Mathematische Annalen manuscript No. (will be inserted by the editor)

State spaces of JB^{∗}-triples^{⋆}

Matthew Neal · Bernard Russo

the date of receipt and acceptance should be inserted later – \begin{matrix}c \\ ◯ \end{matrix} Springer-Verlag 2003

Abstract. An atomic decomposition is proved for Banach spaces which satisfy some affine geometric axioms compatible with notions from the quantum mechanical measuring process. This is then applied to yield, under appropriate assumptions, geometric characterizations, up to isometry, of the unit ball of the dual space of a JB^{∗}-triple, and up to complete isometry, of one-sided ideals in C^{∗}-algebras.

Mathematics Subject Classification (2000): 17C65, 46L07

Introduction

The Jordan algebra of self-adjoint elements of a C^{∗}-algebra A has long been used as a model for the bounded observables of a quantum mechanical system, and the states of A as a model for the states of the system. The state space of this Jordan Banach algebra is the same as the state space of the C^{∗}-algebra A and is a weak^{∗}-compact convex subset of the dual of A. With the development of the structure theory of C^{∗}- algebras, and the representation theory of Jordan Banach algebras, the problem arose of determining which compact convex sets in locally convex spaces are affinely isomorphic to such a state space. In the context of ordered Banach spaces, such a characterization has been given for Jordan algebras in the pioneering paper by Alfsen and Shultz, [1].

After the publication of [1], and the corresponding result for C^{∗}- algebras [4], there began in the 1980s a development of the theory of JB^{∗}-triples which paralleled in many respects the functional analytic aspects of the theory of operator algebras. JB^{∗}-triples, which are characterized by holomorphic properties of their unit ball, form a large class of Banach spaces supporting a ternary algebraic structure which includes C^{∗}-algebras, Hilbert spaces, and spaces of rectangular matrices, to name a few examples. In particular, most of the axioms used by Alfsen and Shultz were shown to have non-ordered analogs in the context of JB^{∗}-triples (see

Denison University, Granville, OH 43023, e-mail: nealm@denison.edu University of California, Irvine, CA 92697-3875, e-mail: brusso@uci.edu ⋆ Both authors are supported by NSF grant DMS-0101153

[13]). By the end of the decade, a framework was proposed by Friedman and Russo in [16] in which to study the analog of the Alfsen-Shultz result for JB^{∗}-triples. A characterization of those convex sets which occur as the unit ball of the predual of an irreducible JBW^{∗}-triple was given in [19] (see Theorem 1.1 below). Since JB^{∗}-triples have only a local order, the result characterizes the whole unit ball, which becomes the “state space” in this non-ordered setting.

Guided by the approach of Alfsen and Shultz in the binary context, it was natural to expect that to prove a geometric characterization of predual unit balls of global (that is, not irreducible) JBW^{∗}-triples would require a decomposition of the space into atomic and non-atomic summands and a version of spectral duality. These goals have remained elusive in the framework of the axioms used in [19]. In the present paper, by introducing the very natural axiom asserting the existence of a Jordan decomposition in the real linear span of every norm-exposed face, we are able to prove the atomic decomposition. In addition, by imposing a spectral axiom every bit as justified as the one in the Alfsen-Shultz theory, we are able to give a geometric characterization of the unit ball of the dual of a JB^{∗}-triple (see Theorem 3.1 below). This gives positive answers to Problems 1,2 and 3 in [19].

In [8], Blecher, Effros and Zarikian define complete one-sided M-ideals for any operator space and show that in the case of a C*-algebra, they coincide with the closed one-sided ideals. Our Theorem 3.1, when combined with the recent characterization of ternary rings of operators (TROs) in terms of its linear matricial norm structure [23] (see Theorem 1.2 below), characterizes those operator spaces which are completely isometric to one-sided ideals in C^{∗}-algebras (see Theorem 3.2 below). This answers a question posed by D. Blecher.

