# Ring isomorphisms of type II∞ locally measurable operator algebras

Michiya Mori  Graduate School of Mathematical Sciences, The University of
Tokyo, 3-8-1 Komaba Meguro-ku Tokyo, 153-8914, Japan; Interdisciplinary
Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, 2-1 Hirosawa
Wako Saitama, 351-0198, Japan<EMAIL_ADDRESS>
###### Abstract.

We show that every ring isomorphism between the algebras of locally measurable
operators for type II∞ von Neumann algebras is similar to a real
∗-isomorphism. This together with previous results by the author and
Ayupov–Kudaybergenov completely describes ring isomorphisms between the
algebras of locally measurable operators as well as lattice isomorphisms
between the projection lattices for a general pair of von Neumann algebras
without finite type I direct summands.

###### Key words and phrases:

ring isomorphism; locally measurable operator; projection lattice; lattice
isomorphism; von Neumann algebra

###### 2020 Mathematics Subject Classification:

Primary 46L10, Secondary 16E50, 47B49, 51D25.

The author was supported by RIKEN Special Postdoctoral Researcher Program and
JSPS KAKENHI Grant Number 22K13934.

##  1\. Introduction

For a von Neumann algebra $M$, let ${\mathcal{P}}(M)$ denote the collection of
all projections of $M$, i.e., ${\mathcal{P}}(M):=\\{p\in M\mid
p=p^{*}=p^{2}\\}$. The importance of this collection in the investigation of
the structure of a von Neumann algebra was first recognized by Murray and von
Neumann in [12] in the 1930s. Recall that ${\mathcal{P}}(M)$ forms a lattice
with respect to the partial order relation $p\leq q$ $\Leftrightarrow$ $pq=p$.
This is why ${\mathcal{P}}(M)$ is called the _projection lattice_ of $M$. When
$M$ is commutative, then $M$ can be identified with the $L_{\infty}$-space for
some measure space. In this case, ${\mathcal{P}}(M)$ is identified with the
collection of measurable sets modulo null sets. If we regard a general von
Neumann algebra $M$ as a noncommutative measure space, then we may say that
${\mathcal{P}}(M)$ is the lattice of, so to speak, noncommutative measurable
sets. In this paper, we consider the following fairly natural question.

######  Question.

What is the general form of lattice isomorphisms between the projection
lattices of a pair of von Neumann algebras?

Note that we only need to consider Question in the case where the two von
Neumann algebras are of the same type (namely, one of In, I∞, II1, II∞ or
III), see [11, Lemma 4.1]. In the finite-dimensional case, the fundamental
theorem of projective geometry gives a partial answer to Question. It was von
Neumann himself who was the first to consider Question for non-type I von
Neumann algebras. He proved, as a byproduct of his theory on complemented
modular lattices and von Neumann regular rings, that lattice isomorphisms
between the projection lattices of a pair of type II1 factors are in one-to-
one correspondence with ring isomorphisms between the algebras of affiliated
operators, see [13, Chapter 2.4] and Appendix 2 of [13, Chapter 2.2].

Feldman [6] and Dye [4] studied lattice isomorphisms of projection lattices
under the orthogonality preserving assumption. Fillmore and Longstaff studied
lattice isomorphisms for the lattice of closed subspaces of a complex normed
space [7], which particularly answers our Question in the case of type I∞
factors. In [11], the author studied lattice isomorphisms in the general
setting of von Neumann algebras, and gave the following result which extends
von Neumann’s result for type II1 factors.

######  Theorem 1.1 ([11, Theorem A]).

Let $M$ and $N$ be two von Neumann algebras. Suppose that $M$ does not have
type I1 nor I2 direct summands, and that
$\Phi\colon{\mathcal{P}}(M)\to{\mathcal{P}}(N)$ is a lattice isomorphism. Then
there exists a unique ring isomorphism $\Psi\colon LS(M)\to LS(N)$ such that
$\Phi(l(x))=l(\Psi(x))$ for all $x\in LS(M)$.

Here $LS(M)$ is the collection of locally measurable operators for a von
Neumann algebra $M$, and $l(x)$ means the left support projection for $x\in
LS(M)$. The detailed definition of undefined terms will be given in Section 2.
Note that the converse of this theorem is easy to prove. Namely, if
$\Psi\colon LS(M)\to LS(N)$ is a ring isomorphism, then the formula
$\Phi(l(x))=l(\Psi(x))$, $x\in LS(M)$, uniquely determines a lattice
isomorphism $\Phi\colon{\mathcal{P}}(M)\to{\mathcal{P}}(N)$ [11, Proposition
3.1]. Therefore, our Question can be answered once we get the description of
ring isomorphisms of locally measurable operator algebras.

The author further studied such ring isomorphisms in [11] in several cases.
Below we consider a ring isomorphism $\Psi\colon LS(M)\to LS(N)$ under the
assumption that $M$ and $N$ are of the same type.

If $M$ and $N$ are of type In ($1\leq n<\infty$), the description can be done
utilizing ring isomorphisms between their centers [11, Proposition 4.2] (see
also [1]). The center of $LS(M)$ is in fact identified with the algebra of all
complex-valued measurable functions on a measure space (modulo the almost
everywhere equivalence relation). Getting a further concrete description of
the general form of the latter ring isomorphisms seems to be out of the reach
of the theory of operator algebras (see the remark right after [11,
Proposition 4.2]).

The author proved that $\Psi\colon LS(M)\to LS(N)$ is similar to a real
∗-isomorphism whenever $M$ and $N$ are type I∞ or III von Neumann algebras
[11, Theorem B]. See [11] for a more detailed overview of the above results
and the history of related topics. In the same paper, the author left the case
of type II as an open question, conjecturing that every ring isomorphism
$\Psi\colon LS(M)\to LS(N)$ is similar to a real ∗-isomorphism when $M$ and
$N$ are of type II [11, Conjecture 5.1]. This conjecture was soon confirmed by
Ayupov and Kudaybergenov in the case of type II1 von Neumann algebras [2,
Theorem 1.4] .

The purpose of this article is to fill in the still blank case of type II∞.
Namely, the main theorem is as follows.

######  Theorem 1.2.

Let $M,N$ be von Neumann algebras of type II∞. If $\Psi\colon LS(M)\to LS(N)$
is a ring isomorphism, then there exist a real ∗-isomorphism $\psi\colon M\to
N$ (which extends to a real ∗-isomorphism from $LS(M)$ onto $LS(N)$) and an
invertible element $y\in LS(N)$ such that $\Psi(x)=y\psi(x)y^{-1}$, $x\in
LS(M)$.

This together with [11, Theorem B] and [2, Theorem 1.4] verifies

######  Theorem 1.3.

Let $M,N$ be von Neumann algebras without type Ifin direct summands. If
$\Psi\colon LS(M)\to LS(N)$ is a ring isomorphism, then there exist a real
∗-isomorphism $\psi\colon M\to N$ (which extends to a real ∗-isomorphism from
$LS(M)$ onto $LS(N)$) and an invertible element $y\in LS(N)$ such that
$\Psi(x)=y\psi(x)y^{-1}$, $x\in LS(M)$.

Furthermore, applying [11, Theorem A], we obtain

######  Theorem 1.4.

Let $M,N$ be von Neumann algebras without type Ifin direct summands. If
$\Phi\colon{\mathcal{P}}(M)\to{\mathcal{P}}(N)$ is a lattice isomorphism, then
there exist a real ∗-isomorphism $\psi\colon M\to N$ and an invertible element
$y\in LS(N)$ such that $\Phi(p)=l(y\psi(p))$, $p\in{\mathcal{P}}(M)$.

The proof of our main theorem rests on the results in the previous papers [11,
2] together with delicate arguments concerning unbounded operators and their
invertibility.

##  2\. Preliminaries

In this paper, we will freely use basic facts about von Neumann algebras as in
[9].

For a Hilbert space $H$, the von Neumann algebra of all bounded linear
operators on $H$ is written as $B(H)$. Let $M\subset B(H)$ be a von Neumann
algebra. We use the symbol $\sim$ (resp. $\prec$) to mean the Murray–von
Neumann equivalence relation (resp. preorder relation) on ${\mathcal{P}}(M)$.
That is, for $p,q\in{\mathcal{P}}(M)$, $p\sim q$ (resp. $p\prec q$) means that
there exists a partial isometry $v\in M$ such that $p=vv^{*}$ and $q=v^{*}v$
(resp. $p\leq vv^{*}$ and $q=v^{*}v$). For $p,q\in{\mathcal{P}}(M)$, we write
$p\perp q$ if $p$ and $q$ are orthogonal, i.e., $pq=qp=0$, or equivalently,
$pH\perp qH$ in the Hilbert space $H$. We use the symbol $p^{\perp}:=1-p$ for
$p\in{\mathcal{P}}(M)$. The symbol ${\mathcal{Z}}(M)=\\{x\in M\mid xy=yx\text{
for all }y\in M\\}$ means the center of $M$. For an operator $x\in M$, let
$z(x)$ denote the central support of $x$, which is the minimal central
projection $r$ in ${\mathcal{P}}({\mathcal{Z}}(M))$ with $rx=x$.

