Theorem 2.1 in [3], is well known and its proof is sketched.

Proposition 3.10**.**

Let $\mathcal{M}$ be a type ${\rm II}_{1}$ factor with trace $\tau$. Let $x$ be an element in $\mathcal{M}$. Consider the following statements:

1. (i)

For any given $\epsilon>0$, for every nonzero projection $p\in\mathcal{M}$, there exists a nonzero sub-projection $q$ of $p$ in $\mathcal{M}$ satisfying

[TABLE] 2. (ii)

$W^{*}(x)^{\prime}\cap\mathcal{M}^{\omega}\neq\mathbb{C}I$.

Then the implication (i) $\Rightarrow$ (ii) holds.

Proof.

Assume that (i) holds. To prove (ii), it suffices to show that, for any $\epsilon>0$, there exists a projection $q$ in $\mathcal{M}$ such that $\tau(q)=1/2$ and $\|[x,q]\|_{2}\leq\epsilon$.

Denote by $\mathcal{P}(\mathcal{M})$ the set of all the projections in $\mathcal{M}$. Define $\mathcal{S}$ to be a subset of $\mathcal{M}$ in the following form:

[TABLE]

For projections $q_{1}$ and $q_{2}$ in $\mathcal{S}$, if $q_{1}$ is a sub-projection of $q_{2}$, then define $q_{1}\preceq q_{2}$. Thus, the binary relation “$\preceq$” is a partial order on $\mathcal{S}$.

The Assumption (i) implies that $\mathcal{S}$ contains a nonzero projection in $\mathcal{M}$. Moreover, since $\tau$ is normal, each totally ordered chain in $\mathcal{S}$ has an upper bound in $\mathcal{S}$. Thus, by Zorn’s lemma, there exists a maximal element $q$ in $\mathcal{S}$.

If $\tau(q)=1/2$, then the proof is completed. Assume, on the contrary, that $\tau(q)<1/2$. Then there exists a positive number $\delta>0$ with $\tau(q)+\delta<1/2$ and with $0<\delta<\epsilon/2$. By applying Assumption (i) to $\delta>0$ and $I-q$, there exists a nonzero sub-projection $q_{0}$ of $I-q$ such that

[TABLE]

Note that $\tau(q)<\tau(q+q_{0})<1/2$ and

[TABLE]

This implies that $q+q_{0}\in\mathcal{S}$. But $q+q_{0}\in\mathcal{S}$ contradicts the fact that $q$ is a maximal element in $\mathcal{S}$. This ends the proof of the proposition. ∎

A type II1 factor $\mathcal{M}$ with separable predual is called a McDuff factor if $\mathcal{M}\simeq\mathcal{M}\hat{\otimes}\mathcal{R}$, where $\mathcal{R}$ is the hyperfinite II1 factor with separable predual. Here we provide an answer to Sherman’s question in Problem 2.11 of [26].

Corollary 3.11**.**

Let $n\geq 2$ be a positive integer and $\mathcal{M}$ a type ${\rm II}_{1}$ factor with separable predual. Then the following statements are equivalent:

1. (i)

$\mathcal{M}$* is a McDuff factor.* 2. (ii)

For any $x\in\mathcal{M}$, $W^{*}(x)^{\prime}\cap\mathcal{M}^{\omega}$ is a type ${\rm II}_{1}$ von Neumann algebra. 3. (iii)

For any $x\in\mathcal{M}$, $W^{*}(x)^{\prime}\cap\mathcal{M}^{\omega}$ unitally contains $\mathcal{M}_{n}(\mathbb{C})$. 4. (iv)

For any $x\in\mathcal{M}$, $W^{*}(x)^{\prime}\cap\mathcal{M}^{\omega}$ is not abelian.

Proof.

(i)$\Rightarrow$ (ii)$\Rightarrow$(iii)$\Rightarrow$(iv) is obvious.

(iv) $\Rightarrow$ (i). The assumption (iv) implies that $W^{*}(x)^{\prime}\cap\mathcal{M}^{\omega}\neq\mathbb{C}I$, for every $x$ in $\mathcal{M}$. By Proposition 3.8, $\mathcal{M}$ has Property $\Gamma$. As $\mathcal{M}$ has a separable predual, there exists an element $y$ in $\mathcal{M}$ such that $W^{*}(y)=\mathcal{M}$ by Theorem 6.2 of [10]. Again from the assumption (iv), $W^{*}(y)^{\prime}\cap\mathcal{M}^{\omega}=\mathcal{M}^{\prime}\cap\mathcal{M}^{\omega}$ is noncommutative. Now (i) follows from Theorem 3 of [18]. ∎

4. An operator with spectral gap in a non-$\Gamma$  type ${\rm II}_{1}$  factor

In this section, we will show the existence of a single operator with spectral gap in a non-$\Gamma$ type ${\rm II}_{1}$ factor. The main result, Theorem 4.9, will be proved by a series of lemmas. Let $\mathcal{M}$ be a type ${\rm II}_{1}$ factor with trace $\tau$.

Definition 4.1**.**

Let $\{q_{k}\}^{\infty}_{k=1}$ be a sequence of nonzero projections in $\mathcal{M}$. Define

[TABLE]

Lemma 4.2**.**

If $\{q_{k}\}^{\infty}_{k=1}$ is a sequence of nonzero projections in $\mathcal{M}$, then $\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$ is a unital ${\rm C}^{*}$-subalgebra of $\mathcal{M}$.

Proof.

Apparently $\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$ contains [math] and $I$. Let $x$ and $y$ be in $\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$ and $\alpha\in\mathbb{C}$. Observe that

[TABLE]

It follows that $\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$ is a $*$-algebra.

Assume that $z\in\mathcal{M}$ is in the operator norm closure of $\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$. For any $\delta>0$, there exists an element $\tilde{z}$ in $\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$ such that $\|z-\tilde{z}\|\leq\delta$. Hence

[TABLE]

It follows that $z\in\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$, which implies that $\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$ is a unital ${\rm C}^{*}$-subalgebra of $\mathcal{M}$. ∎

Lemma 4.3**.**

Let $\{q_{k}\}^{\infty}_{k=1}$ be a sequence of nonzero projections in $\mathcal{M}$. If $p$ is a projection in $\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$, then

[TABLE]

Proof.

For every $x$ in $\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$,

[TABLE]

Since $p,pxp$ are in $\mathcal{A}(\{q_{k}\}^{\infty}_{k=1}))$,

[TABLE]

This completes the proof. ∎

The following Lemma 4.4 is prepared for Lemma 4.5.

