Geometry of the Dual Ball of the Spin Factor

The main result of this paper is a geometric characterization of the unit ball of the dual of a complex spin factor. THEOREM. A strongly facially symmetric space of type I 2 in which every proper norm closed face in the unit ball is norm exposed, and which satisfies 'symmetry of transition probabilities', is linearly isometric to the dual of a complex spin factor. This result is an important step in the authors' program of showing that the class of all strongly facially symmetric spaces satisfying certain natural and physically significant axioms is equivalent isometrically to the class of all predual spaces of JBW*-triples. The result can be interpreted as a characterization of the non-ordered state space of 'two state' physical systems. A new tool for working with concrete spin factors, the so-called facial decomposition, is also developed.

Facially symmetric spaces were introduced and studied in [6] and [7] as a geometrical model for quantum mechanics.They provide an appropriate framework in which to study the problem of characterizing the unit ball of the predual of a JBW*-triple (cf.[5]) in terms of geometric and physically significant properties of a convex set.In this paper we will restrict ourselves to atomic spaces (Definition 2.5).The rank-1 case, together with the axiom (STP) (symmetry of transition probabilities, Definition 2.8) leads easily to a Hilbert space, as will be shown in §2 (Corollary 2.11).The next significant case, that of rank 2, will be treated in detail in § § 3 and 4.This case occurs already as the model for the state space of a spin-\ particle.More generally, this model can occur for any physical system in which each measurement results in at most two distinct outcomes ('two state systems').
A standard algebraic model in this context is the (real or complex) spin factor.Since this is a Jordan algebra (JB-algebra in the real case, and JB*-algebra in the complex case), its state space has been extensively studied.Less well known is the structure of the entire dual ball.In the real case, as will be shown below (see Proposition 1.8), this is the intersection of two cylinders based on real Hilbert balls.These two Hilbert balls are either of the same dimension, or their dimensions differ by 1.In the complex case, the geometry of the spin factor has a much richer structure, an analysis of which is carried out in this paper.
The main result of this paper is a geometric characterization of the predual ball of a complex spin factor.This is the same as the unit ball of the dual since the spin factor is a reflexive Banach space.We show (in Theorem 4.16) that any facially symmetric space of type 7 2 (Definition 3.10) satisfying some physically significant geometric axioms ((STP) and (FE), Definition 3.3) is linearly isometric to the predual of a complex spin factor.This result is an important step in the program whose objective is to show that the dual of any facially symmetric space which satisfies certain natural geometric axioms supports the structure of a JB*-triple.Conversely, it is known that the predual of every JBW*-triple is a neutral strongly facially symmetric space [8,Theorem 3.1] which satisfies (STP) [5,Lemma 2.2] and (FE) [4,Corollary 4.5].
The organization of this paper is as follows.In § 1 we describe the facial structure of the dual ball of the concrete spin factor, using the basis called a spin grid in [3] and [10].This facial structure was revealed from the study of facially symmetric spaces of type I 2 carried out in § § 3 and 4. The first named author wishes to thank Professor Itamar Pitowsky of the Hebrew University for discussions of the connections of this facial structure to models in quantum mechanics.These discussions were instrumental in the formulation of the material of § 1.Although the results of § 1 are presented here primarily to illustrate in a concrete form the abstract results of § § 3 and 4, nevertheless, some of these results are used to simplify the proof of our main theorem.Moreover, the (facial) decomposition introduced here (Proposition 1.11) may provide a new tool for working with spin factors.
In § 2 the definition and some properties of facially symmetric spaces are recalled and some supplementary results from the global theory of facially symmetric spaces are proved.The definition and some properties of atomic spaces are given and the principal geometric axioms are introduced.It is shown that an atomic facially symmetric space satisfying these axioms admits a symmetric sesquilinear form, which leads to a Hilbert space structure in the rank-1 case.
In §3 we show that a rank-2 face in an atomic neutral strongly facially symmetric space satisfying the axioms (FE) and (STP) is affinely isometrically isomorphic to the unit ball of a real Hilbert space, and moreover that the unit ball of the real linear span of this face is a cylinder with the Hilbert ball as its base.This provides a fundamental computational tool for the main result in § 4.
In § 4 we study the complex span of a rank-2 face, which by definition is dense in a facially symmetric space of type I 2 .Starting from an arbitrary orthonormal basis in the real Hilbert space whose unit ball is affinely isomorphic to the given rank-2 face, we construct a total subset (called a dual spin grid) which resembles the 'dual' of a spin grid in a concrete spin factor.In this construction, the imaginary unit enters naturally from geometric considerations.After a reduction to finite dimensions, it is shown by a induction argument that the linear extension of the natural map between the two dual spin grids (abstract and concrete) is isometric, and therefore extends to an isometry of the dual of the spin factor onto the norm closure of the complex span of the rank-2 face.The induction proceeds by two dimensions at a time, so it is necessary to analyse in complete detail the cases in which the Hilbert spaces arising from the rank-2 face are of dimensions 2 and 3.This leads to characterizations of the spin factors of dimensions 3 and 4, namely, the JBW*-triples S^C) of 2 by 2 symmetric complex matrices and A/ 2 (C) of all 2 by 2 complex matrices.
The following symbols will be used in this paper: the closed unit ball of a normed space X will be denoted by X lf U denotes the set of real numbers, C denotes the set of complex numbers, and T denotes the unit circle in C.
The duality between a normed space X and its dual X* will be denoted by (cp, x) or by (x, q>) for (q>, x)eX* x X.The real and imaginary parts of a complex number z will be denoted by dlz and £sz.
For a convex set AT in a normed space Z, ext K denotes the set of extreme points of K.An element/ e K is said to be norm exposed in K if {/} = K C\ H for a suitable hyperplane H.The set of norm exposed points of K will be denoted by exp AT.More generally, FaK is a norm exposed face of K if F = KC\H.In general, exp K <= ext K, with equality holding, for example, in the case that K= U* fl is the unit ball of the predual U* of a JBW*-triple U [5, Proposition 4].The equality 'exp Z x = ext Z / for a normed space Z is a physically meaningful assumption and is analogous to one of the pure state properties of Alfsen and Shultz [1] (cf.Definition 2.8 below).

