Hilbert and Thompson isometries on cones in JB-algebras

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∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^*$$\end{document}-algebra. To obtain the results we develop a variety of new geometric and Jordan algebraic techniques.


Introduction
On the interior A • + of the cone in an order unit space A there exist two important metrics: Hilbert's metric and Thompson's metric.Hilbert's metric goes back to Hilbert [19], who defined a metric δ H on an open bounded convex set Ω in a finite dimensional real vector space V by where a ′ and b ′ are the points of intersection of the line through a and b and ∂Ω such that a is between a ′ and b, and b is between b ′ and a.The Hilbert's metric spaces (Ω, δ 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 [17] 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 • + is not equal to the group of projectivities of A • + 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 ∈ A • + → a −1 ∈ A • + .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 yield 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 a number of earlier works.In 2009 Molnár [37] described the Thompson's and Hilbert's metric isometries on the selfadjoint operators on a Hilbert space of dimension at least three, using geometric means to show that these isometries preserve commutativity.A different approach was taken in 2010 by Hatori and Molnár [18] where they characterized Thompson's metric isometries on the positive cone of a C*-algebra by showing that they induce a linear norm isometry on the selfadjoint elements of the C*-algebra.Similarly, Bosché [6] described Thompson's and Hilbert's metric isometries on a symmetric cone in 2012 by showing that they induce norm isometries on the whole Euclidean Jordan algebra: a Thompson's metric isometry yields a linear JB-norm isometry, and a Hilbert's metric isometry yields a linear variation norm isometry.On JB-algebras, linear variation norm isometries are exactly linear maps preserving the maximal deviation, the quantum analogue of the maximal standard deviation, see [38,39,15].These were charactized on the selfadjoint elements of von Neumann algebras without a type I 2 summand by Molnar [38] in 2010, and in 2012 this result was extended to JBW-algebras without a type I 2 summand by Hamhalter [15].
Our approach is to show that Thompson's and Hilbert's metric isometries on the positive cone of a JB-algebras induce linear norm isometries on the whole JB-algebra: the Thompson's metric isometries yield norm isometries, whereas the Hilbert's metric isometries induce variation norm isometries, see Theorem 2.17.This extends the approach in [6] and [18].By using a characterization of linear norm isometries of JB-algebras due to Isidro and Rodríguez-Palacios [21] we then characterize the Thompson's metric isometries of JB-algebras, generalizing results of [6] and [18].As for Hilbert's metric, we restrict to JBW-algebras.If there is no type I 2 summand, Hamhalter's characterization of the linear variation norm isometries mentioned above yields the desired description of the Hilbert's metric isometry.But in general this result can not be used, so we exploit the fact that in our case the variation norm isometry is induced by a Hilbert's metric isometry to obtain the desired characterization.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 [20,30,35,44].
The structure of the paper is as follows.Section 2 is our preliminary section.We first introduce Thompson's and Hilbert's metrics 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 Thompson's and Hilbert's metric between two elements of a JB-algebra, and we study the interplay between geometric means and the isometries for both metrics.These findings also generalize earlier work done on Euclidean Jordan algebras and C * -algebras, and result in the crucial Theorem 2.17 mentioned above.
In Section 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 Section 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 [12] 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. .
We claim that this result extends Molnar's theorem ([37, Theorem 2]), reformulated below using our notation.Theorem 1.2 (Molnar).Let H be a complex Hilbert space with dim(H) ≥ 3 and let f : + be a bijective Hilbert's metric isometry.Then there is an invertible bounded linear or conjugate linear operator z : H → H and an ε ∈ {±1} such that Indeed, [21,Theorem 2.2] states that all Jordan isomorphisms 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 It remains to show that any invertible (conjugate) linear operator z ∈ B(H) can be written as 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.

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.

Order unit spaces
Let A be a partially ordered real vector space with cone A + .So, A + is convex, λA + ⊆ A + for all λ ≥ 0, A + ∩ −A + = {0}, and the partial ordering ≤ on A is given by a ≤ b if b − a ∈ A + .Suppose that there exists an order unit u ∈ A + , i.e., for each a ∈ A there exists λ > 0 such that −λu ≤ a ≤ λu.Furthermore assume that A is Archimedean, that is to say, if na ≤ u for all n = 1, 2, . .., then a ≤ 0. In that case A can be equipped with the order unit norm, and (A, • u ) is called an order unit space, see [16].It is not hard to show, see for example [29], that A + has nonempty interior A • + in (A, • u ) and A • + = {a ∈ A : a is an order unit of A}.On A • + Hilbert's metric and Thompson's metric are defined as follows.For a, b

Note that as
and Thompson's metric is defined by It is well known (cf.[26,41]) that d T is a metric on A + the set of rays through Ω will be denoted by Ω.