Let us recall some terms about the geometry of ${\mathcal{P}}(M)$. A
projection $p\in{\mathcal{P}}(M)$ is said to be _finite_ if any subprojection
$p\geq q\in{\mathcal{P}}(M)$ with $p\sim q$ satisfies $p=q$. A projection
$p\in{\mathcal{P}}(M)$ is said to be _properly infinite_ if there is $p\geq
q\in{\mathcal{P}}(M)$ with $p\sim q\sim p-q$. It is well-known that for every
$p\in{\mathcal{P}}(M)$ there is a unique subprojection $p\geq
p_{1}\in{\mathcal{P}}(M)$ such that $p_{1}$ is finite, $p-p_{1}$ is properly
infinite, and $z(p_{1})z(p-p_{1})=0$. We call $p_{1}$ the _finite part_ of
$p$, and $p-p_{1}$ the _properly infinite part_ of $p$.

###  2.1. Various isomorphisms of ∗-algebras

For ∗-algebras $A$ and $B$, a (not necessarily linear) bijection $\psi\colon
A\to B$ is called

  * •

a _ring isomorphism_ if it is additive and multiplicative,

  * •

a _real ∗-isomorphism_ if it is real-linear, multiplicative, and satisfies
$\psi(x^{*})=\psi(x)^{*}$ for any $x\in A$,

  * •

a _∗ -isomorphism_ if it is a complex-linear real ∗-isomorphism, and

  * •

a _conjugate-linear ∗-isomorphism_ if it is a conjugate-linear real
∗-isomorphism.

It is well-known that a real ∗-isomorphism between two von Neumann algebras is
decomposed into the direct sum of a (complex-linear) ∗-isomorphism and a
conjugate-linear ∗-isomorphism. See for example [11, Lemma 2.1 (4)]. Two
mappings $\psi_{1}$, $\psi_{2}$ from $A$ into $B$ are said to be _similar_ if
there is an invertible element $y\in B$ such that
$\psi_{2}(x)=y\psi_{1}(x)y^{-1}$ for every $x\in A$.

In this paper, we will frequently use the following fact. Let
$A_{1},A_{2},A_{3}$ be ∗-algebras. If $\psi_{1}\colon A_{1}\to A_{2}$ and
$\psi_{2}\colon A_{2}\to A_{3}$ are bijections that are similar to real
∗-isomorphisms, then $\psi_{2}\circ\psi_{1}\colon A_{1}\to A_{3}$ is also
similar to a real ∗-isomorphism. The proof is easy.

###  2.2. The algebra of locally measurable operators

Let $M\subset B(H)$ be a von Neumann algebra. Let $x$ be a densely-defined
closed operator on $H$. Then $x$ is said to be _affiliated with $M$_ (and we
write $x\eta M$) if $yx\subset xy$ for any $y\in M^{\prime}$. Here
$M^{\prime}:=\\{y\in B(H)\mid ay=ya\text{ for any }a\in M\\}$ denotes the
commutant of $M$. We say $x\eta M$ is a _measurable_ operator of $M$ if the
spectral projection $\chi_{(c,\infty)}(\lvert x\rvert)\in{\mathcal{P}}(M)$ is
a finite projection in $M$ for some real number $c>0$. We say $x\eta M$ is a
_locally measurable_ operator of $M$ if there exists an increasing sequence
$(r_{n})_{n\geq 1}$ of central projections in $M$ such that $r_{n}\nearrow 1$
and $xr_{n}$ is a measurable operator of $M$ for any $n$. In other words,
$x\eta M$ is locally measurable if the central support of the properly
infinite part of the projection $\chi_{(c,\infty)}(\lvert x\rvert)$ tends to
zero as $c\to\infty$.

We use the symbol $LS(M)$ to mean the collection of all locally measurable
operators of $M$. If $x,y\in LS(M)$, then $x^{*}$ and the closures of $xy$,
$x+y$ are in $LS(M)$. More generally, the closure of any algebraic combination
of elements in $LS(M)$ lies in $LS(M)$. These facts enable us to regard
$LS(M)$ as a ∗-algebra that contains $M$. In what follows, an algebraic
combination of operators in $LS(M)$ refers to the corresponding operator in
$LS(M)$ (i.e. its closure), unless otherwise stated.

It is not difficult to see that each element of $LS(M)$ can be written as the
direct sum of bounded operators if $M$ is of type I or III. More precisely,
for each $x\in LS(M)$, there exists an increasing sequence $(r_{n})_{n\geq 1}$
of central projections in $M$ such that $r_{n}\nearrow 1$ and $xr_{n}$ is
bounded for any $n$. Note that the same statement does not hold for a type II
von Neumann algebra $M$. See [15] and [14] for more details of (locally)
measurable operators.

For $x\in LS(M)$, let $l(x)\in{\mathcal{P}}(M)$ denote the left support of
$x$. That is, $l(x):=\bigwedge\\{p\in{\mathcal{P}}(M)\mid px=x\\}$. Similarly,
for the right support, we use the symbol
$r(x):=\bigwedge\\{p\in{\mathcal{P}}(M)\mid x=xp\\}$. Then the condition
$xy=0$ is equivalent to $r(x)l(y)=0$ for a pair of operators $x,y\in LS(M)$,
see the last part of [11, Subsection 2.2].

###  2.3. Two projections

When we play with projections, it is often useful to look at two projections.
The so-called Halmos’s two projection theorem [8] (see also [3]) describes the
relative position of a general pair of projections. Let us briefly recall it
from the viewpoint of von Neumann algebra theory.

Let $M\subset B(H)$ be a von Neumann algebra and $p,q\in{\mathcal{P}}(M)$.
Then we may find a unique pair of mutually orthogonal equivalent projections
$e_{1},e_{2}\in{\mathcal{P}}(M)$ with the following properties.

  * •

The Hilbert space $H$ is orthogonally decomposed as

(2.1) $H=(p\wedge q^{\perp})H\oplus(p^{\perp}\wedge q)H\oplus(p\wedge
q)H\oplus(p^{\perp}\wedge q^{\perp})H\oplus(e_{1}+e_{2})H.$

  * •

By choosing a suitable identification of $(e_{1}+e_{2})M(e_{1}+e_{2})$ with
${\mathbb{M}}_{2}(e_{1}Me_{1})$, $p$ and $q$ are decomposed as

$p=1\oplus 0\oplus 1\oplus 0\oplus\begin{pmatrix}1&0\\\ 0&0\end{pmatrix},\quad
q=0\oplus 1\oplus 1\oplus 0\oplus\begin{pmatrix}a^{2}&ab\\\
ab&b^{2}\end{pmatrix}$

with respect to the Hilbert space decomposition (2.1), where $a$ and $b$ are
positive injective operators in $e_{1}Me_{1}$ such that $a^{2}+b^{2}=e_{1}$.

Assume that $p\wedge q=0$. We say $p$ is _LS-orthogonal_ to $q$ if the
operator $b\in e_{1}Me_{1}$ is invertible in $LS(e_{1}Me_{1})$. More details
can be found in [11].

###  2.4. A few facts on locally measurable operators

In this subsection, let $M$ be a von Neumann algebra. We prove a few facts
about the invertibility of positive operators in $LS(M)$ for later use.

######  Lemma 2.1.

Let $a\in LS(M)$ be a positive and injective operator. Then the following are
equivalent.

  1. (1)

The operator $a$ is not invertible in $LS(M)$.

  2. (2)

There are a strictly decreasing sequence $c_{n}$ of positive real numbers and
a sequence $f_{n}\in{\mathcal{P}}(M)$ of pairwise equivalent nonzero
projections with the following properties: (i) $c_{n}\to 0$ as $n\to\infty$.
(ii) $f_{n}\leq\chi_{(c_{n+1},c_{n}]}(a)$.

  3. (3)

There is a sequence $f_{n}\in{\mathcal{P}}(M)$ of pairwise equivalent nonzero
projections such that ($f_{n}af_{n}$ is bounded for each $n\geq 1$ and)
$\lVert f_{n}af_{n}\rVert\to 0$.

###### Proof.

The implication (2)$\Rightarrow$(3) is clear.

(1)$\Rightarrow$(2) What follows is a reproduction of part of [10, Proof of
Lemma 2.2]. By decomposing $M$ into a direct sum, we may assume that $M$ is of
one of the three types I, II, or III.

