# Hilbert and Thompson isometries on cones in JB-algebras

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## Abstract

Hilbert’s and Thompson’s metric spaces on the interior of cones in JB-algebras are important examples of symmetric Banach-Finsler spaces. In this paper we characterize the Hilbert’s metric isometries on the interiors of cones in JBW-algebras, and the Thompson’s metric isometries on the interiors of cones in JB-algebras. These characterizations generalize work by Bosché on the Hilbert’s and Thompson’s metric isometries on symmetric cones, and work by Hatori and Molnár on the Thompson’s metric isometries on the cone of positive self-adjoint elements in a unital \(C^*\)-algebra. To obtain the results we develop a variety of new geometric and Jordan algebraic techniques.

## Keywords

Hilbert’s metric Thompson’s metric Order unit spaces JB-algebras Isometries Symmetric Banach–Finsler manifolds## Mathematics Subject Classification

Primary 58B20 Secondary 32M15## 1 Introduction

*A*there exist two important metrics: Hilbert’s metric and Thompson’s metric. Hilbert’s metric goes back to Hilbert [20], who defined a metric \(\delta _H\) on an open bounded convex set \(\Omega \) in a finite dimensional real vector space

*V*by

*a*and

*b*and \(\partial \Omega \) such that

*a*is between \(a'\) and

*b*, and

*b*is between \(b'\) and

*a*. The Hilbert’s metric spaces \((\Omega ,\delta _H)\) are Finsler manifolds that generalize Klein’s model of the real hyperbolic space. They play a role in the solution of Hilbert’s Fourth problem [2], and possess features of nonpositive curvature [4, 23]. In recent years there has been increased interest in the geometry of Hilbert’s metric spaces, see [18] for an overview. In this paper we shall work with a slightly more general version of Hilbert’s metric, which is a metric between pairs of the rays in the interior of the cone. It is defined in terms of the partial ordering of the cone and was introduced by Birkhoff [5]. It has found numerous applications in the spectral theory of linear and nonlinear operators, ergodic theory, and fractal analysis, see [26, 27, 33, 36, 41, 42, 43] and the references therein.

Thompson’s metric was introduced by Thompson in [47], and is also a useful tool in the spectral theory of operators on cones. If the order unit space is complete, the resulting Thompson’s metric space is a prime example of a Banach-Finsler manifold. Moreover, if the order unit space is a JB-algebra (which is a simultaneous generalization of both a Euclidean Jordan algebra as well as the selfadjoint elements of a \(C^*\)-algebra), then the Banach-Finsler manifold is symmetric and possesses certain features of nonpositive curvature [3, 10, 11, 24, 25, 32, 40, 42, 48]. This is one of the main reasons why Thompson’s metric is of interest in the study of the geometry of spaces of positive operators.

It appears that understanding the isometries of Hilbert’s and Thompson’s metrics on the interiors of cones in order unit spaces is closely linked with the theory of JB-algebras. Evidence for this link was provided by Walsh [49], who showed, among other things, that for finite dimensional order unit spaces *A*, the Hilbert’s metric isometry group on \(A_+^\circ \) is not equal to the group of projectivities of \(A_+^\circ \) if and only if *A* is a Euclidean Jordan algebra whose cone is not Lorentzian [49, Corollary 1.4]. Moreover, in that case, the group of projectivities has index 2 in the isometry group, and the additional isometries are obtained by adjoining the map induced by \( a\in A_+^\circ \mapsto a^{-1}\in A_+^\circ \). At present it is unknown if this result has an infinite dimensional extension.

The main objective of this paper is to characterize the Hilbert’s metric isometries on the interiors of cones in JBW-algebras (a subclass of JB-algebras that includes both the selfadjoint elements of von Neumann algebras as well as Euclidean Jordan algebras), and the Thompson’s metric isometries on the interiors of cones in JB-algebras. Unfortunately our methods do not give a characterization of the Hilbert’s metric isometries for general JB-algebras, as we require the existence of sufficiently many projections. Our results generalize and complement and number of earlier works. Firstly, the isometries for Thompson’s metric and Hilbert’s metric between the positive cones of the bounded operators on a Hilbert space of dimension at least three were characterized by Molnár in [37]. He exploited the geometric mean to show that these isometries preserve commutativity and applied the characterization of such maps. In [19], Hatori and Molnár described the isometries for Thompson’s metric between the positive cones of *C**-algebras by showing that these isometries yield linear isometries on the whole space. As we shall see in Theorem 2.17 the Hilbert’s metric isometries on cones in JB-algebras induce variation norm preserving isometries on the whole JB-algebra. For von Neumann algebras without a type \(I_2\) summand the variation norm isometries were characterized by Molnár in [38]. His result was extended to JBW-algebras without a type \(I_2\) summand by Hamhalter in [16]. Finally we should mention the work by Bosché [6], who characterized the isometries for Thompson’s metric and Hilbert’s metric on cones in Euclidean Jordan algebras by making essential use of the fact that the symmetric cones are finite dimensional.

Our approach is to show that Hilbert’s metric and Thompson’s metric isometries mapping the identity to the identity induce bijective linear norm isometries; the Thompson’s metric isometries yield norm isometries of the JB-algebra, whereas the Hilbert’s metric isometries induce isometries on the quotient of the JB-algebra by the span of the unit, equipped with the variation norm, see Theorem 2.17. This extends results in [6, 19]. By using a characterization of bijective linear norm isometries of JB-algebras due to Isidro and Rodríguez-Palacios [21] we then characterize the Thompson’s metric isometries of JB-algebras, extending results of [6, 19]. As for Hilbert’s metric, the variation norm isometries induced by Hilbert’s metric isometries can be viewed as linear maps preserving the maximal deviation, the quantum analogue of the maximal standard deviation, see [16, 38, 39]. These have been characterized for JBW-algebras without a type \(I_2\) summand as mentioned above. We exploit the fact that the variation norm isometry is induced by a Hilbert’s metric isometry to obtain the desired characterization without any restriction on the JBW-algebras. This characterization also complements our earlier work [29], in which we considered the order unit space *C*(*K*) consisting of all continuous functions on a compact Hausdorff space *K*. In the same paper we showed that the group of Hilbert’s metric isometries is equal to the group of projectivities if the Hilbert’s metric is uniquely geodesic. Other works on Hilbert’s metric isometries and Thompson’s metric isometries on finite dimensional cones include [12, 30, 35, 44].

The structure of the paper is as follows.

Section 2 is our preliminary section. We first introduce Hilbert’s metric and Thompson’s metric and JB(W)-algebras. We then investigate some properties that will prove to be very useful in characterizing the isometries for both metrics. In particular, we characterize when there exist unique geodesics for Hilbert’s metric and Thompson’s metric between two elements of a JB-algebra, and we study the interplay between geometric means and the isometries for both metrics. Our findings also generalize earlier work done on Euclidean Jordan algebras and \(C^*\)-algebras. These investigations then result in the crucial Theorem 2.17 mentioned above.

In Sect. 3 we characterize the isometries for Thompson’s metric, and we exploit this result to describe the corresponding isometry group of a direct product of simple JB-algebras in terms of the automorphism groups of the components.

Finally, we consider Hilbert’s metric isometries in Sect. 4. Since the extreme points of the unit ball in the quotient coincide with the equivalence classes of nontrivial projections, every Hilbert’s metric isometry induces a bijection on the projections. At this point we restrict to JBW-algebras as they contain a lot of projections in contrast to JB-algebras. By using geometric properties of Hilbert’s metric as well as operator algebraic methods, we obtain that the above bijection on the projections is actually a projection orthoisomorphism: two projections are orthogonal if and only if their images are orthogonal. Dye’s classical theorem [13] shows that every projection orthoisomorphism between von Neumann algebras without a type \(I_2\) summand extends to a Jordan isomorphism on the whole algebra. This was extended by Bunce and Wright [7] to JBW-algebras, and we use this result to extend our projection orthoisomorphism defined outside the type \(I_2\) summand to a Jordan isomorphism. It remains to take care of the type \(I_2\) summand, which we are able to do using a characterization of type \(I_2\) JBW-algebras due to Stacey [45] and the explicit fact that our projection orthoisomorphism comes from a linear map on the quotient. Thus we are able to extends the whole projection orthoisomorphism to a Jordan isomorphism, which then easily yields the main result of our paper, Theorem 4.21, which we repeat below for the reader’s convenience. The set \(\overline{M}_+^\circ \) denotes the set of rays in \(M_+^\circ \), and \(U_b\) denotes the quadratic representation of *b*.

### Theorem 1.1

*M*and

*N*are JBW-algebras, then \(f :\overline{M}_+^\circ \rightarrow \overline{N}_+^\circ \) is a bijective Hilbert’s metric isometry if and only if

Note that Theorem 1.1 follows from [16, Theorem 1.1] if *M* is a JBW-algebra without a type \(I_2\) summand, since the Hilbert’s metric isometry induces a variation norm isometry by Theorem 2.17.

We claim that this result extends Molnar’s theorem [37, Theorem 2], reformulated below using our notation.

### Theorem 1.2

*H*be a complex Hilbert space with \(\dim (H) \ge 3\) and let \(f :B(H)_+^\circ \rightarrow B(H)_+^\circ \) be a bijective Hilbert’s metric isometry. Then there is an invertible bounded linear or conjugate linear operator \(z :H \rightarrow H\) and an \(\varepsilon \in \{\pm 1 \}\) such that

*J*of

*B*(

*H*) are of the form \(Ja = uau^*\), where

*u*is a unitary or anti-unitary (i.e., conjugate linear unitary) operator. Hence

*bu*, with a positive

*b*and (anti-)unitary

*u*. For linear operators this is just the polar decomposition, and by considering a conjugate linear operator to be a linear operator from

*H*to its conjugate Hilbert space, we obtain the same decomposition for conjugate linear operators.

In view of [49, Corollary 1.4] mentioned above we make the following contribution in Proposition 4.23, where we show that the isometry group for Hilbert’s metric on JBW-algebras is not equal to the group of projectivities if and only if the cone is not a Lorentz cone.

## 2 Preliminaries

In this section we collect some basic definitions and recall several useful facts concerning Hilbert’s and Thompson’s metrics and cones in JB-algebras.