The main results of this paper are Theorems 2.1, 3.1, and 3.2, which we state here. Theorem 2.1. Let Z be a neutral, locally base normed, strongly facially symmetric space satisfying FE, JP, and the pure state properties STP and ERP. Then Z = Z_{a} ⊕^{ℓ^{1}} N, where Z_{a} and N are strongly facially symmetric spaces satisfying the same properties as Z, N has no extreme points in its unit ball, and Z_{a} is the norm closed complex span of the extreme points of its unit ball. Furthermore, by Proposition 3.2, (Z_{a})^{∗} is isometric to an atomic JBW^{∗}-triple.

Theorem 3.1. A Banach space X is isometric to a JB*-triple if and only if X^{∗} is an L-embedded, locally base normed, strongly spectral, strongly facially symmetric space which satisfies FE, JP, and the pure state properties STP and ERP.

State spaces of JB^{∗}-triples  3

In [23], it was proved that an operator space A is completely isometric to a TRO if and only if M_{n}(A) is isometric to a JB^{∗}-triple for every n ≥ 2. Combining this fact with Theorem 3.1 gives a facial operator space characterization of TRO’s. Since a one-sided ideal is a TRO, Theorem 3.2 then gives an operator space characterization of one-sided ideals in C^{∗}- algebras.

Theorem 3.2. Let A be a TRO. Then A is completely isometric to a left ideal in a C*-algebra if and only if there exists a convex set C = {x_{λ} : λ ∈ Λ} ⊂ A_{1} such that the collection of faces

F_{λ} := F_{[ \begin{matrix}0 \\ x_{λ}/∥x_{λ}∥ \end{matrix} ]} ⊂ M_{2,1}(A)^{∗},

form a directed set with respect to containment, F := sup_{λ} F_{λ} exists, and

(a) The set \{[ \begin{matrix}0 \\ x_{λ} \end{matrix} ] : λ ∈ Λ\} separates the points of F;

(b) F^{⊥} = 0 (that is, the partial isometry V ∈ (M_{2,1}(A))^{∗∗} with F = F_{V} is maximal);

(c) ⟨F, [ \begin{matrix}0 \\ x_{λ} \end{matrix} ]⟩ ≥ 0 for all λ ∈ Λ;

(d) S_{F}^{∗} ([ \begin{matrix}0 \\ x_{λ} \end{matrix} ]) = [ \begin{matrix}0 \\ x_{λ} \end{matrix} ] for all λ ∈ Λ.

This paper is organized as follows. In section 1 we recall the background on facially symmetric spaces and on JB^{∗}-triples and TRO’s. Section 2 is devoted to a proof of the atomic decomposition. The first subsection contains a result for some contractive projections on facially symmetric spaces and the second subsection introduces and studies the Jordan decomposition property. The third subsection gives a geometric characterization of spin factors (Proposition 2.3), a variation of the main result of [18]. The main result of section 2, the atomic decomposition (Theorem 2.1), is proved in the fourth subsection.

The main applications occur in section 3. After giving a result, interesting in its own right, on contractive projections on Banach spaces in the first subsection (Proposition 3.1), the second subsection then uses all of the machinery developed up to there to give a geometric characterization of Cartan factors (Proposition 3.2), a variation of the main result of [19]. The spectral duality axiom is introduced in the next subsection and used together with the atomic decomposition to give a geometric characterization of the dual ball of a JB^{∗}-triple (Theorem 3.1). The final subsection applies the latter to give an operator space characterization of one-sided ideals in C^{∗}-algebras (Theorem 3.2).

1. Preliminaries

Facially symmetric spaces (see subsection 1.1) were introduced in [15] and studied in [16] and [18]. In [19], the complete structure of atomic facially symmetric spaces was determined, solving a problem posed in [15]. It was shown, more precisely, that an irreducible, neutral, strongly facially symmetric space is linearly isometric to the predual of one of the Cartan factors of types 1 to 6, provided that it satisfies some natural and physically significant axioms, four in number, which are known to hold in the preduals of all JBW^{∗}-triples. As in the study of state spaces of Jordan algebras (see [1] and the books [2],[3]), we shall refer to the axioms involving extreme points as the pure state properties. Since we can regard the entire unit ball of the dual of a JB*-triple as the “state space” of a physical system, cf. [15, Introduction], we have given a geometric characterization of such state spaces.

