Wigner-Type Theorem on transition probability preserving maps in semifinite factors

TL;DR

This paper extends Wigner's theorem to transition probability preserving maps in semifinite factors, providing a new approach to characterize such maps beyond bijections and generalizing previous results.

Contribution

It introduces a novel method to describe transition probability preserving maps in semifinite factors, broadening the scope of Wigner-type theorems.

Findings

Generalizes Molnár and G. P. Geher's results

Provides a new approach to non-bijective maps

Characterizes transition probability preserving maps in semifinite factors

Abstract

The Wigner's theorem, which is one of the cornerstones of the mathematical formulation of quantum mechanics, asserts that every symmetry of quantum system is unitary or anti-unitary. This classical result was first given by Wigner in 1931. Thereafter it has been proved and generalized in various ways by many authors. Recently, G. P. Geh\'{e}r extended Wigner's and Moln\'{a}r's theorems and characterized the transformations on the Grassmann space of all rank-$n$ projections which preserve the transition probability. The aim of this paper is to provide a new approach to describe the general form of the transition probability preserving (not necessarily bijective) maps between Grassmann spaces. As a byproduct, we are able to generalize the results of Moln\'{a}r and G. P. Geh\'{e}r.

Equations

$$\phi :P\in {\mathcal{P}}_c\mapsto \left(\begin{array}{cc}{\phi }_1(P)& \\ & {\phi }_2(P)\end{array}\right)\in {\mathcal{P}}_c$$

$$\tau (\phi (P)\phi (Q))=\tau (PQ),\forall P,Q\in {\mathcal{P}}_c,$$

$$\tau =\tau \circ ({\sigma }_1+{\sigma }_2),$$

$${\mathcal{P}}_S=\{P\in \mathcal{P}:\tau (P)\in S\}.$$

$$\tau (\phi (P)\phi (Q))=\tau (PQ),\forall P,Q\in {\mathcal{P}}_S.$$

$$\tau =\tau \circ ({\sigma }_1+{\sigma }_2),$$

$$\phi :P\in {\mathcal{P}}_S\to {\sigma }_1(P)+{\sigma }_2(P)\in {\mathcal{P}}_S,$$

$$\Phi (\sum _{\alpha }{E}_{\alpha })=\sum _{\alpha }\Phi (E_{\alpha })$$

$$\Phi (P)=\sum _{\alpha }\phi (P_{\alpha }).$$

$$\tau ([\sum _{\alpha }\phi (P_{\alpha })]\phi (E))=\tau ([\sum _{\alpha }{P}_{\alpha }]E)=\tau ([\sum _{\beta }{Q}_{\beta }]E)=\tau ([\sum _{\beta }\phi (Q_{\beta })]\phi (E)).$$

$$\phi (P_{\alpha })\le \sum _{\beta }\phi (Q_{\beta }),\phi (Q_{\beta })\le \sum _{\alpha }\phi (P_{\alpha }),$$

$$\Phi (\sum _{\alpha }{E}_{\alpha })=\Phi (\sum _{\alpha ,\beta }{P}_{\alpha }^{\beta })=\sum _{\alpha ,\beta }\phi (P_{\alpha }^{\beta })=\sum _{\alpha }\Phi (E_{\alpha }).$$

$$\sum _{i=1}^n{\lambda }_i{P}_i\in M_2(\mathbb{C})\mapsto \sum _{i=1}^n{\lambda }_i\phi (P_i)\in M_2(\mathbb{C}),$$

$$\Phi (\sum _{i=1}^n{\lambda }_i{P}_i{)}^2=(\sum _{i=1}^n{\lambda }_i\Phi (P_i){)}^2=\sum _{i=1}^n{\lambda }_i^2\Phi (P_i)=\Phi ((\sum _{i=1}^n{\lambda }_i{P}_i{)}^2).$$

$$\tau (P)=\tau (\phi (P))=\tau ({\sigma }_1(P)+{\sigma }_2(P)),\forall P\in {\mathcal{P}}_S.$$

$$\tau (\phi (Q)[\phi (P_1)+\phi (P_2)]\phi (Q))=\tau (Q[P_1+P_2]Q).$$

$$0=\phi (Q^{\prime })\phi (Q)=\phi (Q^{\prime })[\phi (P)+\phi (P^{\prime })]\phi (Q)=\phi (Q^{\prime })\phi (P^{\prime })\phi (Q).$$

$$c=\tau (P^{\prime }(Q+Q^{\prime }))<\tau (E+F)\le \tau (P^{\prime })=c.$$

$$P\le P_1+P_2,Q\le P_1+P_2,\tau (P_{1}Q)\le c_1,\tau (P_{2}P)\le c_1.$$

$$\phi (P_i)\phi (P)=\phi (P)\phi (P_i),\phi (P_i)\phi (Q)=\phi (Q)\phi (P_i),i=1,2.$$

$$\tau (E_1)\le \tau (\phi (Q)\phi (P_1))=\tau (Q{P}_1),\tau (E_2)\le \tau (\phi (P)\phi (P_2))=\tau (P{P}_2).$$

$$\tau (\phi (P)\phi (Q))\le \tau (E_1+E_2)\le \tau (Q{P}_1)+\tau (P{P}_2)=\tau (PQ).$$

$$P_3\perp P_1,P_4\perp P_1,P_2\le P_1+P_3,P_2\le P_1+P_4,P_3{P}_2=P_4{P}_2=P_3{P}_4.$$

$$\phi (P_2)\le \phi (P_1)+\phi (P_3),\phi (P_2)\le \phi (P_1)+\phi (P_4).$$

$$\tau (\phi (P_1)\vee \phi (P_2))$$

$$=\tau ([\phi (P_1)+\phi (P_3)]\wedge [\phi (P_1)+\phi (P_4)]).$$

$$\phi (Q)\le [\phi (P_1)+\phi (P_3)]\wedge [\phi (P_1)+\phi (P_4)]=\phi (P_1)\vee \phi (P_2).$$

$$\tau (P_1{Q}_1)=a_1,P_2\le P_1\vee Q_1,$$

$$\tau (P_2{Q}_2)=a_2,P_1\le P_2\vee Q_2,$$

$$\phi (P_1)\vee \phi (P_2)\le [\phi (P_1)\vee \phi (Q_1)]\wedge [\phi (P_2)\vee \phi (Q_2)].$$

$$\phi (Q_1)[I-\phi (P_2)]\perp [\phi (P_2)\vee \phi (Q_2)],\phi (Q_2)[I-\phi (P_1)]\perp [\phi (P_1)\vee \phi (Q_1)],$$

Full text

The fourth author is supported by the Youth Innovation Promotion Association, CAS.