### 2.1 Order unit spaces

*A*be a partially ordered real vector space with cone \(A_+\). So, \(A_+\) is convex, \(\lambda A_+\subseteq A_+\) for all \(\lambda \ge 0\), \(A_+\cap -A_+=\{0\}\), and the partial ordering \(\le \) on

*A*is given by \(a\le b\) if \(b-a\in A_+\). Suppose that there exists an

*order unit*\(u\in A_+\), i.e., for each \(a\in A\) there exists \(\lambda >0\) such that \(-\lambda u\le a\le \lambda u\). Furthermore assume that

*A*is

*Archimedean*, that is to say, if \(na\le u\) for all \(n=1,2,\ldots \), then \(a\le 0\). In that case

*A*can be equipped with the

*order unit norm*,

*order unit space*, see [17]. It is not hard to show, see for example [29], that \(A_+\) has nonempty interior \(A_+^\circ \) in \((A,\Vert \cdot \Vert _u)\) and \(A_+^\circ =\{a\in A:a \text{ is } \text{ an } \text{ order } \text{ unit } \text{ of } A\}\).

*Hilbert’s metric*is given by

*Thompson’s metric*is defined by

### 2.2 JB-algebras

*Jordan algebra*\((A, \circ )\) is a commutative, not necessarily associative algebra such that

*JB-algebra*

*A*is a normed, complete real Jordan algebra satisfying,

*A*, equipped with the Jordan product \(a \circ b := (ab + ba)/2\). By the Gelfand-Naimark theorem, this JB-algebra is a norm closed Jordan subalgebra of the self-adjoint bounded operators on a Hilbert space; such an algebra is called a

*JC-algebra*. By [17, Corollary 3.1.7], Euclidean Jordan algebras are another example of JB-algebras. We can think of JB-algebras as a simultaneous generalization of both the self-adjoint elements of \(C^*\)-algebras as well as Euclidean Jordan algebras.

Throughout the paper, we will assume that all JB-algebras are unital with unit *e*.

The set of invertible elements of *A* is denoted by \({{\mathrm{Inv}}}(A)\). The *spectrum* of \(a \in A\), \(\sigma (a)\), is defined to be the set of \(\lambda \in \mathbb {R}\) such that \(a - \lambda e\) is not invertible in JB(*a*, *e*), the JB-algebra generated by *a* and *e* [17, 3.2.3]. There is a continuous functional calculus: JB\((a,e) \cong C(\sigma (a))\). Both the spectrum and the functional calculus coincide with the usual notions in both Euclidean Jordan algebras as well as JC-algebras.

The elements \(a,b \in A\) are said to *operator commute* if \(a \circ (b \circ c) = b \circ (a \circ c)\) for all \(c \in A\). In a JC-algebra, two elements operator commute if and only if they commute in the \(C^*\)-multiplication [1, Proposition 1.49]. In the sequel we shall write the Jordan product of two operator commuting elements \(a,b\in A\) as *ab* instead of \(a\circ b\). The *center* of *A* consists of all elements that operator commute with all elements of *A*, and it is an associative JB-subalgebra of *A*. Every associative JB-algebra is isomorphic to *C*(*K*) for some compact Hausdorff space *K* [17, Theorem 3.2.2].

*A*into an order unit space with order unit

*e*, i.e.,

*Jordan triple product*\(\{ \cdot , \cdot , \cdot \}\) is defined as

*quadratic representation*of

*a*.

*variation seminorm*\(\left\| \cdot \right\| _v\) on

*A*given by,

*e*, and on the quotient space \([A]:=A / {{\mathrm{Span}}}(e)\) it is a norm. To see this we show that if \(\Vert \cdot \Vert _q\) is the quotient norm of \(2\Vert \cdot \Vert \) on [

*A*], then \(\Vert [a]\Vert _q =\Vert [a]\Vert _v\) for all \([a]\in [A]\). Indeed, for \([a]\in [A]\), using \(\inf _{\lambda \in \mathbb {R}}\max \{t-\lambda ,s+\lambda \}=(t+s)/2\), we have that

*A*] given by \(\log \overline{a} = [\log a]\). Its inverse \({\mathrm {Exp}}:[A]\rightarrow \overline{A}_+^\circ \) is given by \(\exp ([a]) = \overline{\exp (a)}\) for \([a]\in [A]\).

A *JBW-algebra* is the Jordan analogue of a von Neumann algebra: it is a JB-algebra which is monotone complete and has a separating set of normal states, or equivalently, a JB-algebra that is a dual space. In JBW-algebras the spectral theorem holds, which implies in particular that the linear span of projections is norm dense. If *p* is a projection, then the complement \(e-p\) will be denoted by \(p^\perp \). Every JBW-algebra decomposes into a direct sum of a type I, II, and III JBW-algebras. A JBW-algebra with trivial center is called a *factor*. Every Euclidean Jordan algebra is a JBW-algebra, and a Euclidean Jordan algebra is simple if and only if it is a factor.

### 2.3 Order isomorphisms

An important result we use is [21, Theorem 1.4], which we state here for the convenience of the reader. A *symmetry* is an element *s* satisfying \(s^2 = e\). Note that *s* is a symmetry if and only if \(p := (s+e)/2\) is a projection, and \(s = p - p^\perp \).

### Theorem 2.1

(Isidro, Rodríguez-Palacios) The bijective linear isometries from *A* onto *B* are the mappings of the form \(a \mapsto s Ja\), where *s* is a central symmetry in *B* and \(J :A \rightarrow B\) a Jordan isomorphism.

This theorem uses the fact that a bijective unital linear isometry between JB-algebras is a Jordan isomorphism, which is [50, Theorem 4]. We use this simpler statement in the following corollary.

### Corollary 2.2

Let *A* and *B* be order unit spaces, and \(T :A \rightarrow B\) be a unital linear bijection. Then *T* is an isometry if and only if *T* is an order isomorphism. Moreover, if *A* and *B* are JB-algebras, then these statements are equivalent to *T* being a Jordan isomorphism.

### Proof

Suppose *T* is an isometry, and let \(a \in A_+\), \(\left\| a \right\| \le 1\). Then \(\left\| e-a \right\| \le 1\), and so \(\left\| e - Ta \right\| \le 1\), showing that *Ta* is positive. So *T* is a positive map, and by the same argument \(T^{-1}\) is a positive map. (This argument is taken from the first part of [50, Theorem 4].)

Conversely, if *T* is an order isomorphism, then \(-\lambda e \le a \le \lambda e\) if and only if \(-\lambda e \le Ta \le \lambda e\), and so *T* is an isometry.

Now suppose that *A* and *B* are JB-algebras. If *T* is an isometry, then *T* is a Jordan isomorphism by [50, Theorem 4]. Conversely, if *T* is a Jordan isomorphism, then *T* preserves the spectrum, and then also the norm since \(\left\| a \right\| = \max |\sigma (a)|\). \(\square \)

This corollary will be used to show the following proposition. For Euclidean Jordan algebras this proposition has been proved in [14, Theorem III.5.1].

### Proposition 2.3

A map \(T :A \rightarrow B\) is an order isomorphism if and only if *T* is of the form \(T = U_b J\), where \(b \in B_+^\circ \) and *J* is a Jordan isomorphism. Moreover, this decomposition is unique and \(b=(Te)^{\frac{1}{2}}\).

### Proof

If *T* is of the above form, then *T* is an order isomorphism as a composition of two order isomorphisms. Conversely, if *T* is an order isomorphism, then \(T = U_{(Te)^{\frac{1}{2}}} U_{(Te)^{-\frac{1}{2}}} T\), and by the above corollary \(U_{(Te)^{-\frac{1}{2}}} T\) is a Jordan isomorphism.

For the uniqueness, if \(T = U_b J\), then \(Te = U_b Je = U_b e = b^2\) which forces \(b = (Te)^{\frac{1}{2}}\). This implies that \(J = U_{(Te)^{-\frac{1}{2}}} T\), so *J* is also unique. \(\square \)

### 2.4 Hilbert’s and Thompson’s metrics on cones in JB-algebras

Suppose *A* is a JB-algebra. For \(c \in A_+^\circ \), the map \(U_c\) is an order isomorphism of *A*, and hence it preserves *M*(*a* / *b*). Thus, \(U_c\) is an isometry under \(d_H\) and \(d_T\). This can be used to derive the following expressions for \(d_H\) and \(d_T\) on cones in JB-algebras.

### Proposition 2.4

*A*is a JB-algebra and \(a,b\in A_+^\circ \), then

### Proof

*A*for \(c \in A_+^\circ \),

The formula for \(d_H\) follows immediately. As \(\left\| c \right\| = \max \{ \max \sigma (c), - \min \sigma (c)\}\) for \(c \in A\), the identity for \(d_T\) holds. \(\square \)

*A*we follow Bosché [6, Proposition 2.6] and Hatori and Molnár [19, Theorem 9], and introduce for \(n \ge 1\) metrics on [

*A*] and

*A*, respectively, by

### Proposition 2.5

*A*is a JB-algebra and \(a,b \in A\), then

### Proof

*a*,

*b*and

*e*is special, so we can think of \(U_{\exp (b/n)^{-\frac{1}{2}}} \exp (a/n)\) as \(\exp (-b/2n) \exp (a/n) \exp (-b/2n)\) for some \(C^*\)-algebra multiplication. Writing out the exponentials in power series yields

*n*that

*n*we have by Proposition 2.4 that

We will also need some basic facts concerning the unique geodesics for \(d_T\) and \(d_H\). Recall that for a metric space (*M*, *d*) a map \(\gamma :I \rightarrow M\), where *I* is a possibly unbounded interval in \(\mathbb {R}\), is a *geodesic path* if there is a \(k\ge 0\) such that \(d(\gamma (s),\gamma (t)) = k|s-t|\) for all \(s,t \in I\). The image of a geodesic path is called a *geodesic*. The following result generalizes [28, Theorems 5.1 and 6.2].

### Theorem 2.6

If *A* is a JB-algebra and \(a,b \in A_+^\circ \) are linearly independent, then there exists a unique Thompson geodesic between *a* and *b* if and only if \(\sigma (U_{a^{-\frac{1}{2}}}b) = \{ \beta ^{-1}, \beta \}\) for some \(\beta > 1\).

