Partial Orders in Non-commutative $L^p$ Spaces Associated with Semi-finite von Neumann Algebras

Abstract

Let ${\mathscr {M}}$ be a semi-finite von Neumann algebra with a faithful normal semi-finite trace $\tau$ and $L^{p}({\mathscr {M}})$ the non-commutative $L^p$ space associated with $({\mathscr {M}},\tau )$. We extend star partial order $\overset{*}{\le }$ and diamond order $\le ^{\diamond }$ to $L^{p}({\mathscr {M}})$ and present some properties about several bounds of these two partial orders. We establish a result about the existence of the star infimum and supremum. We also prove that a subset with an upper bound must have a minimal upper bound in $L^p({\mathscr {M}})$ under the diamond order in the case of finite von Neumann algebra ${\mathscr {M}}$. However, we give an example and show that this result may fail if ${\mathscr {M}}$ is not finite. Moreover, we characterize the forms of all norm closed hereditary subspaces in $L^{p}({\mathscr {M}})$ under these two partial orders.

1 Introduction

In past decades, several partial orders on matrix algebras and operator algebras have been considered. For example, star partial order as well as diamond order were introduced in [2, 7] and studied by many authors(cf. [1, 3, 5, 6, 8, 11, 14, 15]). As we know, the definition of star partial order in the algebra ${\mathscr {B}}({\mathscr {H}})$ of all bounded linear operators on a complex Hilbert space ${\mathscr {H}}$ is given by :

$$\begin{aligned} A\overset{*}{\le }B\Leftrightarrow A^{*}A=A^{*}B,~AA^{*}=BA^{*}\Leftrightarrow A=P_{A}B=BQ_{A}, \end{aligned}$$

where $P_{A}$ and $Q_{A}$ denote the left as well as right support projections of A respectively. In [3, 5], the star supremum and infimum were studied and the forms of star partial order-hereditary subspaces in ${\mathscr {B}}({\mathscr {H}})$ were characterized in [14]. Very recently, we given a type decomposition for operators in ${\mathscr {B}}({\mathscr {H}})$ with respect to the star partial order and characterized the star partial order automorphisms on the class of type 1 operators in [11]. On the other hand, Cīrulis in [2] introduced the diamond order in strong Rickart rings and the definition of it in ${\mathscr {B}}({\mathscr {H}})$ was given by:

$$\begin{aligned}&A \le ^{\diamond } B\Leftrightarrow \overline{R(A)} \subseteq \overline{R(B)},\\&\overline{R(A^{*})} \subseteq \overline{R(B^{*})},~AA^{*}A = AB^{*}A\Leftrightarrow A=P_{A}BQ_{A}=P_{B}AQ_{B}. \end{aligned}$$

We characterized the form of order automorphisms under the diamond order $\le ^{\diamond }$ on the set of all products of two orthogonal projections in [10] and bounds for the diamond order in ${\mathscr {B}}({\mathscr {H}})$ were considered in [13].

We note that these two partial orders may extend to von Neumann algerbas as well as some unbounded operator spaces such as $L^p$ spaces associated with semi-finite von Neumann algebras. For example, the star partial order in a von Neumann algebra was considered in [15]. We now extend these two partial orders to $L^p({\mathscr {M}})(1\le p<\infty )$ for a semi-finite von Neumann algebra ${\mathscr {M}}$ with a faithful normal semi-finite trace $\tau$.

Next, we recall some notions. Let ${\mathscr {M}}$ be a von Neumann algebra on ${\mathscr {H}}$ with a faithful normal semi-finite trace $\tau$ and $L^p({\mathscr {M}})(1\le p<\infty )$ the non-commutative $L^p$ space associated with $\tau$(cf. [12]). Denote by ${\mathscr {M}}_p$ the set of all the projections in ${\mathscr {M}}$. For any $A\in {\mathscr {M}}_p$, we denote by $C_A$ the central carrier of A. Let $E,~F\in {\mathscr {M}}_p$, $[E,F]=\{U\in {\mathscr {M}}: UU^{*}\le E,~U^{*}U\le F\}$. Moreover, For any $A\in L^p({\mathscr {M}})$, we have $\Vert A\Vert =(\tau (|A|^p))^{1/p}$ and $P_{A}$, $Q_{A}$ denote the left as well as right support projections of A in ${\mathscr {M}}$ respectively, that is, $P_A=\inf \{P\in {\mathscr {M}}_p:PA=A\}$ and $Q_A=\inf \{ Q\in {\mathscr {M}}_p: AQ=A\}$. If $A=U|A|$ is the polar decomposition of A, then we have $P_A=UU^*$ and $Q_A=U^*U$.

2 The Star Partial Order in $L^p({\mathscr {M}})$

In this section, we extend the notion of the star partial order to non-commutative $L^p$ spaces. We present some properties about the bounds of star partial order and then characterize forms of all hereditary subspaces with respect to star partial order.

As we know, in ${\mathscr {B}}({\mathscr {H}})$, $A\overset{*}{\le }B\Leftrightarrow A^{*}A=A^{*}B,~AA^{*}=BA^{*}\Leftrightarrow A=P_{A}B=BQ_{A}$. We give the definition of star partial order in $L^p({\mathscr {M}})$ by the same way.

Definition 2.1

Let $A, B\in L^p({\mathscr {M}})$, we say $A\overset{*}{\le }B$ if $A^{*}A=A^{*}B$ and $AA^{*}=BA^{*}$. $\overset{*}{\le }$ is said to be the star partial order of $L^p({\mathscr {M}})$.

We easily know that $\overset{*}{\le }$ is a partial order in $L^p({\mathscr {M}})$ and $A \overset{*}{\le }B \Longleftrightarrow A=P_{A}B=BQ_{A}$. In [1, Lemma 2.6], Antezana gave a important result which has been used in many problems about bounds of star partial order. In non-commutative $L^p$ spaces, we also have a similar result.

Definition 2.2

Let $B\in L^p({\mathscr {M}})$.

• (1) If $P\in {\mathscr {M}}_p$ and $P\le P_{B}$, then $PB\overset{*}{\le }B\Leftrightarrow PBB^{*}=BB^{*}P$.

• (2) If $Q\in {\mathscr {M}}_p$ and $Q\le Q_{B}$, then $BQ\overset{*}{\le }B\Leftrightarrow QB^{*}B=B^{*}BQ$.

Proof

(1) Put $A=PB$. If $A\overset{*}{\le }B$, then $PBB^{*}=AB^{*}=BA^{*}=BB^{*}P$. Conversely, if $PBB^{*}=BB^{*}P$, then $BA^{*}=BB^{*}P=PBB^{*}P=AA^{*}$ and $A^{*}A=B^{*}PPB=B^{*}PB=B^{*}A$. Thus $A\overset{*}{\le }B$.

(2) The proof is similar to (1). $\square$

In [3], the star lower as well as star upper bounds were characterized. Djikic gave a necessary and sufficient condition for the existence of star supremum of two arbitrary operators on a Hilbert space in [5]. In the next theorem, we present some properties about the star infimum and star supremum and prove that if a subset of $L^p({\mathscr {M}})$ has a upper bound, then it must have supremum in $L^p({\mathscr {M}})$. Let ${\mathscr {A}}\subseteq L^p({\mathscr {M}})$ and $D\in L^p({\mathscr {M}})$. If $A\overset{*}{\le }D$ for all $A\in {\mathscr {A}}$, then we say that D is an upper bound of ${\mathscr {A}}$. If D is an upper bound of ${\mathscr {A}}$ and for any upper bound C of ${\mathscr {A}}$, we have that $D\overset{*}{\le }C$, then we say that D is the supremum of ${\mathscr {A}}$ and denote by $\sup {\mathscr {A}}$. We similarly may define lower bounds and the infimum $\inf {\mathscr {A}}$ of ${\mathscr {A}}$.