Lemma 4.4**.**

If $0\leq y\leq 1$ is an element in $\mathcal{M}$, then there exists a projection $e\in W^{*}(y)$ satisfying $\|y-e\|_{2}\leq 2\|y-y^{2}\|_{2}.$

Proof.

By spectral theorem, the faithful, normal, tracial state $\tau$ induces a probability measure $\mu$ on $[0,1]$ such that

[TABLE]

for all bounded Borel function $f$ on $[0,1]$. Then

[TABLE]

Let $e$ be the spectral projection of $y$ corresponding to the interval $(1/2,1]$. It follows that

[TABLE]

Thus, $e$ is a projection in $W^{*}(y)$ satisfying $\|y-e\|_{2}\leq 2\|y-y^{2}\|_{2}$. ∎

Lemma 4.5**.**

Let $\{q_{k}\}^{\infty}_{k=1}$ be a sequence of nonzero projections in $\mathcal{M}$. If $p$ is a projection in $\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$, then there exists a subprojection $e_{k}$ of $p$ for each $k\geq 1$ such that

[TABLE]

Proof.

Note that

[TABLE]

Applying Lemma 4.4 to $y=pq_{k}p$, we obtain a subprojection $e_{k}$ of $p$ such that

[TABLE]

whence

[TABLE]

This completes the proof. ∎

The following Lemma 4.6 is prepared for Lemma 4.7.

Lemma 4.6**.**

Let $\mathcal{N}$, with $I_{\mathcal{M}}\in\mathcal{N}$, be a type ${\rm I}_{m}$ subfactor of $\mathcal{M}$ and $\{p_{ij}\}_{i,j=1}^{m}$ be a system of matrix units of $\mathcal{N}$. For any $x\in\mathcal{N}^{\prime}\cap\mathcal{M}$,

[TABLE]

Proof.

Notice that, for every $1\leq i\leq m$,

[TABLE]

Hence

[TABLE]

This ends the proof. ∎

Lemma 4.7**.**

Let $\{q_{k}\}^{\infty}_{k=1}\subseteq\mathcal{M}$ be a sequence of nonzero projections. Suppose that $\mathcal{N}$, with $I_{\mathcal{M}}\in\mathcal{N}\subseteq\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$, is a subfactor of type ${\rm I}_{m}$ for some positive integer $m$. Let $\{p_{ij}\}_{ij=1}^{m}$ be a system of matrix units of $\mathcal{N}$. If $p$ is a projection in $\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$ with $p\geq p_{11}$, then

[TABLE]

Proof.

Note that $\mathcal{N}$ is a subfactor of type ${\rm I}_{m}$. Let $\mathcal{U}$ be a finite group consisting of unitary elements in $\mathcal{N}$ such that $\mathcal{N}$ is a linear span of $\mathcal{U}$ (see Lemma 2.4.1 in [28]). Define

[TABLE]

where $|\mathcal{U}|$ is the cardinality of $\mathcal{U}$. It is easy to verify that $\tilde{q}_{k}$ is a positive element in $\mathcal{N}^{\prime}\cap\mathcal{M}$ for $k\geq 1$ such that

[TABLE]

whence

[TABLE]

Then

[TABLE]

Hence, $(\ref{may15Equ4.1})$ and $(\ref{may15Equ4.2})$ guarantee

[TABLE]

This completes the proof. ∎

Lemma 4.3, Lemma 4.5, and Lemma 4.7 are applied in the following Proposition 4.8.

Proposition 4.8**.**

Let $\{q_{k}\}^{\infty}_{k=1}$ be a sequence of nonzero projections in $\mathcal{M}$. Suppose that $\mathcal{N}$, with $I_{\mathcal{M}}\in\mathcal{N}\subseteq\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$, is a subfactor of type ${\rm I}_{m}$ for some positive integer $m$. Let $\{p_{ij}\}_{ij=1}^{m}$ be a system of matrix units of $\mathcal{N}$. If $p$ is a projection in $\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$ with $p\geq p_{11}$, then there exists a projection $e_{k}$ in $\mathcal{M}$ for each $k\geq 1$ such that

1. (i)

$e_{k}$* is a nonzero subprojection of $p$ when $k$ is large enough;* 2. (ii)

$\displaystyle\limsup_{k\rightarrow\infty}\frac{\|e_{k}\|_{2}}{\|q_{k}\|_{2}}\leq 1$; and 3. (iii)

$\displaystyle\lim_{k\rightarrow\infty}\frac{\|[pxp,e_{k}]\|_{2}}{\|e_{k}\|_{2}}=0,\qquad\text{ for all }x\in\mathcal{A}(\{q_{k}\}^{\infty}_{k=1}).$**

Proof.

By Lemma 4.5, there exists a subprojection $e_{k}$ of $p$ for each $k\geq 1$ such that

[TABLE]

Thus

[TABLE]

which means that (ii) holds. By Lemma 4.7,

[TABLE]

This means that $e_{k}$ is nonzero when $k$ is large enough. Thus, (i) is true. Moreover, for every $x\in\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$, from Lemma 4.3, the limit in (4.3), and the inequality in (4.4), it follows that

[TABLE]

This completes the proof. ∎

Now we are ready to prove the main result in this section. Recall that a type ${\rm II}_{1}$ factor $\mathcal{M}$ is non-$\Gamma$ if $\mathcal{M}$ is a type ${\rm II}_{1}$ factor without Property $\Gamma$. Meanwhile, by $\mathcal{P}(\mathcal{M})$ we denote the set of all the projections in $\mathcal{M}$.

Theorem 4.9**.**

Let $\mathcal{M}$ be a non-$\Gamma$ type ${\rm II}_{1}$ factor with trace $\tau$. Then there exist two self-adjoint elements $x_{1}$, $x_{2}$ in $\mathcal{M}$ and a positive number $\alpha>0$ such that

[TABLE]

Proof.