† Corresponding author.

1. Introduction and Statment of the Main Result

The celebrated Wigner’s Symmetry Representation Theorem guarantees that every symmetry’s actions on states is induced by a unitary or an anti-unitary operator. If we identify the set of pure states with the Grassmann space $\mathscr{P}_{1}$ of all rank-one orthogonal projections in $\mathcal{B}(\mathcal{H})$, where $\mathcal{B}(\mathcal{H})$ denotes the algebra of all bounded linear opeartor on a Hilbert space $\mathcal{H}$, then Wigner’s theorem can be formulated mathematically as follows:

Theorem** (E. P. Wigner [24]).**

Let $\varphi:\mathscr{P}_{1}\rightarrow\mathscr{P}_{1}$ be a surjective map. If $\varphi$ preserves the transition probability, i.e., $Tr(PQ)=Tr(\varphi(P)\varphi(Q))$ for every $P,Q\in\mathscr{P}_{1}$, where $Tr$ is the canonical trace on $\mathcal{B}(\mathcal{H})$, then there is an either unitary or anti-unitary $U\in\mathcal{B}(\mathcal{H})$ such that $\varphi(P)=UPU^{*}$.

This result is first proved by Wigner in [24]. Thereafter, it has been proved and generalized in various ways by many authors such as [5, 2, 7, 6, 8, 9, 12, 14, 13, 15, 17, 18, 19, 21], to mention but a few. The aim of the present paper is to extend Wigner’s theorem and characterize the transition probability preserving maps between more Grassmann spaces of projections in semifinite factors. In order to achieve the goal, we first reformulate Wigner’s theorem in the language of von Neumann algebras.

Recall that a (concrete) von Neumann algebra is a *-subalgera of $\mathcal{B}(\mathcal{H})$ that is closed in the weak-operator topology. If a von Neumann algebra $\mathfrak{A}$ endowed with a normal faithful tracial weight $\tau$, then $\mathfrak{A}$ is called semifinite. A factor is a von Neumann algebra whose center consists of multiples of the identity. A factor is said to be of type I if it has a minimal projection. And a type II factor is a semifinite factor which contains no minimal projections. We direct the reader to [3, 10, 22] for a general reference on the theory of von Neumann algebras.

Let $\mathfrak{A}$ be a semifinite factor with a normal faithful tracial weight $\tau$ (it is unique up to scalar multiplication) and $\mathscr{P}$ be the set of all orthogonal projections in $\mathfrak{A}$. Denote by $\mathscr{P}_{c}$ the Grassmann space of all projections in $\mathfrak{A}$ with the trace value $c$, i.e., $\mathscr{P}_{c}=\{P\in\mathscr{P}:\tau(P)=c\}$, $c\in[0,\tau(I)]$. A map $\varphi:\mathscr{P}_{c}\rightarrow\mathscr{P}_{c}$ is transition probability preserving if $\tau(PQ)=\tau(\varphi(P)\varphi(Q))$ for every $P,Q\in\mathscr{P}_{c}$.

Note that $\mathcal{B}(\mathcal{H})$ is a semifinite type I factor. And Wigner’s theorem asserts that if $\mathfrak{A}\cong\mathcal{B}(\mathcal{H})$, then every surjective transition probability preserving map $\varphi:\mathscr{P}_{1}\rightarrow\mathscr{P}_{1}$ is induced by either a unitary or an anti-unitary.

Recently, G. P. Gehér generalized Wigner’s and Molnár’s theorem [24, 14, 17] and prsented the following characterisation of transition probability preserving maps on Grassmann space $\mathscr{P}_{n}$ for $\mathfrak{A}\cong\mathcal{B}(\mathcal{H})$ and $2n\neq\dim\mathcal{H}$ in [9]: every (not necessarily bijective) transition probability preserving map $\varphi:\mathscr{P}_{n}\rightarrow\mathscr{P}_{n}$ is induced by a linear or conjugate-linear isomegry $U:\mathcal{H}\rightarrow\mathcal{H}$, i.e., $\varphi(P)=UPU^{*}$ for all $P\in\mathscr{P}_{n}$.111The case when $2n=\dim\mathcal{H}$ is also handled in [9].

A map $\sigma:\mathcal{B}(\mathcal{H}_{1})\rightarrow\mathcal{B}(\mathcal{H}_{2})$ is said to be implemenyed by a unitary (resp. by an anti-unitary) $U:\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}$ if $\sigma(A)=UAU^{*}$ (resp. $\sigma(A)=UA^{*}U^{*})$) for all $A\in\mathcal{B}(\mathcal{H}_{1})$. Since every *-isomorphism (resp. *-anti-isomorphism) $\sigma$ form $\mathcal{B}(\mathcal{H}_{1})$ to $\mathcal{B}(\mathcal{H}_{2})$ is implemented by a unitary (resp. by an anti-unitary) by [1, Theorem 4.27], the result in [9] can be restated as:

Theorem** (G. P. Gehér [9]).**

*Let $\mathfrak{A}=\mathcal{B}(\mathcal{H})$ where $\mathcal{H}$ is a Hilbert space with $\dim\mathcal{H}\geq 2$. Assume that $2n\neq\dim\mathcal{H}$. For every transition probability preserving map $\varphi:\mathscr{P}_{n}\rightarrow\mathscr{P}_{n}$, there exists an orthogonal projection $E\in\mathfrak{A}=\mathcal{B}(\mathcal{H})$ and a *-isomorphism or a -anti-isomorphism $\sigma:\mathfrak{A}\rightarrow E\mathfrak{A}E$ such that $\varphi(P)=\sigma(P)$ for all $P\in\mathscr{P}_{n}$.