Theorem 2.3

Let ${\mathscr {A}}\subseteq L^p({\mathscr {M}})$ be a nonempty subset.

• (1) There exists the infimum $\inf {\mathscr {A}}$ in $L^p({\mathscr {M}})$.

• (2) If there is a upper bound $D\in L^p({\mathscr {M}})$, then there exists the supremum $\sup {\mathscr {A}}$ in $L^p({\mathscr {M}})$.

Proof

(1) Put $P_{0}=\inf \{P_{A}: A\in {\mathscr {A}}\}$, $Q_{0}=\inf \{Q_{A}: A\in {\mathscr {A}}\}$ and

$$\begin{aligned} {\mathscr {P}}=\{P\in {\mathscr {M}}_p: P\le P_{0}, P|A^*|=|A^*|P \text{ and } PA=PB, \forall A, B\in {\mathscr {A}}\}. \end{aligned}$$

For any $P\in {\mathscr {P}}$ and $A\in {\mathscr {A}}$, since $P|A^*|=|A^*|P$, we have $PAA^{*}=AA^{*}P$. It follows that $PAA^{*}P=AA^{*}P$. Put $G=PA$, then $GG^{*}=AG^{*}$, $G^{*}G=A^{*}PPA=A^{*}PA=G^{*}A$. This means that $G=PA \overset{*}{\le }A$ for all $A\in {\mathscr {A}}$. That is, $G=PA$ is a lower bound of ${\mathscr {A}}$. Note that $G=PA$ and $P\in {\mathscr {P}}$, we can see that

$$\begin{aligned} \forall A, B\in {\mathscr {A}},~PA=PB \end{aligned}$$

by the definition of ${\mathscr {P}}$. This implies that G is independent of the choice of A.

Put $P_1=\sup \{P:P\in {\mathscr {P}}\}$. Then $P_1\in {\mathscr {P}}$. Put $D=P_1A$ for all $A\in {\mathscr {A}}$. Similarly, we can see that D is independent of the choice of A and is a lower bound of ${\mathscr {A}}$. Now for any $C\overset{*}{\le }A$ for all $A\in {\mathscr {A}}$, we have that $C=P_{C}A=AQ_{C}$ for all $A\in {\mathscr {A}}$. This implies that $P_{C}\le P_{A}$ and $P_{C}A=P_{C}B$ for all $A, B \in {\mathscr {A}}$ and hence $P_{C}\le P_{0}$. On the other hand, since $CC^*=CA^{*}$, $P_{C}AA^*P_{C}=CC^*=CA^{*}=P_{C}AA^*$ for all $A\in {\mathscr {A}}$. It follows that $P_{C}AA^*=AA^*P_{C}$, that is, $P_{C}|A^*|=|A^*|P_{C}$. Thus $P_{C}\in {\mathscr {P}}$. This implies that $C=P_{C}A=P_{C}P_{1}A=P_{C}D$. It is elementary that $C\overset{*}{\le }D$. Thus $D=\inf {\mathscr {A}}$. (2) Put ${\mathscr {D}}=\{D\in L^p({\mathscr {M}}): A \overset{*}{\le } D,\ \forall A\in {\mathscr {A}}\}$ and $M=\inf {\mathscr {D}}$. We claim that $\sup {\mathscr {A}}=M$. Note that for any $A\in {\mathscr {A}}$, $A \overset{*}{\le }D , \forall D\in {\mathscr {D}}$. This means that for any $A\in {\mathscr {A}}$, A is a lower bound of ${\mathscr {D}}$ and thus $A\overset{*}{\le }M$, $\forall A\in {\mathscr {A}}$. Hence $M\in {\mathscr {D}}$ and $M=\sup {\mathscr {A}}$. $\square$

We recall that a subspace ${\mathscr {A}}\subseteq {\mathscr {M}}$ is said to be star partial order-hereditary if for any $B\in {\mathscr {M}}$ and $A\in {\mathscr {A}}$, we have $B\in {\mathscr {A}}$ whenever $B\overset{*}{\le } A$(cf. [14, 15]). In [15], all forms of $\sigma -$weakly closed star partial order-hereditary subspaces in a von Neumann algebra ${\mathscr {M}}$ were given. We may consider star partial order-hereditary subspaces in $L^p({\mathscr {M}})$ by the same way. Next, we characterize the form of norm closed star partial order-hereditary subspace in $L^{p}({\mathscr {M}})$. We note that the idea follows from [15]. We need to treat unbounded operators in $L^p({\mathscr {M}})$ in this case, so we still give the detailed proof.

Lemma 2.4

Let ${\mathscr {A}}$ be a norm closed star partial order-hereditary subspace of $L^{p}({\mathscr {M}})$. Then $P_{T}L^{p}({\mathscr {M}})Q_{T}\subseteq {\mathscr {A}}$ for all $T\in {\mathscr {A}}$.

Proof

Let $T=W|T|$ be the polar decomposition of T and $|T|=\int _{0}^{+\infty }\lambda dE_{\lambda }$ the spectral decomposition of |T|. Then we have $W\in {\mathscr {M}}$ and $|T|\in L^{p}({\mathscr {M}})$. Note that $\forall \varepsilon > 0$, $\varepsilon E[\varepsilon ,+\infty )\le E[\varepsilon ,+\infty )|T|\le |T|$. It follows that

$$\begin{aligned} \varepsilon ^p\tau ((E[\varepsilon , +\infty ))\le \tau (E[\varepsilon ,+\infty )|T|^p)\le \tau (|T|^p)< +\infty \end{aligned}$$

and therefore $\tau (E[\varepsilon , +\infty ))<\infty$. It follows that $E[\varepsilon ,+\infty )\mathscr {M}E[\varepsilon ,+\infty )\subseteq L^p({\mathscr {M}})$.

For any bounded Borel subset $\Delta \subseteq [\varepsilon , +\infty )$, we have $WE(\Delta )|T|\overset{*}{\le } T$. Then we easily see that $WE(\Delta )|T| \in {\mathscr {A}}$. This implies that $W\sum _{k=1}^ma_kE(\Delta _k)|T|\in {\mathscr {A}}$ for any complex $a_k\in {\mathbb {C}}$ and pairwise disjoint bounded Borel subsets $\Delta _k$ for any $1 \le k\le m$. $\forall a>\varepsilon$ and let f be a continuous function on $[\varepsilon ,a]$. It follows that $WE[\varepsilon , a]f(|T|)|T| \in {\mathscr {A}}$. Put $f(\lambda )=\lambda ^{-1}$ on $[\varepsilon ,a]$, then $WE[\varepsilon , a]\in {\mathscr {A}}$. For any projection $E\in {\mathscr {M}}_p$ such that $E\le E[\varepsilon , a]$, it is easy to know that $WE \overset{*}{\le } WE[\varepsilon , a]$. Then $WE \in {\mathscr {A}}$. This means that $WE[\varepsilon , a]{\mathscr {M}}E[\varepsilon , a]\subseteq {\mathscr {A}}$. Since ${\mathscr {A}}$ is norm closed and $E[\varepsilon , a]{\mathscr {M}}E[\varepsilon , a]$ is dense in $E[\varepsilon , a]L^p({\mathscr {M}})E[\varepsilon , a]$ in norm topology, $WE[\varepsilon , a]L^{p}({\mathscr {M}})E[\varepsilon , a]\subseteq {\mathscr {A}}$. Note that $Q_{T}=W^*W=E(0, +\infty )$. Putting $\varepsilon \rightarrow 0$ and $a\rightarrow +\infty$, we have $WL^{p}({\mathscr {M}})Q_{T}\subseteq {\mathscr {A}}$. Note that $P_T=WW^*$. Thus

$$\begin{aligned} P_{T}L^{p}({\mathscr {M}})Q_{T}=P_{T}WL^{p}({\mathscr {M}})Q_{T}=WL^{p}({\mathscr {M}})Q_{T}\subseteq {\mathscr {A}}. \end{aligned}$$

$\square$

Next, we characterize the form of a norm closed star partial order-hereditary subspace of $L^{p}({\mathscr {M}})$.