JB-algebras
A Jordan algebra (A, •) is a commutative, not necessarily associative algebra such that A JB-algebra A is a normed, complete real Jordan algebra satisfying, for all a, b ∈ A. An important example of a JB-algebra is the set of selfadjoint elements of a C *algebra A, equipped with the Jordan product a•b := (ab+ba)/2.By the Gelfand-Naimark theorem, this JB-algebra is a norm closed Jordan subalgebra of the selfadjoint bounded operators on a Hilbert space; such an algebra is called a JC-algebra.By [16, 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 selfadjoint 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 Inv(A).The spectrum of a ∈ A, σ(a), is defined to be the set of λ ∈ R such that a − λe is not invertible in JB(a, e), the JB-algebra generated by a and e ([16, 3.2.3]).There is a continuous functional calculus: JB(a, e) ∼ = C(σ(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 ∈ A are said to 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 ∈ A as ab instead of a • 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 ([16, Theorem 3.2.2]).
The cone of elements with nonnegative spectrum is denoted by A + , and equals the set of squares by the functional calculus, and its interior A • + consists of all elements with strictly positive spectrum, or equivalently, all elements in A + ∩ Inv(A).This cone turns A into an order unit space with order unit e, i.e., a = inf{λ > 0 : −λe ≤ a ≤ λe}.
Note that the JB-norm is not the same as the usual norm in a Euclidean Jordan algebra.The Jordan triple product {•, •, •} is defined as for a, b, c ∈ A. In a JC-algebra one easily verifies that {a, b, c} = (abc + cba)/2.For a ∈ A, the linear map U a : A → A defined by U a b := {a, b, a} will play an important role and is called the quadratic representation of a.
By the Shirshov-Cohn theorem for JB-algebras [16, Theorem 7.2.5], the unital JB-algebra generated by two elements is a JC-algebra, which shows all but the fifth of the following identities for JB-algebras, since U a b = aba in JC-algebras.(For the rest of the paper, the operator-algebraic reader is encouraged to think of this equality whenever the quadratic representation appears.) (2.3) A proof of the fifth identity can be found in [16, 2.4.18], as well as proofs of the other identities.A JB-algebra A induces an algebra structure on A • + by a • b := a • b, which is well-defined.We can also define a α := a α for α ∈ R. For a ∈ inv(A), the quadratic representation U a is an order isomorphism, and induces a well defined map U a on A When studying Hilbert's metric on A • + in JB-algebras, the variation seminorm • v on A given by, a v := diam σ(a) = max σ(a) − min σ(a), will play an important role.The kernel of this seminorm is the span of e, and on the quotient space [A] := A/ Span(e) it is a norm.To see this we show that if • q is the quotient norm of 2 • on [A], then [a] q = [a] v for all [a] ∈ [A].Indeed, for [a] ∈ [A], using inf λ∈R max{t−λ, s+λ} = (t+s)/2, we have that Note that the map Log : A • + → A given by a → log(a) is a bijection, whose inverse Exp is given by a → exp(a).Furthermore, as log(λa) = log(a) + log(λ)e for all a ∈ A • + and λ > 0, the map Log induces a bijection from 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 ⊥ .Every JBW-algebra decomposes into a direct sum of a type I, II, and III JBWalgebras.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.

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 ⊥ .Theorem 2.1 (Isidro, Rodríguez-Palacios).The bijective linear isometries from A onto B are the mappings of the form a → sJa, where s is a central symmetry in B and J : A → 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 → 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 ∈ A + , a ≤ 1.Then e − a ≤ 1, and so e − T a ≤ 1, showing that T a 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 −λe ≤ a ≤ λe if and only if −λe ≤ T a ≤ λe, 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 a = max |σ(a)|.This corollary will be used to show the following proposition.For Euclidean Jordan algebras this proposition has been proved in [13,Theorem III.5.1].
+ and J is a Jordan isomorphism.Moreover, this decomposition is unique and b = (T e) 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 (T e) 2 T , and by the above corollary . This implies that J = U (T e) − 1 2 T , so J is also unique.

Hilbert's and Thompson's metrics on cones in JB-algebras
Suppose A is a JB-algebra.For c ∈ A • + , 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.
).The formula for d H follows immediately.As c = max{max σ(c), − min σ(c)} for c ∈ A, the identity for d T holds.