At this stage one is tempted to conjecture that the same result holds for every transition probability preserving map between the Grassmann space of projections in any semifinite factor with a fixed trace value. However, as showed by the example below, this naive generalization of the Wigner’s theorem is doomed to fail.

Example 1.1**.**

*Let $\mathfrak{A}=\mathcal{R}\otimes M_{2}(\mathbb{C})$ where $\mathcal{R}$ is the hyperfinite II1 factor. Let $\mathcal{R}_{0}\subsetneq\mathcal{R}$ be a hyperfinite subfactor of $R$. Since $\mathfrak{A}$ is also the hyperfinite II1 factor, there exist a *-isomorphism $\varphi_{1}$ and a -anti-isomorphism $\varphi_{2}$ from $\mathfrak{A}$ to $\mathcal{R}_{0}$. Then the map

[TABLE]

preserves the transition probability.

Keeping in mind the example above, we formulate the main result of the paper as below.

Theorem 1.1**.**

Let $\mathfrak{A}$ be a semifinite factor with a faithful normal tracial weight $\tau$ and $c\in(0,\tau(I))\setminus\{\tau(I)/2\}$. If the map $\varphi:\mathscr{P}_{c}\rightarrow\mathscr{P}_{c}$ satisfies

[TABLE]

*then there exist two orthogonal projections $E_{1}$ and $E_{2}$ in $\mathfrak{A}$ and a $\sigma$-weak continuous *-homomorphism $\sigma_{1}:\mathfrak{A}\rightarrow E_{1}\mathfrak{A}E_{1}$ and a $\sigma$-weak continuous -anti-homomorphism $\sigma_{2}:\mathfrak{A}\rightarrow E_{2}\mathfrak{A}E_{2}$ such that

[TABLE]

and $\varphi(P)=\sigma_{1}(P)+\sigma_{2}(P)$ for every $P\in\mathscr{P}_{c}$. Furthermore,

1. (1)

$\sigma_{i}(I)=E_{i}$, $i=1,2$. And $E_{1}+E_{2}=I$ if $\mathfrak{A}$ is a finite factor. 2. (2)

If $\mathfrak{A}$ is type I, then $\sigma_{1}$ or $\sigma_{2}$ is the zero map.

In order to prove this result, we first characterize the transition probability preserving maps between subsets of $\mathscr{P}$ whose linear span contains $\mathscr{P}$ (see Theorem 2.1 for details) in Section 2. Next, we show that the maps considered in Theorem 1.1 preserve the relation of commutativity and inclusion between the elements of $\mathscr{P}_{c}$ (see Theorem 3.1 and Theorem 3.2) in Section 3. After these preparations, we complete the proof of Theorem 1.1 in Section 4.

2. The transformations from $\mathscr{P}_{S}$ to $\mathscr{P}_{S}$ preserving transition probability

In this section, we assume that $\mathfrak{A}$ is a semifinite factor with a fixed normal faithful tracial weight $\tau$. Let $\mathscr{P}$ be the set of all projections in $\mathfrak{A}$. If $S$ is a subset of $[0,\infty)$, we use $\mathscr{P}_{S}$ to denote the subset of $\mathscr{P}$:

[TABLE]

For $S=\{c\}$, where $c\in[0,\infty)$, we simplify notation and write $\mathscr{P}_{c}$ for $\mathscr{P}_{\{c\}}$.

Recall that a map $\varphi:\mathscr{P}_{S}\rightarrow\mathscr{P}_{S}$ preserves the transition probability if

[TABLE]

Note that $P\perp Q$ if and only if $PQ=QP=0$ and the following result is immediate.

Lemma 2.1**.**

Let $\varphi:\mathscr{P}_{S}\rightarrow\mathscr{P}_{S}$ be a map preserving the transition probability. Then $\varphi$ preserves the orthogonality in both directions, i.e., $P\bot Q$ if and only if $\varphi(P)\bot\varphi(Q)$.

Let $E_{1}$ and $E_{2}$ be two orthogonal projections in $\mathfrak{A}$. If there exist a *-homomorphism $\sigma_{1}:\mathfrak{A}\rightarrow E_{1}\mathfrak{A}E_{1}$ and a *-anti-homomorphism $\sigma_{2}:\mathfrak{A}\rightarrow E_{2}\mathfrak{A}E_{2}$ such that

[TABLE]

then it is not hard to check that the map,

[TABLE]

preserves the transition probability. Conversely, we show that the transition probability preserving map $\varphi:\mathscr{P}_{S}\rightarrow\mathscr{P}_{S}$ always arises from a sum of a *-homomorphism and a *-anti-homomorphism of $\mathfrak{A}$, where $S$ is a subset of $[0,\infty)$ such that every projection $P\in\mathscr{P}$ is a sum of a family of mutaually orthogonal projections in $\mathscr{P}_{S}$. The case when $S=[0,\infty)$ and $\varphi$ is bijective is handled by L. Molnár in [12]. By modifying the argument used in [12], we can prove Theorem 2.1 below.

Throughout the rest of this section, we assume that $\mathscr{P}_{S}$ is a subset of $\mathscr{P}$ such that every projection $P\in\mathscr{P}$ is a sum of a family of mutaually orthogonal projections in $\mathscr{P}_{S}$.

Lemma 2.2**.**

If $\varphi$ is a transformation from $\mathscr{P}_{S}$ to $\mathscr{P}_{S}$ preserving the transition probablity, then $\varphi$ can be extended to a transformation $\Phi:\mathscr{P}\rightarrow\mathscr{P}$ such that

[TABLE]

whenever $\{E_{\alpha}\}$ is an orthogonal family of projections in $\mathscr{P}$. Furthermore, if $\varphi$ is surjective, then $\Phi$ is also surjective.

Proof.

Let $P\in\mathscr{P}$. There exists a family of pairwise orthogonal projections $\{P_{\alpha}\}_{\alpha}$ in $\mathscr{P}_{S}$ such that $P=\sum_{\alpha}P_{\alpha}$. We define

[TABLE]

By Lemma 2.1, $\Phi(P)\in\mathscr{P}$. If $\{Q_{\beta}\}\subset\mathscr{P}_{S}$ also satisfies $P=\sum_{\beta}Q_{\beta}$, then for every $E\in\mathscr{P}_{S}$, we have

[TABLE]

Then it is clear that

[TABLE]

and $\sum_{\alpha}\varphi(P_{\alpha})=\sum_{\beta}\varphi(Q_{\beta})$. Thus $\Phi$ is well-defined.