Theorem 2.5

Let ${\mathscr {A}}$ be a norm closed star partial order-hereditary subspace in $L^{p}({\mathscr {M}})$. Then there exists unique projection pair (E, F) with the same central carriers such that ${\mathscr {A}}=EL^{p}({\mathscr {M}})F$.

Proof

(i) Suppose ${\mathscr {A}}=EL^{p}({\mathscr {M}})F=E_{0}L^{p}({\mathscr {M}})F_{0}$ and $C_{E}=C_{F}, C_{E_{0}}=C_{F_{0}}$, then we have $E(L^{p}({\mathscr {M}})\bigcap {\mathscr {M}})F=E_{0}(L^{p}({\mathscr {M}})\bigcap {\mathscr {M}})F_{0}$. Clearly, $\overline{E(L^{p}({\mathscr {M}})\bigcap {\mathscr {M}})F}^{\sigma -w}=\overline{E_{0}(L^{p}({\mathscr {M}})\bigcap {\mathscr {M}})F_{0}}^{\sigma -w}$, that is, $E{\mathscr {M}}F=E_{0}{\mathscr {M}}F_{0}$. It follows from [15, Theorem 3.2] that $E=E_0$ and $F=F_0$.

(ii) If ${\mathscr {A}}=0$, then the conclusion is obviously true. Now suppose ${\mathscr {A}}\ne 0$. Put

$$\begin{aligned} \Lambda =\{(P,Q): P, Q \in {\mathscr {M}}_p,~ C_{P}=C_{Q},~ PL^{p}({\mathscr {M}})Q\subseteq {\mathscr {A}}\}. \end{aligned}$$

Since $(0,0)\in {\mathscr {M}}$, $\Lambda \ne \emptyset$. Define a partial order on $\Lambda$ by: $(P_{1}, Q_{1})\le (P_{2}, Q_{2})\Leftrightarrow P_{1}\le P_{2}$ and $Q_{1}\le Q_{2}$. Let $\{(P_{\alpha }, Q_{\alpha })\}_{\alpha \in I}$ be a totally ordered family of $\Lambda$, then $\{P_{\alpha }\}_{\alpha \in I}$ and $\{Q_{\alpha }\}_{\alpha \in I}$ are monotone totally ordered sets respectively. Assume that $P_{\alpha }\rightarrow P$ and $Q_{\alpha }\rightarrow Q$ $\sigma$-weakly. It is clear that $\{C_{P_{\alpha }}\}_{\alpha \in I}$ and $\{C_{Q_{\alpha }}\}_{\alpha \in I}$ are also monotone totally ordered sets and $C_{P_{\alpha }}\rightarrow C_{P},~C_{Q_{\alpha }}\rightarrow C_{Q}$ $\sigma$-weakly. Obviously, $C_{P}=C_{Q}$. As $P_{k}L^{p}({\mathscr {M}})Q_{k}\subseteq P_{l}L^{p}({\mathscr {M}})Q_{l}\subseteq {\mathscr {A}},~\forall k< l$, it is easy to see that $PL^{p}({\mathscr {M}})Q\subseteq {\mathscr {A}}$. Thus $(P,Q)\in \Lambda$. By Zorn’s lemma, there is a maximal element (E, F) in $\Lambda$. Then we have $EL^{p}({\mathscr {M}})F\subseteq {\mathscr {A}}$.

Next, we prove that ${\mathscr {A}} \subseteq EL^{p}({\mathscr {M}})F$. We give two assertions firstly.

Assertion 1 Let $(P, Q)\in \Lambda$. If $P\le I-E,~ Q\le I-F$, then $P=Q=0$.

Suppose that $P \ne 0$. Take any monotonically increasing $\tau -$finite projection families $\{P_{i}\}_{i\in I}$ and $\{Q_{i}\}_{i\in I}$ in ${\mathscr {M}}_p$ such that $P_{i} \rightarrow P$ and $Q_{i} \rightarrow Q$ $\sigma -$weakly. If $C_{P_{i} } \ne C_{Q_{i} }$ for some $i\in I$, then $\{P_{i} '\}_{i\in I}$ and $\{Q_{i} '\}_{i\in I}$ are also monotonically increasing $\tau -$finite projection families with $C_{P_{i} '}=C_{Q_{i} '}$ by putting $P_{i} '=P_{i} C_{Q_{i} }$ and $Q_{i} '=Q_{i} C_{P_{i} }$. It is easy to check that $P_{i} '\rightarrow PC_{Q}=PC_{P}=P$ and $Q_{i} '\rightarrow QC_{P}=QC_{Q}=Q$ $\sigma -$weakly. Without loss of generality, we may assume that $C_{P_{i} }= C_{Q_{i} }$ for all $i\in I$. Similarly, let $\{E_{j}\}_{j\in J}$ and $\{F_{j}\}_{j\in J}$ be monotonically increasing $\tau -$finite projection families satisfying $E_{j}\rightarrow E$, $F_{j}\rightarrow F$ $\sigma -$weakly and $C_{E_{j}}= C_{F_{j}}$ for any $j\in J$. Since all of these projections are $\tau -$finite, we have $[P_{i} , Q_{i} ]\subseteq P_i\mathscr {M}Q_i\subseteq P_{i} L^{p}({\mathscr {M}})Q_{i} \subseteq PL^{p}({\mathscr {M}})Q\subseteq {\mathscr {A}}$ and $[E_{j}, F_{j}]\subseteq E_j\mathscr {M}F_j\subseteq E_{j}L^{p}({\mathscr {M}})F_{j}\subseteq EL^{p}({\mathscr {M}})F\subseteq {\mathscr {A}}$.

For any $U\in [E_{j}, F_{j}]$ and $V\in [P_{i} , Q_{i} ]$, we have $U+V\in {\mathscr {A}}$. It follows from Lemma 2.4 that $P_{U+V}L^{p}({\mathscr {M}})Q_{U+V}\subseteq {\mathscr {A}}$. It is easy to see that $P_{U}\le E_{j},~P_{V}\le P_{i} \le P \le I-E\le I-E_{j},~Q_{U}\le F_{j},~Q_{V}\le Q_{i} \le Q \le I-F \le I-F_{j}$, then $P_{U}\bot P_{V},~Q_{U}\bot Q_{V}$. Similarly with [15, Theorem 3.2], $\forall T\in L^{p}({\mathscr {M}})$, one obtains $P_{U}TQ_{V}+P_{V}TQ_{U}\in {\mathscr {A}}$ and

$$\begin{aligned} P_{U}TQ_{V}\overset{*}{\le }P_{U}TQ_{V}+P_{V}TQ_{U}, \\ P_{V}TQ_{U}\overset{*}{\le }P_{U}TQ_{V}+P_{V}TQ_{U}. \end{aligned}$$

Since ${\mathscr {A}}$ is star partial order-hereditary, $P_{U}TQ_{V},~P_{V}TQ_{U}\in {\mathscr {A}}$. By the arbitrariness of U, V, T, one obtains $E_{j}L^{p}({\mathscr {M}})Q_{i} \subseteq {\mathscr {A}},~P_{i} L^{p}({\mathscr {M}})F_{j}\subseteq {\mathscr {A}}$, so $(E_{j}+P_{i} )L^{p}({\mathscr {M}})(F_{j}+Q_{i} )\subseteq {\mathscr {A}}$.