Also note that the inverse map on
), so this is an isometry for both metrics as well.Indeed, using (2.3) we see that Given a JB-algebra A we follow Bosché [6, Proposition 2.6] and Hatori and Molnár [18,Theorem 9], and introduce for n ≥ 1 metrics on [A] and A, respectively, by Proof.We start with some preparations.The JB-algebra generated by a, b and e is special, so we can think of ) for some C * -algebra multiplication.Writing out the exponentials in power series yields which is valid for c < 1, we obtain for sufficiently large n that So, for all sufficiently large n we have by Proposition 2.4 that As the right hand side converges to 0 for n → ∞, the first limit holds.The second limit can be derived in the same way.
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 γ : I → M, where I is a possibly unbounded interval in R, is a geodesic path if there is a k ≥ 0 such that d(γ(s), γ(t)) = k|s − t| for all s, t ∈ I.The image of a geodesic path is called a geodesic.The following result generalizes [28, Theorems 5.1 and 6.2].

Consider the Peirce decomposition
) 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 ⊥ .From [1, Proposition 1.3.8]we know that if a ∈ A + , then U p a = a if and only if U p ⊥ a = 0. Using this result we now prove the following claim.
To show the claim, note that 0 By applying the claim to λp as well as [28,Theorem 4.3], there is a unique geodesic between b and e.
Conversely, suppose that there is a unique geodesic between b and e.Then this is also a unique geodesic in JB(b, e) ∼ = C(σ(b)).For f, g ∈ C(σ(b)) we have by Proposition 2.4 that So, the pointwise logarithm is an isometry from (C(σ(b + are linearly independent, then there exists a unique geodesic between a and b in (A The following special geodesic paths play an important role. ) is similar and is left to the reader.Using the fact that U c λ c µ = c 2λ+µ in the fourth step, we get that

Geometric means in JB-algebras
The cone A • + in a JB-algebra is a symmetric space, see Lawson and Lim [25] and Loos [34].Indeed, for c ∈ A • + one can define maps S c : Clearly S c (c) = c, and Moreover, by the fifth equation in (2.3) we see that for all a ∈ A • + .The map S c is called the symmetry around c, see [34].The equation S c (a) = b has a unique solution in A • + , namely γ b a (1/2).Indeed, using (2.3) and taking the unique positive square root in the third step, we obtain the following equivalent identities: Definition 2.9.For a, b ∈ A • + the unique solution of the equation S c (a) = b is called the geometric mean of a and b.It is denoted by a#b, so We remark that the equation + , 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.Let M, N be metric spaces.Suppose that for each x, y ∈ M there exists an element z xy ∈ M, a bijective isometry ψ xy : M → M and a constant k xy > 1 such that Suppose N satisfies the same requirements.If ϕ : M → 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.If A and B are JB-algebras and f : we already saw that S a#b is an isometry that satisfies the first two properties in Lemma 2.10.To show the third property note that by Proposition 2.4, So, if we take k ab := 2, then all conditions of Lemma 2.10 are satisfied, and its application yields the proposition.
To see that the same result holds for Hilbert's metric isometries on A • + , we need to make a couple of observations.Firstly for c ∈ A • + , the map S c induces a well defined maps S c on A Proposition 2.12.If A and B are JB-algebras and f : A The next proposition will be useful.
It is straightforward to derive a similar identity for Hilbert's metric.
Proof.The proof follows from Proposition 2.13 and 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 b a, which can be easily shown using (2.3).
Corollary 2.15.Let A and B be JB-algebras.If f : Proof.By Propositions 2.13 and 2.11, the first statement holds for all dyadic rationals t ∈ [0, 1].
As the dyadic rationals are dense in [0, 1], it holds for all 0 ≤ t ≤ 1. Suppose f (e) = e.Since γ a e (t) = a t , the first statement yields that f so by uniqueness of the solution of f (a)#c = e, we obtain f (a −1 ) = f (a) −1 .Using (2.3) we also get 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.Proof.We will prove the second assertion.The same arguments can be used to show the statements for Thompson's metric.Using Corollary 2.16, Thus, + 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, • ) and (A • + , d T ), see [29].Likewise, the exponential map yields an isometric isomorphism between ([A], • v ) and (A In the nonassociative case this is no longer true.In fact, it has been shown for finite dimensional order unit spaces A that (A • + , d H ) is isometric to a normed space if and only if A + is a simplicial cone, see [14].For Thompson's metric the same result holds, see [28,Theorem 7.7].