The project of classifying facially symmetric spaces was started in [18], where, using the property FE and the pure state property STP, geometric characterizations of complex Hilbert spaces and complex spin factors were given. The former is precisely a rank 1 JBW^{∗}-triple and a special case of a Cartan factor of type 1, and the latter is the Cartan factor of type 4 and a special case of a JBW^{∗}-triple of rank 2. (For a description of all of the Cartan factors, see subsection 1.2.) The explicit structure of a spin factor naturally embedded in a facially symmetric space was then used in [19] to construct abstract generating sets and complete the classification in the atomic case.

1.1. Facially symmetric spaces

Let Z be a complex normed space. Elements f, g ∈ Z are orthogonal, notation f ⋄ g, if ∥f + g∥ = ∥f − g∥ = ∥f∥ + ∥g∥. A norm exposed face of the unit ball Z_{1} of Z is a non-empty set (necessarily ≠ Z_{1}) of the form F_{x} = {f ∈ Z_{1} : f(x) = 1}, where x ∈ Z^{∗}, ∥x∥ = 1. Recall that a face G of a convex set K is a non-empty convex subset of K such that if g ∈ G and h, k ∈ K satisfy g = λh + (1 − λ)k for some λ ∈ (0, 1), then h, k ∈ G. In particular, an extreme point of K is a face of K. We denote the set of extreme points of K by ext K. An element u ∈ Z^{∗} is called a projective unit if ∥u∥ = 1 and ⟨u, F_{u}^{⋄}⟩ = 0. Here, for any subset S, S^{⋄} denotes the set of all elements orthogonal to each element of S. F and U denote the collections of norm exposed faces of Z_{1} and projective units in Z^{∗}, respectively.

Motivated by measuring processes in quantum mechanics, we defined a symmetric face to be a norm exposed face F in Z_{1} with the following property: there is a linear isometry S_{F} of Z onto Z, with S_{F}^{2} = I (we call such maps symmetries), such that the fixed point set of S_{F} is (\overline{sp}F) ⊕ F^{⋄} (topological direct sum). A complex normed space Z is said to be weakly facially symmetric (WFS) if every norm exposed face in Z_{1} is symmetric. For each symmetric face F we defined contractive projections P_{k}(F), k = 0, 1, 2 on Z as follows. First P_{1}(F) = (I − S_{F})/2 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 \overline{sp}F and F^{⋄} respectively, so that P_{2}(F) + P_{0}(F) = (I + S_{F})/2. A geometric tripotent is a projective unit u ∈ 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. The projections P_{k}(F_{u}) are called geometric Peirce projections.

GT and SF denote the collections of geometric tripotents and symmetric faces respectively, and the map GT ∋ u ↦ F_{u} ∈ SF is a bijection [16, 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}), k = 0, 1, 2. Also we let U := Z^{∗}, Z_{k}(u) = Z_{k}(F_{u}) := P_{k}(u)Z and U_{k}(u) = U_{k}(F_{u}) := P_{k}(u)^{∗}(U), so that we have the geometric Peirce decompositions Z = Z_{2}(u) + Z_{1}(u) + Z_{0}(u) and U = U_{2}(u) + U_{1}(u) + U_{0}(u). A symmetry corresponding to the symmetric face F_{u} will sometimes be denoted by S_{u}. Two geometric tripotents u_{1} and u_{2} are orthogonal if u_{1} ∈ U_{0}(u_{2}) (which implies u_{2} ∈ U_{0}(u_{1})) and colinear if u_{1} ∈ U_{1}(u_{2}) and u_{2} ∈ U_{1}(u_{1}). More generally, 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, i.e., [P_{k}(u), P_{j}(v)] = 0 for k, j ∈ {0, 1, 2}. By [16, Theorem 3.3], this is the case if u ∈ U_{k}(v) for some k = 0, 1, 2. For each G ∈ F, v_{G} denotes the unique geometric tripotent with F_{v_{G}} = G.

A contractive projection Q on a normed space X is said to be neutral if for each ξ ∈ X, ∥Qξ∥ = ∥ξ∥ implies Qξ = ξ. A normed space Z is neutral if for every symmetric face F, the projection P_{2}(F) corresponding to some choice of symmetry S_{F}, is neutral.