First, we consider the case of type I and III. Then the assumption that $a\in
LS(M)$ is not invertible implies that $r:=\bigwedge_{n\geq
1}z(\chi_{(0,1/n]}(a))\neq 0$. Considering the pair $(Mr,ar)$ instead of
$(M,a)$, we may assume $\bigwedge_{n\geq 1}z(\chi_{(0,1/n]}(a))=1$. Take a
normal (tracial) state $\tau$ on ${\mathcal{Z}}(M)$. We may also assume
$\operatorname{supp}(\tau)=1\in{\mathcal{P}}({\mathcal{Z}}(M))$. By our
assumptions, we may take a strictly decreasing sequence $(c_{n})_{n\geq 1}$ of
positive real numbers that satisfies $c_{n}\to 0$ ($n\to\infty$) and
$\tau(z(\chi_{(c_{n+1},c_{n}]}(a)))\geq 1-3^{-n}$, $n\geq 1$. Then
$\tau(\bigwedge_{n\geq 1}z(\chi_{(c_{n+1},c_{n}]}(a)))\geq 1-\sum_{n\geq
1}3^{-n}>0$ and thus $\bigwedge_{n\geq 1}z(\chi_{(c_{n+1},c_{n}]}(a))\neq 0$.
We may assume $\bigwedge_{n\geq 1}z(\chi_{(c_{n+1},c_{n}]}(a))=1$.

By basic properties of projections, we obtain the following: If $M$ is of type
I, we can take an abelian projection $f_{n}\leq\chi_{(c_{n+1},c_{n}]}(a)$ with
$z(f_{n})=1$ for each $n\geq 1$. If $M$ is of type III, by the assumption that
${\mathcal{Z}}(M)$ has a normal faithful state, we can take a countably
decomposable projection $f_{n}\leq\chi_{(c_{n+1},c_{n}]}(a)$ with $z(f_{n})=1$
for each $n\geq 1$. In both cases, using [9, Corollary 6.3.5] and [9,
Proposition 6.4.6(iii)], we see that $(f_{n})_{n\geq 1}$ is a family of
mutually orthogonal equivalent nonzero projections satisfying
$f_{n}\leq\chi_{(c_{n+1},c_{n}]}(a)$ for each $n\geq 1$.

Next, we consider the case where $M$ is of type II. For a projection
$p\in{\mathcal{P}}(M)$, let $z_{\infty}(p)\in{\mathcal{P}}({\mathcal{Z}}(M))$
denote the central support of the properly infinite part of $p$. Assume that
$a\in LS(M)$ is not invertible in $LS(M)$. We may assume that
$\bigwedge_{n\geq 1}z_{\infty}(\chi_{(0,1/n]}(a))=1$ without loss of
generality. Then $M$ is of type II∞. Take a normal semifinite faithful tracial
weight $\tau$ on $M$ and a (finite) projection $p\in{\mathcal{P}}(M)$ with
$\tau(p)=1$. It follows that $\chi_{(0,1/n]}(a)\succ p$ for every $n\geq 1$.
By [10, Lemma 2.3], there exist a strictly decreasing sequence $(c_{n})_{n\geq
1}$ of positive real numbers and a sequence $(p_{n})_{n\geq 1}$ of projections
in $M$ such that $c_{n}\to 0\,\,(n\to\infty)$,
$p_{n}\leq\chi_{(c_{n+1},c_{n}]}(a)$, $p_{n}\prec p$ and $\tau(p_{n})\geq
1-3^{-n}$, $n\geq 1$. Take a projection $\tilde{p}_{n}\in{\mathcal{P}}(M)$
such that $p_{n}\sim\tilde{p}_{n}\leq p$, $n\geq 1$. Put
$\tilde{p}:=\bigwedge_{n\geq 1}\tilde{p}_{n}$. Then
$\tau(\tilde{p})=\tau(\bigwedge_{n\geq 1}\tilde{p}_{n})\geq 1-\sum_{n\geq
1}3^{-n}>0$. Hence $\tilde{p}\neq 0$. Take a projection
$f_{n}\in{\mathcal{P}}(M)$ such that $f_{n}\leq p_{n}$ and
$f_{n}\sim\tilde{p}$, for $n\geq 1$. Then $(f_{n})_{n\geq 1}$ is a family of
mutually orthogonal equivalent nonzero projections satisfying
$f_{n}\leq\chi_{(c_{n+1},c_{n}]}(a)$ for each $n\geq 1$.

(3)$\Rightarrow$(1) Assume that there is a sequence $f_{n}\in{\mathcal{P}}(M)$
of pairwise equivalent nonzero projections such that $\lVert
f_{n}af_{n}\rVert\to 0$. We show that $a$ is not invertible in $LS(M)$. Define
the function $F\colon[0,\infty)\to[0,1]$ by $F(t)=\min\\{t,1\\}$, $t\geq 0$.
Consider the bounded operator $F(a)$. Note that $F(a)\leq a$, hence $\lVert
f_{n}F(a)f_{n}\rVert\leq\lVert f_{n}af_{n}\rVert$. Moreover, it is easy to see
that $a$ is invertible in $LS(M)$ if and only if $F(a)$ is invertible in
$LS(M)$. Therefore, by considering $F(a)$ instead of $a$, we may assume that
$a$ is a bounded operator.

Assume for a contradiction that the bounded operator $a$ is invertible in
$LS(M)$. Let $b$ be the inverse of $a$ in $LS(M)$. Then there is an increasing
sequence $r_{m}$ of central projections in $M$ such that $r_{m}b$ is
measurable for each $m\geq 1$ and $r_{m}\nearrow 1$. We may take an integer
$m\geq 1$ such that $r_{m}f_{n}\neq 0$ for some (or equivalently, every)
$n\geq 1$. Let $\varepsilon>0$ be a positive real number. We may find $n\geq
1$ such that $\lVert f_{n}af_{n}\rVert\leq\varepsilon$. It clearly follows
that $f_{n}\wedge\chi_{(\varepsilon,\infty)}(a)=0$, which implies that
$f_{n}\prec\chi_{(0,\varepsilon]}(a)$. (Here we imitated the argument in [5,
Proof of Proposition 2.2].) It follows that $0\neq r_{m}f_{1}\sim
r_{m}f_{n}\prec\chi_{(0,\varepsilon]}(r_{m}a)$ for every positive real number
$\varepsilon>0$. On the other hand, since $r_{m}a$ has an inverse $r_{m}b$ in
$LS(r_{m}M)$ and $r_{m}b$ is measurable, we see that
$\chi_{(0,c_{0}]}(r_{m}a)$ is a finite projection for some $c_{0}>0$. We also
have $\chi_{(0,c]}(r_{m}a)\to 0$ as $c\to 0$. Thus we arrive at a
contradiction via the following statement: If a decreasing sequence of finite
projections $p_{n}$ with $p_{n}\searrow 0$ and a projection $p$ in a von
Neumann algebra $M$ satisfy $p\prec p_{n}$ for every $n$, then $p=0$. This
statement can be obtained e.g. by considering the center-valued trace of $p$
in the finite von Neumann algebra $(p\vee p_{1})M(p\vee p_{1})$. Indeed, by
the assumption the center-valued trace of $p$ is $0$, thus $p=0$. ∎

######  Corollary 2.2.

Let $y\eta M$ be a positive self-adjoint operator. Then $y\notin LS(M)$ if and
only if there are a strictly increasing sequence $c_{n}$ of positive real
numbers and a sequence $f_{n}\in{\mathcal{P}}(M)$ of pairwise equivalent
nonzero projections with the following properties: (i) $c_{n}\to\infty$ as
$n\to\infty$. (ii) $f_{n}\leq\chi_{[c_{n},c_{n+1})}(y)$.

###### Proof.

Apply Lemma 2.1 (1)$\Leftrightarrow$(2) to $a:=(1+y)^{-1}$, which is a
positive injective (bounded) operator in $M$. ∎

######  Corollary 2.3.

Let $p,q\in{\mathcal{P}}(M)$ satisfy $p\wedge q=0$. Then $p$ is not LS-
orthogonal to $q$ if and only if there is a sequence
$f_{n}\in{\mathcal{P}}(M)$ of pairwise equivalent nonzero subprojections of
$p$ such that $\|f_{n}q^{\perp}f_{n}\rVert\to 0$.

###### Proof.

Applying Subsection 2.3 and using the same symbols as there, we may regard

$p=1\oplus 0\oplus 0\oplus\begin{pmatrix}1&0\\\ 0&0\end{pmatrix},\quad
q=0\oplus 1\oplus 0\oplus\begin{pmatrix}a^{2}&ab\\\ ab&b^{2}\end{pmatrix},$

and thus

$q^{\perp}=1\oplus 0\oplus 1\oplus\begin{pmatrix}b^{2}&-ab\\\
-ab&a^{2}\end{pmatrix}.$

Recall that $p$ is not LS-orthogonal to $q$ if and only if $b\in e_{1}Me_{1}$
is not invertible in $LS(e_{1}Me_{1})$. Clearly, this is equivalent to the
condition $1\oplus b^{2}\in(p\wedge q^{\perp}+e_{1})M(p\wedge
q^{\perp}+e_{1})$ is not invertible in $LS((p\wedge q^{\perp}+e_{1})M(p\wedge
q^{\perp}+e_{1}))$. Now apply Lemma 2.1 (1)$\Leftrightarrow$(3) to obtain the
desired conclusion. ∎

##  3\. Proof of the main theorem

###  3.1. Reduction to a special case

First, let us recall a basic fact about a type II∞ von Neumann algebra. We
give a sketch of a proof for completeness.