Now assume that $\{E_{\alpha}\}$ is an orthogonal family of projections in $\mathscr{P}$. For each $E_{\alpha}$, there exists a family of pairwise orthogonal projections $\{P_{\alpha}^{\beta}\}_{\beta}\subset\mathscr{P}_{S}$ such that $E_{\alpha}=\sum_{\beta}P_{\alpha}^{\beta}$. By the definition of $\Phi$ and note that $P_{\alpha_{1}}^{\beta_{1}}\bot P_{\alpha_{2}}^{\beta_{2}}$ unless $\alpha_{1}=\alpha_{1}$ and $\beta_{1}=\beta_{2}$, we have

[TABLE]

Finally, since any projection in $\mathfrak{A}$ is the sums of projections in $\mathscr{P}_{S}$, the surjectivity of $\varphi$ implies the surjectivity of $\Phi$. ∎

Lemma 2.3**.**

Let $\Phi$ be the map defined in Lemma 2.2. Then $\Phi$ can be extended to a C∗-homomorphism ([20]) from $\mathfrak{A}$ into $\mathfrak{A}$, i.e., a linear self-adjoint map such that $\Phi(H^{2})=\Phi(H)^{2}$ for every self-adjoint operator $H\in\mathfrak{A}$.

Proof.

We claim that $\Phi$ can be extended to a bounded linear operator from $\mathfrak{A}$ into $\mathfrak{A}$. If $\mathfrak{A}$ is not type I2 factor, then [4, Theoerem A] implies that the orthoadditive transformation $\Phi$ has a unique extension to a bounded linear operator from $\mathfrak{A}$ into $\mathfrak{A}$. Assume now that $\mathfrak{A}\cong M_{2}(\mathbb{C})$. Then $\mathscr{P}_{S}$ contains the all rank-one projections. Since every operator in $\mathfrak{A}$ is a linear combination of rank-one projections, it is not hard to check that the following gives a bounded extension of $\Phi$:

[TABLE]

where $P_{i}$ is a rank-one projection, $i=1,\ldots,n$. Thus claim is proved. To simplify the notation, we use $\Phi$ to denote its extension as well.

For every family of pairwise orthogonal projections $\{P_{1},\ldots,P_{n}\}$ and $\lambda_{1},\ldots\lambda_{n}\in\mathbb{R}$, we have

[TABLE]

Since $\Phi$ is bounded, we have $\Phi(H)$ is self-adjoint and $\Phi(H^{2})=\Phi(H)^{2}$ for every self-adjoint operator $H\in\mathfrak{A}$. Therefore, $\Phi$ is a C∗-homomorphism (actually it is not hard to show that $\Phi$ is a Jordan *-homomorphism [11, Proof of Theorem 2]). ∎

Theorem 2.1**.**

*Let $S$ be a subset of $[0,\infty)$ such that every projection $P\in\mathscr{P}$ is a sum of a family of mutaually orthogonal projections in $\mathscr{P}_{S}$ and $\varphi:\mathscr{P}_{S}\rightarrow\mathscr{P}_{S}$ be a map preserving the transition probability. Let $\Phi$ be the C∗-homomorphism defined in Lemma 2.3 and $\mathcal{B}$ be the C∗-algebra generated by $\Phi(\mathscr{P})$. Then $\mathcal{B}$ is a von Neumann subalgebra of $\mathfrak{A}$. Furthermore, there exist two orthogonal central projections $E_{1}$ and $E_{2}$ in $\mathcal{B}$ with $E_{1}+E_{2}=I_{\mathcal{B}}$ (here $I_{\mathcal{B}}$ is the identity of $\mathcal{B}$ and we use $I$ to denote the identity of $\mathfrak{A}$) and a $\sigma$-weak continuous *-homormorphsim $\sigma_{1}:\mathfrak{A}\rightarrow\mathcal{B}E_{1}$ and a $\sigma$-weak continuous -anti-homomorphism $\sigma_{2}:\mathfrak{A}\rightarrow\mathcal{B}E_{2}$ such that:

1. (1)

$\varphi(P)=\sigma_{1}(P)+\sigma_{2}(P)$* for every $P\in\mathscr{P}_{S}$.* 2. (2)

$\tau=\tau\circ(\sigma_{1}+\sigma_{2})$. Therefore $E_{1}+E_{2}=I$ if $\mathfrak{A}$ is a finite factor. 3. (3)

If $\mathfrak{A}$ is a type I factor, then $\sigma_{1}$ or $\sigma_{2}$ is the zero map.

Proof.

[20, Lemma 3.2] implies that there exist central projections $E_{i}$ of $\mathcal{B}$ with $E_{1}+E_{2}=I_{\mathcal{B}}$ such that the map $\sigma_{1}:A\in\mathfrak{A}\rightarrow\Phi(A)E_{1}$ (resp. $\sigma_{2}:A\rightarrow\Phi(A)E_{2}$) is a *-homomorphism (resp. *-anti-homomorphism) and $\Phi(A)=\sigma_{1}(A)+\sigma_{2}(A)$ for every $A\in\mathfrak{A}$.

We now show that $\sigma_{1}$ and $\sigma_{2}$ are $\sigma$-weak continous. For every state $\rho\in\mathfrak{A}_{*}$ where $\mathfrak{A}_{*}$ is the predual of $\mathfrak{A}$, $\rho\circ\sigma_{i}$ is completely additive by [3, Theorem III.2.1.4] and Lemma 2.2. Thus $\rho\circ\sigma_{i}\in\mathfrak{A}_{*}$. Since every normal linear functional is a linear combination of normal states, we have $\sigma_{i}$ is $\sigma$-weakly continuous. Note that $E_{i}\in\mathcal{B}$, $\mathcal{B}E_{i}=\sigma_{i}(\mathfrak{A})$ is a von Neumann subalgebra of $\mathfrak{A}$, $i=1,2$ (see [10, Proposition 7.1.15]) and $\mathcal{B}$ is also a von Neumann algebra.