By Proposition 3.8, there exist two self-adjoint elements $x_{1}$ and $x_{2}$ in $\mathcal{M}$ such that $W^{*}(x_{1}+ix_{2})^{\prime}\cap\mathcal{M}^{\omega}=\mathbb{C}I$. By Proposition 3.10, there exist an $\epsilon>0$ and a nonzero projection $p$ in $\mathcal{M}$ such that for every nonzero subprojection $e$ of $p$,

[TABLE]

If $p=I$, we claim that there exists an $\alpha$ such that $x_{1}$, $x_{2}$ and $\alpha$ have the desired property stated in the theorem. Actually, assume, to the contrary, that such an $\alpha$ doesn’t exist. Thus for each $\alpha=1/k$, there exists a projection $q_{k}$ in $\mathcal{M}$ such that

[TABLE]

Obviously, $0<\tau(q_{k})<1$. Replacing $q_{k}$ by $I-q_{k}$ if needed, we assume that

[TABLE]

Notice that $W^{*}(x_{1}+ix_{2})^{\prime}\cap\mathcal{M}^{\omega}=\mathbb{C}I$. We have that

[TABLE]

Since we have assumed that $p=I$, we obtain that (4.6) and (4.7) contradict with (4.5). This ends the proof in this case.

Now we assume that $p\neq I$. Let $m\in\mathbb{N}$ be such that $2/m<\min\{\tau(p),\tau(I-p)\}$. Let $i_{0}$ be an integer such that $(i_{0}-1)/m\leq\tau(p)<i_{0}/m$. Then $3\leq i_{0}\leq m-2$. Assume that $p$ and $px_{1}p$ are contained in a masa $\mathcal{W}$ of $\mathcal{M}$. Let $\{p_{i}\}_{i=1}^{m}$ be a family of mutually orthogonal projections in $\mathcal{W}$ such that

1. (a)

$p_{1}+\cdots+p_{m}=I$ and $\tau(p_{i})=1/m$ for $1\leq i\leq m$; 2. (b)

$p_{1},p_{i},\ldots,p_{i_{0}-1}$ are subprojections of $p$; 3. (c)

$p_{i_{0}+1},\ldots,p_{m-1},p_{m}$ are subprojections of $I-p$.

Let $\mathcal{N}$ be a type ${\rm I}_{m}$ subfactor of $\mathcal{M}$ with a system of matrix units $\{p_{ij}\}_{i,j=1}^{m}$ such that $p_{ii}=p_{i}$ for $1\leq i\leq m$. Let $\mathcal{P}=\mathcal{N}^{\prime}\cap\mathcal{M}$. Then $\mathcal{M}\cong\mathcal{P}\otimes\mathcal{N}$. By Lemma 2.5, we obtain that $\mathcal{P}$ is also a type ${\rm II}_{1}$ factor without Property $\Gamma$. Proposition 3.8 implies there exist two self-adjoint elements $y_{1},y_{2}$ in $\mathcal{P}$ such that $W^{*}(y_{1},y_{2})^{\prime}\cap\mathcal{P}^{\omega}=\mathbb{C}I$. From Lemma 2.1, it follows that

[TABLE]

Define $p_{i_{0},1}=pp_{i_{0}}$ and $p_{i_{0},2}:=(I-p)p_{i_{0}}$.

Without loss of generality, we assume that $\|px_{1}p\|<1/10,\|y_{1}\|<1/10$ and $\|y_{2}\|<1/10$. Let $z_{1}$ and $z_{2}$ be elements in $\mathcal{M}$ defined as follows:

[TABLE]

Notice that

[TABLE]

By Lemma 2.6, we have that

[TABLE]

From $p_{i}z_{2}p_{m}$ and $pz_{2}p$, we obtain that $\{p_{1m},p_{2m},\ldots,p_{m{m}},px_{2}p\}\subseteq{\rm C}^{*}(z_{1},z_{2})$, whence

[TABLE]

Combining (4.8) and (4.10), we have that

[TABLE]

To finish the proof of the theorem, it suffices to show the following claim.

Claim 4.10**.**

There exists a constant $\alpha>0$ such that, for every projection $e$ in $\mathcal{M}$,

[TABLE]

Proof of the Claim.

Assume, to the contrary, that for each $\alpha=1/k$ there exists a projection $q_{k}$ in $\mathcal{M}$ such that

[TABLE]

Obviously, $0<\tau(q_{k})<1$. Replacing $q_{k}$ by $I-q_{k}$ if needed, we assume that $0<\tau(q_{k})\leq 1/2$. Then it follows that

[TABLE]

Notice from (4.11) that $W^{*}(z_{1}+iz_{2})^{\prime}\cap\mathcal{M}^{\omega}=\mathbb{C}I$. We have that

[TABLE]

Let $\mathcal{A}(\{q_{k}\}^{\infty}_{k=1})$ be as defined in Definition 4.1. From (4.10) and (4.12),

[TABLE]

Applying Proposition 4.8 to $\mathcal{N}$ and $p$, we can find a projection $e_{k}$ in $\mathcal{M}$ for each $k\geq 1$ such that

1. (i)

$e_{k}$ is a nonzero subprojection of $p$ when $k$ is large enough. 2. (ii)

$\displaystyle\limsup_{k\to\infty}\frac{\|e_{k}\|_{2}}{\|q_{k}\|_{2}}\leq 1$ and 3. (iii)

$\displaystyle\lim_{k\to\infty}\frac{\|[px_{i}p,e_{k}]\|_{2}}{\|e_{k}\|_{2}}=0,\qquad\text{ for all }i=1,2.$

Combining (4.13) and (ii), we have

[TABLE]

It is easy to check that (i), (4.14), and (iii) contradict (4.5). ∎

This ends the proof of the claim and the proof of the theorem. ∎

Combining with Marrakchi’s Proposition 2.2 in [19], Theorem 4.9 gives an operator with spectral gap in a non-$\Gamma$ II1 factor. The result could be compared with Theorem 2.1 (c) in [3].

Corollary 4.11**.**

Let $\mathcal{M}$ be a type ${\rm II}_{1}$ factor with trace $\tau$. Then the following statements are equivalent:

1. (i)

$\mathcal{M}$* is non-$\Gamma$, i.e. $\mathcal{M}$ doesn’t have Property $\Gamma$.* 2. (ii)

There exist two self-adjoint elements $x_{1}$ and $x_{2}$ in $\mathcal{M}$ and an $\alpha_{1}>0$ such that

[TABLE] 3. (iii)

There exist an $x$ in $\mathcal{M}$ and an $\alpha_{2}>0$ such that

[TABLE] 4. (iv)

There exist an $x$ in $\mathcal{M}$ and an $\alpha_{3}>0$ such that

[TABLE] 5. (v)

There exist two unitary elements $u_{1}$ and $u_{2}$ in $\mathcal{M}$ and an $\alpha_{4}>0$ such that

[TABLE]

Proof.