A WFS space Z is strongly facially symmetric (SFS) if for every norm exposed face F in Z_{1} and every y ∈ Z^{∗} with ∥y∥ = 1 and F ⊂ F_{y}, we have S_{F}^{∗} y = y, where S_{F} denotes a symmetry associated with F.

The principal examples of neutral strongly facially symmetric spaces are preduals of JBW^{∗}-triples, in particular, the preduals of von Neumann algebras, see [17]. In these cases, as shown in [17], geometric tripotents correspond to tripotents in a JBW^{∗}-triple and to partial isometries in a von Neumann algebra. Moreover, because of the validity of the Jordan decomposition for hermitian functionals on JB*-algebras, sp_{C}F is automatically norm closed (cf. Lemma 2.6).

In a neutral strongly facially symmetric space Z, every non-zero element has a polar decomposition [16, Theorem 4.3]: for 0 ≠ f ∈ Z there exists a unique geometric tripotent v = v(f) = v_{f} with f(v) = ∥f∥ and ⟨v, {f}^{⋄}⟩ = 0. Let M denote the collection of minimal geometric tripotents of U, i.e., M = {v ∈ GT : U_{2}(v) is one dimensional}. If Z is a neutral strongly SFS space satisfying PE (see Definition 1.3 below), then the map f ↦ v(f) is a bijection of ext Z_{1} and M ([18, Prop. 2.4]).

A partial ordering can be defined on the set of geometric tripotents as follows: if u, v ∈ GT , then u ≤ v if F_{u} ⊂ F_{v}, or equivalently ([16, Lemma 4.2]), P_{2}(u)^{∗}v = u or v − u is either zero or a geometric tripotent orthogonal to u. Let I denote the collection of indecomposable geometric tripotents of U, i.e., I = {v ∈ GT : u ∈ GT , u ≤ v ⇒ u = v}. In general, M ⊂ I, and under certain conditions, (Proposition2.2(a) below and [18, Prop. 2.9]), M coincides with I.

We now recall the definitions of the pure state properties and other axioms.

Definition 1.1. Let f and g be extreme points of the unit ball of a neutral SFS space Z. The transition probability of f and g is the number

⟨f|g⟩ := f(v(g)).

A neutral SFS space Z is said to satisfy “symmetry of transition probabilities” STP if for every pair of extreme points f, g ∈ ext Z_{1}, we have

\overline{⟨f|g⟩} = ⟨g|f⟩.

In order to guarantee a sufficient number of extreme points, the following definition was made in [18] and assumed in [19]. For the present paper, this definition is too strong and will be abandoned. It will turn out that the property (b) of Proposition 1.1 will be available to us and suffice for our purposes.

Definition 1.2. A normed space Z is said to be atomic if every symmetric face of Z_{1} has an extreme point.

Definition 1.3. A neutral SFS space Z is said to satisfy property FE if every norm closed face of Z_{1} different from Z_{1} is a norm exposed face. We use the terminology PE for the special case of this that every extreme point of Z_{1} is norm exposed.

The following consequence of atomicity will be more useful to us in this paper.

Proposition 1.1 ([18],Proposition 2.7). If Z is an atomic SFS space satisfying PE, then

(a) U = \overline{sp} M (weak^{∗}-closure), where M is the set of minimal geometric tripotents.

(b) Z_{1} = \overline{co} ext Z_{1} (norm closure).

Definition 1.4. A neutral SFS space Z is said to satisfy the “extreme rays property” ERP if for every u ∈ GT and every f ∈ ext Z_{1}, it follows that P_{2}(u)f is a scalar multiple of some element in ext Z_{1}. We also say that P_{2}(u) preserves extreme rays.

Definition 1.5. A WFS space Z satisfies JP if for any pair u, v of orthogonal geometric tripotents, we have

S_{u}S_{v} = S_{u+v},  (1.1)

where for any geometric tripotent w, S_{w} is the symmetry associated with the symmetric face F_{w}.