Since $\varphi$ preserves the transition probabilty, we have

[TABLE]

By the defintion of $\Phi$, $\tau(Q)=\tau(\Phi(Q))=\tau(\sigma_{1}(Q)+\sigma_{2}(Q))$ for every $Q\in\mathscr{P}$. Note that the finite linear combinations of finite projections in $\mathfrak{A}$ is a $\sigma$-weakly dense *-subalgebra of $\mathfrak{A}$. Then [22, Vol. II, VIII. Proposition 3.15] implies that $\tau=\tau\circ(\sigma_{1}+\sigma_{2})$. Since $\sigma_{i}(I)=E_{i}$, $i=1,2$, $\tau(E_{1}+E_{2})=\tau(I)$ implies that $E_{1}+E_{2}=I$ if $\mathfrak{A}$ is a finite von Neumann algebra.

Assume that $\mathfrak{A}$ is a type I factor. Note that $\sigma_{1}(P)+\sigma_{2}(P)=\Phi(P)$ is rank one if $P$ is a rank one projection. Since $\sigma_{1}$ and $\sigma_{2}$ are $\sigma$-weak continuous, we have $\sigma_{1}$ or $\sigma_{2}$ is the zero map. ∎

3. Auxiliary results for maps $\varphi:\mathscr{P}_{c}\rightarrow\mathscr{P}_{c}$, $c\in(0,\tau(I)/2)$

From now on, we assume that $\mathfrak{A}$ is a semifinite factor which is not type I2 and $\varphi:\mathscr{P}_{c}\rightarrow\mathscr{P}_{c}$ is a map preserving the transition probability, where $c\in(0,\tau(I)/2)$ and $\mathscr{P}_{c}\neq\emptyset$ or $\{0\}$.

The following two results must be well-known to experts (Lemma 3.2 is essentially [14, Lemma 5]) and we sketch the proof for the reader’s convenience.

Lemma 3.1**.**

Let $P_{1},P_{2},Q\in\mathscr{P}_{c}$. If $P_{1}\bot P_{2}$, then $Q\leq P_{1}+P_{2}$ if and only if $\varphi(Q)\leq\varphi(P_{1})+\varphi(P_{2})$.

Proof.

By Lemma 2.1, $\varphi(P_{1})+\varphi(P_{2})$ is a projection. Note that

[TABLE]

Thus $Q\leq P_{1}+P_{2}$ if and only if $\varphi(Q)\leq\varphi(P_{1})+\varphi(P_{2})$. ∎

Lemma 3.2**.**

Let $P,Q\in\mathscr{P}$ and $E$ be the range projection of $PQP$. Assume that $\tau(P)<\infty$. Then $\tau(PQ)\leq\tau(E)$ and $\tau(PQP)=\tau(E)$ if and only if $PQ=QP$.

Proof.

Note that $0\leq PQP\leq E$. Thus $\tau(PQP)\leq\tau(E)$. It is clear that $PQ=QP$ if and only if $PQP=E$. Thus $PQ=QP$ if and only if $\tau(PQP)=\tau(E)$. ∎

Let $Q,P_{1},P_{2}\in\mathscr{P}_{c}$ such that $Q\leq P_{1}\vee P_{2}$ and $P_{1}P_{2}=P_{2}P_{1}$. If $\mathfrak{A}$ is a type I factor and $\mathscr{P}_{c}$ is the set of the minimal projections in $\mathfrak{A}$, then $P_{1}=P_{2}$ or $P_{1}\bot P_{2}$. Therefore Lemma 2.1 implies that $\varphi(P_{1})\varphi(P_{2})=\varphi(P_{2})\varphi(P_{1})$, and $\varphi(Q)\leq\varphi(P_{1})\vee\varphi(P_{2})$ by Lemma 3.1. For the rest of this section, we will prove in Theorem 3.1 and Theorem 3.2 that the same result holds for every semifinite factor $\mathfrak{A}$ which is not type I2 and transition probability preserving map $\varphi:\mathscr{P}_{c}\rightarrow\mathscr{P}_{c}$, $c\in(0,\tau(I)/2)$.

By the discussion above, we will assume that $\mathscr{P}_{c}$ is not the set of minimal projections of $\mathfrak{A}$ for the rest of the section.

Lemma 3.3**.**

Let $c_{0}=\min(\tau(I)-2c,c/2)$ and $P,Q$ be two commutative projections in $\mathscr{P}_{c}$. If $\tau(PQ)\geq c-c_{0}$ or $\tau(PQ)\leq c_{0}$, then $\varphi(P)\varphi(Q)=\varphi(Q)\varphi(P)$. In particular, if $\frac{5c}{2}\leq\tau(I)$, then $PQ=QP$ implies that $\varphi(P)\varphi(Q)=\varphi(Q)\varphi(P)$ for every $P$, $Q\in\mathscr{P}_{c}$.

Proof.

Let $P^{\prime}\in\mathscr{P}_{c}$ such that $P\bot P^{\prime}$ and $Q\leq P+P^{\prime}$. By Lemma 3.1, it is clear that $\varphi(P)\varphi(Q)=\varphi(Q)\varphi(P)$ if and only if $\varphi(P^{\prime})\varphi(Q)=\varphi(Q)\varphi(P^{\prime})$. Thus we may assume that $\tau(PQ)\geq c-c_{0}$.

Since $2c+c_{0}\leq\tau(I)$, we can choose $Q^{\prime}\in\mathscr{P}_{c}$ such that $Q^{\prime}\bot(P\vee Q)$, $Q^{\prime}P^{\prime}=P^{\prime}Q^{\prime}$ and $\tau(P^{\prime}Q^{\prime})=\tau(PQ)$. Note that $\tau(P^{\prime}(Q+Q^{\prime}))=c$ and $P^{\prime}\leq Q+Q^{\prime}$. By Lemma 2.1,

[TABLE]

Thus $[\varphi(P^{\prime})\varphi(Q^{\prime})\varphi(P^{\prime})][\varphi(P^{\prime})\varphi(Q)\varphi(P^{\prime})]=0$.