### Proof

As the map \(U_{a^{-\frac{1}{2}}}\) is a Thompson’s metric isometry, we may assume without loss of generality that \(a = e\). First suppose that \(\sigma (b) = \{\beta ^{-1}, \beta \}\) for some \(\beta > 1\), then \(b = \beta ^{-1}p + \beta p^\perp \) and the line through *b* and *e* intersects \(\partial A_+\) in \(\lambda p\) and \(\mu p^\perp \) for some \(\lambda ,\mu >0\). We wish to apply [28, Theorem 4.3].

Consider the Peirce decomposition \( A = A_1 \oplus A_{1/2} \oplus A_0\) (cf. [17, 2.6.2]) with respect to *p*. We denote the projection onto \(A_i\) by \(P_i\), for \(i=1, 1/2, 0\). Then \(P_1 = U_p\) and \(P_0 = U_{p^\perp }\). From [1, Proposition 1.3.8] we know that if \(a \in A_+\), then \(U_p a = a\) if and only if \(U_{p^\perp } a = 0\). Using this result we now prove the following claim.

*Claim.* Let \(v \in A\). If \(\alpha ,\delta > 0\) and \(p \in A\) is a projection such that \(\alpha p + tv \in A_+\) for all \(|t| < \delta \), then \(v \in A_1\).

To show the claim, note that \(0 \le U_{p^\perp }(\alpha p+tv) = t U_{p^\perp } v\) for all \(|t| < \delta \), so that \(U_{p^\perp } v = 0\), and consequently \(U_{p^\perp }(\alpha p+tv) = t U_{p^\perp } v = 0\) for all \(|t| < \delta \). Let \(0<|t| < \delta \) be arbitrary. It follows that \(\alpha p + tv = U_p( \alpha p + tv) = \alpha p + t U_p v\) and so \(v = U_p v = P_1 v\), i.e., \(v \in A_1\).

By applying the claim to \(\lambda p\) as well as \(\mu p^\perp \), it follows that if \(v \in A\) is such that \(\lambda p + tv \in A_+\) and \(\mu p^\perp + tv \in A_+\) for all \(|t| < \delta \), then \(v \in A_1 \cap A_0 = \{0\}\). Hence, by [28, Theorem 4.3], there is a unique geodesic between *b* and *e*.

*b*and

*e*. Then this is also a unique geodesic in JB\((b,e) \cong C(\sigma (b))\). For \(f,g \in C(\sigma (b))\) we have by Proposition 2.4 that

*e*to the zero function and

*b*to the function \(k \mapsto \log k\).

Note that for \(f\in C(\sigma (b))\) the images of both \(t\mapsto (t\Vert f\Vert \wedge |f|){\mathrm {sgn}}f\) and \(t\mapsto t f\) are geodesics connecting 0 and *f*, which are different if and only if there is a point \(k\in \sigma (b)\) such that \(|f(k)|\not =\Vert f\Vert \). Hence \(k \mapsto |\log (k)|\) is constant. So, if \(\alpha ,\beta \in \sigma (b)\), then \(|\log \beta | = |\log \alpha |\), and hence \(\alpha =\beta \) or \(\alpha =\beta ^{-1}\). This shows that \(\sigma (b) \subseteq \{ \beta ^{-1}, \beta \}\), and since *b* and *e* are linearly independent we must have equality. \(\square \)

From Theorem 2.6 we can derive in the same way as in [28, Theorem 5.2] the following characterization for Hilbert’s metric.

### Theorem 2.7

If *A* is a JB-algebra and \(a,b \in A_+^\circ \) are linearly independent, then there exists a unique geodesic between \(\overline{a}\) and \(\overline{b}\) in \((\overline{A}_+^\circ , d_H)\) if and only if \(\sigma (U_{a^{-\frac{1}{2}}}b) = \{ \alpha , \beta \}\) for some \(\beta> \alpha > 0\).

Recall that the straight line segment \(\{\overline{(1-t)a+tb}:0\le t\le 1\}\) is a geodesic in \((\overline{A}_+^\circ ,d_H)\) for all \(a,b\in A_+^\circ \).

The following special geodesic paths play an important role.

### Definition 2.8

*a*and

*b*in \((A_+^\circ ,d_T)\). The argument to show that \(\gamma _{\overline{a}}^{\overline{b}}\) is a geodesic in \((\overline{A}_+^\circ , d_H)\) is similar and is left to the reader. Using the fact that \(U_{c^\lambda } c^\mu = c^{2 \lambda + \mu }\) in the fourth step, we get that

### 2.5 Geometric means in JB-algebras

*symmetry around c*, see [34].

### Definition 2.9

*geometric mean*of

*a*and

*b*. It is denoted by \(a \# b\), so

We remark that the equation \(S_c(b)=U_c b^{-1} = a\), which has the unique solution \(c=b \# a\), is equivalent to the equation \(S_c(a)= U_ca^{-1} =b\). Thus, \(a \# b = b \# a\), and hence \(S_{a\# b}(a)=b\) and \(S_{a\# b}(b)=a\). Note also that, as \(S_c(a) =a\) implies that \(c = a\#a =a\), the map \(S_c\) has a unique fixed point *c* in \(A_+^\circ \). Moreover, \(S_c\) is an isometry under both Hilbert’s metric and Thompson’s metric on \(A_+^\circ \), since it is the composition of two isometries.

The idea is now to show that the geometric means are preserved under bijective Hilbert’s metric and Thompson’s metric isometries. The proof relies on properties of the maps \(S_{a\# b}\) and the following lemma. This lemma and its proof are similar to [37, lemma p. 3852], the only difference being that we consider two metric spaces here.

### Lemma 2.10

*M*,

*N*be metric spaces. Suppose that for each \(x,y \in M\) there exists an element \(z_{xy} \in M\), a bijective isometry \(\psi _{xy} :M \rightarrow M\) and a constant \(k_{xy} > 1\) such that

- (i)
\(\psi _{xy}(x) = y, \quad \psi _{xy}(y) = x\);

- (ii)
\(\psi _{xy}(z_{xy}) = z_{xy}\);

- (iii)
\(d(u, \psi _{xy}(u)) \ge k_{xy} d(u, z_{xy}) \quad \text{ for } \text{ all } u \in M\).

*N*satisfies the same requirements. If \(\varphi :M \rightarrow N\) is a bijective isometry, then

Applying this lemma to the maps \(S_{a\# b}\) we derive the following proposition for Thompson’s metric.

### Proposition 2.11

*A*and

*B*are JB-algebras and \(f :A_+^\circ \rightarrow B_+^\circ \) is a bijective Thompson’s metric isometry, then

### Proof

To see that the same result holds for Hilbert’s metric isometries on \(\overline{A}_+^\circ \), we need to make a couple of observations. Firstly for \(c\in A_+^\circ \), the map \(S_c\) induces a well defined maps \(S_{\overline{c}}\) on \(\overline{A}_+^\circ \) by letting \(S_{\overline{c}}(\overline{a}):= \overline{S_c(a)}\). Furthermore, for \(a,b\in A_+^\circ \) and \(\lambda ,\mu >0\) we have that the equation \(U_c(\lambda a) =\mu b\) has unique solution \(c=(\lambda a)\# (\mu b) = \sqrt{\lambda \mu }(a\# b)\). Thus, the equation \(U_{\overline{c}} \overline{a}^{-1}= \overline{U_ca^{-1}}=\overline{b}\) has a unique solution \(\overline{a\#b}\) in \(\overline{A}_+^\circ \) for \(\overline{a},\overline{b}\in \overline{A}_+^\circ \), and we can define the projective geometric mean by \(\overline{a}\#\overline{b} := \overline{a\#b}\) in \(\overline{A}_+^\circ \). Note that \(\overline{a}\#\overline{b}=\gamma _{\overline{a}}^{\overline{b}}(1/2)\). It is now straightforward to verify that the Hilbert’s metric isometries \(S_{\overline{a}\#\overline{b}}\) on \(\overline{A}_+^\circ \) satisfy the requirements of Lemma 2.10 with \(k_{ab} = 2\) and derive the following result.

### Proposition 2.12

*A*and

*B*are JB-algebras and \(f :\overline{A}_+^\circ \rightarrow \overline{B}_+^\circ \) is a bijective Hilbert’s metric isometry, then

The next proposition will be useful.

### Proposition 2.13

### Proof

It is straightforward to derive a similar identity for Hilbert’s metric.

### Proposition 2.14

### Proof

By combining Propositions 2.11 and 2.13 we derive the following corollary. The proof uses the fact that the equation \(a\# c=b\) has a unique solution \(c=U_ba\), which can be easily shown using (2.3).

### Corollary 2.15

*A*and

*B*be JB-algebras. If \(f :A_+^\circ \rightarrow B_+^\circ \) is a bijective Thompson’s metric isometry, then

- (a)
*f*maps \(\gamma _a^b(t)\) to \(\gamma _{f(a)}^{f(b)}(t)\) for all \(a,b\in A_+^\circ \) and \(t\in [0,1]\). - (b)
If \(f(e)=e\), then \(f(a^t) = f(a)^t\) for all \(t\in [0,1]\). Moreover, we have \(f(a^{-1}) = f(a)^{-1}\) and \(f(U_b a) = U_{f(b)} f(a)\).

### Proof

By Propositions 2.13 and 2.11, the first statement holds for all dyadic rationals \(t \in [0,1]\). As the dyadic rationals are dense in [0, 1], it holds for all \(0 \le t \le 1\).

Suppose \(f(e)=e\). Since \(\gamma _e^a(t) = a^t\), the first statement yields that \(f(a^t) = f(a)^t\) for all \(0 \le t \le 1\).

*f*(

*b*) is a solution to \(S_c(f(a)^{-1}) = f(U_ba)\), i.e., \(U_{f(b)} f(a) = f(U_b a)\). \(\square \)

Again, a similar result holds for Hilbert’s metric. The proof is analogous to the one for Thompson’s metric in Corollary 2.15 and is left to the reader.

### Corollary 2.16

*A*and

*B*be JB-algebras. If \(f :\overline{A}_+^\circ \rightarrow \overline{B}_+^\circ \) is a bijective Hilbert’s metric isometry, then

- (a)
*f*maps \(\gamma _{\overline{a}}^{\overline{b}}(t)\) to \(\gamma _{f(\overline{a})}^{f(\overline{b})}(t)\) for all \(\overline{a},\overline{b}\in \overline{A}_+^\circ \) and \(t\in [0,1]\). - (b)
If \(f(\overline{e})=\overline{e}\), then \(f(\overline{a}^t) = f(\overline{a})^t\) for all \(t\in [0,1]\). Moreover, we have \(f(\overline{a}^{-1}) = f(\overline{a})^{-1}\) and \(f(U_{\overline{b}}\overline{a}) = U_{f(\overline{b})} f(\overline{a})\).