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.Suppose that A is a product of order unit space A i for i ∈ I.
Proof.The proposition follows immediately from 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] In this case b = f (e) Proof.The last statement follows from taking a = e, which yields b 2 = f (e).
For the sufficiency, note that the central projection p yields a decomposition B = pB ⊕ p ⊥ 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 : Note that a Jordan isomorphism is an order isomorphism and hence an isometry under Thompson's metric.The inversion and the quadratic representations also preserve Thompson's metric, and so Thompson's metric is preserved on both parts.By Proposition 3.1 Thompson's metric is preserved on the product as well.Now suppose that f : 2 f (a), we obtain that g is a Thompson's metric isometry mapping e to e.By Theorem 2.17 the map S : A → B defined by Sa := log g(exp(a)) is a bijective linear • -isometry.From Theorem 2.1 it follows that there is a central projection p ∈ B and a Jordan isomorphism J : A → B such that Sa = (p − p ⊥ )Ja.We now have for a ∈ A, It follows that, for a ∈ A • + , g(a) = pJa + p ⊥ Ja −1 .The theorem now follows from 2 g(a).

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 [13].
Theorem 3.3.Suppose a JB-algebra A can be decomposed as a direct product where 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 where Aut(A i+ ) denotes the automorphism group of the cone A i+ , i.e., the order isomorphisms of A i into itself, C 2 denotes the cyclic group of order 2 generated by the inverse map ι, and S(n i ) denotes the group of permutations of n i .
Proof.By Theorem 3.2 any bijective Thompson's metric isometry is a composition of a quadratic representation, a Jordan isomorphism and taking inverses on some components.Quadratic representations and taking inverses leave each component invariant, and Jordan isomorphisms leave the Jordan isomorphism classes invariant.This shows that and the other inclusion follows from Proposition 3.1, so we have equality.We will now investigate Isom((A i may permute the components, so it follows that each Thompson's metric isometry of (A n i i ) • + 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 π equals the componentwise operator permuted by π.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.
By Proposition 2.3, any order isomorphism is the product of a quadratic representation and a Jordan isomorphism.If we denote the inverse map by ι = ι −1 , then conjugating an order isomorphisms with the inverse map gives which yields another order isomorphism.So, the product of the group of order isomorphism and the inversion group C 2 is a semidirect product, where the order isomorphisms form the normal subgroup.We conclude that Remark 3.4.If 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 Furthermore, for any i ∈ I the conjugation action (3.1) on an order isomorphism in Aut(A n i i+ ) also shows that Isom((A 2 , so we can write the isometry group as It follows that the automorphism group Aut(A + ) is normal in Isom(A • + , d T ), and its quotient is isomorphic to i∈I C n i 2 .Suppose now that both 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 = i∈I n i is the total number of different components.This is a correction of [6, Remark 4.9], which has the wrong index.