The property JP was defined and needed in [19] only for minimal geometric tripotents u and v. The more restricted definition given here is needed only in Proposition 2.2(b), where ironically, the involved geometric tripotents turn out to be minimal. (The assumption of JP is used in subsection 2.1 only for minimal geometric tripotents.) As in Remark 4.2 of [19],with identical proofs, JP implies the following important joint Peirce rules for orthogonal geometric tripotents u and v:

Z_{2}(u + v) = Z_{2}(u) + Z_{2}(v) + Z_{1}(u) ∩ Z_{1}(v),
Z_{1}(u + v) = Z_{1}(u) ∩ Z_{0}(v) + Z_{1}(v) ∩ Z_{0}(u),
Z_{0}(u + v) = Z_{0}(u) ∩ Z_{0}(v).

Definitions 1.1,1.3 and 1.4 are analogs of physically meaningful axioms in [1]. In the Hilbert space model for quantum mechanics, property JP for minimal geometric tripotents is interpreted as follows. Choose ξ ⊗ ξ to be the state exposed by a yes/no question v and η ⊗ η to be the state exposed by another u, and complete ξ, η to an orthonormal basis. For any state vector ζ expressed in this basis, the symmetry S_{u} (resp. S_{v}) changes the sign of the coefficient of ξ (resp. η) and S_{u+v} changes the sign of both coefficients.

We need the concept of L-embeddedness for the proofs of Proposition 3.2 and Theorem 3.1. This is defined as follows. A linear projection P on a Banach space X is called an L-projection if ∥x∥ = ∥P x∥+∥(I −P)x∥ for every x ∈ X. The range of an L-projection is called an L-summand. The space X is said to be an L-embedded space if it is an L-summand in its second dual. These concepts are studied extensively in [20, Chapter IV]. The predual of a JB^{∗}-triple is an example of an L-embedded space ([6]) and every L-embedded space is weakly sequentially complete ([20, Theorem 2.2,page 169].

The following is the main result of [19]. We have added the assumption of L-embeddedness, which seems to have been overlooked in [19]. This omission was discovered in the process of proving Proposition 3.2. More precisely, our Proposition 3.1 is needed in the proofs of [19, Lemmas 5.5 and 6.6]. In addition, our Proposition 2.1 is needed for [19, Theorem 3.12], and our Corollary 3.1 is needed three times in [19, Proposition 4.11]. Cartan factors are defined in the next subsection.

Theorem 1.1 ([19],Theorem 8.3). Let Z be an atomic neutral strongly facially symmetric space satisfying FE, STP, ERP, and JP. If Z is Lembedded, then Z = ⊕_{α}^{ℓ^{1}}J_{α} where each J_{α} is isometric to the predual of a Cartan factor of one of the types 1-6. Thus Z^{∗} is isometric to an atomic JBW^{∗}-triple. If Z is irreducible, then Z^{∗} is isometric to a Cartan factor.

One of our main objectives in this paper is to be able to drop the assumption of atomicity in this result, i.e. to find a non-ordered analog of the main theorem of Alfsen-Shultz [1]. This will be achieved in our Theorem 3.1 below, but at the expense of some other axioms.

1.2. JB^{∗}-triples and ternary rings of operators

A Jordan triple system is a complex vector space V with a triple product {·, ·, ·} : V × V × V ⟶ V which is symmetric and linear in the outer variables, conjugate linear in the middle variable and satisfies the Jordan triple identity

{a, b, {x, y, z}} = {{a, b, x}, y, z} − {x, {b, a, y}, z} + {x, y, {a, b, z}}.

A complex Banach space A is called a JB^{∗}-triple if it is a Jordan triple system such that for each z ∈ A, the linear map

D(z) : v ∈ A ↦ {z, z, v} ∈ A

is Hermitian, that is, ∥e^{itD(z)}∥ = 1 for all t ∈ R, with non-negative spectrum in the Banach algebra of operators generated by D(z) and ∥D(z)∥ = ∥z∥^{2}. A summary of the basic facts about JB*-triples can be found in [25] and some of the references therein, such as [22],[13], and

A JB^{∗}-triple A is called a JBW^{∗}-triple if it is a dual Banach space, in which case its predual is unique, denoted by A_{∗}, and the triple product is separately weak* continuous. The second dual A^{∗∗} of a JB^{∗}-triple is a JBW^{∗}-triple.