Now we can prove an essential ingredient for characterizing bijective Hilbert’s metric and Thompson’s metric isometries of cones of JB-algebras. Recall that \([A] = A / {{\mathrm{Span}}}(e)\).

### Theorem 2.17

*A*and

*B*be JB-algebras.

- (a)If \(f :A_+^\circ \rightarrow B_+^\circ \) is a bijective Thompson’s metric isometry with \(f(e) = e\), then \(S :A \rightarrow B\) given byis a bijective linear \(\left\| \cdot \right\| \)-isometry.$$\begin{aligned} Sa := \log f( \exp (a)), \end{aligned}$$
- (b)If \(f:\overline{A}_+^\circ \rightarrow \overline{B}_+^\circ \) is a bijective Hilbert’s isometry with \(f(\overline{e}) = \overline{e}\), then \(S :[A] \rightarrow [B]\) given byis a bijective linear \(\left\| \cdot \right\| _v\)-isometry.$$\begin{aligned} S[a] := \log f( \exp ([a])), \end{aligned}$$

### Proof

*S*is a bijective \(\left\| \cdot \right\| _v\)-isometry. As \(f(\overline{e})= \overline{e}\), we have that \(S[0] = [0]\), and hence

*S*is linear by the Mazur-Ulam theorem. \(\square \)

### Remark 2.18

The map \({{\mathrm{Exp}}}:A \rightarrow A_+^\circ \) is a bijection. In the associative case, where \(A = C(K)\) for some compact Hausdorff space *K*, one can show that this bijection induces an isometric isomorphism between the spaces \((A, \left\| \cdot \right\| )\) and \((A_+^\circ , d_T)\), see [29]. Likewise, the exponential map yields an isometric isomorphism between \(([A], \left\| \cdot \right\| _v)\) and \((\overline{A}_+^\circ ,d_H)\) if \(A=C(K)\). In the nonassociative case this is no longer true. In fact, it has been shown for finite dimensional order unit spaces *A* that \((\overline{A}_+^\circ ,d_H)\) is isometric to a normed space if and only if \(A_+\) is a simplicial cone, see [15]. For Thompson’s metric the same result holds, see [28, Theorem 7.7].

## 3 Thompson’s metric isometries of JB-algebras

The next basic property of Thompson’s metric on products of cones will be useful.

### Proposition 3.1

*A*is a product of order unit space \(A_i\) for \(i \in I\). If \(d_T^i\) denotes the Thompson’s metric on \(A_{i+}^\circ \) and \(a= (a_i), b = (b_i)\in A_+^\circ \), then

### Proof

With the above preparations we can now obtain the following theorem. The proof, as well as the statement, is a direct generalization of [6, Section 4] and [37, Theorem 9].

### Theorem 3.2

*A*and

*B*be unital JB-algebras. A map \(f :A_+^\circ \rightarrow B_+^\circ \) is a bijective Thompson’s metric isometry if and only if there exist \(b \in B_+^\circ \), a central projection \(p \in B\), and a Jordan isomorphism \(J :A \rightarrow B\) such that

*f*is of the form

### Proof

The last statement follows from taking \(a=e\), which yields \(b^2 = f(e)\).

*p*yields a decomposition \(B = pB \oplus p^\perp B\), which is left invariant by \(U_b\). This decomposition can be pulled back by

*J*, which yields the following representation of the map \(f :(J^{-1}pB)_+^\circ \times (J^{-1}p^\perp B)_+^\circ \rightarrow (pB)_+^\circ \times (p^\perp B)_+^\circ \):

*g*is a Thompson’s metric isometry mapping

*e*to

*e*. By Theorem 2.17 the map \(S :A \rightarrow B\) defined by

### 3.1 The Thompson’s metric isometry group of a JB-algebra

In the case where a JB-algebra is the direct product of simple JB-algebras, we can explicitly compute its Thompson’s metric isometry group in terms of the Jordan automorphism groups of the simple components. Each Euclidean Jordan algebra satisfies this requirement, and the automorphism groups of the simple Euclidean Jordan algebras are known, see [14].

### Theorem 3.3

*A*can be decomposed as a direct product

*I*is an index set, the \(n_i\) are arbitrary cardinals and the \(A_i\) are mutually nonisomorphic simple JB-algebras. Then the Thompson’s metric isometry group of

*A*equals

### Proof

A Jordan isomorphism of \(A^{n_i}_i\) may permute the components, so it follows that each Thompson’s metric isometry of \((A^{n_i}_i)^\circ _+\) is a composition of a permutation of components, a componentwise possible inversion, a componentwise Jordan isomorphism, and a componentwise quadratic representation. So, all the operators except the permutation will act componentwise, and the componentwise operators form a subgroup. It is easy to compute that a componentwise operator conjugated by a permutation \(\pi \) equals the componentwise operator permuted by \(\pi \). This shows that the componentwise operators and the permutation group form a semidirect product, where the componentwise operators are the normal subgroup. It remains to examine the componentwise operators.

### Remark 3.4

*A*is a JB-algebra as given in the above theorem, then we can use an analogous argument to show that the automorphism group of the cone \(A_+\) equals

*I*and \(n_i\) are finite (i.e.,

*A*is a Euclidean Jordan algebra). Then the index of the automorphism group in the isometry group for Thompson’s metric is \(2^m\), where \(m = \sum _{i \in I} n_i\) is the total number of different components. This is a correction of [6, Remark 4.9], which has the wrong index.

## 4 Hilbert’s metric isometries of JBW-algebras

If *A* and *B* are JB-algebras and \(f :\overline{A}_+^\circ \rightarrow \overline{B}_+^\circ \) is a bijective Hilbert’s metric isometry mapping \(\overline{e}\) to \(\overline{e}\), then by Theorem 2.17 the map \(S :[A] \rightarrow [B]\) defined by, \(S[a] := \log f(\exp ([a]))\), is a bijective linear \(\left\| \cdot \right\| _v\)-isometry. Every bijective linear isometry maps extreme points of the unit ball to extreme points of the unit ball, which is what we will exploit here. Let us first identify these extreme points. For JBW-algebras this is [16, Proposition 2.2].

### Lemma 4.1

The extreme points of the unit ball in \(([A], \left\| \cdot \right\| _v)\) are the equivalence classes [*p*], where \(p\in A\) is a nontrivial projection.

### Proof

*p*] is an extreme point of the unit ball in \(([A],\Vert \cdot \Vert _v)\).

Conversely, if \([a]\in [A]\) with \(\Vert [a]\Vert _v=1\) does not contain a projection, then a representative *a* with \(\sigma (a)\subseteq [0,1]\) must have \(\lambda \in \sigma (a)\) with \(0<\lambda <1\). Now consider JB\((a,e) \cong C(\sigma (a))\). By elementary topology there exists a nonnegative function \(g\in C(\sigma (a))\) with \(g\ne 0\) such that the ranges of \(a+g\) and \(a-g\) are contained in [0, 1]. Since \(a=\frac{1}{2}(a-g)+\frac{1}{2}(a+g)\), it follows that [*a*] can be written as \(\frac{1}{2}([b]+[c])\) with \([b]\ne [c]\) and \(\Vert [b]\Vert _v =\Vert [c]\Vert _v =1\), and hence [*a*] cannot be an extreme point of the unit ball. \(\square \)

To be able to exploit the extreme points we will restrict ourselves to cones in JBW-algebras, as JB-algebras may not have nontrivial projections, e.g. *C*([0, 1]). For a JBW-algebra *M* we will denote its set of projections by \(\mathcal {P}(M)\).

Let *M* be a JBW-algebra. By Lemma 4.1 we can define a map \(\theta :\mathcal {P}(M) \rightarrow \mathcal {P}(N)\) by letting \(\theta (0)=0\), \(\theta (e)=e\), and \(\theta (p)\) be the unique nontrivial projection in the class *S*[*p*], otherwise. Thus, for each bijective Hilbert’s metric isometry \(f:\overline{M}_+^\circ \rightarrow \overline{N}_+^\circ \) with \(f(\overline{e})=\overline{e}\), we get a bijection \(\theta :\mathcal {P}(M)\rightarrow \mathcal {P}(N)\). We say that \(\theta \) is *induced* by *f*. Note that its inverse \(\theta ^{-1}\) is induced by \(f^{-1}\). The map \(\theta \) will be the key in understanding *f*.

We call a bijection \(\theta :\mathcal {P}(M)\rightarrow \mathcal {P}(N)\) an *orthoisomorphism* if \(p,q\in \mathcal {P}(M)\) are orthogonal if and only if \(\theta (p)\) and \(\theta (q)\) are orthogonal. Our goal will be to prove that the map \(\theta \) induced by either *f* or \(\iota \circ f\), where \(\iota (\overline{a}) =\overline{a}^{-1}\) is the inversion, is in fact an orthoisomorphism. For this we need to investigate certain unique geodesics starting from the unit *e*.

We introduce the following notation: \((\overline{a},\overline{b})\) denotes the open line segment \(\{t\overline{a} + (1-t)\overline{b}: 0< t < 1\}\) in \(\overline{M}_+\) for \(\overline{a},\overline{b}\in \overline{M}_+\). The segments \([\overline{a},\overline{b}]\) and \([\overline{a},\overline{b})\) are defined similarly. Furthermore, we denote the affine span of a set *S* by \({\mathrm {aff}}\,(S)\).

### Lemma 4.2

*M*such that \(p_1+\cdots +p_k=e\), then the boundary of \({\mathrm {conv}}(p_1,\ldots ,p_k)\) is contained in \(\partial M_+\) and so

### Proof

### Lemma 4.3

Let *M* and *N* be JBW-algebras and \(f:\overline{M}_+^\circ \rightarrow \overline{N}_+^\circ \) be a bijective Hilbert’s metric isometry with \(f(\overline{e})=\overline{e}\). Let \(p\in \mathcal {P}(M)\) be nontrivial. The geodesic segment \([\overline{e},\overline{p})\) is mapped to the geodesic segment \([\overline{e},\overline{q})\) by *f* for some \(q\in \mathcal {P}(N)\). Moreover, \(S[p]=[q]\) and so \(\theta (p)=q\).