Put $E_{j}'=(E_{j}+P_{i} )C_{(F_{j}+Q_{i} )},~F_{j}'=(F_{j}+Q_{i} )C_{(E_{j}+P_{i} )}$, then $E_{j}'L^{p}({\mathscr {M}})F_{j}'\subseteq {\mathscr {A}}$. Since $E_{j}\le C_{E_{j}}=C_{F_{j}}\le C_{(F_{j}+Q_{i} )}$ and $F_{j}\le C_{F_{j}}=C_{E_{j}}\le C_{(E_{j}+P_{i} )}$, it follows that $E_{j}\le E_{j}',~F_{j}\le F_{j}'$. Assume that $E_{j}'\rightarrow E'$ and $F_{j}'\rightarrow F'$ $\sigma -$weakly. Then $E\le E'$ and $F\le F'$. As $E_{j}'L^{p}({\mathscr {M}})F_{j}'\subseteq {\mathscr {A}}$, it is easy to see that $E'L^{p}({\mathscr {M}})F'\subseteq {\mathscr {A}}$. A direct calculation yields $C_{E'}=C_{(E+P)}C_{(F+Q)}=C_{F'}$, so $(E', F')\in \Lambda$ and $(E,F)\le (E',F')$. We have $(E,F)=(E',F')$ because (E, F) is the maximal element in $\Lambda$. Note that

$$\begin{aligned} E'=(E+P)C_{(F+Q)}= & {} EC_{(F+Q)}+PC_{(F+Q)}\ge EC_{F}+PC_{Q}\\= & {} EC_{E}+PC_{P}=E+P, \end{aligned}$$

then $E' > E$, which is a contradiction. Thus $P=0$ and $Q=0$.

Assertion 2 Let $(P,Q)\in \Lambda$. If $P\le E,~Q\le I-F$ or $P\le I-E,~Q\le F$, then $P=Q=0$.

Suppose that $P\le E,~Q\le I-F$. Put $P'=E-P$, then $E=P\oplus P'$. If $P=E$, then $C_{E}=C_{F}=C_{P}=C_{Q}=C_{(F+Q)}$. It is easy to check that $(E, F)\le (E, F+Q)\in \Lambda$, then $Q=0$, which is a contradiction. Thus $P< E$.

If $P'C_{Q}L^{p}({\mathscr {M}})C_{P'}Q=P'L^{p}({\mathscr {M}})Q=0$, similarly with [15, Theorem 3.2], we easily get $Q=0$, $P=0$. If $P'C_{Q}L^{p}({\mathscr {M}})C_{P'}Q=P'L^{p}({\mathscr {M}})Q\ne 0$, then we may take monotonically increasing $\tau -$finite projection families $\{P_{i}\}_{i\in I}$ and $\{Q_{i} \}_{i\in I}$ in ${\mathscr {M}}$ such that $P_{i} \rightarrow P$ and $Q_{i}\rightarrow Q$ $\sigma -$weakly with $C_{P_{i}}=C_{Q_{i}}$ for any $i\in I$. Similarly, we take monotonically increasing $\tau -$finite projection families $\{P_{j}'\}_{j\in J}$ such that $P_{j}'\rightarrow P'$ $\sigma -$weakly. Since $C_{P_{j}'}\le C_{P'}\le C_{E}=C_{F}$, there exists a $j_0$ such that $P_{j}'L^{p}({\mathscr {M}})C_{P_{j}'}F=P_{j}'L^{p}({\mathscr {M}})F\ne 0$ for all $j\ge j_0$. By [9, Proposition 5.5.3], we know, if T is a central projection in a von Neumann algebra ${\mathscr {M}}$, then $TC_{A}=C_{TA}$ for each A in ${\mathscr {M}}$. As we know, $T=C_{P_{j}'}$ is a central projection, then we have $C_{C_{P_{j}'}F}=C_{P_{j}'}C_{F}$. Note that $C_{P_{j}'}\le C_{P'}\le C_{E}=C_{F}$, then $C_{P_{j}'}\le C_{F}$, thus we have $C_{C_{P_{j}'}F}=C_{P_{j}'}C_{F}=C_{P_{j}'}$.

For any $U \in [P_{i}, Q_{i}],~V\in [P_{j}',C_{P_{j}'}F]$, we easily see that $U+V\in P_{i}L^{p}({\mathscr {M}})Q_{i}+P_{j}'L^{p}({\mathscr {M}})C_{P_{j}'}F\subseteq PL^{p}({\mathscr {M}})Q+EL^{p}({\mathscr {M}})F\subseteq {\mathscr {A}}$. It follows from Lemma 2.4 that $P_{U+V}L^{p}({\mathscr {M}})Q_{U+V}\subseteq {\mathscr {A}}$. It is obvious that $P_{U+V}=P_{U}+P_{V},~Q_{U+V}=Q_{U}+Q_{V}$. Using similar method in Assertion 1, we can prove that $P_{V}L^{p}({\mathscr {M}})Q_{U}\subseteq {\mathscr {A}}$. By the arbitrariness of U, V, we have $P_{j}'L^{p}({\mathscr {M}})Q_{i}\subseteq {\mathscr {A}}$. Therefore, $P'L^{p}({\mathscr {M}})Q\subseteq {\mathscr {A}}$. Then $EL^{p}({\mathscr {M}})Q=(P+P')L^{p}({\mathscr {M}})Q\subseteq {\mathscr {A}}$. Similarly, one obtains $Q=0$, $P=0$.

If $P\le I-E,~Q\le F$, we have the same conclusion.

We next prove that ${\mathscr {A}}\subseteq EL^{p}({\mathscr {M}})F$. $\forall T\in {\mathscr {A}}$, by Lemma 2.4 we can see that $P_{T}L^{p}({\mathscr {M}})Q_{T}\subseteq {\mathscr {A}}$ and $C_{P_{T}}=C_{Q_{T}}$, so $(P_{T}, Q_{T})\in \Lambda$. Note that $T\in P_{T}L^{p}({\mathscr {M}})Q_{T}$. So we just need to prove that $P_{T}L^{p}({\mathscr {M}})Q_{T}\subseteq EL^{p}({\mathscr {M}})F$.

Let $\{P_{i}\}_{i\in I}$ and $\{Q_{i}\}_{i\in I}$ be monotonically increasing $\tau -$finite projection families with $C_{P_{i}}=C_{Q_{i}}$ in ${\mathscr {M}}$ such that $P_{i}\rightarrow P_{T}$ and $Q_{i}\rightarrow Q_{T}$ $\sigma -$weakly. Then $P_{i}L^{p}({\mathscr {M}})Q_{i}\subseteq P_{T}L^{p}({\mathscr {M}})Q_{T}\subseteq {\mathscr {A}}$, so $(P_{i}, Q_{i})\in \Lambda$. We can easily see that if we have proven that $P_{i}L^{p}({\mathscr {M}})Q_{i}\subseteq EL^{p}({\mathscr {M}})F$ for all $i\in I$, then the conclusion follows. Thus without loss of generality, we may assume that $P=P_T$ and $Q=Q_T$ are $\tau$-finite projections.

Let $L_{1}=R(E)\cap R(P),~L_{2}=R(E)\cap N(P),~L_{3}=N(E)\cap R(P )$, $L_{4}=N(E)\cap N(P),~L_{5}=R(E)\ominus (L_{1}\oplus L_{2}),~L_{6}={\mathscr {H}}\ominus (\oplus _{i=1}^{5}L_{i})$. Denote by $P_{kk}=P_{L_{k}}$ and let $I_{k}$ be the identity operator on $L_{k}(1\le k \le 6)$. Let

$$\begin{aligned} P_{56}=0\oplus 0 \oplus 0 \oplus 0 \oplus \left( \begin{array}{cc} P_{0} &{} (P_{0}(I_{5}-P_{0}))^{\frac{1}{2}}D \\ D^{*}(P_{0}(I_{5}-P_{0}))^{\frac{1}{2}} &{} D^{*}(I_{5}-P_{0})D \\ \end{array} \right) , \end{aligned}$$

where $P_{0}\in {\mathscr {B}}(L_{5})$ such that $P_{0}$ and $I_{5}-P_{0}$ are injective positive contraction operators, $D\in {\mathscr {B}}(L_{6}, L_{5})$ is a unitary operator. Note that $E,P\in {\mathscr {M}}$. Thus $P_0,P_{ii}, P_{56},D\in {\mathscr {M}}$. By [4], we have that ${\mathscr {H}}=\bigoplus _{k=1}^6L_k$ and

$$\begin{aligned} E=P_{11}+P_{22}+P_{55},~P=P_{11}+P_{33}+P_{56}. \end{aligned}$$