Facial structure of the dual ball of the spin factor
The spin factor, or 'spinors', occur naturally in different areas of mathematics and physics.Several (equivalent) definitions exist and are used.In this paper, the definition of spin factor will be based on algebraic properties of a natural basis called a spin grid, which is shown in [3] and [10] to generate the spin factor linearly and topologically.It is known (cf. for example [9]) that the complex spin factor can be represented as a norm closed subspace of the bounded operators on a complex Hilbert space which is stable under the operation a*-^aa*a and which generates the Clifford algebra (CAR algebra) as a C*-algebra.The spin factor has the structure of a complex Banach Jordan *-algebra (JB*-algebra) and is thus an example of a JB*-triple (cf.[5]).It is the complexification of the real spin factor, which is a JB-algebra.
We begin by recalling the recent description (see [3]) of the algebraic structure of a spin factor or so-called Cartan factor of type 4, which we shall refer to as a concrete spin factor.Although the concrete spin factor can be realized as a norm closed subspace of #($?), our approach will be via spin grids.This approach simplifies our calculations and makes possible an elementary description of the facial structure of the unit ball of the dual space.DEFINITION 1.1.A Banach space U over C is said to be a JB*-triple if it is equipped with a continuous triple product (a, b, c)*-* {abc} mapping U x U x U to U such that (i) {abc} is linear in a and c and conjugate linear in b; (ii) {abc} is symmetric in the outer variables, that is, {abc} = {cba}; (iii) for any x e U, the operator 6(x) from U to U defined by d(x)y = {xxy}, y eU, is hermitian (that is, exp it6 is an isometry for all real t) with non-negative spectrum; (iv) the triple product satisfies the following identity, called the 'main identity': (v) the following norm condition holds: A non-zero element a in a JB*-triple is called a tripotent if a = {aaa}, and we say that a is a minimal tripotent if it cannot be written as a sum of two orthogonal tripotents, where elements a and b are said to be orthogonal if {abx} =0 for every x e U.For example, in the above-mentioned representation of a spin factor as bounded operators on a Hilbert space $?, the triple product is given by {xyz} = \{xy*z + zy*x) for x, y, z eB(%€), so the tripotents are precisely the partial isometries, and orthogonality of tripotents corresponds to orthogonality of operators.We shall not make use of this representation anywhere in this paper.
We are now going to construct the triple product and norm in a concrete spin factor in an elementary way by using properties of a spin grid.The assumptions that we make in the following definition are known to hold for a spin grid in a spin factor [3]. DEFINITION 1.2.Let / be an index set of arbitrary cardinality.A basis, or spin grid is a collection ^of linearly independent elements {u i} M/} ie/ or {u 0> u h u,} /e/ .Define a triple product {uvw} for elements of the basis by 1. {uuu} = u for all M 6 ^ (the basis will consist of tripotents); 2. for distinct non-zero i, j , {UiUiUj} = {UiUiUj} = \llj, {UjUjUi} = {UjUjUi} = \u h {UiUiUj} = \u jf and {«,«,£,} = \ui (iij will be colinear with u, and with u jt and u, will be colinear with w ; ), and (the quadruple («,, w,, u if Hj) will be an odd quadrangle); 3. in the case where u 0 exists, for each i =£ 0, (M 0 governs M, and «,), and 4. {uvw} = {wvu} for all u, v, w e % 5. all other products {uvw} where u, v, w are from the basis, are zero; in particular, for each i ¥= 0, {ufiiu} = 0 = {HiUjU} for all u e (u h ^ will be orthogonal).It follows from these properties that the set of all scalar multiples of basis elements is closed under the triple product {•,-,•}.Hence, the triple product {•,-,•} can be extended to the real or complex span of the basis elements to be linear in the outer variables and (in the complex case) conjugate linear in the middle variable.
Define an inner product on sp ^ by If / is finite with n elements, the dimension of % is In or In + 1.Otherwise, % is infinite-dimensional, and (3) is then a convergent sum.The norm on the spin factor ^ which will make it into a JB*-triple is not the Hilbert space norm used in the definition, since that norm does not satisfy (2).In order to define the correct norm, which will be equivalent to the Hilbert norm, we introduce the following concepts.
Define a conjugation # on basis elements by uf = u it uf = u h and u$ = u 0 , and extend this to the linear span in a conjugate linear way.
The connection between the triple product, inner product, and conjugation is given by For each element a of <#, the notion of determinant is defined by where convergence is in ||-|| (M 0 m a y n ot exist).The space % will now be denoted by U. The Banach space dual U* (which is the same as the predual U* since U is reflexive) can be identified with U via the conjugate linear (Riesz) map JZ: U* s q>*-*a <p = Ji{(p) e U, where In the rest of this section, by abuse of notation, we shall sometimes use the same notation for <p e U* and a v e U. Thus (b, x) = (x \ b) forbe U*, x eU, and if a is given by (10), then n~\a) = 2 aiJt~\ui) + 2 fli^" 1 ^.-) + a o jt~l(u o ), deta = det(jr 1 (a)), and the inner product (3) transferred to ^ satisfies Since the spin factor U is a JBW*-triple, we have from [5, Propositions 4 and 8] a one-to-one correspondence between tripotents and norm exposed faces in the unit ball of the predual U*, in which minimal tripotents correspond to minimal faces, that is, extreme points of the unit ball.We also have the following corollary.We now use (13) to describe the dual ball of the real spin factor.PROPOSITION 1.8.// U is a spin factor over U, then the unit ball U* tl of the predual £/* is the intersection of two cylinders with base a real Hilbert ball.The dimensions of these Hilbert spaces differ by at most one.More precisely, U* x is the unit ball of the l°°-sum of these two Hilbert spaces.
Since by Proposition 1.5 any tripotent v in U is either minimal or a sum of two orthogonal minimal tripotents, each non-trivial norm exposed face of Z, is either a single point (an extremal point of Z x , corresponding to a minimal tripotent) or a (so-called) face of rank 2. For any rank-2 face defined by a tripotent v, define an element of F v called the centre of F v by % = \JZ~1{V).Note that by our convention, we could write % = \v and still have v e U and § e i v> c= Z.We shall usually write F^ for F v , since § also determines the face.Later in this section we shall show that F ? is affinely isomorphic to the unit ball of a real Hilbert space in such a way that the centre § corresponds to the origin.This should now be compared with Theorem 3.8.DEFINITION 1.9.Let Z be the predual of a concrete spin factor.An element £ 6 Z is said to be unitary if ||£|| z = 1 and § # = A£ for some A e T. For O ^a e Z define the phase £(a) of a to be detfl/|deta| ifdeta^O, 1 if det a = 0.
If deta=£0, applying (7) to n(a), we have a e F V C\F e+f which is therefore a rank-2 face contained in each of F v and F e+f .Therefore F v and F e+f both equal this intersection, and hence are equal and v = e+f.Now, using ( 8) and ( 9) applied to Jt(a), we obtain Similarly, v,-= e t +fi = a t + Aa, and v 0 = e 0 + f 0 = d 0 + Xa 0 , The following proposition is the facial decomposition of an arbitrary element of the predual of a spin factor.Proo/.Let A = £(a) and set £ = \{a + ka*) and h = \{a-Xa*).Then § # = A£ and h u = -Xh.This proves (i). Since Similarly, det/i = -2?(«)II^Hi and €:(/») = -tifl).This proves (ii).
This proves the uniqueness.
Let A = ^(a).Then In Proposition 1.11, if ||a|| z = 1, then a and £ belong to the face F § containing a whereas h = a -£ is 'parallel' to F.Moreover, by this proposition, the face F% is a real Hilbert ball defined by