### Proof

The geodesic segments \([\overline{e},\overline{p})\) is unique by Lemma 4.2. Thus, \(f([\overline{e},\overline{p}))\) is also a unique geodesic segments starting at \(\overline{e}\), since \(f(\overline{e})=\overline{e}\).

*f*is \([\overline{e},\overline{r})\).

*q*is a nontrivial projection and \(0< t<1\), then by using Proposition 2.4 it is easy to verify that \(d_H(tq+(1-t)e,e) =-\log (1-t)\). As

*f*is an isometry that fixes \(\overline{e}\), we find that

We can now show that \(\theta \) preserves operator commuting projections.

### Proposition 4.4

If \(p,q\in \mathcal {P}(M)\) operator commute, then \(\theta (p),\theta (q)\in \mathcal {P}(N)\) operator commute.

### Proof

*p*and

*q*operator commute, then \(e+p\) and \(e+q\) operator commute. It follows that \(U_{(e+p)^{1/2}}(e+q)=U_{(e+q)^{1/2}}(e+p)\), so \(U_{\overline{e+p}^{1/2}}\overline{e+q}=U_{\overline{e+q}^{1/2}}\overline{e+p}\). By Corollary 2.16 and Eq. (4.1) in the proof of Lemma 4.3,

*e*is a JC-algebra by [17, Theorem 7.2.5]. So, we can think of \(U_{(e+\theta (p))^{1/2}}(e+\theta (q))\) and \(U_{(e+\theta (q))^{\frac{1}{2}}}(e+\theta (p))\) as

*C**-algebra multiplication. The equality in (4.2) implies that

*N*by [1, Proposition 1.47]. \(\square \)

This allows us to show that \(\theta \) preserves orthogonal complements.

### Lemma 4.5

\(\theta (p^\perp )=\theta (p)^\perp \) for all \(p\in \mathcal {P}(M)\).

### Proof

We may assume that *p* is nontrivial by definition of \(\theta \). Since \(S[p]+S[p^\perp ] =S[e] =[e]\), we obtain \(\theta (p)+\theta (p^\perp )=\lambda e\) for some \(\lambda \in \mathbb {R}\). As *p* and \(p^\perp \) operator commute, the projections \(\theta (p)\) and \(\theta (p^\perp )\) operator commute by Proposition 4.4. By [1, Proposition 1.47], \(\theta (p)\) and \(\theta (p^\perp )\) are contained in an associative subalgebra, which is isomorphic to a *C*(*K*)-space. In a *C*(*K*)-space it is obvious that \(\lambda =1 \) or \(\lambda =2\). Now note that \(\lambda =2\) implies that \(\theta (p)=\theta (p^\perp )=e\) which contradicts the injectivity of *S*, and hence \(\theta (p)+\theta (p^\perp )=e\), which shows that \(\theta (p^\perp )=\theta (p)^\perp \). \(\square \)

We will proceed to show that if \(f:\overline{M}_+^\circ \rightarrow \overline{N}_+^\circ \) is a bijective Hilbert’s metric isometry with \(f(\overline{e})=\overline{e}\), then for either *f* or \( \iota \circ f\), the induced map \(\theta \) maps orthogonal noncomplementary projections to orthogonal projections. For this we need to look at special simplices in the cone \(M_+\).

### 4.1 Orthogonal simplices

*M*with \(p_1+p_2+p_3=e\), we call

*orthogonal simplex*in \(\overline{M}_+^\circ \). The next lemma shows that a bijective Hilbert’s metric isometry

*f*maps orthogonal simplices onto orthogonal simplices.

### Lemma 4.6

- (i)
\(q_1 + q_2 + q_3 = e\), and then \(f(\Delta (p_1,p_2,p_3))= \Delta (q_1,q_2,q_3)\), or

- (ii)
\(q_1^\perp + q_2^\perp + q_3^\perp = e\), and then \(f(\Delta (p_1,p_2,p_3))= \Delta (q^\perp _1,q^\perp _2,q^\perp _3)\).

*i*), \(\theta \) preserves the orthogonality of \(p_1,p_2,p_3\). Moreover, if the map \(\theta \) induced by

*f*satisfies the assumptions of case (

*ii*), then the map \(\theta \) induced by the isometry \(\iota \circ f\) satisfies the conditions of case (

*i*).

### Proof

*S*is linear, \(S[p_1]+S[p_2]+S[p_3] =S[e] =[e]\), and hence

*C*(

*K*)-space. Note that this subalgebra also contains \(\lambda e\) and hence also \(q_3\) by (4.3). In a

*C*(

*K*)-space it is obvious that \(\lambda \in \{1, 2\}\) in (4.3). In fact, the case \(\lambda = 1\) corresponds with the pairwise orthogonality of \(q_1\), \(q_2\) and \(q_3\), whereas the case \(\lambda =2\) corresponds to pairwise orthogonality of \(q_1^\perp \), \(q_2^\perp \) and \(q_3^\perp \), and \(q_1^\perp +q_2^\perp +q_3^\perp = e\).

*f*maps \(\Delta (p_1,p_2,p_3)\) onto \(\Delta (q_1,q_2,q_3)\) in case \(q_1+q_2+q_3 = e\). Let \(a\in {\mathrm {conv}}(p_1,p_2,p_3)\cap M_+^\circ \) be a point not lying on any \((p_i,p_i^\perp )\) for \(i=1,2,3\). We know that \([\overline{a},\overline{p}_i)\) is a unique geodesic by Lemma 4.2. Let \((\overline{a}',\overline{p}_1)\) be the line segment through \(\overline{p}_1\) and \(\overline{a}\) with \(a'\) in the boundary of \({\mathrm {conv}}(p_1,p_2,p_3)\). This unique geodesic intersects \((\overline{p}_2,\overline{p}_2^\perp )\) and \((\overline{p}_3,\overline{p}_3^\perp )\) in 2 distinct points, say \(\overline{b}_2\) and \(\overline{b}_3\) respectively, see Fig. 1.

Since it must be mapped to a line segment, it follows that \(f(\overline{a})\) lies on the line segment through \(f(\overline{b}_2)\) and \(f(\overline{b}_3)\), which is contained in \(\Delta (q_1,q_2,q_3)\). By the invertibility of *f*, we conclude that \(f(\Delta (p_1,p_2,p_3))= \Delta (q_1,q_2,q_3)\). The same argument can be used to show that \(f(\Delta (p_1,p_2,p_3))= \Delta (q^\perp _1,q^\perp _2,q^\perp _3)\) if \(q_1^\perp +q_2^\perp +q_3^\perp =e\).

*f*with the inversion \(\iota \), we obtain

*ii*). \(\square \)

*f*to \(\Delta (p_1,p_2,p_3)\) is a Hilbert’s metric isometry onto either \(\Delta (\theta (p_1),\theta (p_2),\theta (p_3))\) or \(\Delta (\theta (p_1)^\perp ,\theta (p_2)^\perp ,\theta (p_3)^\perp )\). The Hilbert’s metric isometries between simplices have been characterized, see [12] or [30], and yields the following dichotomy, as \(f(\overline{e})=\overline{e}\). The isometry

*f*maps \(\Delta (p_1,p_2,p_3)\) onto \(\Delta (\theta (p_1),\theta (p_2),\theta (p_3))\) in Lemma 4.6 if and only if the restriction of

*f*to \(\Delta (p_1,p_2,p_3)\) is of the form,

*f*to \(\Delta (p_1,p_2,p_3)\) is projectively linear. On the other hand, the isometry

*f*maps \(\Delta (p_1,p_2,p_3)\) onto \(\Delta (\theta (p_1)^\perp ,\theta (p_2)^\perp ,\theta (p_3)^\perp )\) in Lemma 4.6 if and only if the restriction of

*f*to \(\Delta (p_1,p_2,p_3)\) is of the form,

### Corollary 4.7

Let \(f:\overline{M}_+^\circ \rightarrow \overline{N}_+^\circ \) be a bijective Hilbert’s metric isometry with \(f(\overline{e}) =\overline{e}\) and let \(\Delta (p_1,p_2,p_3)\) be an orthogonal simplex in \(\overline{M}_+^\circ \). Then either *f* or \(\iota \circ f\) is projectively linear on \(\Delta (p_1,p_2,p_3)\), and its induced map \(\theta \) preserves the orthogonality of \(p_1,p_2\) and \(p_3\).

*f*is projectively linear on one simplex if and only if it projectively linear on the other one. The proof uses, among other things, the following well known fact. If \(a,b\in M_+^\circ \) are such that the line through

*a*and

*b*intersect \(\partial M_+\) in \(a'\) and \(b'\) such that

*a*is between

*b*and \(a'\),

*b*is between

*a*and \(b'\), then

### Proposition 4.8

Let \(f:\overline{M}_+^\circ \rightarrow \overline{N}_+^\circ \) be a bijective Hilbert’s metric isometry with \(f(\overline{e}) =\overline{e}\). Let \(\Delta (p_1,p_2,p_3)\) and \(\Delta (p_4,p_5,p_6)\) be two distinct orthogonal simplices in \(\overline{M}_+^\circ \) such that either \(p_3=p_6\) or \(p_3=p^\perp _6\), so they share the segment \((\overline{p}_3,\overline{p}_3^\perp )\). Then *f* is projectively linear on \(\Delta (p_1,p_2,p_3)\) if and only if it is projectively linear on \(\Delta (p_4,p_5,p_6)\).

### Proof

*f*is projectively linear \(\Delta (p_1,p_2,p_3)\), but not on \(\Delta (p_4,p_5,p_6)\). Denote the image of \(\Delta (p_1,p_2,p_3)\) by \(\Delta (q_1,q_2,q_3)\), and the image of \(\Delta (p_4,p_5,p_6)\) by \(\Delta (q_4^\perp ,q_5^\perp ,q_6^\perp )\) as in Lemma 4.6. There are 2 cases to consider: \(p_3=p_6\) and \(p_3=p_6^\perp \). Let us first assume that \(p_3=p_6\).