(1)

Using the same method, let $L_{1}'=R(F)\cap R(Q),~L_{2}'=R(F)\cap N(Q),~L_{3}'=N(F)\cap R(Q)$, $L_{4}'=N(F)\cap N(Q),~L_{5}'=R(F)\ominus (L_{1}'\oplus L_{2}'),~L_{6}'={\mathscr {H}}\ominus (\oplus _{i=1}^{5}L_{i}')$. Denote by $Q_{kk}=P_{L_{k}'}$ and let $I_{k}$ be the identity operator on $L_{k}'(1\le k \le 6)$. Let

$$\begin{aligned} Q_{56}=0\oplus 0 \oplus 0 \oplus 0 \oplus \left( \begin{array}{cc} Q_{0} &{} (Q_{0}(I_{5}-Q_{0}))^{\frac{1}{2}}K \\ K^{*}(Q_{0}(I_{5}-Q_{0}))^{\frac{1}{2}} &{} K^{*}(I_{5}-Q_{0})K \\ \end{array} \right) , \end{aligned}$$

where $Q_{0}\in {\mathscr {B}}(L_{5}')$ such that $Q_{0}$ and $I_{5}-Q_{0}$ are injective positive contraction operators and $K\in {\mathscr {B}}(L_{6}', L_{5}')$ is a unitary operator. In this case we have

$$\begin{aligned} F=Q_{11}+Q_{22}+Q_{55},~Q=Q_{11}+Q_{33}+Q_{56}. \end{aligned}$$

(2)

Since P and Q are $\tau$-finite projections, $P_{ii}$, $Q_{ii}(i=1,3)$, $P_{56}$ and $Q_{56}$ are all $\tau$-finite projections. It is known that

$$\begin{aligned} {\mathscr {A}}\supseteq PL^{p}({\mathscr {M}})Q=P_{11}L^{p}({\mathscr {M}})Q_{11}+P_{11}L^{p}({\mathscr {M}})Q_{33}+P_{11}L^{p}({\mathscr {M}})Q_{56}+\nonumber \\ P_{33}L^{p}({\mathscr {M}})Q_{11}+P_{33}L^{p}({\mathscr {M}})Q_{33}+P_{33}L^{p}({\mathscr {M}})Q_{56}+\nonumber \\ P_{56}L^{p}({\mathscr {M}})Q_{11}+P_{56}L^{p}({\mathscr {M}})Q_{33}+P_{56}L^{p}({\mathscr {M}})Q_{56}. \end{aligned}$$

(3)

• (1) It is easy to see that $P_{11}L^{p}({\mathscr {M}})Q_{11}\subseteq EL^{p}({\mathscr {M}})F$.

• (2) $P_{33}C_{Q_{33}}L^{p}({\mathscr {M}})C_{P_{33}}Q_{33}=P_{33}L^{p}({\mathscr {M}})Q_{33}\subseteq {\mathscr {A}}$, so $(P_{33}C_{Q_{33}}, C_{P_{33}}Q_{33})\in \Lambda$. It follows from Assertion 1 that $P_{33}L^{p}({\mathscr {M}})Q_{33}=0$.

• (3) Similarly, by $P_{11}L^{p}({\mathscr {M}})Q_{33}\subseteq {\mathscr {A}}$, $P_{33}L^{p}({\mathscr {M}})Q_{11}\subseteq {\mathscr {A}}$ and Assertion 2, we have $P_{11}L^{p}({\mathscr {M}})Q_{33}=0$ and $P_{33}L^{p}({\mathscr {M}})Q_{11}=0$.

• (4) Suppose that $P_{56}L^{p}({\mathscr {M}})Q_{56}\ne 0$. Since $P_{55}L^{p}({\mathscr {M}})Q_{55}\subseteq EL^{p}({\mathscr {M}})F\subseteq {\mathscr {A}}$, for any $T_{55}\in P_{55}L^{p}({\mathscr {M}})Q_{55}$, we have $P_{56}T_{55}Q_{56}\in {\mathscr {A}}$ and

  $$\begin{aligned} P_{56}T_{55}Q_{56}&=P_{55}P_{0}T_{55}Q_{0}Q_{55} +P_{55}P_{0}T_{55}(Q_{0}(I_{5}-Q_{0}))^{\frac{1}{2}}KQ_{66}\nonumber \\&\quad + P_{66}D^{*}(P_{0}(I_{5}-P_{0}))^{\frac{1}{2}}T_{55}Q_{0}Q_{55}\\&\quad + P_{66}D^{*}(P_{0}(I_{5}-P_{0}))^{\frac{1}{2}}T_{55}(Q_{0}(I_{5}-Q_{0}))^{\frac{1}{2}}KQ_{66}.\end{aligned}$$

  Let

  $$\begin{aligned} G= & {} P_{55}P_{0}T_{55}(Q_{0}(I_{5}-Q_{0}))^{\frac{1}{2}}KQ_{66}\\&+P_{66}D^{*}(P_{0}(I_{5}-P_{0}))^{\frac{1}{2}}T_{55}(Q_{0}(I_{5}-Q_{0}))^{\frac{1}{2}}KQ_{66}, \\ H= & {} P_{66}D^{*}(P_{0}(I_{5}-P_{0}))^{\frac{1}{2}}T_{55}Q_{0}Q_{55}\\&+P_{66}D^{*}(P_{0}(I_{5}-P_{0}))^{\frac{1}{2}}T_{55}(Q_{0}(I_{5}-Q_{0}))^{\frac{1}{2}}KQ_{66}, \end{aligned}$$

  and let

  $$\begin{aligned} J&=-P_{55}(I_{5}-P_{0})T_{55}Q_{0}Q_{55}+P_{55}P_{0}T_{55}(Q_{0}(I_{5}-Q_{0}))^{\frac{1}{2}}KQ_{66}\nonumber \\&\quad +P_{66}D^{*}(P_{0}(I_{5}-P_{0}))^{\frac{1}{2}}T_{55}Q_{0}Q_{55}+P_{66}D^{*}(P_{0}(I_{5}-P_{0}))^{\frac{1}{2}}\\&\quad T_{55}(Q_{0}(I_{5}-Q_{0}))^{\frac{1}{2}}KQ_{66}, \end{aligned}$$
  $$\begin{aligned} K&=-P_{55}P_{0}T_{55}(I_{5}-Q_{0})Q_{55}+P_{55}P_{0}T_{55}(Q_{0}(I_{5}-Q_{0}))^{\frac{1}{2}}KQ_{66}\nonumber \\&\quad +P_{66}D^{*}(P_{0}(I_{5}-P_{0}))^{\frac{1}{2}}T_{55}Q_{0}Q_{55}+P_{66}D^{*}(P_{0}(I_{5}-P_{0}))^{\frac{1}{2}}\\&\quad T_{55}(Q_{0}(I_{5}-Q_{0}))^{\frac{1}{2}}KQ_{66}. \end{aligned}$$