######  Lemma 3.1.

Let $M$ be a type II∞ von Neumann algebra. Let $p\in{\mathcal{P}}(M)$ be a
nonzero finite projection. Then there is a family $Z$ of pairwise orthogonal
nonzero central projections with sum $z(p)$ satisfying the following
properties: For each $r\in Z$, there is a family $(q_{i})_{i\in I}$ of
mutually orthogonal equivalent projections such that $rp\in\\{q_{i}\mid i\in
I\\}$ and $\sum_{i\in I}q_{i}=r$.

###### Proof.

Take a maximal family of mutually orthogonal projections $(p_{i})_{i\in I}$
with $p_{i}\sim p$ for each $i\in I$ and $p\in\\{p_{i}\mid i\in I\\}$. By the
comparison theorem, we may find a central projection
$r\in{\mathcal{P}}({\mathcal{Z}}(M))$ such that $(1-\bigvee_{i\in
I}p_{i})r\prec pr$ and $(1-\bigvee_{i\in I}p_{i})r^{\perp}\succ pr^{\perp}$.
The maximality of $(p_{i})_{i\in I}$ implies that $r\neq 0$. Note that $I$ is
an infinite set since $Mr$ is also a type II∞ von Neumann algebra. Thus we may
retake a family of mutually orthogonal equivalent projections $(q_{i})_{i\in
I}$ with the properties $rp\in\\{q_{i}\mid i\in I\\}$ and $\sum_{i\in
I}q_{i}=r$. (This retaking can be done in the following manner: Let $i_{0}\in
I$ be the element with $p_{i_{0}}=p$. Fix an element $i_{0}\neq j\in I$, and
take a bijection $\sigma\colon I\setminus\\{i_{0},j\\}\to
I\setminus\\{i_{0}\\}$. For each $i\in I\setminus\\{i_{0},j\\}$, we may take a
projection $\check{p}_{i}\leq p_{i}$ with $\check{p}_{i}r\sim(1-\bigvee_{i\in
I}p_{i})r$ since $(1-\bigvee_{i\in I}p_{i})r\prec pr\sim p_{i}r$. Then we may
set $q_{i}=(p_{i}-\check{p}_{i}+\check{p}_{\sigma(i)})r$ for each $i\in
I\setminus\\{i_{0},j\\}$, $q_{i_{0}}=pr=rp$ and
$q_{j}=(p_{j}-\check{p}_{j}+1-\bigvee_{i\in I}p_{i})r$.) Now an application of
Zorn’s lemma leads us to the desired conclusion. ∎

Let $M,N$ be von Neumann algebras of type II∞, and $\Psi\colon LS(M)\to LS(N)$
a ring isomorphism. The goal of this article is to show that $\Psi$ is similar
to a real ∗-isomorphism. The aim of this subsection is to reduce our task to a
special case. Note that $\Psi$ maps a central projection in $M$ to a central
projection in $N$ because an idempotent in the center of $LS(M)$ is a
projection. First, we show the following claim.

######  Claim 3.2.

There is a family of mutually orthogonal central projections
$Z\subset{\mathcal{P}}({\mathcal{Z}}(M))$ with $\sum_{r\in Z}r=1$ satisfying
the following properties: For each $r\in Z$, there are von Neumann algebras
$M_{0},N_{0}$ of type II1, Hilbert spaces $H,K$, ∗-isomorphisms $\pi\colon
rM\to M_{0}{\mathbin{\overline{\otimes}}}B(H)$, $\rho\colon\Psi(r)N\to
N_{0}{\mathbin{\overline{\otimes}}}B(K)$, and projections
$e_{0}\in{\mathcal{P}}(B(H))$, $f_{0}\in{\mathcal{P}}(B(K))$ of rank one such
that $l(\rho\circ\Psi\circ\pi^{-1}(1\otimes e_{0}))=1\otimes f_{0}$.

###### Proof.

Recall that the lattice isomorphism
$\Phi\colon{\mathcal{P}}(M)\to{\mathcal{P}}(N)$ determined by
$\Phi(p)=l(\Psi(p))$, $p\in{\mathcal{P}}(M)$, maps a finite projection to a
finite projection (see [11, Proof of Lemma 4.1]). Fix a finite projection
$p\in{\mathcal{P}}(M)$ with $z(p)=1$. Applying the preceding lemma to
$p\in{\mathcal{P}}(M)$ and $\Phi(p)\in{\mathcal{P}}(N)$, we obtain families
$Z_{1}\subset{\mathcal{P}}({\mathcal{Z}}(M))$,
$Z_{2}\subset{\mathcal{P}}({\mathcal{Z}}(N))$ of mutually orthogonal central
projections with sum $1$ in $M$ and $N$, respectively. Set
$Z=\\{z\Psi^{-1}(w)\mid z\in Z_{1},w\in Z_{2}\\}$. For each element
$r=z\Psi^{-1}(w)\in Z$ with $z\in Z_{1}$, $w\in Z_{2}$, we see that $\Psi$
restricts to a ring isomorphism from $rLS(M)=LS(rM)$ onto
$\Psi(r)LS(N)=LS(\Psi(r)N)$. Observe that $M_{0}:=prMp$ and
$N_{0}:=\Phi(p)\Psi(r)N\Phi(p)$ are von Neumann algebras of type II1. By Lemma
3.1, we may take ∗-isomorphisms $\pi\colon rM\to
M_{0}{\mathbin{\overline{\otimes}}}B(H)$, $\rho\colon\Psi(r)N\to
N_{0}{\mathbin{\overline{\otimes}}}B(K)$ such that $\pi(pr)=1\otimes e_{0}$,
$\rho(\Phi(p)\Psi(r))=1\otimes f_{0}$ for some projections
$e_{0}\in{\mathcal{P}}(B(H))$, $f_{0}\in{\mathcal{P}}(B(K))$ of rank one. Then
we have

$l(\rho\circ\Psi\circ\pi^{-1}(1\otimes
e_{0}))=l(\rho\circ\Psi(pr))=\rho(l(\Psi(pr)))=\rho(\Phi(pr)).$

Since $r$ is central we see that $\Phi(pr)=\Phi(p)\Psi(r)$, and we obtain
$\rho(\Phi(pr))=\rho(\Phi(p)\Psi(r))=1\otimes f_{0}$. ∎

In order to show Theorem 1.2, it suffices to show that the restriction of
$\Psi$ to $rLS(M)=LS(rM)$ is similar to a real ∗-isomorphism for each $r\in Z$
in Claim 3.2. Therefore, refreshing the notations, we just need to consider
the following situation. Assume that $M_{0},N_{0}$ are von Neumann algebras of
type II1, $H,K$ are infinite-dimensional Hilbert spaces, and that
$e_{0}\in{\mathcal{P}}(B(H))$, $f_{0}\in{\mathcal{P}}(B(K))$ are projections
of rank one. Assume further that $\Psi\colon LS(M)\to LS(N)$ is a ring
isomorphism satisfying $l(\Psi(p_{0}))=q_{0}$, where
$M:=M_{0}{\mathbin{\overline{\otimes}}}B(H)$,
$N_{0}:=N_{0}{\mathbin{\overline{\otimes}}}B(K)$, and $p_{0}:=1\otimes e_{0}$,
$q_{0}:=1\otimes f_{0}$. Under this setting, we have the following claim.

######  Claim 3.3.

There is a mapping $\psi\colon LS(N_{0}{\mathbin{\overline{\otimes}}}B(K))\to
LS(M_{0}{\mathbin{\overline{\otimes}}}B(K))$ that is similar to a real
∗-isomorphism with $\psi\circ\Psi(x\otimes e_{0})=x\otimes f_{0}$ for every
$x\in LS(M_{0})$.

###### Proof.

We see that the operator $\Psi(p_{0})$ is an idempotent with
$l(\Psi(p_{0}))=q_{0}$. Thus we may take an invertible operator $y_{1}\in
LS(N)$ such that $y_{1}\Psi(p_{0})y_{1}^{-1}=q_{0}$. (Indeed, we may set
$y_{1}:=\Psi(p_{0})+q_{0}^{\perp}$. Note that
$0=\Psi(p_{0}^{\perp}p_{0})=\Psi(p_{0}^{\perp})\Psi(p_{0})$, which implies
$0=\Psi(p_{0}^{\perp})l(\Psi(p_{0}))=(1-\Psi(p_{0}))q_{0}$. Using this
equality, we may easily obtain $y_{1}^{-1}=-\Psi(p_{0})+1+q_{0}$ and
$y_{1}\Psi(p_{0})y_{1}^{-1}=q_{0}$.) Then the mapping
$\Psi_{1}=y_{1}\Psi(\cdot)y_{1}^{-1}$ satisfies $\Psi_{1}(p_{0})=q_{0}$.