(i) $\Rightarrow$ (ii). Assume that $\mathcal{M}$ is non-$\Gamma$. It follows from Theorem 4.9 that there exist two self-adjoint elements $x_{1}$ and $x_{2}$ in $\mathcal{M}$ and a positive number $\alpha$ such that

[TABLE]

By Proposition 2.2 in [19], there exists a positive number $\alpha_{1}$ such that

[TABLE]

(ii) $\Rightarrow$ (i). It follows directly from the definition of Property $\Gamma$.

(ii) $\Leftrightarrow$ (iii). Let $x=x_{1}+ix_{2}$ where $x_{1},x_{2}$ are self-adjoint elements in $\mathcal{M}$. Then

[TABLE]

This means that the biconditional “(ii) $\Leftrightarrow$ (iii)” is obvious.

(iii) $\Rightarrow$ (iv). It is trivial.

(iv) $\Rightarrow$ (iii). Assume that (iv) is true. Let $y=y_{1}+iy_{2}$ be in $\mathcal{M}$ where $y_{1},y_{1}$ are self-adjoint. Without loss of generality, we assume $\tau(y)=0$, thus $\tau(y_{1})=\tau(y_{2})=0$. Then

[TABLE]

I.e. (iii) is true.

(ii) $\Rightarrow$ (v). Assume that (ii) is true. Let $\lambda>\max\{\|x_{1}\|,\|x_{2}\|\}$ and

[TABLE]

Then $u_{1}$, $u_{2}$ are unitary elements in $\mathcal{M}$ such that $u_{i}^{*}+u_{i}=2x_{i}/\lambda$ for $i=1,2$. Hence

[TABLE]

Thus, (v) is true.

(v) $\Rightarrow$ (iv). Assume that (v) is true. From Theorem 5.2.5 of [14], we have $u_{1}=e^{ix_{1}}$ and $u_{2}=e^{ix_{2}}$ for some positive elements $x_{1},x_{2}$ in $\mathcal{M}$. We show that there exists an $\alpha^{\prime}>0$ such that

[TABLE]

Assume, to the contrary, that for any $\alpha^{\prime}=1/k$ there exists a self-adjoint element $y_{k}$ such that (1) $\tau(y_{k})=0$; (2) $\|y_{k}\|_{2}=1$; and (3) $\|[x_{1}+ix_{2},y_{k}]\|_{2}<1/k$. Define

[TABLE]

Similar to Lemma 4.2, we obtain that $\mathcal{B}(\{y_{k}\}^{\infty}_{k=1})$ is a unital ${\rm C}^{*}$-algebra containing $x_{1}+ix_{2}$. Thus $u_{1},u_{2}\in\mathcal{B}(\{y_{k}\}^{\infty}_{k=1})$, which contradicts Assumption (v). ∎

5. Reducible operators in non-$\Gamma$ type ${\rm II}_{1}$ factors

Let $\mathcal{M}$ be a type ${\rm II}_{1}$ factor with trace $\tau$. Recall that, for an element $x$ in $\mathcal{M}$, the von Neumann subalgebra generated by $x$ in $\mathcal{M}$ is denoted by $W^{*}(x)$.

Definition 5.1**.**

An element $x\in\mathcal{M}$ is reducible if there is a nontrivial projection $p\in\mathcal{M}$ such that $xp=px$, equivalently, $W^{*}(x)^{\prime}\cap\mathcal{M}\neq\mathbb{C}I$. If an element $x\in\mathcal{M}$ is not reducible, then $x$ is irreducible, equivalently, $W^{*}(x)^{\prime}\cap\mathcal{M}=\mathbb{C}I$. The set of all the reducible operators in $\mathcal{M}$ is denoted by $\operatorname{Red}(\mathcal{M})$ and the operator-norm closure of $\operatorname{Red}(\mathcal{M})$ is denoted by $\operatorname{Red}(\mathcal{M})^{\|\cdot\|}$.

Proposition 5.2**.**

Suppose that $\mathcal{M}$ is a type ${\rm II}_{1}$ factor with separable predual. Then $\operatorname{Red}(\mathcal{M})$ is not operator norm closed in $\mathcal{M}$. In particular, $\operatorname{Red}(\mathcal{M})\neq\mathcal{M}$.

Proof.

As a special case of Corollary 4.1 of [22], there exists an irreducible, hyperfinite subfactor $\mathcal{R}$ of $\mathcal{M}$, i.e., $\mathcal{R}^{\prime}\cap\mathcal{M}=\mathbb{C}I$.

Notice that there exists a unital CAR subalgebra $\mathcal{A}$ of $\mathcal{R}$ such that $\mathcal{R}$ is the ${\rm weak}^{*}$-closure of $\mathcal{A}$. The reader is referred to Example III.2.4 of [6] for the definition of CAR algebras. Thus there exists an increasing sequence $\{\mathcal{A}_{n}\}^{\infty}_{n=1}$ of full matrix algebras such that

1. (1)

$\mathcal{A}_{n}$ is $\ast$-isomorphic to $M_{2^{n}}(\mathbb{C})$ for each $n\geq 1$; 2. (2)

$\cup^{\infty}_{n=1}\mathcal{A}_{n}$ is dense in $\mathcal{A}$ in the operator norm.

In terms of Theorem of [29], there exists a single generator $a\in\mathcal{A}$. It follows that $W^{*}(a)=\mathcal{R}$. This entails that $a$ is irreducible in $\mathcal{M}$, or $a\notin\operatorname{Red}(\mathcal{M})$. The fact $\mathcal{A}$ is a CAR algebra implies that there exists a sequence $\{a_{n}\}^{\infty}_{n=1}$ of operators in $\mathcal{A}$ with $a_{n}\in\mathcal{A}_{n}$ for each $n\geq 1$ such that

[TABLE]

That $a_{n}\in\mathcal{A}_{n}$ entails that $a_{n}$ is reducible in $\mathcal{M}$ for every $n\geq 1$. Hence $a\in\operatorname{Red}(\mathcal{M})^{\|\cdot\|}$. Thus $\operatorname{Red}(\mathcal{M})$ is not closed in $\mathcal{M}$ in the operator norm and $\operatorname{Red}(\mathcal{M})\neq\mathcal{M}$. ∎

Suppose $\mathcal{N}$ is a separable type II1 factor with Property $\Gamma$. It is straight forward to see that $\operatorname{Red}(\mathcal{N}^{\omega})=\mathcal{N}^{\omega}$. In fact, if $(x_{n})_{\omega}\in\mathcal{N}^{\omega}$, then there exists a sequence $\{q_{n}\}_{n=1}^{\infty}$ of projections in $\mathcal{N}$ with $\tau(q_{n})=1/2$ and $\|[x_{n},q_{n}]\|_{2}\leq 1/n$ for each $n\geq 1$. Then $(q_{n})_{\omega}$ is a nontrivial projection in $\mathcal{N}^{\omega}$ that commutes with $(x_{n})_{\omega}$. Hence, $\operatorname{Red}(\mathcal{N}^{\omega})=\mathcal{N}^{\omega}$. Here we present another type of examples of nonseparable type ${\rm II}_{1}$ factors $\mathcal{M}$ such that $\operatorname{Red}(\mathcal{M})=\mathcal{M}$.