This implies that
from which we conclude that λ = 0.By the same argument as in [ 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 P(M).
Let M be a JBW-algebra.By Lemma 4.1 we can define a map θ : P(M) → P(N) by letting θ(0) = 0, θ(e) = e, and θ(p) be the unique nontrivial projection in the class S[p], otherwise.Thus, for each bijective Hilbert's metric isometry f : , we get a bijection θ : P(M) → P(N).We say that θ is induced by f .Note that its inverse θ −1 is induced by f −1 .The map θ will be the key in understanding f .We call a bijection θ : P(M) → P(N) an orthoisomorphism if p, q ∈ P(M) are orthogonal if and only if θ(p) and θ(q) are orthogonal.Our goal will be to prove that the map θ induced by either f or ι • f , where ι(a) = 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: if and only if k i=1 λ i = 0. Hence the relative boundary of conv(p 1 , . . ., p k ) in aff(p 1 , . . ., p k ) lies in ∂M + , which proves the equality.
Note that if a = µ and hence σ( and hence the image of the [e, p) under f is [e, r).
If 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 e, we find that for all 0 ≤ t < 1.Using the spectral decomposition p = 1p + 0p ⊥ , we now deduce that and hence q := θ(p) = r.
We can now show that θ preserves operator commuting projections.
We will proceed to show that if f : + is a bijective Hilbert's metric isometry with f (e) = e, then for either f or ι • f , the induced map θ maps orthogonal noncomplementary projections to orthogonal projections.For this we need to look at special simplices in the cone M + .
In case (i), θ preserves the orthogonality of p 1 , p 2 , p 3 .Moreover, if the map θ induced by f satisfies the assumptions of case (ii), then the map θ induced by the isometry ι • f satisfies the conditions of case (i).
Proof.First remark that, as p 1 + p 2 + p 3 = e and S is linear, and hence As p 1 + p 2 < e, we know that p 1 and p 2 are orthogonal by [1, Proposition 2.18], and hence p 1 and p 2 operator commute by [1,Proposition 1.47].We also know from Proposition 4.4 that q 1 = θ(p 1 ) and q 2 = θ(p 2 ) operator commute.By [1, Proposition 1.47], q 1 and q 2 are contained in an associative subalgebra, which is isomorphic to a C(K)-space.Note that this subalgebra also contains λe and hence also q 3 by (4.3).In a C(K)-space it is obvious that λ ∈ {1, 2} in (4.3).In fact, the case λ = 1 corresponds with the pairwise orthogonality of q 1 , q 2 and q 3 , whereas the case λ = 2 corresponds to pairwise orthogonality of q ⊥ 1 , q ⊥ 2 and q ⊥ 3 , and q ⊥ 1 + q ⊥ 2 + q ⊥ 3 = e.We will now show that f maps ∆(p 1 , p 2 , p 3 ) onto ∆(q 1 , q 2 , q 3 ) in case q 1 + q 2 + q 3 = e.Let a ∈ conv(p 1 , p 2 , p 3 ) ∩ M • + be a point not lying on any (p i , p ⊥ i ) for i = 1, 2, 3. We know that [a, p i ) is a unique geodesic by Lemma 4.2.Let (a ′ , p 1 ) be the line segment through p 1 and a with a ′ in the boundary of conv(p 1 , p 2 , p 3 ).This unique geodesic intersects (p 2 , p ⊥ 2 ) and (p 3 , p ⊥ 3 ) in 2 distinct points, say b 2 and b 3 respectively, see Figure 4. Since it must be mapped to a line segment, it follows that f (a) lies on the line segment through f (b 2 ) and f (b 3 ), which is contained in ∆(q 1 , q 2 , q 3 ).By the invertibility of f , we conclude that f (∆(p 1 , p 2 , p 3 )) = ∆(q 1 , q 2 , q 3 ).The same argument can be used to show that To prove the final statement remark that if we compose f with the inversion ι, we obtain So, the map θ induced by ι • f satisfies θ(p 1 ) + θ(p 2 ) + θ(p 3 ) = q ⊥ 1 + q ⊥ 2 + q ⊥ 3 = e, as the q ⊥ i are pairwise orthogonal in case (ii).
It follows from Lemma 4.6 that if ∆(p 1 , p 2 , p 3 ) is an orthogonal simplex, then the restriction of f to ∆(p 1 , p 2 , p 3 ) is a Hilbert's metric isometry onto either ∆(θ(p 1 ), θ(p 2 ), θ(p 3 )) or ∆(θ(p 1 ) ⊥ , θ(p 2 ) ⊥ , θ(p 3 ) ⊥ ).The Hilbert's metric isometries between simplices have been characterized, see [20] or [30], and yields the following dichotomy, as f (e) = e.The isometry f maps ∆(p 1 , p 2 , p 3 ) onto ∆(θ(p 1 ), θ(p 2 ), θ(p 3 )) in Lemma 4.6 if and only if the restriction of f to ∆(p 1 , p 2 , p 3 ) is of the form, which is equivalent to saying that the restriction of f to ∆(p 1 , p 2 , p 3 ) is projectively linear.On the other hand, the isometry f maps ∆(p 1 , p 2 , p 3 ) onto ∆(θ(p 1 ) ⊥ , θ(p 2 ) ⊥ , θ(p 3 ) ⊥ ) in Lemma 4.6 if and only if the restriction of f to ∆(p 1 , p 2 , p 3 ) is of the form, which is equivalent to saying that the restriction of ι • f to ∆(p 1 , p 2 , p 3 ) is projectively linear.The above discussion yields the following corollary.Our next proposition states that if two orthogonal simplices have a line in common, then 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 ∈ M • + are such that the line through a and b intersect ∂M + in a ′ and b ′ such that a is between b and a ′ , b is between a and b ′ , then A proof can be found in [26,Chapter 2].Proof.Suppose for the sake of contradiction that f is projectively linear ∆(p 1 , p 2 , p 3 ), but not on ∆(p 4 , p 5 , p 6 ).Denote the image of ∆(p 1 , p 2 , p 3 ) by ∆(q 1 , q 2 , q 3 ), and the image of ∆(p 4 , p 5 , p 6 ) by ∆(q ⊥ 4 , q ⊥ 5 , q ⊥ 6 ) as in Lemma 4.6.There are 2 cases to consider: p 3 = p 6 and p 3 = p ⊥ 6 .Let us first assume that p 3 = p 6 .
Let c, a ′ n , and b ′ n be in the boundary of conv(q 1 , q 2 , q ⊥ 4 , q ⊥ 5 ) as in Figure 7. Then the triangles with vertices b n , b ′ n and q ⊥ 5 are similar for all n ≥ 1. Hence there exists a constant C > 0 such that Now using (4.4) we deduce that Thus, there exists a constant As f −1 is an isometry and f (e) = e, we get that lim sup By construction, however, for some sequences (t n ) and (s n ) in [0, 1) with t n , s n → 1, which contradicts (4.6).Thus, 1 2 (p ⊥ 2 + p ⊥ 5 ) ∈ ∂M + and hence conv(p 1 , p 3 , p 4 ) ⊆ ∂M + .The same argument works for the other faces containing p 3 .The square face is also contained in ∂M + , as it contains 1  2 p ⊥ 3 .This proves (4.5).