The JB^{∗}-triples form a large class of Banach spaces which include C^{∗}-algebras, Hilbert spaces, spaces of rectangular matrices, and JB*- algebras. The triple product in a C*-algebra A is given by

\{x, y, z\} = \frac{1}{2} (xy^{∗}z + zy^{∗}x).

In a JB*-algebra with product x ∘ y, the triple product is given by {x, y, z} = (x ∘ y^{∗}) ∘ z + z ∘ (y^{∗} ∘ x) − (x ∘ z) ∘ y^{∗}. An element e in a JB*-triple A is called a tripotent if {e, e, e} = e in which case the map D(e) : A ⟶ A has eigenvalues 0, \frac{1}{2} and 1, and we have the following decomposition in terms of eigenspaces

A = A_{2}(e) ⊕ A_{1}(e) ⊕ A_{0}(e)

which is called the Peirce decomposition of A. The \frac{k}{2}-eigenspace A_{k}(e) is called the Peirce k-space. The Peirce projections from A onto the Peirce k-spaces are given by

P_{2}(e) = Q^{2}(e), P_{1}(e) = 2(D(e) − Q^{2}(e)), P_{0}(e) = I − 2D(e) + Q^{2}(e)

where Q(e)z = {e, z, e} for z ∈ A. The Peirce projections are contractive.

For any tripotent v, the space A_{2}(v) is a JB*-algebra under the product x · y = {x v y} and involution x^{♯} = {v x v}. JBW*-triples have an abundance of tripotents. In fact, given a JBW*-triple A and f in the predual A_{∗}, there is a unique tripotent v_{f} ∈ A, called the support tripotent of f, such that f ∘ P_{2}(v_{f}) = f and the restriction f|_{A_{2}(v_{f})} is a faithful positive normal functional.

An important class of JBW*-triples are the following six types of Cartan factors (see [9, pp. 292-3]) :

type 1 B(H, K), with triple product \{x, y, z\} = \frac{1}{2}(xy^{∗}z + zy^{∗}x), type 2 {z ∈ B(H, H) : z^{t} = −z}, type 3 {z ∈ B(H, H) : z^{t} = z}, type 4 spin factor (defined below), type 5 M_{1,2}(O) with triple product \{x, y, z\} = \frac{1}{2}(x(y^{∗}z) + z(y^{∗}x)), type 6 M_{3}(O)

where O denotes the 8 dimensional complex Octonians, B(H, K) is the Banach space of bounded linear operators between complex Hilbert spaces H and K, and z^{t} is the transpose of z induced by a conjugation on H. Cartan factors of type 2 and 3 are obviously subtriples of B(H, H), the latter notation is shortened to B(H), while type 4 can be embedded as a subtriple of some B(H). The type 3 and 4 are Jordan algebras with the usual Jordan product x ∘ y = \frac{1}{2}(xy + yx). Abstractly, a spin factor is a Banach space that is equipped with a complete inner product ⟨·, ·⟩ and a conjugation j on the resulting Hilbert space, with triple product

\{x, y, z\} = \frac{1}{2}(⟨x, y⟩z + ⟨z, y⟩x − ⟨x, jz⟩jy)

such that the given norm and the Hilbert space norm are equivalent.

An important example of a JB^{∗}-triple is a ternary ring of operators (TRO). This is a subspace of B(H) which is closed under the product xy^{∗}z. Every TRO is (completely) isometric to a corner pA(1 − p) of a C*-algebra A. TRO’s play an important role in the theory of quantized Banach spaces (operator spaces), see [12] for the general theory and [11] for the role of TRO’s. For one thing, as shown by Ruan [24], the injectives in the category of operator spaces are TRO’s (corners of injective C*- algebras) and not, in general, operator algebras. If A is a TRO and v is a partial isometry in A, then A_{2}(v) is a C^{∗}-algebra under the product (x, y) ↦ xv^{∗}y and involution x ↦ vx^{∗}v.