Example 5.3**.**

For each $\alpha\in\mathbb{R}$, let $G_{\alpha}$ be a free group on two generators. Define a group $G$ in the following form

[TABLE]

to be the direct sum of $\{G_{\alpha}\}_{\alpha\in\mathbb{R}}.$ Then $G$ is a discrete uncountable ICC (infinite conjugacy class) group. Define $l^{2}(G)$ to be a Hilbert space associated with $G$ in the form

[TABLE]

and let $L(G)$ be the corresponding group von Neumann algebra, acting on $l^{2}(G)$, associated with $G$. It is easy to observe that $L(G)$ is a type ${\rm II}_{1}$ factor. Naturally, we can view $L(G)\subseteq l^{2}(G)$ and $L(H)\subseteq L(G)$ for any subgroup $H$ of $G$ (see Section $6.7$ in [15]).

Suppose that $a$ is an element in $L(G)$. From $a\in l^{2}(G)$, it follows that there is a countable subgroup $G^{\prime}$ of $G$ such that $a\in L(G^{\prime})$, i.e. $a$ is supported on $G^{\prime}$. Recall that $G$ is a direct sum of uncountably many subgroups $G_{\alpha}$ where $\alpha\in\mathbb{R}$. Thus there is some $\beta\in\mathbb{R}$ such that $G^{\prime}$ and $G_{\beta}$ commute with each other, whence $a$ commutes with $L(G_{\beta})$. Note that each $L(G_{\beta})$ is a free group factor on two generators. Hence $a\in\operatorname{Red}(L(G))$, which implies that $\operatorname{Red}(L(G))=L(G)$.

Remark 5.4**.**

It is worthwhile pointing out that $L(G)$ in Example 5.3 has Property $\Gamma$ of Murray and von Neumann. In the forthcoming Theorem 5.11, we will prove that if $\mathcal{M}$ is a non-$\Gamma$ type ${\rm II}_{1}$ factor, then $\operatorname{Red}(\mathcal{M})^{\|\cdot\|}\neq\mathcal{M}$.

In Proposition 5.2, we have seen that the set of reducible operators in a separable type II1 factors is not operator norm closed. Example 5.3 gives us an example of a non-separable type II1 factor $\mathcal{M}$ with Property $\Gamma$ and with $\operatorname{Red}(\mathcal{M})=\mathcal{M}$. In the next result, we will show that, in a (separable or nonseparable) non-$\Gamma$ type II1 factor, the set of reducible operators is always not closed in the operator norm topology.

Proposition 5.5**.**

The following statements are true:

1. (i)

Let $\mathcal{N}$ be a separable type ${\rm II}_{1}$ factor. If $x$ is an element in $\mathcal{N}$ satisfying $W^{*}(x)=\mathcal{N}$, then there exists an element $y$ in $\mathcal{N}$ such that $W^{*}(y)=\mathcal{N}$ and $y\in\operatorname{Red}(\mathcal{N})^{\|\cdot\|}$. 2. (ii)

Let $\mathcal{M}$ be a non-$\Gamma$* type ${\rm II}_{1}$ factor. Then $\operatorname{Red}(\mathcal{M})$ is not operator norm closed in $\mathcal{M}$.*

Proof.

(i). Assume that $x=x_{1}+ix_{2}$ for some self-adjoint elements $x_{1},x_{2}$ in $\mathcal{N}$. Suppose that $x_{1}$ is contained in a masa $\mathcal{A}$ in $\mathcal{N}$. Assume that $y_{1}$ is a self-adjoint generator of $\mathcal{A}$. Let $\{p_{n}\}_{n=1}^{\infty}$ be an increasing sequence of nontrivial projections in $\mathcal{A}$ such that $\lim_{n\rightarrow\infty}\|I-p_{n}\|_{2}=0$. Define

[TABLE]

Thus $y_{2}$ is a self-adjoint element in $\mathcal{N}$ such that $\lim_{m\rightarrow\infty}\|y_{2}-z_{m}\|=0$. From the fact that $z_{m}p_{m}=z_{m}=p_{m}z_{m}$, it follows that $y_{1}+iz_{m}\in\operatorname{Red}(\mathcal{N})$, whence $y_{1}+iy_{2}\in\operatorname{Red}(\mathcal{N})^{\|\cdot\|}$.

Define $y=y_{1}+iy_{2}$. We next show that $W^{*}(y)=\mathcal{N}$. In fact, by the choice of $y_{1}$, we have that $x_{1}$ and $\{p_{n}\}_{n=1}^{\infty}$ are in $\mathcal{A}\subseteq W^{*}(y)$. By the construction of $y_{2}$, we have

[TABLE]

and

[TABLE]

Therefore we obtain that $p_{m}x_{2}p_{m}\in W^{*}(y)$ for $m\geq 1$. This implies that $x_{2}\in W^{*}(y),$ as $\lim_{m\rightarrow\infty}\|I-p_{m}\|_{2}=0$. It follows that $W^{*}(y)=\mathcal{N}$.