  It is easy to see that $J, K\in {\mathscr {A}}$. By an elementary calculation, we have $G\overset{*}{\le }J,~H\overset{*}{\le }K$. Then $G, H, H-G\in {\mathscr {A}}$. Let $G_{0}=-P_{55}P_{0}T_{55}(Q_{0}(I_{5}-Q_{0}))^{\frac{1}{2}}KQ_{66}$, $H_{0}=P_{66}D^{*}(P_{0}(I_{5}-P_{0}))^{\frac{1}{2}}T_{55}Q_{0}Q_{55}$. Since $G_{0}\overset{*}{\le }H-G,~H_{0}\overset{*}{\le }H-G$, we have $G_{0}, H_{0}, G+G_{0}\in {\mathscr {A}}$. Note that $G_{0}\in EL^{p}({\mathscr {M}})(I-F),~H_{0}\in (I-E)L^{p}({\mathscr {M}})F,~G+G_{0}\in (I-E)L^{p}({\mathscr {M}})(I-F)$. By Assertion 1 and Assertion 2, we have $G_{0}=H_{0}=G+G_{0}=0$. Since $P_{0}, (Q_{0}(I_{5}-Q_{0}))^{\frac{1}{2}}$ are injective positive operators and K is unitary operator, $T_{55}=0$, that is, $P_{55}L^{p}({\mathscr {M}})Q_{55}=0$. Thus $C_{P_{55}}C_{Q_{55}}=0$. Note that $C_{P_{55}}=C_{P_{66}},~C_{Q_{55}}=C_{Q_{66}}$, then

  $$\begin{aligned} P_{56}L^{p}({\mathscr {M}})Q_{56}\subseteq (P_{55}+P_{66})L^{p}({\mathscr {M}})(Q_{55}+Q_{66})=0, \end{aligned}$$

  which is a contradiction. Therefore, $P_{56}L^{p}({\mathscr {M}})Q_{56}=0$.

• (5) Since $P_{56}L^{p}({\mathscr {M}})Q_{11}=(P_{55}+P_{66})P_{56}L^{p}({\mathscr {M}})Q_{11}$ and $P_{55}P_{56}L^{p}({\mathscr {M}})Q_{11}\subseteq EL^{p}({\mathscr {M}})F\subseteq {\mathscr {A}}$, one obtains $P_{66}P_{56}L^{p}({\mathscr {M}})Q_{11}\subseteq {\mathscr {A}}$. It follows from Assertion 2 that $P_{66}P_{56}L^{p}({\mathscr {M}})Q_{11}=0$. Then for any $T_{51}\in P_{55}L^{p}({\mathscr {M}})Q_{11}$, we have $D^{*}(P_{0}(I_{5}-P_{0}))^{\frac{1}{2}}T_{51}=0$. This implies that $T_{51}=0$. Then $0=C_{P_{55}}C_{Q_{11}}=C_{P_{66}}C_{Q_{11}}=C_{P_{56}}C_{Q_{11}}$. Thus we have $P_{56}L^{p}({\mathscr {M}})Q_{11}=0$. Symmetrically, we can see that $P_{11}L^{p}({\mathscr {M}})Q_{56}=0$.

• (6) Suppose that $P_{56}L^{p}({\mathscr {M}})Q_{33}\ne 0$. Without loss of generality, we may assume that $C_{P_{56}}=C_{Q_{33}}$. At this time, $C_{P_{55}}=C_{P_{56}}=C_{P_{66}}=C_{Q_{33}}\ne 0$. Then there exists a nonzero partial isometry U such that $U^{*}U\le Q_{33},~UU^{*}\le P_{55}$. Since $Q_{33}$ is $\tau$-finite, $U\in P_{55}L^{p}({\mathscr {M}})Q_{33}$. Obviously, $0\ne A=P_{56}UQ_{33}\in P_{A}L^{p}({\mathscr {M}})Q_{A}\subseteq P_{56}L^{p}({\mathscr {M}})Q_{33}\subseteq {\mathscr {A}}$. Similarly with [15, Theorem 3.2], we may assume that $P_{A}$ is equivalent to a subprojection of F.

Using the same method, E and $P_{A}$ can be represented in the form of equation (1). $E=P_{11}'+P_{22}'+P_{55}',~P_{A}=P_{11}'+P_{33}'+P_{56}'$. Then

$$\begin{aligned} {\mathscr {A}}\supseteq P_{A}L^{p}({\mathscr {M}})Q_{A}=P_{11}'L^{p}({\mathscr {M}})Q_{A}+P_{33}'L^{p}({\mathscr {M}})Q_{A}+P_{56}'L^{p}({\mathscr {M}})Q_{A}. \end{aligned}$$

Thus $P_{11}'L^{p}({\mathscr {M}})Q_{A},~P_{33}'L^{p}({\mathscr {M}})Q_{A},~P_{56}'L^{p}({\mathscr {M}})Q_{A}\subseteq {\mathscr {A}}$. Since $P_{11}'\le E,~Q_{A}\le I-F,~P_{33}'\le I-E$, we have $P_{11}'L^{p}({\mathscr {M}})Q_{A}=0,~P_{33}'L^{p}({\mathscr {M}})Q_{A}=0$ by Assertion 1 and Assertion 2. Thus $P_{56}'L^{p}({\mathscr {M}})Q_{A}\ne 0$.

Let $P_{0}'$ and $D'$ be the injective positive contraction operators and unitary operator which corresponding to $P_{56}'$, respectively. Similarly with [15, Theorem 3.2], we may assume that there exist $\alpha , \beta \in (0, 1)$ such that $0<\alpha \le P_{0}' \le \beta <1$.

Since $P_{56}'\le P_{A}\thicksim Q_{A}$ and $P_{A}$ is equivalent to a subprojection of F, there exist $Q_{33}'\le Q_{A},~F_{1}\le F$ such that $P_{55}' \thicksim P_{56}' \thicksim Q_{33}' \thicksim F_{1}$. Then there exists a nonzero partial isometry U such that $U^{*}U= Q_{33}',~UU^{*}= P_{55}'$. Because $Q_{A}\le Q$ and Q is $\tau -$finite, $P_{55}',~P_{56}',~ Q_{33}'$ and $F_{1}$ are all $\tau -$finite. Then $U\in P_{55}'L^{p}({\mathscr {M}})Q_{33}'$. Clearly, $A'=P_{56}'UQ_{33}'\ne 0$ and $A'\in P_{56}'L^{p}({\mathscr {M}})Q_{33}'\subseteq P_{A}L^{p}({\mathscr {M}})Q_{A}\subseteq {\mathscr {A}}$. It is easy to check that $A'=P_{55}'P_{0}'UQ_{33}'+P_{66}'D'^{*}(P_{0}'(I_{5}'-P_{0}'))^{\frac{1}{2}}UQ_{33}'$. Note that $P_{55}'\sim F_1$. Then there exist some invertible elements in $P_{55}'\mathscr {M}F_1$. Since $0<\alpha \le P_{0}' \le \beta <1$, for any invertible $X\in P_{55}'\mathscr {M}F_1\subseteq P_{55}'L^{p}({\mathscr {M}})F_{1}\subseteq {\mathscr {A}}$, $B_{X}=X+A'\in {\mathscr {A}}$ is also invertible and

$$\begin{aligned} B_{X}=P_{55}'XF_{1}+P_{55}'P_{0}'UQ_{33}'+P_{66}'D'^{*}(P_{0}'(I_{5}'-P_{0}'))^{\frac{1}{2}}UQ_{33}'. \end{aligned}$$

It is easy to see that for any $0<\lambda <1$,

$$\begin{aligned} E_{\lambda }=\frac{1}{1+\lambda ^{2}}(P_{55}'+\lambda P_{55}'D'P_{66}'+\lambda P_{66}'D'^{*}P_{55}'+\lambda ^{2}P_{66}') \end{aligned}$$

is a projection. It follows form Lemma 2.2 that if $E_{\lambda }B_{X}B_{X}^{*}=B_{X}B_{X}^{*}E_{\lambda }$, then $E_{\lambda }B_{X}\overset{*}{\le } B_{X}$.