Let $E$ and $F$ be the range projections of $\varphi(P^{\prime})\varphi(Q)\varphi(P^{\prime})$ and $\varphi(P^{\prime})\varphi(Q^{\prime})\varphi(P^{\prime})$ respectively. $[\varphi(P^{\prime})\varphi(Q^{\prime})\varphi(P^{\prime})][\varphi(P^{\prime})\varphi(Q)\varphi(P^{\prime})]=0$ implies that $EF=0$. If $\varphi(P^{\prime})\varphi(Q)\neq\varphi(Q)\varphi(P^{\prime})$, Lemma 3.2 implies that $\tau(E)>\tau(\varphi(P^{\prime})\varphi(Q))=\tau(P^{\prime}Q)$. Note that $\tau(F)\geq\tau(P^{\prime}Q^{\prime})$ and $E\vee F=E+F\leq\varphi(P^{\prime})$. Thus we have

[TABLE]

Hence $\varphi(P)\varphi(Q)$ must equals $\varphi(Q)\varphi(P)$. ∎

Lemma 3.4**.**

Let $c_{1}\in(0,c)$ such that $\mathscr{P}_{c_{1}}\neq\emptyset$. Suppose that for every $P$, $Q\in\mathscr{P}_{c}$ such that $PQ=QP$ and $\tau(PQ)\leq c_{1}$, we have $\varphi(P)\varphi(Q)=\varphi(Q)\varphi(P)$. Then $PQ=QP$ and $\tau(PQ)\leq 2c_{1}$ also implies that $\varphi(P)\varphi(Q)=\varphi(Q)\varphi(P)$.

Proof.

We choose two orthogonal projections $P_{1}$, $P_{2}$ in $\mathscr{P}_{c}$ such that $P_{i}P=PP_{i}$, $P_{i}Q=QP_{i}$, $i=1,2$ and $PQ=PP_{2}+QP_{1}$,

[TABLE]

By Lemma 3.1 and the assumption, we have

[TABLE]

Thus $\varphi(P_{1})\varphi(Q)$ and $\varphi(P_{2})\varphi(P)$ are two projections whose traces are $\tau(P_{1}Q)$ and $\tau(P_{2}P)$ respectively. Let $E_{i}$ be the range projection of $\varphi(P)\varphi(Q)\varphi(P)\varphi(P_{i})$, $i=1,2$. It is clear that $E_{1}E_{2}=0$ and

[TABLE]

Note that the range projection of $\varphi(P)\varphi(Q)\varphi(P)$ is $E_{1}+E_{2}$ and

[TABLE]

Therefore $\tau(E_{1}+E_{2})=\tau(\varphi(P)\varphi(Q))$ and Lemma 3.2 implies that $\varphi(P)\varphi(Q)=\varphi(Q)\varphi(P)$. ∎

Theorem 3.1**.**

Let $P,Q\in\mathscr{P}_{c}$. If $PQ=QP$ then $\varphi(P)\varphi(Q)=\varphi(Q)\varphi(P)$.

Proof.

If $5c/2\leq\tau(I)$, the theorem is proved by Lemma 3.3. Assume $2c<\tau(I)<5c/2$, let $c_{0}=\tau(I)-2c<c/2$. Note that $\mathscr{P}_{c_{0}}\neq\emptyset$. Then Lemma 3.4 implies the result. ∎

Lemma 3.5**.**

Let $c_{0}=\min(\tau(I)-2c,c)$ and $P_{1},P_{2},Q\in\mathscr{P}_{c}$. Suppose that $P_{1}P_{2}=P_{2}P_{1}$ and $\tau(P_{1}P_{2})\leq c_{0}$. Then $Q\leq P_{1}\vee P_{2}$ implies $\varphi(Q)\leq\varphi(P_{1})\vee\varphi(P_{2})$.

Proof.

Since $2c+\tau(P_{1}P_{2})\leq\tau(I)$, we can choose $P_{3},P_{4}\in\mathscr{P}_{c}$ such that $\{P_{1},P_{2},P_{3},P_{4}\}$ are mutually commutative and

[TABLE]

By Lemma 3.1, we have

[TABLE]

Since $\varphi(P_{1})\vee\varphi(P_{2})\leq[\varphi(P_{1})+\varphi(P_{3})]\wedge[\varphi(P_{1})+\varphi(P_{4})]$, Theorem 3.1 implies that

[TABLE]

Thus $\varphi(P_{1})\vee\varphi(P_{2})=[\varphi(P_{1})+\varphi(P_{3})]\wedge[\varphi(P_{1})+\varphi(P_{4})]$. By Lemma 3.1,

[TABLE]

∎

Lemma 3.6**.**

Let $c_{1}\in(0,c)$ such that $\mathscr{P}_{c_{1}}\neq\emptyset$. Suppose that for every $P_{1},P_{2},Q\in\mathscr{P}_{c}$ such that $P_{1}P_{2}=P_{2}P_{1}$, $\tau(P_{1}P_{2})\leq c_{1}$ and $Q\leq P_{1}\vee P_{2}$, we have $\varphi(Q)\leq\varphi(P_{1})\vee\varphi(P_{2})$. Then $P_{1}P_{2}=P_{2}P_{1}$, $\tau(P_{1}P_{2})\leq 2c_{1}$ and $Q\leq P_{1}\vee P_{2}$ also implies that $\varphi(Q)\leq\varphi(P_{1})\vee\varphi(P_{2})$.

Proof.

We assume that $P_{1}\neq P_{2}$ and choose $Q_{1},Q_{2}\in\mathscr{P}_{c}$ such that $\{P_{1},P_{2},Q_{1},Q_{2}\}$ is a family of mutually commutative projections, $Q_{1}\bot Q_{2}$, $P_{1}\vee P_{2}\leq Q_{1}+Q_{2}$, $P_{1}Q_{1}+P_{2}Q_{2}=P_{1}P_{2}$,

[TABLE]

where $0<a_{i}\leq c_{1}$ and $a_{1}+a_{2}=\tau(P_{1}P_{2})$. Therefore, we have

[TABLE]

And Theorem 3.1 implies that

[TABLE]

and

[TABLE]

Since $\varphi(Q_{1})[I-\varphi(P_{2})]\bot\varphi(Q_{2})[I-\varphi(P_{1})]$ and

[TABLE]

we have

[TABLE]

Therefore, $\varphi(P_{1})\vee\varphi(P_{2})=[\varphi(P_{1})\vee\varphi(Q_{1})]\wedge[\varphi(P_{2})\vee\varphi(Q_{2})]$. Note that $Q\leq P_{1}\vee P_{2}\leq[P_{1}\vee Q_{1}]\wedge[P_{2}\vee Q_{2}]$. And the assumption implies that

[TABLE]

∎

Theorem 3.2**.**

Let $P_{1},P_{2},Q\in\mathscr{P}_{c}$. Assume that $P_{1}P_{2}=P_{2}P_{1}$. Then $Q\leq P_{1}\vee P_{2}$ implies $\varphi(Q)\leq\varphi(P_{1})\vee\varphi(P_{2})$.