*f*is the segment \([\overline{e},\overline{q}_2)\), and the image of \([\overline{e},\overline{p}_5^\perp )\) is \([\overline{e},\overline{q}_5^\perp )\). Let us now consider representations of these segments in \({\mathrm {conv}}(q_1,q_2,q_4^\perp ,q_5^\perp )\). It is easy to verify that \(\frac{1}{4}e\), \(\frac{1}{3}q_2^\perp \) and \(\frac{1}{3}q_5\) lie inside \({\mathrm {conv}}(q_1,q_2,q_4^\perp ,q_5^\perp )\). Now for \(n\ge 1\) select \(a_n\) from the segment \([\frac{1}{4}e,\frac{1}{3}q_2^\perp )\) and \(b_n\) from the segment \([\frac{1}{4}e,q_5^\perp )\) such that \(a_n\rightarrow \frac{1}{3} q_2^\perp \), \(b_n\rightarrow q_5^\perp \), and the segment \([a_n,b_n]\) is parallel to the segment \([\frac{1}{3}q_2^\perp ,q_5^\perp ]\).

*c*, \(a'_n\), and \(b'_n\) be in the boundary of \({\mathrm {conv}}(q_1,q_2,q_4^\perp ,q_5^\perp )\) as in Fig. 4. Then the triangles with vertices \(b_n\), \(b'_n\) and \(q^\perp _5\) are similar for all \(n\ge 1\). Hence there exists a constant \(C>0\) such that

Thus, \(\frac{1}{2} (p_2^\perp +p_5^\perp )\in \partial M_+\) and hence \({\mathrm {conv}}(p_1,p_3,p_4)\subseteq \partial M_+\). The same argument works for the other faces containing \(p_3\). The square face is also contained in \(\partial M_+\), as it contains \(\frac{1}{2}p_3^\perp \). This proves (4.5).

*c*is a point on the segment \((q_2,q_5^\perp )\). The triangle \({\mathrm {conv}}(c,q_1,q_4^\perp )\) intersects the triangles \({\mathrm {conv}}(q_1,q_2,q_3)\) and \({\mathrm {conv}}(q_3^\perp ,q_4^\perp ,q_5^\perp )\) in a line segment, say \(\gamma _1\) and \(\gamma _2\) respectively, see Figure 5. Now suppose that \(a\in {\mathrm {conv}}(c,q_1,q_4^\perp )\cap N_+^\circ \) and let

*b*be the point of intersection of the line segment from

*c*through

*a*with \({\mathrm {conv}}(q_3^\perp ,q_4^\perp ,q_5^\perp )\).

*c*,

*b*) intersects \(\gamma _1\) and \(\gamma _2\) in two distinct points. In that case it follows that the pre-image of \((\overline{c},\overline{b})\) lies inside \(\overline{{\mathrm {conv}}(p_1,p_2,p_3,p_4,p_5)}\). The collection of the points \(\overline{a}\) for which we obtain such a pre-image forms a dense set of \(\overline{{\mathrm {conv}}(c,q_1,q_4^\perp )}\). So, by continuity of \(f^{-1}\) we conclude that

*f*is projectively linear on \(\Delta (p_4,p_5,p_6)\) as well.

In case \(p_3=p_6^\perp \) and *f* is not projectively linear on \(\Delta (p_4,p_5,p_6)\), then analogously we find that \({\mathrm {conv}}(p_1,p_2,p_3,p_4,p_5,p_6)\) is the interior of a 3-simplex and \({\mathrm {conv}}(q_1,q_2,q_3,q_4^\perp ,q_5^\perp ,q_6^\perp )\) is the interior of a pyramid. Now applying the same arguments above to \(f^{-1}\) yields the desired contradiction, which completes the proof. \(\square \)

### Theorem 4.9

Let \(\Delta (p_1,p_2,p_3)\) and \(\Delta (p_4,p_5,p_6)\) be orthogonal simplices in \(\overline{M}_+^\circ \). A bijective Hilbert’s metric isometry \(f:\overline{M}_+^\circ \rightarrow \overline{N}_+^\circ \) with \(f(\overline{e})=\overline{e}\) is projectively linear on \(\Delta (p_1,p_2,p_3)\) if and only if it is projectively linear on \(\Delta (p_4,p_5,p_6)\).

Theorem 4.9 is a simple consequence from the following lemma, which uses the following concept. If *p* and *q* are nonmaximal nontrivial projections, then by \(p \approx q\) we mean that there exists a sequence of nonmaximal projections \(p=p_1, \ldots , p_n = q\) such that \(p_i \perp p_{i+1}\) and \(p_i + p_{i+1} < e\) for \(1 \le i < n\). This defines an equivalence relation on the nonmaximal nontrivial projections in \(\mathcal {P}(M)\).

### Lemma 4.10

If *p* and *q* are nonmaximal nontrivial projections in a JBW-algebra *M*, then \(p\approx q\).

If we assume Lemma 4.10 for the moment, the proof of Theorem 4.9 goes as follows.

### Proof of Theorem 4.9

By Proposition 4.8, if two orthogonal simplices have a projection in common, then *f* is projectively linear on one of them if and only if it is projectively on the other. So, it suffices to connect any two orthogonal simplices with a chain of orthogonal simplices each having one projection in common. Note that orthogonal simplices are determined by two nonmaximal nontrivial projections \(p_1\) and \(p_2\) such that \(p_1 \perp p_2\) and \(p_1 + p_2 < e\): the third projection is then \((p_1 + p_2)^\perp \). Hence a chain of orthogonal simplices having one projection in common, connecting the projections *p* and *q*, corresponds to a sequence of nonmaximal nontrivial projections \(p=p_1, \ldots , p_n = q\) such that \(p_i \perp p_{i+1}\) and \(p_i + p_{i+1} < e\) for \(1 \le i < n\). By Lemma 4.10 we know that such a sequence always exist, and hence we are done. \(\square \)

The proof of Lemma 4.10 is quite technical and will be given in the next section. However, for particular JB-algebras such as \(B(H)_{{\mathrm {sa}}}\) and Euclidean Jordan algebras, it is fairly easy to show that Lemma 4.10 holds. To do this we make the following basic observation.

### Lemma 4.11

*M*be a JBW-algebra and \(p,q \in \mathcal {P}(M)\) be nonmaximal and nontrivial.

- (i)
If \(p \perp q\), then \(p \approx q\).

- (ii)
If \(p \le q\), then \(p \approx q\).

- (iii)
If

*p*and*q*operator commute, then \(p \approx q\).

### Proof

For the first assertion, note that if \(q \ne p^\perp \) we are done. Also, if \(q = p^\perp \), then by nonmaximality of *q* and *p*, there exist projections \(0<p_0< p\) and \(0<q_0<q\), so that \(p \approx q_0 \approx p_0 \approx q\). The second assertion follows from (*i*), as \(p \approx q^\perp \approx q\). To prove the last one recall that the JBW-algebra generated by *p* and *q* is associative by [1, Proposition 1.47], and hence it is isomorphic to *C*(*K*) for some compact Hausdorff space *K*. By part (*i*) we may assume \(pq \ne 0\), and then \(p \approx pq \approx q\) by part (*ii*). \(\square \)

Let us now show that Lemma 4.10 holds in case \(M=B(H)_{{\mathrm {sa}}}\). If \(\dim H\le 2\), then all projections in \(\mathcal {P}(M)\) are maximal. So, assume \(\dim H\ge 3\). In that case, any two distinct rank 1 projections *p* and *q* are equivalent, because the orthogonal complements of the ranges of *p* and *q* have codimension 1, and hence their intersection is nonempty. Let *r* be the orthogonal projection on the intersection. Note that *r* is nonmaximal, as the range of *r* has codimension at least 2. Then \(p\perp r\) and \(r\perp q\) and hence \(p\approx r\approx q\) by Lemma 4.11(*i*). To compete the proof we remark that any nonmaximal projection *p* with rank at least 2 is equivalent to a rank 1 projection. Simply take \(x\in H\) in the range of *p*. Then the orthogonal projection \(p_x\) on the span of *x* satisfies \(p_x\le p\), and hence \(p_x\approx p\) by Lemma 4.11(*ii*).

*iii*) that if the center

*Z*(

*M*) is nontrivial, then any nontrivial projection \(z\in Z(M)\) yields \(p\approx z\approx q\). Indeed, in this case \(z^\perp \) also operator commutes with

*p*and

*q*, and we are done if either

*z*or \(z^\perp \) is nonmaximal. Suppose that they are both maximal. Then they are also both minimal, and therefore \(pz\le z\), forcing \(pz\in \{0,z\}\), and \(pz^\perp \le z^\perp \), forcing \(pz^\perp \in \{0,z^\perp \}\). Combining these identities yields

*p*. So, we may assume that

*Z*(

*M*) is trivial, i.e.,

*M*is a factor. Thus, the verify that Lemma 4.10 holds for Euclidean Jordan algebras, we only need to check the simple ones.

### Lemma 4.12

If *M* is a simple Euclidean Jordan algebra of rank at least 3 and \(p,q \in \mathcal {P}(M)\) are nonmaximal and nontrivial, then \(p\approx q\).

### Proof

Using the classification of simple Euclidean Jordan algebras we know that \(M= H_n(R)\) where \(n\ge 3\) and \(R=\mathbb {R}\), \(\mathbb {C}\) or \(\mathbb {H}\), or \(M = H_3(\mathbb {O})\).

By Lemma 4.11(*ii*) we may assume that *p* and *q* are primitive. It suffices to show the existence of a nontrivial nonmaximal \(z\in \mathcal {P}(M)\) that operator commutes with *p* and *q* by the above remarks. We know from [14, Corollary IV.2.4] that there exists \(w\in M\) such that \(w^2=e\) and \(U_w(p) =e_{11}\). Note that \(U_w e =w^2 =e\), and hence it is a Jordan isomorphism by Corollary 2.2. So, we may also assume that \(p = e_{11}\). The Jordan algebra generated by *p* and *q* is isomorphic to \(H_2(\mathbb {R})\) by [14, Proposition 1.6] and the isomorphism in the proof of [14, Proposition 1.6] sends \(e_{11} \in H_2(\mathbb {R})\) to \(p = e_{11} \in M\).