In fact, put $P(\lambda )=\frac{1-\lambda ^{2}}{\lambda }(P_{0}')^{\frac{3}{2}}(I_{5}'-P_{0}')^{\frac{1}{2}}+P_{0}'-2(P_{0}')^{2}$. Since $0<\alpha \le P_{0}' \le \beta <1$, there exits $\lambda >0$ such that $P(\lambda )$ is positive invertible operator. Note that $P_{55}'$ and $F_{1}$ are $\tau -$finite projections. Let W be a partial isometry with $WW^{*}=P_{55}',~W^{*}W=F_{1}$ and let $X=(P(\lambda ))^{\frac{1}{2}}W$. Similarly with [15, Theorem 3.2], we can easily see that X and $\lambda$ satisfy the equation $E_{\lambda }B_{X}B_{X}^{*}=B_{X}B_{X}^{*}E_{\lambda }$, so $E_{\lambda }B_{X}\overset{*}{\le } B_{X}$. Since $B_{X}\in {\mathscr {A}}$, we have

$$\begin{aligned} {\mathscr {A}}\ni (1+\lambda ^{2})E_{\lambda }B_{X}&=P_{55}'XF_{1}+P_{55}'(P_{0}'U+\lambda (P_{0}'(I_{5}'-P_{0}'))^{\frac{1}{2}}U)Q_{33}'\nonumber \\&\quad +\lambda P_{66}' D'^{*}XF_{1}+P_{66}'(\lambda D'^{*}P_{0}'U+\lambda ^{2}D'^{*}(P_{0}'(I_{5}'-P_{0}'))^{\frac{1}{2}}U)Q_{33}'. \end{aligned}$$

Let

$$\begin{aligned} A_{1}=\lambda P_{66}' D'^{*}XF_{1}+P_{66}'(\lambda D'^{*}P_{0}'U+\lambda ^{2}D'^{*}(P_{0}'(I_{5}'-P_{0}'))^{\frac{1}{2}}U)Q_{33}', \\ Y=-P_{55}'(P_{0}'+\lambda (P_{0}'(I_{5}'-P_{0}'))^{\frac{1}{2}})^{2}X^{*-1}F_{1}\in P_{55}'L^{p}({\mathscr {M}})F_{1}\subseteq {\mathscr {A}}. \end{aligned}$$

Then $B_{1}=(1+\lambda ^{2})E_{\lambda }B_{X}+Y-X\in {\mathscr {A}}$. It is easy to check that $A_{1}\overset{*}{\le }B_{1}$, which implies that $A_{1}\in {\mathscr {A}}$. Thus $B_{1}-A_{1}-Y\in {\mathscr {A}}$. In fact, $B_{1}-A_{1}-Y\in P_{55}'L^{p}({\mathscr {M}})Q_{33}'$, by Assertion 2 we have $B_{1}-A_{1}-Y=0$. Then

$$\begin{aligned} 0=(P_{0}'+\lambda (P_{0}'(I_{5}'-P_{0}'))^{\frac{1}{2}})U, \end{aligned}$$

which implies $U=0$. It is a contradiction. Therefore, $P_{56}L^{p}({\mathscr {M}})Q_{33}= 0$. Symmetrically, we easily get $P_{33}L^{p}({\mathscr {M}})Q_{56}= 0$.

From $(1)\thicksim (6)$, we know that $PL^{p}({\mathscr {M}})Q=P_{11}L^{p}({\mathscr {M}})Q_{11}\subseteq EL^{p}({\mathscr {M}})F$. Thus we have ${\mathscr {A}}\subseteq EL^{p}({\mathscr {M}})F$. Therefore, ${\mathscr {A}}= EL^{p}({\mathscr {M}})F$. $\square$

If ${\mathscr {A}}\subseteq L^p({\mathscr {M}})$ is a norm closed star partial order-hereditary subspace of $L^{p}({\mathscr {M}})$, then we have that ${\mathscr {A}}\cap {\mathscr {M}}$ is a star partial order-hereditary liner manifold in ${\mathscr {M}}$. By Theorem 2.5, we have that $\overline{{\mathscr {A}}\cap {\mathscr {M}}}^{\sigma -w}=E\mathscr {M}F$ for some $E,F\in {\mathscr {M}}_p$ with the same central carriers. In general, let ${\mathfrak {M}}\subseteq {\mathscr {M}}$ be a star partial order-hereditary liner manifold in ${\mathscr {M}}$. Is the $\sigma$-weak closure $\overline{{\mathfrak {M}}}^{\sigma -w}$ of ${\mathfrak {M}}$ a star partial order-hereditary subspace?

3 The Diamond Order in $L^{p}({\mathscr {M}})$

In this section, we consider the diamond order in non-commutative $L^p$ spaces. The definition is similar to the case in ${\mathscr {B}}({\mathscr {H}})$.

Definition 3.1

Let $A, B\in L^p({\mathscr {M}})$. Then we say that $A \le ^{\diamond } B$ if $P_{A}\le P_{B},~ Q_{A}\le Q_{B},~AA^{*}A = AB^{*}A$ and $\le ^{\diamond }$ is said to be diamond order.

As in ${\mathscr {B}}({\mathscr {H}})$, it is also known that $A \le ^{\diamond } B\Leftrightarrow A=P_{A}BQ_{A}=P_{B}AQ_{B}$. We now may consider the upper bounds as well as lower bounds of a subset ${\mathscr {A}}\subseteq L^p({\mathscr {M}})$. However, there is no supremum and infimum in general. It is known that there exist minimal upper bounds for a subset with an upper bound in ${\mathscr {B}}({\mathscr {H}})$ (cf. [13]). Now for a finite von Neumann algebra ${\mathscr {M}}$, we also have the following result.

Theorem 3.2

Let ${\mathscr {M}}$ be a finite von Neumann algebra with a faithful normal finite trace $\tau$ and $L^p({\mathscr {M}})$ the non-commutative $L^p$-space associated with $\tau$. If ${\mathscr {A}}\subseteq L^p({\mathscr {M}})$ has an upper bound, then ${\mathscr {A}}$ has a minimal upper bound in $L^p({\mathscr {M}})$.

Proof

Let ${\mathscr {A}}$ be a nonempty subset in $L^p({\mathscr {M}})$ and $B\in L^p({\mathscr {M}})$ an upper bound of ${\mathscr {A}}$ with respect to the diamond order. Put $P=\bigvee \{P_A: A\in {\mathscr {A}}\}$ and $Q=\bigvee \{Q_A:A\in {\mathscr {A}}\}$. Then $P\le P_B$ and $Q\le Q_B$. Put $B_{11}=PBQ$, $P_1=P_{B_{11}}$ and $Q_1=Q_{B_{11}}$. Then $P_1\le P\le P_B$ and $Q_1\le Q\le Q_B$. It follows that $B_{11}=P_1B_{11}Q_1=P_1BQ_1=P_{B_{11}}BQ_{B_{11}}$. Thus $B_{11}\le ^{\diamond } B$.

If $P_1=P$ and $Q_1=Q$, then it is easy to see that $B_{11}$ is a minimal upper bound of ${\mathscr {A}}$. Otherwise, we let $Q_2=I-Q$, $Q_3=Q-Q_1$, $P_2=P-P_1$ and $P_3=I-P$. Put $B_{ij}=P_iBQ_j$ for $1\le i,j\le 3$. Since $PBQ=P_1BQ_1$, we have $B_{13}=0$, $B_{21}=0$ and $B_{23}=0$. Note that $P\le P_B$ as well as $Q\le Q_B$. We also have that $P_{B_{22}}=P_2$ and $Q_{B_{33}}=Q_3$. Next we may assume that $P_{1}<P$ and $Q_{1}<Q$. The case that either $P_{1}=P$ or $Q_{1}=Q$ is similar. Let $B_{22}=U|B_{22}|$ and $B_{33}=V|B_{33}|$ be the polar decompositions of $B_{22}$ and $B_{33}$. Then $UU^*=P_2$, $U^*U\le Q_2$ and $V^*V=Q_3$, $VV^*\le P_3$. Put $D=B_{11}+U+V$. Then $D\in L^p({\mathscr {M}})$ since ${\mathscr {M}}\subseteq L^p({\mathscr {M}})$. We next show that D is a minimal upper bound of ${\mathscr {A}}$.