Atomic facially symmetric spaces
In this section we shall recall some basic facts about facially symmetric spaces from [7] and prove several new results which supplement those in [7] and are needed in the present paper.We then establish some properties of atomic facially symmetric spaces, including a Hilbert space structure on those atomic facially symmetric spaces which satisfy two geometric axioms.
In this and the next section, Z will be a weakly or strongly facially symmetric space over the real or complex field.In § 4 we shall restrict our study to complex spaces.
Let Z be a real or complex normed space.Elements f, g eZ are orthogonal, notation /<>£> if II/ + &II = ll/~£ll = 11/11 + II&II-A norm exposed face of the unit ball Z x of Z is a non-empty set (necessarily not equal to Z x ) of the form F x = {/ e Z x : f(x) = 1}, where x e Z*, \\x\\ = 1.An element ueZ* is called a projective unit if ||M|| = 1 and (u, F < j?)=0.We use 8F and °U to denote the collections of norm exposed faces of Z x and projective units in Z*, respectively.
Motivated by measuring processes in quantum mechanics, we define a symmetric face to be a norm exposed face F in Z x with the following property: there is a linear isometry S F of Z onto Z, with S F = I, such that the fixed point set of S F is (sp F) 0 F° (topological direct sum).A real or complex normed space Z is said to be weakly facially symmetric (WFS) if every norm exposed face in Zj is symmetric.For each symmetric face F we define contractive projections P k {F), with k = 0, 1, 2 on Z as follows.First P X (F) = \{I -S F ) is the projection on the -1 eigenspace of S F .Next we define P 2 {F) and P 0 (F) as the projections of Z onto sp F and F° respectively, so that P 2 (F) + P 0 (F) = | ( / + S F ).These projections were called generalized Peirce projections in [6] and [7].A generalized tripotent is a projective unit u e°U with the property that F: = F u is a symmetric face and S F u = u for some choice of symmetry S F corresponding to F. In this paper we shall call such u geometric tripotents, and the projections P k (u) will be called geometric Peirce projections.
We use ^ST and SfSF to denote the collections of geometric tripotents and symmetric faces respectively, and the map ^9 « ^F u e ^ is a bijection [7,Proposition 1.6].For each geometric tripotent u in the dual of a WFS space Z, we shall denote the geometric Peirce projections by P k (u) = P k (F u ), for k = 0, 1, 2. Also we let (u).A symmetry corresponding to the symmetric face F u will sometimes be denoted by S u .
Elements a and b of U are orthogonal if one of them belongs to U 2 (u) and the other to U 0 (u) for some geometric tripotent u.Two geometric tripotents u and v are said to be compatible if their associated geometric Peirce projections commute, that is [P k (u), Pj(v)] = 0 for k, j e {0, 1, 2}.
A contractive projection Q on a normed space X is said to be neutral if for each %eX, \\Q%\\ = ||£|| implies Q£ = £.A normed space Z is neutral if for every symmetric face F, the projection /^(F) corresponding to some choice of symmetry S F , is neutral.
If Y is a closed subspace of a normed space Z, the collections of norm exposed faces and symmetric faces in dY u will be denoted by 9> Y and SfSFy respectively.We define % y , <33V similarly.
A WFS space Z is strongly facially symmetric (SFS) if for every norm exposed face F in Z x and every y e Z* with || v|| = 1 and F c:F y , we have S£y =y, where 5/7 denotes a symmetry associated with F. Let Z be a strongly facially symmetric space and suppose that u, v e '^S'z-If F u cF v , we write u^v.
In a neutral strongly facially symmetric space Z, every non-zero element has a polar decomposition [7,Theorem 4. (17) Another property of the polar decomposition is that if / e Z k (w) for some w e < 3&', then v(f) e U k (w).This follows from the uniqueness of the polar decomposition as we now show.
LEMMA 2.3.Let Z be a neutral WFS space, and let w and u be a pair of orthogonal geometric tripotents.Then and therefore Let M denote the collection of minimal geometric tripotents of U, that is, M = {v e C S3'\ U 2 (v) is one-dimensional}.In the setting of a JBW*-triple and its predual, there is a one-to-one correspondence between minimal tripotents and extreme points of the unit ball of the predual.Moreover, these extreme points are all norm exposed (cf.[5,Proposition 4]).The following proposition is a generalization of this correspondence.Proof.If / e exp Z x , then {/} = F v for some unique v e ( §S', which we denote by rj(f).Since spiv, is one-dimensional, U 2 (v) = (spF v )* is one-dimensional, so that v eM.
Conversely, let veM.Then spF v is one-dimensional, so that F v is a point.Hence, rj is surjective.
To prove the last statement, note that since P 2 (v)*x = k(x)v and P 2 (v)g = p(g)f for some scalars k(x), fi(g), we have DEFINITION 2.5.A normed space Z is said to be atomic if every symmetric face of Z x has an extreme point.
Let 3> denote the collection of indecomposable geometric tripotents of U, that is, $ = {v e ^ZT: u e In general, M <=.$>, and we shall show in Proposition 2.9 below that under a mild assumption, equality holds.DEFINITION 2.6.A neutral SFS space Z is said to satisfy the axiom (PE) (for It is easy to see that being atomic is equivalent, in SFS spaces satisfying (PE), to the assertion that every u e WST is greater than or equal to some v in M. Proof Observe first that if Z is an atomic SFS space, then the set {x eU: (x,f) = \\x\\ * 0 for some / e ext Z x ) is norm dense in U. Indeed, by the Bishop-Phelps theorem for the bounded closed convex set Z x [2, p. 45], the set of elements xe U := Z* which assume their norm on some element/ e Z l7 that is, (x,f) = \\x\\, is norm dense in U.But for all such x, F x , nxn =£ 0 so there exists cp e ext Z x H F x/iixn with (x, op) = ||JC||.
Let K denote coextZj.If K=tZ x , let / eZ x ,f $K.Choose y € U and real numbers c,c + e, such that
We now introduce the principal axioms, which are geometric and physically significant properties of the state space of a physical system.