*z*under this isomorphism, then

*z*operator commutes with

*p*and

*q*and we are done. We will show that it is impossible that \(I_2 \in H_2(\mathbb {R})\) corresponds to \(e \in M\). In that case, the element \(s = e_{12}+e_{21} \in H_2(\mathbb {R})\) is in the Peirce 1/2 eigenspace of \(e_{11}\) and satisfies \(s^2 = I_2\). However, in \(H_n(R)\), elements in the Peirce 1/2 eigenspace of

*p*are of the form

### 4.2 Proof of Lemma 4.10

*iii*), it suffices to find a nontrivial projection \(z \in \mathcal {P}(M)\) that operator commutes with both

*p*and

*q*. Hence we may assume that

*p*or \(p^\perp \) and

*q*or \(q^\perp \), and hence it operator commutes with

*p*and

*q*.

The idea of the rest of the proof is to use the theory of von Neumann algebras, and so we would like to view *M* as the set of self-adjoint elements of a von Neumann algebra. Note that if *M* is of type \(I_2\), then [17, Theorem 6.1.8] implies that *M* is a spin factor \(H\oplus \mathbb {R}\). However, in a spin factor all nonzero projections are maximal, so *M* is not of type \(I_2\). As mentioned, the procedure will be divided into several steps. In the case where *M* is the self-adjoint part of a von Neumann algebra, the proof of this lemma is given in Step 2.

**Step 1:**We can assume that

*M*is not isomorphic to \(H_3(\mathbb {O})\) by Lemma 4.12. Then by [17, Theorem 7.2.7] we have that

*M*is a

*JW*-algebra, that is, it can be represented as a \(\sigma \)-weakly closed Jordan subalgebra of the self-adjoint operators on a complex Hilbert space. By [17, Theorem 7.3.3], it follows that

*M*is a subset of a von Neumann algebra, but the \(*\)-anti-automorphism \(\alpha \) is a problem, which we will eliminate.

Let \(R := \{ x \in W^*(M): \alpha (x) = x^* \}\). Then \(M = R_{sa}\), and by [17, Theorem 7.3.2] we have that *R* is a \(\sigma \)-weakly closed real \(*\)-algebra and \(W^*(M) = R \oplus i R\). It follows from [31, Definition 6.1.1] that *R* is a *real* \(W^*\)*-algebra*. By [31, Proposition 6.1.2], *R* is isomorphic to a *real von Neumann algebra*, that is, a \(\sigma \)-weakly closed \(*\)-subalgebra of *B*(*H*), where *H* is a *real* Hilbert space. Or equivalently, a \(*\)-subalgebra of *B*(*H*) which has a pre-dual. So, we have succeeded at viewing *M* as the self-adjoint elements of a von Neumann algebra. Unfortunately, it is a real von Neumann algebra instead of a complex one, which will pose some additional difficulties.

**Step 2:**Let \(N \subseteq R\) be the real von Neumann algebra generated by

*p*and

*q*. In the case where

*M*is the self-adjoint part of a von Neumann algebra, the reader can regard

*N*as the von Neumann algebra generated by

*p*and

*q*, and \(R=M\oplus i M\) here. We denote by \(N'\) the commutant of

*N*. That is,

*z*and \(z^\perp \) commute with

*p*and

*q*, and hence operator commute with

*p*and

*q*by [1, Proposition 1.49]. Similarly to the discussion preceding Step 1, we can conclude that either

*z*or \(z^\perp \) is nonmaximal. So, we may assume that \(N' \cap N\) contains no nontrivial projections. We will now generalize the proof of [46, Theorem V.1.41], so that it will also be applicable to the real von Neumann algebra case. From equation (4.7), we obtain that \(p^\perp q p\) maps

*pH*injectively onto a dense subspace of \(p^\perp H\). Let

*uh*be the polar decomposition of \(p^\perp qp\). By [31, Proposition 4.3.4] we have that \(u,h \in N\). Then

*u*is a partial isometry with initial space

*pH*and final space \(p^\perp H\), and so \(u^* u = p\) and \(uu^* = p^\perp \). We will use this partial isometry

*u*to make a matrix unit \(\{ e_{11}, e_{12}, e_{21}, e_{22} \}\). That is, the set of elements \(\{e_{11},e_{12},e_{21},e_{22}\}\) satisfies the properties

*M*is an algebra with projection \(p \in M\), then we denote the subalgebra

*pMp*by \(M_p\). Furthermore, by \(\mathbb {M}_2(M_p)\) we mean the \(2\times 2\) matrices whose entries are elements of \(M_p\).

### Lemma 4.13

If *M* is a (real) von Neumann algebra with a matrix unit \(\{ e_{11}, e_{12}, e_{21}, e_{22} \}\), then \(M \cong \mathbb {M}_2( M_{e_{11}})\).

### Proof

The reader can easily verify that the map \(\varphi :M \rightarrow \mathbb {M}_2(M_{e_{11}})\) given by \(\varphi (x)_{ij} := e_{1i} x e_{j1}\) is a \(*\)-homomorphism with inverse \(\theta :\mathbb {M}_2(M_{e_{11}}) \rightarrow M\) defined by \(\theta (y_{ij}) := \sum _{i,j=1}^2 e_{i1} y_{ij} e_{1j}\). \(\square \)

*R*which dominates

*p*, and has to be of the form

*M*is the self-adjoint part of a von Neumann algebra, we can apply [46, Theorem V.1.41(ii)] to conclude that

*N*is of type \(I_2\), and since \(N' \cap N\) contains no nontrivial projections, the spectral theorem implies that \(N' \cap N\) is trivial and hence

*N*is a factor. Therefore, we must have \(N \cong \mathbb {M}_2(\mathbb {C})\). Since we also have that \(N \cong \mathbb {M}_2(N_p)\), it follows that \(N_p \cong \mathbb {C}\). In the case where \(N\subseteq R\) in a real von Neumann algebra, we have to do some more work to show that \(N_p \cong \mathbb {R}\).

**Step 3:** We will need the following lemma.

### Lemma 4.14

\(N_p\) is generated by *p* and *pqp*.

### Proof

Taking products of *p* and *q* repeatedly yields expressions of the form \( \cdots pqpqpq\cdots \). For \(r,s \in \{p,q\}\), let *Q*(*r*, *s*) be the set of such expressions that start with *r* and end with *s*. It follows that *N* is the closed linear span of \(Q(p,p) \cup Q(p,q) \cup Q(q,p) \cup Q(q,q)\). Hence \(N_p\) is the closed linear span of *Q*(*p*, *p*). Since \((pqp)^n = (pq)^{n-1} (pqp)\), it follows that \(Q(p,p) = \{p\} \cup \{(pqp)^n: n \ge 1 \}\). \(\square \)

*p*and

*pqp*. Since

*p*is the identity on \(N_p\), it is commutative and contains \(C_\mathbb {R}(\sigma (pqp))\), the continuous real-valued functions on \(\sigma (pqp)\), by the continuous functional calculus for real von Neumann algebras [31, Proposition 5.1.6(2)]. Therefore, we have that \(N_p \subseteq N'_p\), and so

### 4.3 Characterization of Hilbert isometries on JBW-algebras

Using Theorem 4.9 we can now deduce the desired result.

### Corollary 4.15

If *M* and *N* are JBW-algebras and \(f:\overline{M}_+^\circ \rightarrow \overline{N}_+^\circ \) is a bijective Hilbert’s metric isometry with \(f(\overline{e})=\overline{e}\), then either for *f* or for \(\iota \circ f\) the induced map \(\theta :\mathcal {P}(M)\rightarrow \mathcal {P}(N)\) is an orthoisomorphism.

### Proof

Suppose that \(p_1,p_2\in \mathcal {P}(M)\) are orthogonal projections. By Lemma 4.5 we may assume that \(p_1+p_2<e\). Let \(p_3:=(p_1+p_2)^\perp \). After possibly composing *f* with the inversion \(\iota \) we may assume that *f* is projectively linear on \(\Delta (p_1,p_2,p_3)\) and so \(\theta \) preserves the orthogonality of \(p_1,p_2\) and \(p_3\) by Corollary 4.7. Hence \(\theta (p_1)\) and \(\theta (p_2)\) are orthogonal. By Theorem 4.9, *f* is projectively linear on all other orthogonal simplices as well, so \(\theta \) preserves the orthogonality of all noncomplementary orthogonal projections in \(\mathcal {P}(M)\). Applying the same argument to \(f^{-1}\) shows that \(\theta ^{-1}\) also preserves orthogonality.

By the proof of [13, Lemma 1], \(\theta \) is an order isomorphism and preserves products of operator commuting projections. Our next goal is to show that \(\theta \) extends to a Jordan isomorphism. If *M* and *N* are Euclidean Jordan algebras, this can be done with a similar argument as used in [6], see Remark 4.20. We will now explain how to proceed in the general case of JBW-algebras. The reader only interested in the von Neumann algebra case should follow this argument, but instead of the representations (4.9), each type \(I_2\) von Neumann algebra is isomorphic to \(L^\infty (\Omega , \mathbb {M}_2(\mathbb {C}))\).

We can write \(M=M_2\oplus \tilde{M}\) and \(N=N_2\oplus \tilde{N}\) where \(M_2\) and \(N_2\) are type \(I_2\) direct summands, and \(\tilde{M}\) and \(\tilde{N}\) are JBW-algebras without type \(I_2\) direct summands. See [17, Theorem 5.1.5, Theorem 5.3.5]. Suppose \(\tilde{p}\in \mathcal {P}(M)\) and \(\tilde{q}\in \mathcal {P}(N)\) are the central projections such that \(\tilde{p}M=\tilde{M}\) and \(\tilde{q}N=\tilde{N}\). Since \(\theta \) is an order isomorphism, the restriction \(\theta |_{\mathcal {P}(\tilde{M})}:\mathcal {P}(\tilde{M})\rightarrow \mathcal {P}(\theta (\tilde{p})N)\) is an orthoisomorphism. As \(\tilde{M}\) has no type \(I_2\) direct summand, we can use the following result.

### Theorem 4.16

(Bunce, Wright) Let *M* and *N* be JBW-algebras such that *M* has no type \(I_2\) direct summand. If \(\theta :\mathcal {P}(M)\rightarrow \mathcal {P}(N)\) is an orthoisomorphism, then \(\theta \) extends to a Jordan isomorphism \(J:M\rightarrow N\).