Proof of Lemma 4.10
The proof of Lemma 4.10 requires a number of steps.First note that by Lemma 4.11(iii), it suffices to find a nontrivial projection z ∈ P(M) that operator commutes with both p and q.Hence we may assume that p ∧ q = p ∧ q ⊥ = p ⊥ ∧ q = p ⊥ ∧ q ⊥ = 0. (4.7) Indeed, suppose one of them is nonzero, then it operator commutes with p or p ⊥ and q or q ⊥ , 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 selfadjoint elements of a von Neumann algebra.Note that if M is of type I 2 , then [16,Theorem 6.1.8]implies that M is a spin factor H ⊕ 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 selfadjoint 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 (O) by Lemma 4.12.Then by [16,Theorem 7.2.7]we have that M is a JW -algebra, that is, it can be represented as a σ-weakly closed Jordan subalgebra of the selfadjoint operators on a complex Hilbert space.By [16, Theorem 7. 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 σ-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 selfadjoint 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 ⊆ R be the real von Neumann algebra generated by p and q.In the case where M is the selfadjoint part of a von Neumann algebra, the reader can regard N as the von Neumann algebra generated by p and q, and R = M ⊕ iM here.We denote by N ′ the commutant of N.That is, N ′ := {x ∈ B(H) : xy = yx for all y ∈ N} .
It suffices to find a nontrivial projection z ∈ N ′ ∩ R, because then both z and z ⊥ commute with p and q, and hence operator commute with p and q by [1, Proposition Proof.The reader can easily verify that the map ϕ : M → M 2 (M e 11 ) given by ϕ(x) ij := e 1i xe j1 is a * -homomorphism with inverse θ : M 2 (M e 11 ) → M defined by θ(y ij ) := 2 i,j=1 e i1 y ij e 1j .We now apply Lemma 4.13 for M = N and M = R, which yields that N ∼ = M 2 (N p ) and R ∼ = M 2 (R p ).Moreover, since we used the same matrix unit, the inclusion N ⊆ R corresponds to the natural embedding M 2 (N p ) ⊆ M 2 (R p ).It is straightforward to verify that The projection p = e 11 is nonmaximal, so there exists a nontrivial projection in R which dominates p, and has to be of the form p 0 0 z for some nontrivial projection z ∈ P(R p ).We claim that it now suffices to show that N p is a trivial von Neumann algebra.Indeed, in that case N ′ p ∩ R p = R p , and so by (4.8), is a nontrivial projection, as desired.In the case where M is the selfadjoint 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 ′ ∩ N contains no nontrivial projections, the spectral theorem implies that N ′ ∩ N is trivial and hence N is a factor.Therefore, we must have N ∼ = M 2 (C).Since we also have that N ∼ = M 2 (N p ), it follows that N p ∼ = C.In the case where N ⊆ R in a real von Neumann algebra, we have to do some more work to show that N p ∼ = 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 • • • pqpqpq • • • .For r, s ∈ {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) ∪ Q(p, q) ∪ Q(q, p) ∪ 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} ∪ {(pqp) n : n ≥ 1}.
By the above lemma, N p is generated by p and pqp.Since p is the identity on N p , it is commutative and contains C R (σ(pqp)), the continuous real-valued functions on σ(pqp), by the continuous functional calculus for real von Neumann algebras [31,Proposition 5.1.6(2)].Therefore, we have that N p ⊆ N ′ p , and so Since N ∩ N ′ contains no trivial projections, we obtain that N p contains no trivial projections.