VTgJ=(g\f),
where in the case of complex scalars, the bar denotes conjugation.
For example, in a concrete spin factor, the transition probability coincides with the restriction of the inner product (12) to extremal elements.PROPOSITION 2.9.Let Z be an atomic neutral SFS space satisfying (PE).Then M = $.If furthermore Z satisfies (STP), then the map n\ sp ext Z x -+ sp M defined by is well-defined and a conjugate-linear bijection.
Denoting this extension also by JZ, we find that the scalar-valued map is a continuous symmetric sesquilinear form.
Proof.Let u e i and let q> be an extreme point of F v .Then <p is an extreme point of Z x and so {q?} = F u for some ueM, by Proposition 2.4.Since cpeF v> F u c= F v so u =£ v and therefore u = v, that is, sp F v is one-dimensional, and v e M. Thus 3 = M.
By (STP), for all ip, q> x , ..., cp n eextZ, and scalars z,, ..., z n , Therefore V e e x t z i ^> 2 S/v(Vy) = °-Finally, by (STP), the map (• | •) is symmetric, that is, (f \g) = (g\f), on the linear span of extreme points, and since it is obviously sesquilinear and continuous, the last statement follows.
As a consequence of the sesquilinear form just introduced, we are now able to handle the rank-1 case.DEFINITION 2.10.A normed space Z is said to be of rank 1 if no two non-zero elements of Z are orthogonal.
In an atomic neutral SFS space of rank 1, satisfying (PE), every geometric tripotent is indecomposable, and hence minimal.Thus, every element/in such a space is a multiple of an extreme point of Z\.It follows that the sesquilinear form (• | •) is positive definite and moreover, if / e Z is non-zero, then f/\\f\\ z is an extreme point and so has norm 1 in the inner product space determined by < • | •).Therefore, ||/||z= H/H2 and Z is a Hilbert space.Furthermore, for ||/|| = 1, by Proposition 2.4, P 2 (f) is the orthogonal projection onto the span of / in this Hilbert space.
Two geometric tripotents u, w are said to be colinear if each of them belongs to the geometric Peirce 1-space of the other, that is, u e U^w) and w e U x (u).Thus, in a rank-1 space, since all geometric Peirce O-projections are zero, colinearity of v(f) and v(g) is equivalent to P2(f)g = 0, that is, orthogonality in the Hilbert space structure.Therefore orthonormal bases in the Hilbert space structure correspond to maximal families of mutually colinear geometric tripotents.Hence we have: COROLLARY 2.11.Let Z be an atomic neutral SFS space satisfying (PE) and (STP).Assume that Z is of rank 1.Then Z is linearly isometric with a real or complex Hilbert space.Moreover, there is a one-to-one correspondence between orthonormal bases in this Hilbert space and maximal families of mutually colinear geometric tripotents.