Motivated by a characterization for JB*-triples as complex Banach spaces whose open unit ball is a bounded symmetric domain, we gave in [23] a holomorphic operator space characterization of TRO’s up to complete isometry. As a consequence, we obtained a holomorphic operator space characterization of C*-algebras as well. Since a closed left ideal in a C*-algebra is a TRO, Theorem 1.2 below will allow us, in our facial operator space characterization of left ideals (Theorem 3.2) to restrict to TROs from the beginning. The following is the main result of [23].

Theorem 1.2 ([23],Theorem 4.3). Let A ⊂ B(H) be an operator space and suppose that M_{n}(A)_{0} is a bounded symmetric domain for some n ≥ 2. Then A is n-isometric to a ternary ring of operators (TRO). If M_{n}(A)_{0} is a bounded symmetric domain for all n ≥ 2, then A is ternary isomorphic and completely isometric to a TRO.

2. Atomic decomposition of facially symmetric spaces

2.1. Contractive projections on facially symmetric spaces

In this subsection, we shall assume that Z is a strongly facially symmetric space with dual U = Z^{∗}. If {v_{i}} is a countable family of mutually orthogonal minimal geometric tripotents, then v = sup v_{i} exists as it is the support geometric tripotent of ∑ 2^{−i}f_{i}, where v_{i} = v_{f_{i}}. This fact will be used in the proof of the following lemma.

Lemma 2.1. Let {v_{i}} be a countable family of mutually orthogonal minimal geometric tripotents, with v =: sup v_{i}. Then v = ∑_{i} v_{i} (w*-limit).

Proof. Note first that by [16, Cor. 3.4(a) and Lemma 1.8], for each n ≥ 1,

Π_{1}^{n}(P_{2}(v_{i}) + P_{0}(v_{i})) = ∑_{1}^{n} P_{2}(v_{i}) + P_{0}(∑_{1}^{n} v_{i}),

so by [16, Cor. 3.4(b)],

P_{2}(∑_{1}^{n} v_{i})Π_{1}^{n}(P_{2}(v_{i}) + P_{0}(v_{i})) = ∑_{1}^{n} P_{2}(v_{i}),

and hence ∑_{1}^{n} P_{2}(v_{i}) is a contractive projection. For φ ∈ Z, by orthogonality,

∑_{1}^{n} ∥P_{2}(v_{i})φ∥ = ∥ ∑_{1}^{n} P_{2}(v_{i})φ∥ ≤ ∥φ∥,

so that ∑_{1}^{∞} ∥P_{2}(v_{j})φ∥ ≤ ∥φ∥ and with Q_{n} := ∑_{1}^{n} P_{2}(v_{i}) and for m ≥ n,

∥Q_{m}φ − Q_{n}φ∥ = ∥Q_{n}φ∥ − ∥Q_{m}φ∥

so that Q_{n}φ converges to a limit, call it Qφ, and Q is a contractive projection.

For each x ∈ U, Q_{n}^{∗}x converges in the weak*-topology to Q^{∗}x. Applying this with x = v and recalling that P_{2}(v_{i})^{∗}v = v_{i}, we obtain ∑_{1}^{n} v_{i} = Q_{n}^{∗}v → y in the weak*-topology for some y ∈ U_{2}(v). On the other hand, since ⟨y, ∑ 2^{−i}f_{i}⟩ = 1, by [16, Theorem 4.3(c)], we have F_{v} ⊂ F_{y} and therefore by strong facial symmetry, y = v + y_{0}, where y_{0} ∈ U_{0}(v). Since y ∈ U_{2}(v) we must have y_{0} = 0 and hence y = v. ⊓ ⊔

Lemma 2.2. Suppose that Z is neutral and satisfies JP. Let {v_{i}} be a countable family of mutually orthogonal minimal geometric tripotents, with v =: sup v_{i}. Then ∪_{i=1}^{∞}[Z_{2}(v_{i})∪Z_{1}(v_{i})] is norm total in Z_{2}(v)+Z_{1}(v).