Proof.

By Lemma 3.5, we assume that $0\neq c_{0}=\tau(I)-2c<c$. It is clear that $\mathscr{P}_{c_{0}}\neq\emptyset$. Then Lemma 3.6 implies the result. ∎

Let $2\leq m\in\mathbb{N}$ such that $\mathscr{P}_{c/m}\neq\emptyset$.

Lemma 3.7**.**

For every $Q\in\mathscr{P}_{c/m}$, there exist $m+1$ mutually commutative projections $P_{i}\in\mathscr{P}_{c}$, $i=0,\ldots,m$, such that

[TABLE]

Proof.

Note that $2c\leq\tau(I)$, there exists a family of orthogonal projections $\{Q_{i}\}_{i=0}^{m}\subset\mathscr{P}_{c/m}$ such that $Q=Q_{0}$. Let $P_{i}=\sum_{j=0}^{m}Q_{j}-Q_{i}$. Then we have

[TABLE]

and $P_{i}\in\mathscr{P}_{c}$, $i=0,\ldots,m$. ∎

Lemma 3.8**.**

Let $Q=\frac{1}{m}[P_{1}+\cdots+P_{m}-(m-1)P_{0}]\in\mathscr{P}_{c/m}$, where $\{P_{i}\}_{i=0}^{m}$ is a family of projections in $\mathscr{P}_{c}$. Then there exists a family of orthogonal projections $\{Q_{i}\}_{i=0}^{m}\subset\mathscr{P}_{c/m}$ such that $Q=Q_{0}$ and $P_{i}=\sum_{j=0}^{m}Q_{j}-Q_{i}$.

Proof.

Note that

[TABLE]

This implies that $Q\leq P_{i}$, $i=1,\ldots,m$ and $Q\bot P_{0}$. Let $P_{i}=Q+P_{i}^{\prime}$, $i=1,\ldots,m$. Then we have $P_{1}^{\prime}+\cdots+P_{m}^{\prime}=(m-1)P_{0}$ and

[TABLE]

Since $\tau(P_{i}^{\prime})=(m-1)c/m$, $P_{i}^{\prime}\leq P_{0}$, $i=1,\ldots m$. Let $Q_{i}=P_{0}-P_{i}^{\prime}$, $i=1,\ldots m$. Note that $\tau(Q_{i})=c/m$,

[TABLE]

We have $Q_{i}\leq P_{j}^{\prime}$ if $i\neq j$, $i,j\in\{1,\ldots,m\}$. For $i\neq j$,

[TABLE]

Therefore, $Q_{i}\bot Q_{j}$ for $i\neq j$ and $P_{i}=\sum_{j=0}^{m}Q_{j}-Q_{i}$. ∎

Lemma 3.9**.**

Let $Q=\frac{1}{m}[P_{1}+\cdots+P_{m}-(m-1)P_{0}]\in\mathscr{P}_{c/m}$, where $\{P_{i}\}_{i=0}^{m}$ is a family of projections in $\mathscr{P}_{c}$. Then $\frac{1}{m}[\varphi(P_{1})+\cdots+\varphi(P_{m})-(m-1)\varphi(P_{0})]\in\mathscr{P}_{c/m}$.

Proof.

By Lemma 3.8, we assume that $P_{i}=\sum_{j=0}^{m}Q_{j}-Q_{i}$ where $\{Q_{i}\}_{i=0}^{m}\subset\mathscr{P}_{c/m}$ is a family of orthogonal projections such that $Q=Q_{0}$. In particular, $P_{i}P_{j}=P_{j}P_{i}$. Thus $\varphi(P_{i})\varphi(P_{j})=\varphi(P_{j})\varphi(P_{i})$ by Theorem 3.1. Note that

[TABLE]

Also Theorem 3.2 implies that $\varphi(P_{i})\leq\varphi(P_{j})+\varphi(P_{k})-\varphi(P_{j})\varphi(P_{k})$ if $j\neq k$. Let $i,j\in\{1,\ldots m\}$. If $i\neq j$ and $i\cdot j\neq 0$,

[TABLE]

We have $\tau(\varphi(P_{0})\varphi(P_{i})\varphi(P_{j})))=\frac{(m-2)c}{m}$. Let $E_{i}=\varphi(P_{0})-\varphi(P_{0})\varphi(P_{i})$, $i=1,\ldots m$. Note that $\tau(E_{i})=c/m$ and

[TABLE]

Thus $E_{i}\bot E_{j}$ and $\varphi(P_{0})=\sum_{i=1}^{m}E_{i}$. For $i,j\in\{1,\ldots,m\}$ and $i\neq j$,

[TABLE]

Therefore, $E_{j}\leq\varphi(P_{i})$ and $\varphi(P_{i})=F_{i}+\sum_{j=1}^{m}E_{j}-E_{i}$, where $F_{i}\in\mathscr{P}_{c/m}$ and $F_{i}\varphi(P_{0})=0$. Note that

[TABLE]

we have $F_{i}=F_{j}$. Let $E_{0}=F_{1}$, we have $\varphi(P_{i})=\sum_{i=0}^{m}E_{i}-E_{i}$ and $\frac{1}{m}[\varphi(P_{1})+\cdots+\varphi(P_{m})-(m-1)\varphi(P_{0})]=E_{0}\in\mathscr{P}_{c/m}$. ∎

Lemma 3.10**.**

Let $Q=\frac{1}{m}[P_{1}+\cdots+P_{m}-(m-1)P_{0}]$ and $Q^{\prime}=\frac{1}{m}[P_{1}^{\prime}+\cdots+P_{m}^{\prime}-(m-1)P_{0}^{\prime}]\in\mathscr{P}_{c/m}$, where $\{P_{i}\}_{i=0}^{m}$ and $\{P_{i}^{\prime}\}_{i=0}^{m}$ are two families of projections in $\mathscr{P}_{c}$. Then

[TABLE]

In particular, if $Q=Q^{\prime}$ then

[TABLE]

Proof.