### Proof

The theorem is exactly [7, Corollary 2] but for JBW-algebras instead of JW-algebras. This corollary follows from [7, Proposition p. 91], and the crucial ingredient here is that any quantum measure on the projection lattice of a JW-algebra extends to a state. But this statement is also true for JBW-algebras by [8, Theorem 2.1]. \(\square \)

So \(\theta |_{\mathcal {P}(\tilde{M})}\) extends to a Jordan isomorphism \(\tilde{J}:\tilde{p}M\rightarrow \theta (\tilde{p})N\). Moreover, \(\theta (\tilde{p})=\tilde{q}\). Indeed, the image of \(\tilde{p}M\) under \(\tilde{J}\) in *N* contains no type \(I_2\) direct summand, hence \(\tilde{J}(\tilde{p}M)\subseteq \tilde{q}N\). This implies that \(\theta (\tilde{p})\le \tilde{q}\). Applying the same argument to \(\theta ^{-1}\) shows that \(\theta ^{-1}(\tilde{q})\le \tilde{p}\), so \(\tilde{p}=\tilde{q}\).

*k*,

*l*are cardinals, \(\Omega _k,\Xi _l\) are measure spaces, \(V_i=H_i\oplus \mathbb {R}\) are spin factors with \(\dim H_i=i\). We denote the unit in each \(V_k\) by

*u*. Let \(\Omega := \bigsqcup _k \Omega _k\) be the disjoint union of the \(\Omega _k\)’s. By identifying \(f\in L^\infty (\Omega )\) with \(\omega \mapsto f(\omega ) u\), we can view \(L^\infty (\Omega )\) as lying inside \(M_2\). It follows that \(Z(M_2)=L^\infty (\Omega )\) and if \(p := \mathbf {1}_A \in Z(M_2)\), then \(Z(pM_2) = L^\infty (A)\). Since \(\theta \) preserves operator commutativity, it preserves the center, and it is straightforward to see that \(\theta |_{\mathcal {P}(Z(M_2))} :\mathcal {P}(Z(M_2)) \rightarrow \mathcal {P}(Z(N_2))\) extends to a Jordan isomorphism \(T :Z(M_2) \rightarrow Z(N_2)\).

### Lemma 4.17

\(p\in \mathcal {P}(M_2)\) is a.e. rank 1 if and only if \(qp\ne 0\) and \(qp^\perp \ne 0\) for all nonzero central projections \(q\in \mathcal {P}(M_2)\).

### Proof

Let \(A\subseteq \Omega \) be measurable and suppose that \(p(\omega )=0\) a.e. on *A*. Then \(\mathbf {1}_A\in \mathcal {P}(M_2)\) is a central projection and \(\mathbf {1}_Ap=0\). Similarly, if \(B\subseteq \Omega \) is a measurable set such that \(p(\omega )=u\) a.e. on *B*, then \(\mathbf {1}_Bp^\perp =0\).

Conversely, if \(p\in \mathcal {P}(M_2)\) is a.e. rank 1, then neither \(\mathbf {1}_Ap=0\) nor \(\mathbf {1}_Ap^\perp =0\) for all nonzero measurable \(A\subseteq \Omega \), which are precisely the nonzero central projections of \(\mathcal {P}(M_2)\). \(\square \)

*T*extends \(\theta |_{\mathcal {P}(Z(M_2))}\),

*S*on \([M_2]\). To that end, let \(a \in M_2\) be such that \(a=\alpha +\beta p+\gamma p^\perp \) where \(\alpha =\sum _i\alpha _i\mathbf {1}_{A_i},\beta =\sum _j\beta _j\mathbf {1}_{B_j}\), and \(\gamma =\sum _k\gamma _k\mathbf {1}_{C_k}\) are step functions. Since \(\theta \) preserves products of operator commuting projections and the fact that

*T*maps step functions to step functions,

*T*is an isometry, so both norms can be made arbitrarily small. This implies that

### Corollary 4.18

If \(f:\overline{M}_+^\circ \rightarrow \overline{N}_+^\circ \) is a bijective Hilbert’s metric isometry with \(f(\overline{e})=\overline{e}\) such that its induced map \(\theta :\mathcal {P}(M)\rightarrow \mathcal {P}(N)\) is an orthoisomorphism, then \(\theta \) extends to a Jordan isomorphism \(J:M\rightarrow N\).

We will now show that the quotient map induced by the Jordan isomorphism *J* above coincides with *S*.

### Lemma 4.19

Let \(J :M \rightarrow N\) be a Jordan isomorphism that extends \(\theta \). Then *J* induces the quotient map \(\overline{J} :[M] \rightarrow [N]\) defined by \(\overline{J}([a]) := [J(a)]\), which satisfies \(\overline{J} = S\).

### Proof

*b*be as above such that \(\left\| a-b \right\| < \varepsilon \). Then \(\left\| Ja - Jb \right\| < \varepsilon \), and since

*S*is a \(\left\| \cdot \right\| _v\)-isometry and \(\left\| \cdot \right\| _v \le 2 \left\| \cdot \right\| \),

### Remark 4.20

If *M* and *N* are Euclidean Jordan algebras and \(\theta :\mathcal {P}(M) \rightarrow \mathcal {P}(N)\) is an orthoisomorphism, then an easier argument shows that \(\theta \) extends to a Jordan isomorphism. Indeed, every \(a \in M\) has a unique spectral decomposition \(a = \lambda _1 p_1 + \ldots + \lambda _n p_n\), and so we can define \(J(a) := \lambda _1 \theta (p_1) + \ldots \lambda _n \theta (p_n)\). Then \(J(a + \mu e) = J(a) + \mu e\), so *J* induces a map \(\overline{J} :[M] \rightarrow [N]\) by \(\overline{J}([a]) := [J(a)]\). By (4.11), \(\overline{J} = S\) is linear. Let \(M_0\) and \(N_0\) be the kernels of the traces in *M* and *N* respectively, then \([M] \cong M_0\) and \([N] \cong N_0\). It is clear from the definition of *J* that it maps \(M_0\) into \(N_0\), and so (4.10) implies that \(\overline{J} \cong J|_{M_0}\) is linear, thus \(J = J|_{M_0} \oplus {\mathrm {Id}}_{\mathbb {R}e}\) is linear. Since the spectrum and hence the norm is preserved, *J* is a Jordan isomorphism by Corollary 2.2.

We can now prove the following characterization of the Hilbert’s metric isometries on cones in JBW-algebras.

### Theorem 4.21

*M*and

*N*are JBW-algebras, then \(f :\overline{M}_+^\circ \rightarrow \overline{N}_+^\circ \) is a bijective Hilbert’s metric isometry if and only if

### Proof

*g*or \(\iota \circ g\) has the property that the induced map \(\theta :\mathcal {P}(M)\rightarrow \mathcal {P}(N)\) is an orthoisomorphism. Let \(h\in \{g,\iota \circ g\}\) be the map with this property and

*J*be the Jordan isomorphism from Corollary 4.18. Note that

*J*induces a map from \(\overline{M}^\circ _+\) to \(\overline{N}_+^\circ \). Let \(a \in M_+^\circ \), then \(a = \exp (c)\) for some \(c \in M\), and so by Lemma 4.19,

*h*coincides with

*J*on \(\overline{M}_+^\circ \). Since \(h\in \{g,\iota \circ g\}\), for either \(\varepsilon =1\) or \(\varepsilon =-1\) we have that

Theorem 4.21 has the following direct consequence.

### Corollary 4.22

Let *M* and *N* be JBW-algebras. The metric spaces \((\overline{M}_+^\circ ,d_H)\) and \((\overline{N}_+^\circ ,d_H)\) are isometric if and only if *M* and *N* are Jordan isomorphic.

*J*a Jordan isomorphism. So, if we let \(\iota _0:\overline{a}\in \overline{M}_+^\circ \mapsto \overline{a^{-1}}\in \overline{N}_+^\circ \), then

*M*contains no nontrivial projections, then \(M=\mathbb {R}\) and \(\iota _0\) is clearly projectively linear here. If

*M*contains a nontrivial projection, then it is minimal and maximal. So, if \(p\in M\) is a nontrivial central projection, then \(M=M_p\oplus M_{p^\perp }\). Since both \(M_p\) and \(M_{p^\perp }\) are JBW-algebras which contain no nontrivial projections, we conclude that \(M_p\cong M_{p^\perp }\cong \mathbb {R}\) and \(M\cong \mathbb {R}^2\). On \((\mathbb {R}^2_+)^\circ \) the inversion map satisfies \(\iota (x,y)=(x^{-1},y^{-1})=(xy)^{-1}(y,x)\), which shows that \(\iota _0\) belongs to \({\mathrm {Proj}}(M_+)\). Finally, suppose that all nontrivial projections in

*M*are not central. Then

*M*is a factor, and for any nontrivial projection

*p*, it follows that \(M_p\cong \mathbb {R}\) by the minimality of

*p*. This means that all nontrivial projections in

*M*are abelian and their maximality implies that they have central cover

*e*. Since we can write \(e=p+p^\perp \), we find that

*M*is of type \(I_2\). By [17, Theorem 6.1.8] we have that

*M*is a spin factor, so \(M_+\) is strictly convex. For an order unit space with strictly convex cone there always exists a strictly positive state, thus by [29, Remark 3.5] all bijective Thompson’s metric isometries on \(M_+^\circ \) are projective linear order isomorphisms. This implies that \(\iota _0\in {\mathrm {Proj}}(M_+)\). We have shown that if

*M*is a JBW-algebra such that \(\overline{M_+^\circ }\) does not contain an orthogonal simplex, then \(M_+\) must be a Lorentz cone (i.e., the cone of a spin factor or \(\mathbb {R}^2_+\)). To summarize we have the following result.

### Proposition 4.23

*M*be a JBW-algebra. If \(\overline{M}_+^\circ \) contains an orthogonal simplex, then the group of bijective Hilbert’s metric isometries \({\mathrm {Isom}}(\overline{M}_+^\circ ,d_H)\) satisfies

We believe that the results in this section could be extended to general JB-algebras. However, our arguments rely in a crucial way on the existence of nontrivial projections, which may not be present in a JB-algebra. It would also be interesting to know whether it is true that if the Hilbert’s metric isometry group of a cone *C* in a complete order unit space is not equal to the group of projectivities of *C*, then the order unit space is a JB-algebra. To date no counter example to this statement is known.

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