(ii). By Proposition 3.8, there exists an element $x$ in $\mathcal{M}$ such that $W^{*}(x)^{\prime}\cap\mathcal{M}^{\omega}=\mathbb{C}I$, so $W^{*}(x)^{\prime}\cap\mathcal{M}=\mathbb{C}I$. Define $\mathcal{N}=W^{*}(x)$. Then $\mathcal{N}$ is an irreducible subfactor of $\mathcal{M}$. By part (i), there exists an operator $y$ in $\mathcal{N}$ such that $W^{*}(y)=\mathcal{N}$ and $y\in\operatorname{Red}(\mathcal{N})^{\|\cdot\|}$. It follows that $y$ is an irreducible operator in $\mathcal{M}$ with $y\in\operatorname{Red}(\mathcal{M})^{\|\cdot\|}$. This finishes the proof of (ii). ∎

Recall that

[TABLE]

and

[TABLE]

Then $c_{0}(\mathcal{M})$ is a norm closed two sided ideal of $l^{\infty}(\mathcal{M})$ and

[TABLE]

is also a unital ${\rm C}^{*}$-algebra. An element in $l^{\infty}(\mathcal{M})/c_{0}(\mathcal{M})$ is denoted by $[(a_{n})_{n}]$, if no confusion arises. Moreover, there is a natural embedding from $\mathcal{M}$ into $l^{\infty}(\mathcal{M})/c_{0}(\mathcal{M})$ by sending $a$ in $\mathcal{M}$ to $[(a)_{n}]$ in $l^{\infty}(\mathcal{M})/c_{0}(\mathcal{M})$. So we view $\mathcal{M}\subseteq l^{\infty}(\mathcal{M})/c_{0}(\mathcal{M})$.

We need a well-known technical lemma for ${\rm C}^{\ast}$-algebras in the following. The proof is skipped.

Lemma 5.6**.**

Let $x$ be a self-adjoint element in $\mathcal{M}$ such that $\|x-x^{2}\|<1/8$. Then there is a projection $p\in{\rm C}^{*}(x)$ such that $\|x-p\|\leq\sqrt{\|x-x^{2}\|}$.

Proposition 5.7**.**

Let $x$ be an element in a type ${\rm II}_{1}$ factor $\mathcal{M}$. The following statements are equivalent:

1. (i)

$x\in\operatorname{Red}(\mathcal{M})^{\|\cdot\|}$. 2. (ii)

There exists a sequence $\{p_{n}\}_{n=1}^{\infty}$ of projections in $\mathcal{M}$ such that

[TABLE] 3. (iii)

There exists a sequence $\{p_{n}\}_{n=1}^{\infty}$ of projections in $\mathcal{M}$ such that

[TABLE] 4. (iv)

${\rm C}^{*}(x)^{\prime}\cap\left(l^{\infty}(\mathcal{M})/c_{0}(\mathcal{M})\right)$* contains a non-trivial projection.*

Proof.

(i) $\Rightarrow$ (ii). Suppose that $x\in\operatorname{Red}(\mathcal{M})^{\|\cdot\|}$. Then there exists a sequence $\{x_{n}\}^{\infty}_{n=1}$ of operators in $\operatorname{Red}(\mathcal{M})$ such that $\lim_{n\to\infty}\|x_{n}-x\|=0$. Thus, for each reducible operator $x_{n}$ in $\mathcal{M}$ there exists a nontrivial projection $p_{n}$ in $\mathcal{M}$ such that $p_{n}x_{n}=x_{n}p_{n}$ for every $n\in\mathbb{N}$. Replacing $p_{n}$ by $I-p_{n}$ if $\tau(p_{n})>1/2$, we assume that $0<\tau(p_{n})\leq 1/2$. Note that

[TABLE]

This completes the proof of the implication (i) $\Rightarrow$ (ii).

(ii) $\Rightarrow$ (i). Assume that $\{p_{n}\}^{\infty}_{n=1}$ is a sequence of nontrivial projections in $\mathcal{M}$ such that $\lim_{n\to\infty}\|[x,p_{n}]\|=0$. Define $x_{n}=p_{n}xp_{n}+(I-p_{n})x(I-p_{n})$. Since $p_{n}$ is nontrivial, we obtain that $x_{n}$ is reducible in $\mathcal{M}$. Thus $\|x_{n}-x\|=\|xp_{n}-p_{n}x\|$. This completes the proof of the implication (ii) $\Rightarrow$ (i).

(iii) $\Leftrightarrow$ (ii). Note that the implication of (iii) $\Rightarrow$ (ii) is trivial. Assume that (ii) holds. Define $\mathcal{A}$ to be a set in the following form:

[TABLE]

It is routine to verify that $\mathcal{A}$ is a unital ${\rm C}^{\ast}$-algebra containing $x$. It follows that ${\rm C}^{\ast}(x)\subseteq\mathcal{A}$. This completes the proof of the implication (ii) $\Rightarrow$ (iii).

The implication (iii) $\Rightarrow$ (iv) is trivial. The implication (iv) $\Rightarrow$ (i) follows from Lemma 5.6. This ends the proof. ∎

The following lemmas are useful.

Lemma 5.8**.**

Let $x_{1}$ and $x_{2}$ be self-adjoint elements in $\mathcal{M}$. If there exist a positive number $\alpha>0$ and a projection $p\in\mathcal{M}$ with $\tau(p)>0$, satisfying

[TABLE]

then

[TABLE]

Proof.

By the definition of the operator norm, we have

[TABLE]

Since the equality $\|(I-p)x_{i}p\|^{2}_{2}=\|px_{i}(I-p)\|^{2}_{2}$ holds for $i=1,2$, it follows that

[TABLE]

Inequality (5.2) and equality (5.3) entail that

[TABLE]

Inequalities (5.1) and (5.4) guarantee that

[TABLE]

This completes the proof. ∎

Lemma 5.9**.**

Let $u_{1}$ and $u_{2}$ be unitary elements in $\mathcal{M}$. If there exist a positive number $\alpha>0$ and a projection $p\in\mathcal{M}$ with $\tau(p)>0$, satisfying

[TABLE]

then

[TABLE]

Proof.

By the definition of the operator norm, we have that

[TABLE]

Note that, for $i=1,2$, the equality

[TABLE]

yields that

[TABLE]

It follows that

[TABLE]

Inequality (5.6) and equality (5.7) entail that

[TABLE]

Inequalities (5.5) and (5.8) guarantee that

[TABLE]

This completes the proof. ∎

In next example, we construct an operator in a free group factor such that it is not in the operator norm closure of reducible ones.

Example 5.10**.**

Let $\mathbb{F}_{2}$ be a free group on two generators and $L(\mathbb{F}_{2})$ the free group factor associated to $\mathbb{F}_{2}$. Denote by $u_{1}$ and $u_{2}$ the unitary operators in $L(\mathbb{F}_{2})$ corresponding to the two generators of $\mathbb{F}_{2}$.