Characterization of Hilbert isometries on JBW-algebras
Using Theorem 4.9 we can now deduce the desired result.Proof.Suppose that p 1 , p 2 ∈ 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 ) ⊥ .After possibly composing f with the inversion ι we may assume that f is projectively linear on ∆(p 1 , p 2 , p 3 ) and so θ preserves the orthogonality of p 1 , p 2 and p 3 by Corollary 4.7.Hence θ(p 1 ) and θ(p 2 ) are orthogonal.By Theorem 4.9, f is projectively linear on all other orthogonal simplices as well, so θ preserves the orthogonality of all noncomplementary orthogonal projections in P(M).Applying the same argument to f −1 shows that θ −1 also preserves orthogonality.
By the proof of [12, Lemma 1], θ is an order isomorphism and preserves products of operator commuting projections.Our next goal is to show that θ 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 ∞ (Ω, M 2 (C)).
We can write M = M 2 ⊕ M and N = N 2 ⊕ Ñ where M 2 and N 2 are type I 2 direct summands, and M and Ñ are JBW-algebras without type I 2 direct summands.See [16,Theorem 5.1.5,Theorem 5.3.5].Suppose p ∈ P(M) and q ∈ P(N) are the central projections such that pM = M and qN = Ñ .Since θ is an order isomorphism, the restriction θ| P( M ) : P( M ) → P(θ(p)N) is an orthoisomorphism.As M has no type I 2 direct summand, we can use the following result.
For µ ∈ R and the unit e 2 ∈ M 2 we have that J 2 (a+µe 2 ) = J 2 (a)+µe 2 , so J 2 induces the quotient map J 2 : [M 2 ] → [N 2 ] defined by J 2 ([a]) := [J 2 a].We claim that J 2 coincides with S on [M 2 ].To that end, let a ∈ M 2 be such that a = α + βp + γp ⊥ where α = i α i 1 A i , β = j β j 1 B j , and γ = k γ k 1 C k are step functions.Since θ preserves products of operator commuting projections and the fact that T maps step functions to step functions, as T is an isometry, so both norms can be made arbitrarily small.This implies that Having this, we will now proceed to show that J 2 is linear.Let Ξ := l Ξ l be the disjoint union of the Ξ l 's, and let ϕ be a state on Z(N 2 ) = L ∞ (Ξ).Then T * ϕ is a state on Z(M 2 ) = L ∞ (Ω), and define the functionals tr ⊗ T * ϕ ∈ M * • + is a bijective Hilbert's metric isometry with f (e) = e such that its induced map θ : P(M) → P(N) is an orthoisomorphism, then θ extends to a Jordan isomorphism J : M → N.
We will now show that the quotient map induced by the Jordan isomorphism J above coincides with S.

Theorem 1 . 1 .
The set M • + denotes the set of rays in M • + , and U b denotes the quadratic representation of b.If M and N are JBW-algebras, then f :

Theorem 2 . 6 .
If A is a JB-algebra and a, b ∈ A • + are linearly independent, then there exists a unique Thompson geodesic between a and b if and only which sends e to the zero function and b to the function k → log k.Note that for f ∈ C(σ(b)) the images of both t → (t f ∧ |f |)sgnf and t → tf are geodesics connecting 0 and f , which are different if and only if there is a point k ∈ σ(b) such that |f (k)| = f .Hence k → | log(k)| is constant.So, if α, β ∈ σ(b), then | log β| = | log α|, and hence α = β or α = β −1 .This shows that σ(b) ⊆ {β −1 , β}, and since b and e are linearly independent we must have equality.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 ∈ A • which has the unique solution c = b#a, is equivalent to the equation S c (a) = U c a −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 • + .Moreover, S c is an isometry under both Hilbert's metric and Thompson's metric on A •

•
+ by letting S c (a) := S c (a).Furthermore, for a, b ∈ A • + and λ, µ > 0 we have that the equation U c (λa) = µb has unique solution c = (λa)#(µb) = √ λµ(a#b).Thus, the equation U c a −1 = U c a −1 = b has a unique solution a#b in A • + for a, b ∈ A • + , and we can define the projective geometric mean by a#b := a#b in A • + .Note that a#b = γ b a (1/2).It is now straightforward to verify that the Hilbert's metric isometries S a#b on A • + satisfy the requirements of Lemma 2.10 with k ab = 2 and derive the following result.