Rank-two faces in facially symmetric spaces
We now begin the study of rank-2 faces, culminating in Theorem 3.8 (the Hilbert ball property).This property will be a basic tool in the following section.DEFINITION 3.1.Let v and v be orthogonal minimal geometric tripotents in a neutral strongly facially symmetric space Z.The norm exposed face F 0 + c determined by the geometric tripotent v + v will be called a face of rank 2. Let v = v(f) and v = v(f) for orthogonal extreme points /, / of Z x .Then £: = \(f +f) is called the centre of F v+ij and we shall write F u+C = F §.
It will be shown in Theorem 3.8 that F u + 0 is a Hilbert ball with centre £.The following proposition justifies the terminology 'rank 2', that is, meaning that there are no more than two orthogonal extreme points in such a face.If the face represents the state space of a quantum mechanical system, the proposition implies that for such systems no observable can assume more than two distinct values.PROPOSITION

Let v and v be orthogonal minimal geometric tripotents in an atomic neutral strongly facially symmetric space Z, and assume the axioms (PE) and (STP). Let v = v(f) and v = v(f) for orthogonal extreme points f,f of Z x . If p and o are orthogonal elements of F u+0 , then v(p) + v(o) = v + v, p and o are extreme points, and p+o=f+f
Moreover, each norm exposed face of Z x , which is properly contained in F v+d , is a point, and if p is an extreme point of By (STP), a = p(v) =f(v(p)) = a and 1 -a = b, so that Similarly and Since p<C>o, v(p) + v(o) e ^^" and therefore has norm 1.Since and since p2(v)* is contractive, we have tt+y^sl and similarly 2-a -y s l .Hence ^ + 7 = 1 and We can now easily complete the proof of the proposition.Since Z is atomic, there exist extreme points g and g such that F vig) c F u(p) and F vii) c F u(a) .By the previous paragraph, v + v = v(g) + v(g) ^ v(p) + v(o) ^ v + v, and p and o are extreme points.
If F w cF v+ v is not a point, then it contains an extreme point <p, which by assumption is norm exposed.Since v(q))^w, we have F w _ v(<p) <£>F viv) .Thus by the previous paragraph, F w = F v+Ti .
Finally, v + v -v(p) is the unique minimal geometric tripotent orthogonal to v(p).Therefore, JZ~\V + v-v(p))=f+f -p is the unique extreme point orthogonal to p, where n is defined in Proposition 2.9.
To proceed further, we need to replace the property (PE) by an apparently stronger property (FE).It is very likely that (PE) is equivalent to (FE) in facially symmetric spaces, since (FE) holds in the case of the predual of a JBW*-triple [4], but we have been unable to prove this.DEFINITION 3.3.A neutral SFS space Z is said to satisfy property (FE) if every norm closed face of Z x different from Z, is a norm exposed face.
An important consequence of this axiom is the following Krein-Milman type result.
PROPOSITION 3.4.Let Z be an atomic neutral WFS space which satisfies the axiom (FE), and let u be a geometric tripotent.Then Proof.Let K:= F u and K x :=co ext F u czK.If K x =£ K then there is an x e U 2 (u) and ceU such that s\ip3i(x,K l )<c<supm(x,K).The set G := F u fl {(p e Z 2 (u): $l(y, cp) = r} is a non-empty norm closed face in F u , and is therefore norm exposed and symmetric by our assumptions on Z. Since Z is atomic, G has an extreme point, which is then automatically an extreme point of F u .Because (21) holds with y in place of*, this extreme point is not in K x , a contradiction.COROLLARY 3.5.Let Z be an atomic neutral strongly facially symmetric space satisfying (FE) and (STP), and let F^ be a rank-2 face.Then F% - § is a symmetric convex set.
Proof.For any g e F % , define S(g - §) = -(g - §).From Proposition 3.2, for any extreme point p e F%, there exists an extreme point pe F%, orthogonal to p such that S(p -£) = p -£.By the proposition, S is a symmetry of F% -£.
The following theorem is the main result of this section.It gives the structure of rank-2 faces.We shall use the following lemma in its proof.LEMMA  In the following corollary, we obtain an abstract analogue of the facial decomposition ((i), (iii) and (iv) of Proposition 1.11).In the abstract setting of Lemma 3.6 a centre of any rank-2 face F will be called a unitary element of Z.For example, \{f + / ) is a unitary if / and / are any two orthogonal extreme points of F.  (l/V2)P|| y = (l/V2)P|| z .