From here we apply the result by Ayupov and Kudaybergenov. Observe that now
$\Psi_{1}$ restricts to a ring isomorphism from $p_{0}LS(M)p_{0}$ onto
$q_{0}LS(N)q_{0}$. Let us identify $p_{0}LS(M)p_{0}$ and $q_{0}LS(N)q_{0}$
with $LS(M_{0})$ and $LS(N_{0})$, respectively, in a natural manner. By [2,
Theorem 1.4], we may find a real ∗-isomorphism $\psi_{0}\colon M_{0}\to N_{0}$
(which also determines a real ∗-isomorphism from $LS(M_{0})$ onto $LS(N_{0})$)
and an invertible operator $y_{0}\in LS(N_{0})$ such that
$\Psi_{1}(x)=y_{0}\psi_{0}(x)y_{0}^{-1}$ for every $x\in LS(M_{0})$. Note that
$\psi_{0}\colon M_{0}\to N_{0}$ determines a real ∗-isomorphism
$\psi_{1}\colon M_{0}{\mathbin{\overline{\otimes}}}B(K)\to
N_{0}{\mathbin{\overline{\otimes}}}B(K)$ such that $\psi_{1}(a\otimes
b)=\psi_{0}(a)\otimes b$ for any $a\in M_{0}$, $b\in B(K)$ (which in turn
determines a real ∗-isomorphism from
$LS(M_{0}{\mathbin{\overline{\otimes}}}B(K))$ onto
$LS(N_{0}{\mathbin{\overline{\otimes}}}B(K))$). Put $y_{00}:=y_{0}\otimes
e_{0}+1\otimes e_{0}^{\perp}\in LS(M_{0}{\mathbin{\overline{\otimes}}}B(K))$.
Then the mapping $LS(M_{0}{\mathbin{\overline{\otimes}}}B(H))\ni
x\mapsto\psi_{1}^{-1}(y_{00}^{-1}\Psi_{1}(x)y_{00})\in
LS(M_{0}{\mathbin{\overline{\otimes}}}B(K))$ sends $x\otimes e_{0}$ to
$x\otimes f_{0}$ for every $x\in LS(M_{0})$, so the claim is settled. ∎

Then our task reduces to considering $\psi\circ\Psi$ instead of $\Psi$ in
Claim 3.3. Let us again refresh the notations. We need to consider a von
Neumann algebra $M_{0}$ of type II1, infinite-dimensional Hilbert spaces
$H,K$, projections $e_{0}\in{\mathcal{P}}(B(H))$,
$f_{0}\in{\mathcal{P}}(B(K))$ of rank one, and a ring isomorphism $\Psi\colon
LS(M_{0}{\mathbin{\overline{\otimes}}}B(H))\to
LS(M_{0}{\mathbin{\overline{\otimes}}}B(K))$ satisfying $\Psi(x\otimes
e_{0})=x\otimes f_{0}$ for every $x\in LS(M_{0})$. By considering $\Psi^{-1}$
instead of $\Psi$ if necessary, we may assume that $\dim K\leq\dim H$. Fix a
linear isometry $w$ from $K$ into $H$ with $wf_{0}w^{*}=e_{0}$. By the
embedding $x\mapsto(1\otimes w)x(1\otimes w^{*})$, we may identify
$LS(M_{0}{\mathbin{\overline{\otimes}}}B(K))$ with
$LS(M_{0}{\mathbin{\overline{\otimes}}}B(wK))$, which may be regarded as a
subalgebra of $LS(M_{0}{\mathbin{\overline{\otimes}}}B(H))$. Note that
$(1\otimes w)q_{0}(1\otimes w^{*})=p_{0}$. To consider Theorem 1.2, it
suffices to work with the map $LS(M_{0}{\mathbin{\overline{\otimes}}}B(H))\ni
x\mapsto(1\otimes w)\Psi(x)(1\otimes w^{*})\in
LS(M_{0}{\mathbin{\overline{\otimes}}}B(wK))$ instead of $\Psi$.

###  3.2. Proof of the main theorem

By the preceding subsection, to verify Theorem 1.2, we just need to prove the
following statement. We refresh our notations once again.

######  Proposition 3.4.

Let $L$ be a Hilbert space and $M_{0}\subset B(L)$ a von Neumann algebra of
type II1. Let $H$ be an infinite-dimensional Hilbert space, $K\subset H$ an
infinite-dimensional closed subspace, and let $e_{0}\in{\mathcal{P}}(B(K))$ be
a rank-one projection. Set $M:=M_{0}{\mathbin{\overline{\otimes}}}B(H)$ and
$N:=M_{0}{\mathbin{\overline{\otimes}}}B(K)$. Put $p_{0}:=1\otimes
e_{0}\in{\mathcal{P}}(N)(\subset{\mathcal{P}}(M))$. Let $\Psi\colon LS(M)\to
LS(N)$ be a ring isomorphism such that $\Psi(x\otimes e_{0})=x\otimes e_{0}$
for each $x\in LS(M_{0})$. Then $\Psi$ is similar to a real ∗-isomorphism.

From now on we give a proof of Proposition 3.4. Let
$\Phi\colon{\mathcal{P}}(M)\to{\mathcal{P}}(N)$ denote the lattice isomorphism
associated with $\Psi$. Namely,
$\Phi\colon{\mathcal{P}}(M)\to{\mathcal{P}}(N)$ is the lattice isomorphism
that is determined by the formula $\Phi(p)=l(\Psi(p))$, $p\in{\mathcal{P}}(M)$
(see [11, Proposition 3.1]). We define an unbounded operator $\check{y}$ on
the Hilbert space $L\otimes H$. We will show that it is closable and the
closure brings us to the solution of our task.

Let ${\mathcal{P}}(M)_{0}$ denote the collection of all projections
$p\in{\mathcal{P}}(M)$ with $p\sim p_{0}$. Let ${\mathcal{P}}(M)_{1}$ denote
the collection of all projections $p\in{\mathcal{P}}(M)$ such that there are
$n\geq 1$ and mutually orthogonal projections
$p_{1},\ldots,p_{n}\in{\mathcal{P}}(M)_{0}$ with $p=p_{1}+\cdots+p_{n}$. To
define $\check{y}$, we first construct a family of operators
$(y_{p})_{p\in{\mathcal{P}}(M)_{1}}\subset LS(M)$.

Let $p=p_{1}+\cdots+p_{n}\in{\mathcal{P}}(M)_{1}$ with
$p_{1},\ldots,p_{n}\in{\mathcal{P}}(M)_{0}$. For each $i\in\\{1,\ldots,n\\}$,
take a partial isometry $v_{i}\in M$ such that $v_{i}^{*}v_{i}=p_{0}$,
$v_{i}v_{i}^{*}=p_{i}$. Because $\Psi$ is a ring isomorphism, we have
$r(\Psi(x_{1}))\leq r(\Psi(x_{2}))$ for every pair $x_{1},x_{2}\in LS(M)$ with
$r(x_{1})\leq r(x_{2})$. (To show this, imitate the proof of [11, Proposition
3.1].) It follows that $\Psi(v_{i})\in LS(N)$ is an operator with
$r(\Psi(v_{i}))=r(\Psi(p_{0}))=p_{0}\in N\subset M$. Thus
$\Psi(v_{i})v_{i}^{*}\in LS(M)$ satisfies $r(\Psi(v_{i})v_{i}^{*})=p_{i}$. Put
$y_{p}:=\Psi(v_{1})v_{1}^{*}+\cdots+\Psi(v_{n})v_{n}^{*}\in LS(M)$.

######  Claim 3.5.

The operator $y_{p}\in LS(M)$ is well-defined, and satisfies
$r(y_{p})=p,l(y_{p})=\Phi(p)$ and $y_{p+q}=y_{p}+y_{q}$ for any pair
$p,q\in{\mathcal{P}}(M)_{1}$ with $q\perp p$.

###### Proof.

We first show that $y_{p}$ is well-defined. Let
$p=p^{\prime}_{1}+\cdots+p^{\prime}_{n}$ with
$p^{\prime}_{1},\ldots,p^{\prime}_{n}\in{\mathcal{P}}(M)_{0}$ be another
decomposition of $p$. Let $v^{\prime}_{i}\in M$ satisfy
${v^{\prime}_{i}}^{*}v^{\prime}_{i}=p_{0}$,
$v^{\prime}_{i}{v^{\prime}_{i}}^{*}=p^{\prime}_{i}$ for
$i\in\\{1,\ldots,n\\}$. For each $i\in\\{1,\ldots,n\\}$, we obtain

$\begin{split}\Psi(v_{i})v_{i}^{*}=\Psi(v_{i})v_{i}^{*}p&=\Psi(v_{i})v_{i}^{*}(v^{\prime}_{1}{v^{\prime}_{1}}^{*}+\cdots+v^{\prime}_{n}{v^{\prime}_{n}}^{*})\\\
&=\Psi(v_{i})(v_{i}^{*}v^{\prime}_{1}){v^{\prime}_{1}}^{*}+\cdots+\Psi(v_{i})(v_{i}^{*}v^{\prime}_{n}){v^{\prime}_{n}}^{*}\\\
&=\Psi(v_{i}v_{i}^{*}v^{\prime}_{1}){v^{\prime}_{1}}^{*}+\cdots+\Psi(v_{i}v_{i}^{*}v^{\prime}_{n}){v^{\prime}_{n}}^{*},\end{split}$

where the third equality is a consequence of $v_{i}^{*}v^{\prime}_{j}\in
p_{0}Mp_{0}=\\{x\otimes e_{0}\mid x\in M_{0}\\}$ for each $j$. Therefore,
$\Psi(v_{1})v_{1}^{*}+\cdots+\Psi(v_{n})v_{n}^{*}$ is equal to

$\sum_{1\leq i,j\leq
n}\Psi(v_{i}v_{i}^{*}v^{\prime}_{j}){v^{\prime}_{j}}^{*}=\sum_{j=1}^{n}\Psi\left(\sum_{i=1}^{n}v_{i}v_{i}^{*}v^{\prime}_{j}\right){v^{\prime}_{j}}^{*}=\sum_{j=1}^{n}\Psi\left(pv^{\prime}_{j}\right){v^{\prime}_{j}}^{*}=\sum_{j=1}^{n}\Psi(v^{\prime}_{j}){v^{\prime}_{j}}^{*}.$

This ensures that $y_{p}$ is well-defined. It clearly follows from the
definition that $y_{p+q}=y_{p}+y_{q}$ for any pair
$p,q\in{\mathcal{P}}(M)_{1}$ with $q\perp p$.