For every element $y$ in $L(\mathbb{F}_{2})$, from Theorem $6.7.8$ in [15] it follows that

[TABLE]

Observe that if $p$ is a projection in $L(\mathbb{F}_{2})$ with $0<\tau(p)\leq 1/2$, then

[TABLE]

Combining it with Inequality (\ref{u_1&u_2}) and Lemma 5.9, for a nontrivial projection $p$ in $L(\mathbb{F}_{2})$ with $0<\tau(p)\leq 1/2$, it follows that

[TABLE]

In the following, we construct an operator $x$ in $L(\mathbb{F}_{2})$ such that $x$ is in $L(\mathbb{F}_{2})\backslash\operatorname{Red}(L(\mathbb{F}_{2}))^{\|\cdot\|}$. By virtue of Theorem $5.2.5$ of [14], there exist two positive operators $a_{1}$ and $a_{2}$ in $L(\mathbb{F}_{2})$ such that

[TABLE]

We claim that, for $x=a_{1}+ia_{2}$, there exists an $\epsilon_{0}>0$ such that

[TABLE]

On the contrary, suppose that there exists a sequence of projections $\{p_{n}\}^{\infty}_{n=1}$ in $L(\mathbb{F}_{2})$ such that the inequalities

[TABLE]

hold for each integer $n\geq 1$. Define a subset of $L(\mathbb{F}_{2})$ as follows:

[TABLE]

A calculation shows that $\mathcal{A}$ is a unital ${\rm C}^{\ast}$-subalgebra of $L(\mathbb{F}_{2})$. Since $x\in\mathcal{A}$, we have $u_{k}\in\mathcal{A}$ for $k=1,2$. This contradicts the inequality in (\ref{u_i&p-5}). This ends the proof of (\ref{u_i&p-6}).

More generally, in a non-$\Gamma$ type II1 factor, there always exist operators not in the operator norm closure of reducible ones.

Theorem 5.11**.**

Let $\mathcal{M}$ be a non-$\Gamma$ type ${\rm II}_{1}$ factor. Then $\operatorname{Red}(\mathcal{M})^{\|\cdot\|}\neq\mathcal{M}$.

Proof.

It follows from Theorem 4.9 that there exist two self-adjoint elements $x_{1}$ and $x_{2}$ in $\mathcal{M}$ and a positive number $\alpha$ such that

[TABLE]

By Lemma 5.8,

[TABLE]

From Proposition 5.7, it follows that $x_{1}+ix_{2}\notin\operatorname{Red}(\mathcal{M})^{\|\cdot\|}$, i.e. $\operatorname{Red}(\mathcal{M})^{\|\cdot\|}\neq\mathcal{M}$. ∎

6. Nowhere-dense-property of reducible operators in non-$\Gamma$ type ${\rm II}_{1}$ factors

The goal of this section is to prove that the set of reducible operators in each non-$\Gamma$ type ${\rm II}_{1}$ factor is nowhere dense, in the operator norm topology. For this purpose, we introduce the following definition.

Definition 6.1**.**

Let $\mathcal{B}$ be a unital ${\rm C}^{*}$-algebra with an identity $I_{\mathcal{B}}$, and let $\mathcal{A}\subseteq\mathcal{B}$ be a ${\rm C}^{*}$-subalgebra containing $I_{\mathcal{B}}$. Then an element $x\in\mathcal{A}$ is called reducible in $\mathcal{B}$ if there exists a nontrivial projection $p$ in $\mathcal{B}$, i.e., $p\neq 0$ or $I_{\mathcal{B}}$, such that $xp=px$.

Define $\operatorname{Red}(\mathcal{A}:\mathcal{B})$ to be the set of all these elements of $\mathcal{A}$ that are reducible in $\mathcal{B}$, i.e.

[TABLE]

Remark 6.2**.**

Let $\mathcal{M}$ be a type ${\rm II}_{1}$ factor. If $\mathcal{A}=\mathcal{B}=\mathcal{M}$, then $\operatorname{Red}(\mathcal{M}:\mathcal{M})=\operatorname{Red}(\mathcal{M})$, where $\operatorname{Red}(\mathcal{M})$ is introduced in Definition 5.1.

Remark 6.3**.**

Let $\mathcal{H}$ be an infinite dimensional, complex, separable Hilbert space and $\mathcal{B}(\mathcal{H})$ the algebra of all the bounded linear operators on $\mathcal{H}$. Then, as an application of Voiculescu’s non-commutative Weyl-von Neumann theorem in [30], it is well-known that

[TABLE]

Remark 6.4**.**

Let $\mathcal{M}$ be a type ${\rm II}_{1}$ factor and $\omega$ a free ultrafilter on $\mathbb{N}$. Then, by Proposition 3.8, $\mathcal{M}$ has Property $\Gamma$ if and only if

[TABLE]

Proposition 6.5**.**

Let $\mathcal{B}$ be a unital ${\rm C}^{*}$-algebra with an identity $I_{\mathcal{B}}$, and let $\mathcal{M}$ be a factor of type ${\rm II}_{1}$ such that $I_{\mathcal{B}}\in\mathcal{M}\subseteq\mathcal{B}$. If $p\mathcal{M}p\setminus\operatorname{Red}(p\mathcal{M}p:p\mathcal{B}p)\neq\emptyset$ for any nonzero projection $p$ in $\mathcal{M}$, then $\mathcal{M}\setminus\operatorname{Red}(\mathcal{M}:\mathcal{B})$ is, in the operator norm topology, dense in $\mathcal{M}$.

Proof.

Suppose that $x=x_{1}+ix_{2}$ is an element in $\mathcal{M}$, where $x_{1}$ and $x_{2}$ are two self-adjoint operators in $\mathcal{M}$. Let $\epsilon$ be a positive number.