(Hilbert ball property). Let v and v be orthogonal minimal geometric tripotents in an atomic neutral strongly facially symmetric space Z, and assume the properties (FE), and (STP). Let v = v(f) and v = v(f) for orthogonal extreme points f,f of Z x . Then F § - § is the unit ball of a real subspace Y of Z on which the sesquilinear form (20) is a real inner product whose associated
We complete the proof by computing the inner product, thereby showing it to be real and to satisfy (24).For h l ,h 2 eY, with hj/\\hj\\ Y = <pj -% for extreme points xpj eF ? ) we have and therefore This proves (24) and by ( 15) and ( 16), the inner product is real.Finally, for | | COROLLARY 3.9.The unit ball (in Z) of the real span of F u+C is a cylinder with base the Hilbert ball Y. Precisely, Proof.Let F denote F u+C .Then <p e sp R F if and only if for some a, b^O and a, r e F, where, with 1/; = a(x-^) -6 ( a -£), we have set /i = V/II'V'II* j8 = ||Vll» and a = a -b.Moreover, with p ± = %±h, The following notation will be used in the sequel.The real Hilbert space Y determined in Theorem 3.8 by the face F u + 0 will be denoted by ^fu+i5 or by $? §, since it is uniquely determined by £ or by v + v.The Hilbert ball F u + 0 -£, which is the unit ball of this Hilbert space, will be denoted by B v+i or B%.An orthonormal set in $? § will be denoted by {e a } or {e x , e 2 ,...}.DEFINITION 3.10.A neutral strongly facially symmetric space is said to be of type I 2 if it is of the form Z 2 (F), where F is a face of rank 2 (cf.Definition 3.1).
Our main theorem, Theorem 4.16 below, states that an atomic neutral strongly facially symmetric space over C of type I 2 which satisfies (FE) and (STP) is linearly isometric to the predual of a spin factor.
The following remark follows from the fact that the inner product above is determined globally by use of the mapping n.REMARK 3.12.Formula (24) has the following physical interpretation.Consider the state space of a spin-* particle, where we are concerned only with the directions of spin, which are unit vectors in IR 3 .Identify the Hilbert space 2i? with U 3 .If spin is measured in the direction p -£ and is positive, then the pure state p represents the state of the particle after the measurement.In this case, by (24), the transition probability between states p and T is given by <p | T> = p(v(x)) = i(l + (P " 11 T - §)) = i(l + cos 6) = cos 2 id, where 6 is the angle between the two directions.This is consistent with the known behaviour of spin-!particles in quantum mechanics.(This implies also that a O r and that the decomposition is unique.)Hence ext(sp R F)x = F U -F.

The complex span of a rank-2 face
In order to describe the complex span of a rank-2 face F%, we shall construct, from an orthonormal basis of $? § and the centre £, a natural basis analogous to the dual of a spin grid in a concrete spin factor.This construction is entirely similar to the method of obtaining the spin grid in a concrete spin factor.The main difficulty here, however, is to show that the natural correspondence between the concrete and the constructed grids extends to an isometry.This task, which requires most of this section to carry out, is accomplished by a reduction to the finite-dimensional case.Since the odd and even dimensional spinors behave differently, the cases of dimension 2 and 3 both need to be treated first.This is done in Theorems 4.6 and 4.8.
We assume in the remainder of this paper that Z is a linear space over the complex field C. We shall make use of the main result of § 3, namely Theorem 3.8.We therefore make the following standing assumption: Z is an atomic neutral SFS space satisfying (FE) and (STP). (25) We begin by considering an arbitrary pair of orthogonal unit vectors {e x , e 2 } in the Hilbert space 2i? § defined by a rank-2 face F §, and describe the real span y o := sp R {e u e 2 , £} c Z (26) of these two vectors and the centre £ of the face.By using the main result of § 3, in the following proposition we construct a part of the dual spin grid and show that the extreme points of F § lying in Y have the same description as their analogues in the dual of a concrete spin factor.Proof.By Corollary 3.7, / x and f t are extreme points of the face F M .Since their average is the centre, by Proposition 3.2 there is an element \j> of the face F% such that ^O / i a n d /i + ^ =/i +/i-Therefore ip^fi, which proves that /i and /j are orthogonal.
Since the unit ball of any normed space is built up from its faces, to show that a map is an isometry, it is necessary to obtain information about faces different from the original one.A natural face to consider first is the one with centre h arising in the previous proposition.Note that by (33), h is unitary and hence is the centre of a face.The next lemma gives a partial result about this face.The full description of this face will be given in Proposition 4.4.
It remains to prove (41).We first show that there is some q> e F u with /3 # 0. By (39), with h' e Zx(u) D Z^O) D F u .Now suppose that /3 = 0 for all (p e F M .Then for all such cp, and S w (p = aUr +/i) + S w h', so that <p -S w (p = h' -S w h'. But 5^/1' €Z^Vx), since S^/i' G F U C Z 2 ( ^I + V\) implies S w h' = (S w h', v l )f l + (S w h' t O / t + PiO and, for example, (S w h', fi) = (/i', 5*u t ) = (h', v x ) =0.Therefore, P x {w)h' = PI(UI)PI(W)/I', and we have Since <J5 e F u is arbitrary, this implies P l (w)P 2 (u) = P x (v l )P x (w)P 2 (u).Now/, € Z 2 («) by Lemma 4.2, so that .Pi(w)/i = P\(v x )P l (w)f u and thus a contradiction.Thus (41) is true for some q> eF u .
We can now complete the proof of (41).We start with a cpeF u , with /J^O.Writing (again)
The following corollary reduces to Proposition 4.1 in the case where ju = 1.For arbitrary \i, the 'phase' of F ( M / I + ^I)/2 remains equal to 1.The centre h{\if\ + p-f x ) remains in the span of £ and e x , and so is orthogonal to e 2 .Since 2 (ju/, -Pfi) is anti-self-adjoint and orthogonal to e 2 , it provides a natural candidate for the second vector (cf.(34)).and with §" := £(JU/, + Jif x ), The following theorem characterizes the spin factor of dimension 3.
THEOREM 4.6.Let Z be an atomic neutral SFS space satisfying the properties (FE) and (STP).Let {e t , e 2 ] be an orthonormal set in the Hilbert space $? § defined by a rank-2 face F%.Define f t = e x + f, f\ = % -e x , and let h = e 2 .Then Y := spcl/x, fi, h) is linearly isometric with S 2 (C)*, the predual of the JBW*-triple of two-by-two symmetic complex matrices.In particular, if Z = Z 2 {F^) and %£% has dimension 2, then Z = S 2 (C)#.( The following example illustrates Proposition 4.7 and will be used in Lemma 4.9.The -1 in the definition of f 2 is due to the fact that spin grids contain odd quadrangles [3,Corollary,p. 313].