From the definition again, it is easy to see that $r(y_{p})\leq p$ for any
$p\in{\mathcal{P}}(M)_{1}$. Let $q\in{\mathcal{P}}(M)$ be any nonzero
projection with $q\leq p$. We may find a nonzero projection $q_{1}\leq q$ with
$q_{1}\prec p_{0}$. Then we may in turn take mutually orthogonal
$p_{1},\ldots,p_{n}\in{\mathcal{P}}(M)_{0}$ such that $q_{1}\leq p_{1}$ and
$p=p_{1}+\cdots+p_{n}$ because $M$ is properly infinite and $p_{0}$ is finite.
For each $i\in\\{1,\ldots,n\\}$, take a partial isometry $v_{i}\in M$ such
that $v_{i}^{*}v_{i}=p_{0}$, $v_{i}v_{i}^{*}=p_{i}$. Then we obtain
$y_{p}=\Psi(v_{1})v_{1}^{*}+\cdots+\Psi(v_{n})v_{n}^{*}$. It follows that
$y_{p}q_{1}=\Psi(v_{1})v_{1}^{*}q_{1}$, which is nonzero because
$\Psi(v_{1}^{*})\Psi(v_{1})v_{1}^{*}q_{1}=\Psi(v_{1}^{*}v_{1})v_{1}^{*}q_{1}=\Psi(p_{0})v_{1}^{*}q_{1}=p_{0}v_{1}^{*}q_{1}=v_{1}^{*}q_{1}\neq
0$. Thus we obtain $y_{p}q\neq 0$ for any nonzero $q\leq p$, which leads to
the equality $r(y_{p})=p$.

Let $p=p_{1}+\cdots+p_{n}\in{\mathcal{P}}(M)_{1}$ with
$p_{1},\ldots,p_{n}\in{\mathcal{P}}(M)_{0}$ again, and for each
$i\in\\{1,\ldots,n\\}$, take a partial isometry $v_{i}\in M$ such that
$v_{i}^{*}v_{i}=p_{0}$, $v_{i}v_{i}^{*}=p_{i}$. Then we have

$l(y_{p})\leq\bigvee_{1\leq i\leq n}l(\Psi(v_{i}))=\bigvee_{1\leq i\leq
n}\Phi(l(v_{i}))=\bigvee_{1\leq i\leq n}\Phi(p_{i})=\Phi\left(\bigvee_{1\leq
i\leq n}p_{i}\right)=\Phi(p).$

On the other hand, we see

$l(y_{p})\sim
r(y_{p})=p=p_{1}+\cdots+p_{n}=r(y_{p_{1}})+\cdots+r(y_{p_{n}})\succ
l(y_{p_{1}})\vee\cdots\vee l(y_{p_{n}}).$

The last order relation $\succ$ is obtained by e.g. comparing the center-
valued trace on the finite von Neumann algebra $eMe$ for
$e=(r(y_{p_{1}})+\cdots+r(y_{p_{n}}))\vee l(y_{p_{1}})\vee\cdots\vee
l(y_{p_{n}})$. Since
$l(y_{p_{i}})=l(\Psi(v_{i})v_{i}^{*})=l(\Psi(v_{i})v_{i}^{*}v_{i})=l(\Psi(v_{i})p_{0})=l(\Psi(v_{i}))=\Phi(p_{i})$
for each $i$, we obtain $l(y_{p})\succ\bigvee_{1\leq i\leq
n}\Phi(p_{i})=\Phi(p)$. This together with the fact that $\Phi(p)$ is finite
(see [11, Proof of Lemma 4.1]) implies that $l(y_{p})=\Phi(p)$. ∎

Assume that $p,q\in{\mathcal{P}}(M)_{1}$ satisfy $p\leq q$. By the equations
$y_{q}=y_{p}+y_{q-p}$, $r(y_{p})=p$, and $r(y_{q-p})=q-p$, we obtain the
equality $y_{q}p=y_{p}$ in $LS(M)$. On the other hand, it is clear that the
product $y_{q}p$ as an unbounded operator is already closed. Therefore, we
have ${\mathcal{D}}(y_{p})\cap p(L\otimes H)={\mathcal{D}}(y_{q})\cap
p(L\otimes H)$ and $y_{p}h=y_{q}h$ for every $h\in{\mathcal{D}}(y_{p})\cap
p(L\otimes H)$.

Let $\check{y}$ be the unbounded linear operator on $L\otimes H$ determined by
the following conditions: The domain ${\mathcal{D}}(\check{y})$ is the union
of ${\mathcal{D}}(y_{p})\cap p(L\otimes H)$ for $p\in{\mathcal{P}}(M)_{1}$.
(Here we consider each $y_{p}\in LS(M)$ as an unbounded closed operator on
$L\otimes H$.) If $h\in{\mathcal{D}}(y_{p})\cap p(L\otimes H)$ for
$p\in{\mathcal{P}}(M)_{1}$, then we define $\check{y}h:=y_{p}h$. Let $h\in
L\otimes H$ satisfy $h\in{\mathcal{D}}(y_{p_{i}})\cap{p_{i}}(L\otimes H)$ for
$p_{i}\in{\mathcal{P}}(M)_{1}$, $i=1,2$. Then we may find a projection
$q\in{\mathcal{P}}(M)_{1}$ with $p_{1}\vee p_{2}\leq q$. Thus the preceding
paragraph implies that $y_{p_{1}}h=y_{q}h=y_{p_{2}}h$, and this verifies that
$\check{y}$ is well-defined. The same argument also shows that the equality
$\check{y}p=y_{p}$ holds as unbounded operators for any
$p\in{\mathcal{P}}(M)_{1}$. The domain of $\check{y}$ contains dense subspace
of the range of $p$ for every $p\in{\mathcal{P}}(M)_{1}$, so $\check{y}$ is
densely-defined.

######  Claim 3.6.

The operator $\check{y}$ is closable.

###### Proof.

It suffices to verify that if a sequence $h_{n}\in{\mathcal{D}}(\check{y})$
satisfies $h_{n}\to 0$ and $\check{y}h_{n}\to k$ for some $k\in L\otimes K$ in
norm then $k=0$. We know that ${\mathcal{D}}(y_{p})\cap p(L\otimes
H)\subset{\mathcal{D}}(y_{q})\cap q(L\otimes H)$ if
$p,q\in{\mathcal{P}}(M)_{1}$ satisfy $p\leq q$. Hence we may find a sequence
of mutually orthogonal projections $(p_{n})_{n\geq
1}\subset{\mathcal{P}}(M)_{1}$ such that
$h_{n}\in{\mathcal{D}}(y_{p_{1}+\cdots+p_{n}})\cap(p_{1}+\cdots+p_{n})(L\otimes
H)$ for each $n\geq 1$. Since
$y_{p_{1}+\cdots+p_{n}}=y_{p_{1}+\cdots+p_{n}}(p_{1}+\cdots+p_{n})=\overline{y_{p_{1}}+\cdots+y_{p_{n}}}$
as unbounded operators, we may assume with no loss of generality that there
exist $h_{n,1}\in{\mathcal{D}}(y_{p_{1}})\cap p_{1}(L\otimes H)$, …,
$h_{n,n}\in{\mathcal{D}}(y_{p_{n}})\cap p_{n}(L\otimes H)$ such that
$h_{n}=h_{n,1}+\cdots+h_{n,n}$. Note that
$\check{y}h_{n,m}=y_{p_{m}}h_{n,m}\in\Phi(p_{m})(L\otimes K)$ for
$m\in\\{1,\ldots,n\\}$. Fix $j\geq 1$. For $n\geq j$, we obtain

$\begin{split}\Phi\left(\bigvee_{l\geq
j+1}p_{l}\right)^{\perp}\check{y}h_{n}&=\Phi\left(\bigvee_{l\geq
j+1}p_{l}\right)^{\perp}\check{y}(h_{n,1}+\cdots+h_{n,j})\\\
&=\Phi\left(\bigvee_{l\geq
j+1}p_{l}\right)^{\perp}\check{y}(p_{1}+\cdots+p_{j})h_{n}.\end{split}$