Corollary 2 . 16 .
Let A and B be JB-algebras.If f : A• + → B • + is a bijective Hilbert's metric isometry, then (a) f maps γ b a (t) to γ f (b) f (a) (t) for all a, b ∈ A • + and t ∈ [0, 1].(b) If f (e) = e, then f (a t ) = f (a) t for all t ∈ [0, 1].Moreover, we have f (a −1 ) = f (a) −1 and f (U b a) = U f (b) f (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/ Span(e).
, [b]).By Proposition 2.5 the left-hand side of the above equation converges to S[a] − S[b] v and the right-hand side converges to [a] − [b] v as n → ∞.Hence S is a bijective • v -isometry.As f (e) = e, we have that S[0] = [0], and hence S is linear by the Mazur-Ulam theorem.Remark 2.18.The map Exp : A → A •

IfLemma 4 . 1 .
A and B are JB-algebras and f : A • + → B • + is a bijective Hilbert's metric isometry mapping e to e, then by Theorem 2.17 the map S : [A] → [B] defined by, S[a] := log f (exp([a])), is a bijective linear • 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 [15, Proposition 2.2].The extreme points of the unit ball in ([A], • v ) are the equivalence classes [p], where p ∈ A is a nontrivial projection.Proof.Let p ∈ A be a nontrivial projection and suppose that [p] = t[a] + (1 − t)[b] for some 0 < t < 1, and [a], [b] ∈ [A] with [a] v = [b] v = 1.There exist λ ∈ R, a ∈ [a], and b ∈ [b] such that p = ta + (1 − t)b + λe and {0, 1} ⊆ σ(a), σ(b) ⊆ [0, 1].
(a, b) denotes the open line segment {ta + (1 − t)b : 0 < t < 1} in M + for a, b ∈ M + .The segments [a, b] and [a, b) are defined similarly.Furthermore, we denote the affine span of a set S by aff (S).

Corollary 4 . 15 .
If M and N are JBW-algebras and f : M • + → N • + is a bijective Hilbert's metric isometry with f (e) = e, then either for f or for ι • f the induced map θ : P(M) → P(N) is an orthoisomorphism.
• + , but d H is not, as d H (λa, µb) = d H (a, b) for all λ, µ > 0 and a, b ∈ A • + .However, d H (a, b) = 0 for a, b ∈ A • + ifand only if a = λb for some λ > 0, so that d H is a metric on the set of rays in A • + , which we shall denote by A • + .Elements of A • + will be denoted by a, and if Ω ⊆ A • and [37, Theorem 9].Theorem 3.2.Let A and B be unital JB-algebras.A map f : A • + → B • + is a bijective Thompson's metric isometry if and only if there exist b ∈ B • + , a central projection p ∈ B, and a Jordan isomorphism [31,].Similarly to the discussion preceding Step 1, we can conclude that either z or z ⊥ is nonmaximal.So, we may assume that N ′ ∩ N contains no nontrivial projections.We will now generalize the proof of[46,  Theorem V.1.41],sothatit will also be applicable to the real von Neumann algebra case.From equation (4.7), we obtain that p ⊥ qp maps pH injectively onto a dense subspace of p ⊥ H. Let uh be the polar decomposition of p ⊥ qp.By[31, Proposition 4.3.4]wehave that u, h ∈ N. Then u is a partial isometry with initial space pH and final space p ⊥ H, and so u * u = p and uu * = p ⊥ .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 e 11 + e 22 = e, e * ij = e ji , and e ij e kl = δ jk e il for 1 ≤ i, j, k, l ≤ 2. Let e 11 := p, e 21 := u, e 12 := u * , e 22 := p ⊥ , We will use the following notation.If M is an algebra with projection p ∈ M, then we denote the subalgebra pMp by M p .Furthermore, by M 2 (M p ) we mean the 2 × 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 ∼ = M 2 (M e 11 ).
[31,ver, unlike the case of a von Neumann algebra, a real von Neumann algebra without any nontrivial projections need not be trivial (i.e., C, H).But by[31, Proposition 4.3.4(3)], the linear span of the projections is dense in (N p ) sa , and so (N p ) sa must be trivial.Since C R (σ(pqp)) ⊆ (N p ) sa , this can only happen if σ(pqp) consists of a single element, which implies that N p ∼ = R, as desired.This completes the proof of Lemma 4.10.