Proof
EXAMPLE.Let Z = M 2 (C)*, let £,-,-be the standard matrix unit in A/ 2 (C)*, and let Then F% is the state space ff of the C*-algebra M 2 (C).We claim that {e,, e 2 , e 3 } is an orthonormal set in $?g, where gj = \{j x -f x ), e 2 = h, e 3 = \i(f 2 +/ 2 ).For this it is required to show that e x + | , e 2 + %, e 3 + % all belong to F^ and each pair in this family has transition probability \.Now and Therefore each of these vectors belongs to F^, and moreover, .0oJU iJJ" 2 ' and similarly for the others.
We now recall the description of the state space 5^ of the C*-algebra A/ 2 (C).Let C denote the centre £ of the face Sf.Since A/^C)* is the predual of a JBW*-triple, it is an atomic neutral strongly facially symmetric space satisfying (FE) and (STP), so by Proposition 4.7, Let P = JIM 1 2 (C)A P )> Q = JI: MI(C)XQ) be the extreme points in M 2 (C)* tl corresponding to the elements P, Q respectively.Then j(P + Q) is the centre of the norm exposed face F P+Q = S/ > and the Hilbert ball where E 2 , E' 3 are elements of ffl c .By (56), the map v takes this centre and basis into the centre 2 (p + q) of F § and an orthonormal set in ffl% of the form {\{p-q),e' 2 ,e' 3 }, where e 2 , e' 2 are elements of 3i?g.For any ju eT, by Corollary 4.5 and Remark 3.11, v takes the orthonormal basis {\{liP -JiQ), E 2 , E'Ji in into the orthonormal set -Jlq), e' 2> e' 3 } in Therefore, as above, v maps F^+JXQ onto F^+j^ n Y' in such a way that extremality and orthogonality are preserved.
A special case of the following lemma, which concerns a concrete spin factor, is needed in the proof of Theorem 4.8.To show that 5 M (/) is an isometry on Z 2 (F g ), let / be an extreme point of F% orthogonal to / such that £ = 2 ( / + / ) , and pick an orthonormal basis {ej} jel for $f § containing e x = \{f -/ ) .For <p e sp c {e y , | } y e / we have with z, w £ C and a jt ftelR.
Note that this definition is invariant under surjective isometries (the 'isomorphisms' in the category of facially symmetric spaces).Also, the dual basis of a spin grid in a concrete spin factor (cf. [3]) is an example of a dual spin grid.
Trivial modifications complete the proof in the other case.Now which is norm dense in Z = Z 2 (F%) by Remark 4.12.Therefore, the map T defined by Tgj=fj, Tgj=fj extends linearly to a map from a dense subspace of ^ to a dense subspace of Z.We shall show that T is an isometry.
For notation's sake let % and %' denote the dual spin grids chosen above for Z and <#* respectively.Let g e sp c < §".We shall show that ||7g|| z = \\g\\<e.which will prove the theorem.First of all, if g is a linear combination of g } and g y for a fixed /, then by orthogonality of g jt g y and of fjjj, it is clear that ||7g|| 2 = ||g||«.. To handle the other cases, we shall prove, by induction on n, the following assertion: If n 2=2 and if {g h |,}, e y is any dual spin grid for <#, then T is an isometry from sp c {g!, g u ..., g n , g n } into Z.
For n = 2 this is proved in Theorem 4.8.Suppose that n > 2 and let g belong to spc{gi>gi>~->gn>gn}' Write g = h + k where h = a x g x + d x g x + a 2 g 2 + a 2 g 2 and k = £j-3 (ajgj + ajgj).Since sp c {gi, gi, gi, gi} is isometric to A/ 2 (C)*, /i has a spectral decomposition (cf.Proposition 1.5) h = t x p x + t 2 p x where t u t 2 e C and by Lemma 4.9 (with Z = <#*) and Corollary 4.11, p, e ext F^P +fi Q for some orthogonal minimal partial isometries P, Q in ^ (corresponding to orthogonal minimal projections in M 2 (C)) and some ^ e T.
Choose p 2 , p 2 such that {p x , p x , p 2 , p 2 } form a dual spin grid of Y".Since p x , Pi € F^p+jiQ, by Lemma 4.15, {p x , p x , p 2 , p 2 } U %' Q U %\ is a dual spin grid of (a | b > = 2 Oibi + 2 aft + 2a o b o , (3) where a = £ fl,M, + £ a,w, + a o u o and b = £ 6,u, + £ 6/M, + b o u o are two elements of sp <£ DEFINITION 1.3.The completion of sp ^ with respect to the norm |||| 2 determined by the inner product (3) is called a (concrete) spin factor, and will be denoted by %.