Since $y_{p_{1}+\cdots+p_{j}}\in LS(M)$, the unbounded operator
$\Phi\left(\bigvee_{l\geq
j+1}p_{l}\right)^{\perp}\check{y}(p_{1}+\cdots+p_{j})=\Phi\left(\bigvee_{l\geq
j+1}p_{l}\right)^{\perp}y_{p_{1}+\cdots+p_{j}}$ is closable. On the other
hand, we know $h_{n}\to 0$ and $\Phi\left(\bigvee_{l\geq
j+1}p_{l}\right)^{\perp}\check{y}h_{n}\to\Phi\left(\bigvee_{l\geq
j+1}p_{l}\right)^{\perp}k$. Thus we obtain $\Phi\left(\bigvee_{l\geq
j+1}p_{l}\right)^{\perp}k=0$. We have shown that $k$ lies in
$\Phi\left(\bigvee_{l\geq j+1}p_{l}\right)(L\otimes K)$ for each $j\geq 1$. On
the other hand, we know $\bigwedge_{j\geq 1}\bigvee_{l\geq j+1}p_{l}=0$, which
together with the fact that $\Phi$ is a lattice isomorphism shows that
$\bigwedge_{j\geq 1}\Phi\left(\bigvee_{l\geq
j+1}p_{l}\right)=\Phi\left(\bigwedge_{j\geq 1}\bigvee_{l\geq
j+1}p_{l}\right)=0$. Thus we finally arrive at the conclusion $k=0$, and we
see that $\check{y}$ is closable. ∎

Let $\hat{y}$ denote the closure of $\check{y}$. Since $y_{p}\eta M$ for every
$p\in{\mathcal{P}}(M)_{1}$, we may easily obtain $\hat{y}\eta M$. For each
$p\in{\mathcal{P}}(M)_{1}$, we see that $y_{p}$ and $\hat{y}p$ are densely-
defined closed operators with $y_{p}\subset\hat{y}p$. Since $p$ is finite, we
obtain $y_{p}=\hat{y}p$. (This is a consequence of the fact that a closed
densely-defined operator affiliated with a finite von Neumann algebra has no
proper closed extension, see e.g. [9, Exercise 6.9.54].)

For a projection $p\in{\mathcal{P}}(M)$ with $p\prec p_{0}$, we may find a
projection $p_{1}\in{\mathcal{P}}(M)_{0}$ with $p\leq p_{1}$. Fix a partial
isometry $v_{1}\in M$ with $v_{1}^{*}v_{1}=p_{0}$ and $v_{1}v_{1}^{*}=p_{1}$.
Since $y_{p_{1}}=\Psi(v_{1})v_{1}^{*}$, we see that

$l(\hat{y}p)=l(y_{p_{1}}p)=l(\Psi(v_{1})v_{1}^{*}p)=l(\Psi(v_{1})v_{1}^{*}pv_{1})=l(\Psi(v_{1}v_{1}^{*}pv_{1}))=l(\Psi(pv_{1}))=\Phi(p).$

Thus we obtain $l(\hat{y}p)=\Phi(p)$ for every $p\in{\mathcal{P}}(M)$ with
$p\prec p_{0}$.

Let $\hat{y}=v\lvert\hat{y}\rvert=\lvert\hat{y}^{*}\rvert v$ be the polar
decomposition of $\hat{y}$. Then $v$ is a partial isometry in $M$. The fact
that $r(y_{p})=p$ and $l(y_{p})=\Phi(p)$ for $p\in{\mathcal{P}}(M)_{1}$ shows
that $v$ satisfies $v^{*}v=1\in M$ and $vv^{*}=1\otimes p_{K}\in
M_{0}{\mathbin{\overline{\otimes}}}B(H)$, where $p_{K}\in{\mathcal{P}}(B(H))$
denotes the projection onto $K$. Thus $v$ determines a ∗-isomorphism $M\ni
x\mapsto vxv^{*}\in N$.

######  Claim 3.7.

$\lvert\hat{y}\rvert\in LS(M)$.

###### Proof.

Assume that $\lvert\hat{y}\rvert$ is not in $LS(M)$. By Corollary 2.2, we may
take a strictly increasing sequence $c_{n}$ of positive real numbers and a
sequence $f_{n}\in{\mathcal{P}}(M)$ of pairwise equivalent nonzero projections
with the following properties: (i) $c_{n}\to\infty$ as $n\to\infty$. (ii)
$f_{n}\leq\chi_{[c_{n},c_{n+1})}(\lvert\hat{y}\rvert)$. By taking
subprojections of $f_{n}$ if necessary, we may additionally assume that
$f_{n}\prec p_{0}$ for every $n\geq 1$. We may take a strictly increasing
sequence of positive integers $(n_{k})_{k\geq 1}\subset\mathbb{N}$ such that
$c_{n_{2k+1}}/c_{n_{2k}+1}\to\infty$. Take a sequence of partial isometries
$v_{k}\in M$ such that $v_{k}^{*}v_{k}=f_{n_{2k}}$ and
$v_{k}v_{k}^{*}=f_{n_{2k+1}}$ for each $k\geq 1$.

Set $g_{k}:=(f_{n_{2k}}+v_{k}+v_{k}^{*}+f_{n_{2k+1}})/2$ and
$e_{k}:=f_{n_{2k+1}}$. Then $(g_{k})_{k\geq 1}$, $(e_{k})_{k\geq 1}$ are
sequences of mutually orthogonal equivalent projections. Set
$g:=\bigvee_{k\geq 1}g_{k}$ and $e:=\bigvee_{k\geq 1}e_{k}$. It is clear that
$g$ and $e$ are mutually LS-orthogonal. We claim that $\lVert
l(\lvert\hat{y}\rvert g_{k})l(\lvert\hat{y}\rvert
e_{k})^{\perp}l(\lvert\hat{y}\rvert g_{k})\rVert\to 0$ as $k\to\infty$. To see
this, observe that each unit vector of the range of the operator
$\lvert\hat{y}\rvert g_{k}$ is of the form $\lvert\hat{y}\rvert
h+\lvert\hat{y}\rvert v_{k}h$ for some $h\in f_{n_{2k}}(L\otimes H)$. Since
$\lvert\hat{y}\rvert
h=\lvert\hat{y}\rvert\chi_{[c_{n_{2k}},c_{n_{2k}+1})}(\lvert\hat{y}\rvert)h$
and $\lvert\hat{y}\rvert
v_{k}h=\lvert\hat{y}\rvert\chi_{[c_{n_{2k+1}},c_{n_{2k+1}+1})}(\lvert\hat{y}\rvert)v_{k}h$,
we obtain $\lvert\hat{y}\rvert h\perp\lvert\hat{y}\rvert v_{k}h$,
$\lVert\lvert\hat{y}\rvert h\rVert\leq c_{n_{2k}+1}\rVert h\rVert$, and
$\lVert\lvert\hat{y}\rvert v_{k}h\rVert\geq c_{n_{2k+1}}\rVert h\rVert$.
Therefore, if $c_{n_{2k+1}}/c_{n_{2k}+1}$ is large, then the unit vector
$\lvert\hat{y}\rvert h+\lvert\hat{y}\rvert v_{k}h$ is close to the unit vector
$\lvert\hat{y}\rvert v_{k}h/\lVert\lvert\hat{y}\rvert v_{k}h\rVert$ which lies
in the range of $\lvert\hat{y}\rvert e_{k}$. This together with an application
of Subsection 2.3 to the pair $l(\lvert\hat{y}\rvert
g_{k}),l(\lvert\hat{y}\rvert e_{k})$ shows that $\lVert l(\lvert\hat{y}\rvert
g_{k})l(\lvert\hat{y}\rvert e_{k})^{\perp}l(\lvert\hat{y}\rvert
g_{k})\rVert\to 0$ as $k\to\infty$.