We note that $P_D=P_1+P_2+VV^*=P+VV^*\ge P_A$ and $Q_D=Q_1+U^*U+Q_3=Q+U^*U\ge Q_A$ for all $A\in {\mathscr {A}}$. Then $A=P_DAQ_D$ for all $A\in {\mathscr {A}}$. On the other hand, for any $A\in {\mathscr {A}}$, $P_A\le P$ and $Q_A\le Q$. It follows that $P_ADQ_A=P_A(PBQ+U+V)Q_A=P_APBQQ_A=P_ABQ_A=A$. Thus $A\le ^{\diamond }D$.

Now for any $C\in L^p({\mathscr {M}})$ with $A\le ^{\diamond }C\le ^{\diamond }D$ for any $A\in {\mathscr {A}}$, we have $P\le P_C$ and $Q\le Q_C$. These mean that $P_C=P+E$ and $Q_C=Q+F$ for some projections $E,F \in {\mathscr {M}}_p$ with $E\le I-P$ and $F\le I-Q$. Note that $C\le ^{\diamond }D$. We have $C=P_CDQ_C=B_{11}+UF+EV$. It is known that both UF and $V^*E$ have dense ranges since $P\le P_C$ and $Q\le Q_C$. By an elementary calculation, we have $UF\le ^{\diamond }U$ and $EV\le ^{\diamond }V$. It follows that $UF=U$ and $EV=V$ by [13, Corollary 2.4] since both U and $V^{*}$ are surjective. Hence D is a minimal upper bound of ${\mathscr {A}}$. $\square$

However, if ${\mathscr {M}}$ is not finite, then the result in Theorem 3.2 may fail. We note that for a separable infinite dimensional Hilbert space ${\mathscr {H}}$, ${\mathscr {B}}({\mathscr {H}})$ is a semi-finite von Neumann algebra with the classical faithful normal semi-finite trace $\tau$ and the non-commutative $L^p$ space associated with ${\mathscr {B}}({\mathscr {H}})$ is the Schatten p-class $C_p({\mathscr {H}})=\{ T\in {\mathscr {B}}({\mathscr {H}}): \tau (|T|^p)<\infty \}$. We next give a subset ${\mathscr {A}}\subseteq C_p({\mathscr {H}})$ with an upper bound but has no minimal upper bounds.

Example 3.3

Let ${\mathscr {K}}$ be a complex separable infinite dimensional Hilbert space with an orthonormal basis $\{e_n:n\ge 1\}$ and ${\mathscr {H}}={\mathscr {K}}\oplus {\mathscr {K}}$. Let $H_1=\vee \{e_1\}$ and $T_1e_1=e_1$ on $H_1$. For any $n\ge 2$, let $K_{n-1}=\vee \{e_{2^{n-2}+j}:1\le j\le 2^{n-2}\}$ and $H_{n}=H_{n-1}\oplus K_{n-1}$. We define $B_{n-1}e_{2^{n-2}+j}=ne_j$ for $1\le j\le 2^{n-2}$ from $K_{n-1}$ to $H_{n-1}$ and

$$\begin{aligned} T_n=\left( \begin{array}{cc} T_{n-1} &{} B_{n-1} \\ 0 &{} I \\ \end{array} \right) \end{aligned}$$

on $H_n$. Then $T_n$ is an invertible operator on $H_n$. Let $P_n$ and $Q_n$ be the projections from ${\mathscr {H}}$ onto $H_n\oplus \{0\}$ and $M_n=\{x\oplus T_nx:x\in H_n\}$ respectively. Note that both $\{P_n\}$ and $\{Q_n\}$ are increasing sequences. It is elementary that $\lim _nP_n=I\oplus 0=P$. On the other hand, for any $j\ge 1$, put $x_n^j=0\oplus \frac{1}{n}e_{2^{n-2}+j}$ for $n\ge 2$. we have $T_nx_n^j=e_j+ \frac{1}{n}e_{2^{n-2}+j}$. Then $\lim _n(x_n^j\oplus T_nx_n^j)=0\oplus e_j$. It follows that $\{0\}\oplus {\mathscr {K}}\subseteq \vee \{M_n:n\ge 1\}$. Thus $\lim _nQ_n=I\oplus I$.

We now take a positive injective operator $A\in C_p({\mathscr {H}})$ such that $P_nA=AP_n$ for all n. Let $A_n=P_nAQ_n=AP_nQ_n$ for $n\ge 1$ and ${\mathscr {A}}=\{A_n:n\ge 1\}$. Note that $T_n$ and $A_n|_{P_n({\mathscr {H}})}$ are invertible on $H_n$ and $H_n\oplus \{0\}$ for all n. Then $P_{A_n}=P_n$ as well as $Q_{A_n}=Q_n$. It follows that $A_n\le ^{\diamond }A$ in $C_p({\mathscr {H}})$ for all n and thus A is an upper bound of ${\mathscr {A}}$. It is clear that $\lim _nA_n=PA=AP=PAP$. Take any upper bound $D\in C_p({\mathscr {H}})$ of ${\mathscr {A}}$. Then $P\le P_D$, $Q_D=I$ and $A_n=P_nDQ_n$. Thus $PA=PD=PDP$. This means that

$$\begin{aligned} A=\left( \begin{array}{cc} A_{11} &{} 0 \\ 0 &{} A_{22} \\ \end{array} \right) \text{ and } D=\left( \begin{array}{cc} A_{11} &{} 0 \\ D_{21} &{} D_{22} \\ \end{array} \right) . \end{aligned}$$

Note that $D_{22}$ is injective and $P_D=P_{A_{11}}\oplus P_{[D_{21},D_{22}]}$. Since $R(D_{22}^*)$ is not closed and dense in ${\mathscr {K}}$, there exists an injective operator $G_{22}$ on ${\mathscr {K}}$ such that $G_{22}\le ^{\diamond }D_{22}$ by [13, Lemma 2.5]. Thus $G_{22}=P_{G_{22} }D_{22}$. Put

$$\begin{aligned} G=\left( \begin{array}{cc} I &{} 0 \\ 0 &{} P_{G_{22}} \\ \end{array} \right) \left( \begin{array}{cc} A_{11} &{} 0 \\ D_{21} &{} D_{22} \\ \end{array} \right) =\left( \begin{array}{cc} A_{11} &{} 0 \\ P_{G_{22}}D_{21} &{} G_{22} \\ \end{array} \right) . \end{aligned}$$

Thus $P\le P_G \le P_D$ and $Q_G=Q_D=I$. By an elementary calculation, we have $A_n\le ^{\diamond }G\le ^{\diamond }D$ for all $n\ge 1$ in $C_p({\mathscr {H}})$. Note that $G\ne D$. Thus ${\mathscr {A}}$ has not any minimal upper bound.

We now may consider the diamond order-hereditary subspaces in $L^{p}({\mathscr {M}})$.

Proposition 3.4

Let ${\mathscr {M}}$ be a semi-finite von Neumann algebra with a faithful normal semi-finite trace $\tau$ and $L^p({\mathscr {M}})$ the non-commutative $L^p$ space associated with $\tau$. If ${\mathscr {A}}$ is a norm closed diamond order-hereditary subspace in $L^{p}({\mathscr {M}})$, then there exists unique projection pair (E, F) with same central carriers such that ${\mathscr {A}}=EL^{p}({\mathscr {M}})F$.

Proof

Suppose that ${\mathscr {A}}$ is a norm closed diamond order-hereditary subspace in $L^{p}({\mathscr {M}})$. For any $A\in L^p({\mathscr {M}})$ and $B\in {\mathscr {A}}$, if $A\overset{*}{\le }B$, then we easily see that $A\le ^{\diamond }B$. Thus we have $A\in {\mathscr {A}}$. Therefore, a diamond order-hereditary subspace must be a star partial order-hereditary subspace. The conclusion follows from Theorem 2.5. $\square$