PROPOSITION 2 . 1 .
Let Z be a neutral SFS space and let v e ^ST.Then for k e {0,1,2}, Proof Let u 6 ^^" z fl U k (v).By [7, Theorem 3.6, Proposition 4.1], Z k (v) is neutral and SFS.Hence to prove that u e ^^^^ we need only show that

PROPOSITION 2 . 4 .
If Z is WFS, then there is a bijection 77 of exp Z x onto M. If Z is neutral and SFS, then rj(f) = v(f), the geometric tripotent occurring in the polar decomposition off.For v = rj(f) e M, with f e exp Z x , P 2 (v)*x=f(x)v forxeU, and P 2 (v)g = g(v)f for g e Z.

DEFINITION 2 . 8 .
Let / and g be extreme points of the unit ball of a neutral SFS space Z.The transition probability of/and g is the numbertf I «>:=/("(*))•A neutral SFS space Z is said to satisfy 'symmetry of transition probabilities' (STP) if for every pair of extreme points/, g e ext Z l5 we have is a minimal geometric tripotent.Proof We show first that if p and o are orthogonal extreme points of the rank-2 face F w+5 , then v(p) + v(o) = v + v. Since p e F u+C , we have by Lemma 2.3, for some h eZ x {v) C\ Z x (v).Moreover, by (16), a:= p(u)s=0.By Lemma 2.3, since v(p) e U 2 (v + v), v(p) = av + bv + v x , for some v x e U x {v) n U x (v), and scalars a, b.
Fv + z = U {[£, P\-P e ext F u + 0 }.(22) Proof.By Corollary 3.5, F% -% is the closed unit ball Y x of a real space Y c Z with respect to the norm ||-|| y given by the Minkowski functional of Fg -| .Let <p be an arbitrary point of F §, and let p be the point where the ray containing q> and emanating from £ leaves Fg, that is, with q>, = tcp + (1 -1)% for t ^ 0, p = <p, 0 for some f 0 (necessarily greater than or equal to 1) with q>, e F| if and only if f =s t 0 .By the definition of the Minkowski functional, ||p -£|| y = 1.Let <I > be a continuous linear functional on Y such that, with Y o equal to the open unit ball of Y, <J>(Y 0 )<c:=<D(p-£), (23) and let / be the hyperplane in Y determined by c, that is, J = {oeY: <J>(CT) = c}.Then p -£ e G : = / D (F § -£) and G is a closed face of F ? - §. Thus p e G + § and G + § is a closed face of F §. Therefore, G + § is a closed face of Z t , which by our assumption, is norm exposed.By Proposition 3.2, either G + § = {p} so p is an extreme point, or G + £ = F §.In the latter case, Fg -£ = G = / D (F § -£) c= / , contradicting (23).Thus p is an extreme point, and q> e [£, p].

REMARK 3 .
11. Let F% and F v be faces of rank 2 so that $? § and 2£, are defined.If a, b € Sfcg D S^rj, then the inner product (a | b) is the same whether computed in %$ or in $£,.

COROLLARY 4 . 5 .
With the notation and assumptions of Proposition 4.4, for each p e l , {5(^/1 -ju/i), e 2 ) is an orthonormal set in 2e (fi fi+M)>2(42) Let M = S^C) and define the map K: M*->Z by ([y *([y where a, (3, y e C. The map K is a linear isomorphism of M onto Y.We shall show the following: t tl )cextZu (44) in extM*,!=> K(R)O*(S).

LEMMA 4 . 10 .
Let F^ be a rank-2 face with %* = % which spans the predual Z of a concrete spin factor.For any extreme point q)eZ x , there exist jtteT and orthogonal extreme points f, f in Fg such that cp e Proof.Since q) is an extreme point, r) := \{cp + <p # ) is the centre of a rank-2 face containing q>.Now 77 and % are vectors in the Hilbert space Z and the vector y:= rj-2(»7 I §) § is orthogonal to | .If y = 0, then |2(r/| §)| = l.But 2(rj I §) =2di((p I £) and therefore 17 = ± §.By definition of face, <p e F ±g , so the lemma follows with ju = ±1.

and fi = 2 2 (-^fli + ^1«,)> 1 2
) are extreme points in F^ with transition probability |.The first vector is ju/i which is obviously in ext F^.The second vector is h + £(#/j + ju^) which is in ext F h by (37).Since F^ =/^U l+M c 1 , this vector lies in F^.Moreover, We now apply Proposition 4.1 to the data in (42).By (28) we have (43).

)
Suppose that we have proved (44) and (45).Every ReM* has the form R = X X R X + k 2 R 2 with /?! and R 2 orthogonal extreme points in M* and X lt k 2 e C. Now apply Proposition 4.1 to the orthonormal set {£, ie 3 } in the Hilbert space df€ e2 .Then, with f 2 :=ie 3 + e 2 and f 2 :=-ie 3 -e 2 , (49) f 2 and -ji are orthogonal extreme points of F e2 , and (47) is satisfied.From this (48) is immediate.

which is an orthonormal set, say {e x , e 2 , e 3 } in the Hilbert space $C% determined
By the above example, the state space Sf of the C*-algebra M 2 (C) is a norm exposed face F c with centre C = 2 (E n + £22) and &-C is a Hilbert ball with orthonormal basis \{E n -E 22 ), k(E n + E 2x ), \i{E X2 -E 2x ).(ext Sf) c ext F% and (by Theorem 3.8 and Proposition 4.1) v maps orthogonal elements of £f into orthogonal elements of Z. Thus p, q are orthogonal minimal geometric tripotents.