Since $g_{k}\prec p_{0}$, we obtain $l(\lvert\hat{y}\rvert
g_{k})=v^{*}l(v\lvert\hat{y}\rvert g_{k})v=v^{*}\Phi(g_{k})v$, and similarly,
$l(\lvert\hat{y}\rvert e_{k})=v^{*}\Phi(e_{k})v$ for each $k$. Therefore, we
obtain

$\begin{split}\lVert\Phi(g_{k})\Phi(e_{k})^{\perp}\Phi(g_{k})\rVert&=\lVert
v^{*}\Phi(g_{k})v\cdot(v^{*}\Phi(e_{k})v)^{\perp}\cdot
v^{*}\Phi(g_{k})v\rVert\\\ &=\lVert l(\lvert\hat{y}\rvert
g_{k})l(\lvert\hat{y}\rvert e_{k})^{\perp}l(\lvert\hat{y}\rvert
g_{k})\rVert\to 0.\end{split}$

Since $\Phi(e)\geq\Phi(e_{k})$, we obtain

$0\leq\Phi(g_{k})\Phi(e)^{\perp}\Phi(g_{k})\leq\Phi(g_{k})\Phi(e_{k})^{\perp}\Phi(g_{k}),$

which shows that $\lVert\Phi(g_{k})\Phi(e)^{\perp}\Phi(g_{k})\rVert\to 0$ as
$k\to\infty$. Note that $(\Phi(g_{k}))_{k\geq 1}$ is a sequence of pairwise
equivalent subprojections of $\Phi(g)$. (Indeed, for a pair $i\neq j$ of
positive integers, we may find a partial isometry $w\in M$ such that
$w^{*}w=g_{i}$ and $ww^{*}=g_{j}$. Then
$g^{\prime}=(g_{i}+w+w^{*}+g_{j})/2+(1-g_{i}-g_{j})\in{\mathcal{P}}(M)$
satisfies $g_{i}\wedge g^{\prime}=g_{j}\wedge g^{\prime}=0$ and $g_{i}\vee
g^{\prime}=g_{j}\vee g^{\prime}=1$. It follows that
$\Phi(g_{i})\wedge\Phi(g^{\prime})=\Phi(g_{j})\wedge\Phi(g^{\prime})=0$ and
$\Phi(g_{i})\vee\Phi(g^{\prime})=\Phi(g_{j})\vee\Phi(g^{\prime})=1$.
Subsection 2.3 applied to the pair $\Phi(g_{i}),\Phi(g^{\prime})$ implies
$\Phi(g_{i})\sim\Phi(g^{\prime})^{\perp}$, and similarly, we obtain
$\Phi(g_{j})\sim\Phi(g^{\prime})^{\perp}$, thus $\Phi(g_{i})\sim\Phi(g_{j})$.)
Moreover, by [11, Lemma 3.6], the projections $\Phi(g)$, $\Phi(e)$ in $N$ are
LS-orthogonal. This contradicts Corollary 2.3. Therefore, we arrive at the
conclusion that $\lvert\hat{y}\rvert\in LS(M)$. ∎

By applying Lemma 2.1 and imitating the above discussion, we may also show
that $\lvert\hat{y}\rvert$ is invertible in $LS(M)$. Thus
$\lvert\hat{y}\rvert$ is an operator in $LS(M)$ that is invertible in $LS(M)$.
Therefore, we may define a ring isomorphism $\Psi_{1}\colon LS(M)\to LS(N)$ by
$\Psi_{1}(x):=v\lvert\hat{y}\rvert x\rvert\hat{y}\rvert^{-1}v^{*}$. Then the
corresponding lattice isomorphism $\Phi_{1}$ satisfies
$\Phi_{1}(p)=l(v\lvert\hat{y}\lvert p)=l(\hat{y}p)$ for every
$p\in{\mathcal{P}}(M)$. It follows that $\Phi(p)=l(\hat{y}p)=\Phi_{1}(p)$ for
every $p\in{\mathcal{P}}(M)$ with $p\prec p_{0}$. Since every projection in
${\mathcal{P}}(M)$ can be written as the least upper bound of some family in
$\\{p\in{\mathcal{P}}(M)\mid p\prec p_{0}\\}$, the two lattice isomorphisms
$\Phi$ and $\Phi_{1}$ coincide on ${\mathcal{P}}(M)$. Therefore, the
corresponding ring isomorphisms $\Psi$ and $\Psi_{1}$ coincide, i.e.,
$\Psi(x)=v\lvert\hat{y}\rvert x\lvert\hat{y}\rvert^{-1}v^{*}$ for every $x\in
LS(M)$. It follows that $\Psi$ is similar to a ∗-isomorphism. This completes
the proof of Proposition 3.4 and hence that of Theorem 1.2.

##  4\. Remarks

Note that the operator $y$ in Theorems 1.2–1.4 has a polar decomposition
$y=u\lvert y\rvert=\lvert y^{*}\rvert u$ such that $u\in N$ is a unitary
operator. Thus we may take the pair $(u\psi(\cdot)u^{*},\lvert y^{*}\rvert)$
instead of $(\psi,y)$. This implies that one may take _positive_ invertible
$y\in LS(N)$ in these theorems. Now let us discuss the uniqueness of
$(\psi,y)$ with positive $y$.

######  Lemma 4.1.

Let $M,N$ be von Neumann algebras. Let $\psi_{1},\psi_{2}\colon M\to N$ be
real ∗-isomorphisms, and let $y_{1},y_{2}$ be positive invertible operators in
$LS(N)$. Then the equality
$y_{1}\psi_{1}(x)y_{1}^{-1}=y_{2}\psi_{2}(x)y_{2}^{-1}$ holds for every $x\in
LS(M)$ if and only if $\psi_{1}=\psi_{2}$ and there is a positive invertible
operator $z\in LS({\mathcal{Z}}(N))$ such that $y_{2}=zy_{1}$.

###### Proof.

The equality $y_{1}\psi_{1}(x)y_{1}^{-1}=y_{2}\psi_{2}(x)y_{2}^{-1}$ means

$y_{2}^{-1}y_{1}\psi_{1}(x)(y_{2}^{-1}y_{1})^{-1}=\psi_{2}(x)=\psi_{2}\circ\psi_{1}^{-1}(\psi_{1}(x)).$

Put $y:=y_{2}^{-1}y_{1}$, and $\psi:=\psi_{2}\circ\psi_{1}^{-1}\colon N\to N$.
Then we have $\psi(a)=yay^{-1}$ for every $a\in LS(N)$. Let $y=u\lvert
y\rvert$ be the polar decomposition of $y$. Then $u$ is a unitary operator in
$N$. We show that $\lvert y\rvert$ lies in the center $LS({\mathcal{Z}}(N))$
of $LS(N)$. If $\lvert y\rvert$ is not central, then the spectral theorem
implies that $\chi_{[0,c)}(\lvert y\rvert)$ is not central for some $c>0$. We
may take a nonzero partial isometry $v\in N$ such that
$vv^{*}\leq\chi_{[0,c)}(\lvert y\rvert)$ and $v^{*}v\leq\chi_{[0,c)}(\lvert
y\rvert)^{\perp}=\chi_{[c,\infty)}(\lvert y\rvert)$. Then $\psi(v)=u\lvert
y\rvert v\lvert y\rvert^{-1}u^{*}$ is not a partial isometry. Indeed, we have
$u^{*}\psi(v)u=\lvert y\rvert v\lvert y\rvert^{-1}=\lvert
y\rvert\chi_{[0,c)}(\lvert y\rvert)v\chi_{[c,\infty)}(\lvert y\rvert)\lvert
y\rvert^{-1}$, and we may check that this is a nonzero bounded operator with
norm at most one such that $\lVert u^{*}\psi(v)uh\rVert<\lVert h\rVert$ for
every nonzero vector $h$. This contradicts the assumption that
$\psi_{1},\psi_{2}$ are real ∗-automorphisms. (Recall that a real
∗-isomorphism is the direct sum of a ∗-isomorphism and a conjugate-linear
∗-isomorphism.) Thus we see that $\lvert y\rvert$ is central. By the
uniqueness of the polar decomposition of $y_{1}=y_{2}y=(y_{2}\lvert y\rvert)u$
together with the assumption that $y_{1},y_{2}$ are positive, we obtain $u=1$
and $y=\lvert y\rvert\in LS({\mathcal{Z}}(N))$. It follows that $\psi$ is the
identity mapping, and we obtain the desired conclusion. ∎

We stated Theorem 1.4 in terms of real ∗-isomorphisms, but we may equivalently
use Jordan ∗-isomorphisms. Recall that a complex-linear bijection $J\colon
A\to B$ between ∗-algebras $A,B$ is called a _Jordan ∗-isomorphism_ if it
satisfies $J(x^{*})=J(x)^{*}$ and $J(x^{2})=J(x)^{2}$ for every $x\in A$. It
is well-known that a Jordan ∗-isomorphism between two von Neumann algebras can
be decomposed into the direct sum of a ∗-isomorphism and a ∗-antiisomorphism.
More precisely, if $J\colon M\to N$ is a Jordan ∗-isomorphism between von
Neumann algebras $M,N$, then there are central projections
$r\in{\mathcal{P}}({\mathcal{Z}}(M))$ and
$z\in{\mathcal{P}}({\mathcal{Z}}(N))$, a ∗-isomorphism $\psi_{1}\colon rM\to
zN$, and a ∗-antiisomorphism $\psi_{2}\colon r^{\perp}M\to z^{\perp}N$ such
that $J(x)=\psi_{1}(rx)+\psi_{2}(r^{\perp}x)$ for every $x\in M$, see e.g. [9,
Exercise 10.5.26]. For such a decomposition, notice that the map $\psi\colon
M\ni x\mapsto\psi(x):=\psi_{1}(rx)+\psi_{2}(r^{\perp}x)^{*}\in N$ is a real
∗-isomorphism. This gives a one-to-one correspondence between Jordan
∗-isomorphisms and real ∗-isomorphisms. Note that $J$ and $\psi$ coincide on
the self-adjoint part of $M$, and in particular, on ${\mathcal{P}}(M)$.
Therefore, we may replace the map $\psi$ in the statement of Theorem 1.4 with
a Jordan ∗-isomorphism.

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