It is clear that eq. 1 holds. If $Q=Q^{\prime}$, Lemma 3.9 implies that $\frac{1}{m}[\varphi(P_{1})+\cdots+\varphi(P_{m})-(m-1)\varphi(P_{0})]=\frac{1}{m}[\varphi(P_{1}^{\prime})+\cdots+\varphi(P_{m}^{\prime})-(m-1)\varphi(P_{0}^{\prime})]$. ∎

We are now in a position to prove Theorem 1.1.

4. Proof of Theorem 1.1

Proof of Theorem 1.1.

We first assume that $c\in(0,\tau(I)/2)$ and claim that there is a subset $S$ of $[0,c]$ and a map $\Phi:\mathscr{P}_{S}\rightarrow\mathscr{P}_{S}$ satisfying the following conditions:

1. (1)

$c\in S$ and every projection $P\in\mathscr{P}$ is a sum of a family of pairwise orthogonal projections in $\mathscr{P}_{S}$, 2. (2)

$\Phi$ preserves the transition probablility and $\Phi|_{\mathscr{P}_{c}}=\varphi$.

If the claim holds, we can prove the theorem in this case by invoking Theorem 2.1. To prove the claim, we consider the following two cases.

Case I: $\mathfrak{A}$ is a type I factor. If $\mathscr{P}_{c}$ is the set of minimal projections of $\mathfrak{A}$, then $S=\{0,c\}$ and $\varphi$ satisfy the condition (1) and (2).

Otherwise, there exists $2\leq m\in\mathbb{N}$ such that $\mathscr{P}_{c/m}$ is the set of minimal projections of $\mathfrak{A}$. Let $S=\{0,c/m,c\}$ and

[TABLE]

where $\varphi_{1}$ is defined by

[TABLE]

where $\{P_{i}\}$ is a familiy of mutually commutative projections in $\mathscr{P}_{c}$. By Lemma 3.7, Lemma 3.9 and Lemma 3.10, $S$ and $\Phi$ satisfy the conditions.

Case II: $\mathfrak{A}$ is a type II factor. Let $S=\{0\}\cup\{\frac{c}{2^{n}}:n=0,1,2,\ldots\}$. Then it is easy to see that $\mathscr{P}_{S}$ satisfies condtion (1). Let $\varphi_{0}=\varphi$. We can define a map $\varphi_{n}:\mathscr{P}_{c/2^{n}}\rightarrow\mathscr{P}_{c/2^{n}}$ which preserves the transition probability for every $n\in\mathbb{N}$ inductively by

[TABLE]

where $\{P_{i}\}_{i=0}^{2}$ is a familiy of mutually commutative projections in $\mathscr{P}_{c/2^{n-1}}$. Let $\Phi$ be the map from $\mathscr{P}_{S}$ to $\mathscr{P}_{S}$ given by $\Phi(P)=\varphi_{n}(P)$ for $P\in\mathscr{P}_{c/2^{n}}$, $n=0,1,2\ldots$ and $\Phi(0)=0$. Suppose that $Q\in\mathscr{P}_{c/2^{k}}$ and $Q^{\prime}\in\mathscr{P}_{c/2^{l}}$, $k,l\in\mathbb{N}$, we have

[TABLE]

where $\{P_{i}\}$ and $\{P_{j}^{\prime}\}$ are two families of projections in $\mathscr{P}_{c}$ and $\lambda_{i}$, $\beta_{j}\in\{1,-1\}$ such that $Q=\frac{1}{2^{k}}\sum_{i}\lambda_{i}P_{i}$ and $Q^{\prime}=\frac{1}{2^{l}}\sum_{j}\lambda_{j}P_{j}^{\prime}$. Then it is clear that $\tau(\Phi(Q)\Phi(Q^{\prime}))=\tau(QQ^{\prime})$.

Now we assume that $\tau(I)<2c$. Without loss of generality, we assume that $\tau(I)=1$. It is clear that

[TABLE]

is a map preserving the transition probability. Note that $2(1-c)<\tau(I)$. The discussion above implies there exists a projection $E\in\mathscr{P}$ and a $\sigma$-weak continuous *-homomorphims $\sigma_{1}:\mathfrak{A}\rightarrow E\mathfrak{A}E$ and a $\sigma$-weak continuous *-anti-homomorphism $\sigma_{2}:\mathfrak{A}\rightarrow(I-E)\mathfrak{A}(I-E)$ such that $\psi(P)=\sigma_{1}(P)+\sigma_{2}(P)$ for every $P\in\mathscr{P}_{1-c}$ and $\sigma_{1}$ or $\sigma_{2}$ is zero map if $\mathfrak{A}$ is a type I factor. Therefore

[TABLE]

∎

Corollary 4.1**.**

*Let $\mathfrak{A}$ be a semifinite factor and $c\in(0,\tau(I))\setminus\{\tau(I)/2\}$. If $\varphi:\mathscr{P}_{c}\rightarrow\mathscr{P}_{c}$ is a surjective map preserving the transition probability, then there exists a *-automorphism or a -anti-automorphism $\sigma$ of $\mathfrak{A}$ such that, $\varphi(P)=\sigma(P)$ for every $P\in\mathscr{P}_{S}$.

Proof.

If $\mathscr{P}_{c}=\emptyset$, the corollary holds trivially. Assume that $\mathscr{P}_{c}\neq\emptyset$. Note that if $A\in\mathfrak{A}$ commutes with every $P\in\mathscr{P}_{c}$, then $A$ is in the center of $\mathfrak{A}$. Since $\mathfrak{A}$ is a factor, Theorem 1.1 implies the result. ∎