By spectral theory for $x_{1}$, there exist an $m\in\mathbb{N}$, an orthogonal family of projections $\{q_{k}\}^{m}_{k=1}$ in $\mathcal{M}$ with sum $I$ and a family of real numbers $\{\lambda_{k}\}^{m}_{k=1}$ such that

[TABLE]

For each $1\leq k\leq m$, by spectral theory for $q_{k}x_{2}q_{k}$, there exist an $n_{k}\in\mathbb{N}$, an orthogonal family of subprojections $\{q_{k,j}\}^{n_{k}}_{j=1}$ of $q_{k}$ with sum $q_{k}$ and a family of real numbers $\{\eta_{k,j}\}^{n_{k}}_{j=1}$ such that $\|q_{k}x_{2}q_{k}-\sum\nolimits_{j=1}^{n_{k}}\eta_{k,j}q_{k,j}\|\leq\epsilon/16,$ in particular

[TABLE]

Let $n=n_{1}+\cdots+n_{m}$ and list $\{q_{k,j}:1\leq j\leq n_{k},1\leq k\leq m\}$ as $\{p_{k}\}_{k=1}^{n}$. By the inequalities (6.1) and (6.2), with a small perturbation, we can assume that there exist families of distinct real numbers $\{\alpha_{k}\}^{n}_{k=1}$ and $\{\beta_{k}\}^{n}_{k=1}$ such that, for $1\leq k\leq n$,

1. (a)

$\alpha_{k}\neq 0$ ; 2. (b)

$\|x_{1}-\sum_{k}\alpha_{k}p_{k}\|\leq\epsilon/8;$ 3. (c)

$\|p_{k}x_{2}p_{k}-\beta_{k}p_{k}\|\leq{\epsilon}/{8}.$

In terms of the condition that $p\mathcal{M}p\setminus\operatorname{Red}(p\mathcal{M}p:p\mathcal{B}p)\neq\emptyset$, there exists a family of elements $\{a_{k}+ib_{k}\in p_{k}\mathcal{M}p_{k}\backslash\operatorname{Red}(p_{k}\mathcal{M}p_{k}:p_{k}\mathcal{B}p_{k})\}^{n}_{k=1}$ with $a_{k}$ and $b_{k}$ being self-adjoint such that

1. (d)

$\|a_{k}\|\leq{\epsilon}/{8}\ \mbox{ and }\ \|b_{k}\|\leq{\epsilon}/{8},$ for $1\leq k\leq n$; 2. (e)

for $1\leq j\neq k\leq n$,

[TABLE]

Note that $\mathcal{M}$ is a type ${\rm II}_{1}$ factor. For each $1\leq k\leq n-1$, since $p_{k}\mathcal{M}p_{k+1}\neq\{0\}$, we let $z_{k}$ be an element in $p_{k}\mathcal{M}p_{k+1}$ such that

1. (f)

$\|z_{k}\|\leq\epsilon/(8n)$; 2. (g)

$z_{k}+p_{k}x_{2}p_{k+1}\neq 0.$

Define self-adjoint operators $y_{1}$ and $y_{2}$ in $\mathcal{M}$ in the following form

[TABLE]

From (b), (c), (d) and (f) , it follows that $\|(x_{1}+ix_{2})-(y_{1}+iy_{2})\|\leq\epsilon$.

To complete the proof, we need only to show that $y_{1}+iy_{2}$ belongs to $\mathcal{M}\setminus\operatorname{Red}(\mathcal{M}:\mathcal{B})$. Suppose that $q$ is a projection in $\mathcal{B}$ such that $[q,y_{1}+iy_{2}]=0$. To prove $y_{1}+iy_{2}\in\mathcal{M}\setminus\operatorname{Red}(\mathcal{M}:\mathcal{B})$, it suffices to prove that $q$ is trivial.

Now (e) and Lemma 2.6 entail that $p_{1},\ldots,p_{n},a_{1},\ldots,a_{n}\text{ are in }{\rm C}^{\ast}(y_{1},).$ By computing $p_{k}y_{2}p_{k}$ for $1\leq k\leq n$, we obtain that $b_{1},\ldots,b_{n}\text{ are in }{\rm C}^{\ast}(y_{1},y_{2}).$ From $[q,y_{1}+iy_{2}]=0$, it follows that $q=q_{1}+\cdots+q_{n}$ and $q_{k}a_{k}=a_{k}q_{k}$, where $q_{k}=p_{k}q$ is a sub-projection of $p_{k}$ for $1\leq k\leq n$. That $[q,y_{1}+iy_{2}]=0$ also implies that $q_{k}b_{k}=b_{k}q_{k}$ for $1\leq k\leq n$. Therefore, $[a_{k}+ib_{k},q_{k}]=0$.

Since $a_{k}+ib_{k}\in p_{k}\mathcal{M}p_{k}\backslash\operatorname{Red}(p_{k}\mathcal{M}p_{k}:p_{k}\mathcal{B}p_{k})$, the equality $[a_{k}+ib_{k},q_{k}]=0$ implies that $q_{k}=0$ or $q_{k}=p_{k}$. For $1\leq k\leq n-1$, notice that $p_{k}y_{2}p_{k+1}\in{\rm C}^{\ast}(y_{1},y_{2})$. It follows that

[TABLE]

Now the inequality (g)

[TABLE]

ensures either $q_{1}=\cdots=q_{n}=0$ or $q_{k}=p_{k}$ for each $k=1,\dots,n$. Therefore $q=0$ or $q=I_{\mathcal{B}}$. ∎

Theorem 6.6**.**

Let $\mathcal{M}$ be a non-$\Gamma$ type ${\rm II}_{1}$ factor. Then, in the operator norm topology, the set of reducible operators in $\mathcal{M}$ is nowhere dense and not closed in $\mathcal{M}$.

Proof.

It has been shown in Proposition 5.5 that, in the operator norm topology, $\operatorname{Red}(\mathcal{M})$ is not closed in $\mathcal{M}$. We prove next that $\operatorname{Red}(\mathcal{M})$ is nowhere dense in $\mathcal{M}$.

Let $\mathcal{B}=l^{\infty}(\mathcal{M})/c_{0}(\mathcal{M})$. From Proposition 5.7, we have

[TABLE]

and, for any nonzero projection $p$ in $\mathcal{M}$,

[TABLE]

Lemma 2.5 implies that $p\mathcal{M}p$ doesn’t have Property $\Gamma$ for each nonzero projection $p$ in $\mathcal{M}$. From Theorem 5.11, we have that $\operatorname{Red}(p\mathcal{M}p)^{\|\cdot\|}\neq p\mathcal{M}p$. This is equivalent to say that

[TABLE]

From Proposition 6.5, we obtain that $\mathcal{M}\setminus\operatorname{Red}(\mathcal{M}:\mathcal{B})$ is dense in $\mathcal{M}$ in the operator norm topology. So $\mathcal{M}\setminus\operatorname{Red}(\mathcal{M})^{\|\cdot\|}$ is dense in $\mathcal{M}$ in the operator norm topology. This guarantees that $\operatorname{Red}(\mathcal{M})$ is a nowhere dense subset of $\mathcal{M}$ in the operator norm topology. ∎