Proof. Let W be the norm closure of the complex span of ∪_{i=1}^{∞}[Z_{2}(v_{i}) ∪ Z_{1}(v_{i})]. We first show that

Z_{2}(v) + Z_{1}(v) ⊂ W.  (2.1)

If φ ∈ Z_{2}(v) + Z_{1}(v) and φ ∉ W, then there exists x ∈ U, ∥x∥ ≤ 1 with ⟨x, φ⟩ ≠ 0 and ⟨x, W ⟩ = 0. We’ll show that x ∈ U_{0}(v). Since φ ∈ Z_{2}(v) + Z_{1}(v) implies ⟨x, φ⟩ = 0, this is a contradiction, proving (2.1).

Let s_{n} = ∑_{1}^{n} v_{j} and for ρ ∈ Z of norm one, let ρ = ρ_{2} + ρ_{1} + ρ_{0} be its geometric Peirce decomposition with respect to s_{n}. By JP (for minimal geometric tripotents), ρ_{2}, ρ_{1} ∈ W. Therefore

⟨s_{n} ± x, ρ⟩ = ⟨s_{n}, ρ_{2}⟩ ± ⟨x, ρ_{0}⟩ = ⟨P_{2}(s_{n})s_{n} ± P_{0}(s_{n})x, ρ⟩

so that

|⟨s_{n} ± x, ρ⟩| ≤ ∥P_{2}(s_{n})s_{n} ± P_{0}(s_{n})x∥ = max(∥P_{2}(s_{n})s_{n}∥, ∥P_{0}(s_{n})x∥) = 1.

Thus ∥ ∑_{1}^{n} v_{i}±x∥ = 1 so by Lemma 2.1, ∥v±x∥ ≤ 1. By [16, Theorem 4.6], v+U_{0}(v)_{1} is a face in the unit ball of U, and since v = (v+x)/2+(v−x)/2, v ± x ∈ v + U_{0}(v)_{1}, proving x ∈ U_{0}(v) and hence (2.1).

To show that equality holds in (2.1), note first that it is obvious that Z_{2}(v_{i}) ⊂ Z_{2}(v), and if φ ∈ Z_{1}(v_{i}), then by compatibility, P_{0}(v)φ = P_{0}(v)P_{1}(v_{i})φ ∈ Z_{1}(v_{i}). But P_{0}(v)φ = P_{0}(v)P_{0}(v_{i})φ ∈ Z_{0}(v_{i}) so that P_{0}(v)φ = 0, as required. ⊓ ⊔

Corollary 2.1. P_{0}(v) = Π_{i=1}^{∞}P_{0}(v_{i}) (strong limit).

Proof. Let Q_{n} = Π_{1}^{n}P_{0}(v_{i}). Let φ ∈ Z have geometric Peirce decomposition φ_{2} + φ_{1} + φ_{0} with respect to v. Since Z_{0}(v) ⊂ Q_{n}(Z), Q_{n}φ_{0} = φ_{0} → φ_{0} = P_{0}(v)φ. It remains to show that Q_{n}(φ_{2} +φ_{1}) → 0. By Lemma 2.2, it suffices to prove that Q_{n}ψ → 0 for every i and every ψ ∈ Z_{2}(v_{i}) ∪ Z_{1}(v_{i}). But for any ψ ∈ Z_{k}(v_{i}) for k = 2, 1, Q_{n}ψ = Q_{n}P_{k}(v_{i})ψ = 0 as soon as n ≥ i. ⊓ ⊔

Proposition 2.1. Suppose that Z is neutral and satisfies JP. Let {u_{i}}_{i∈I} be an arbitrary family of mutually orthogonal minimal geometric tripotents. Then Q := Π_{i∈I}P_{0}(u_{i}) exists as a strong limit and Q is a contractive projection with range ∩_{i∈I}Z_{0}(u_{i}).

Proof. Fix f ∈ Z. For each countable set λ ⊂ I, let g_{λ} := Π_{i∈λ}P_{0}(u_{i})f, which exists as a norm limit by Corollary 2.1.

With α := inf ∥g_{λ}∥, where λ runs over the countable subsets of I, we can find a sequence λ_{n} of countable sets such that α = lim ∥g_{λ_{n}}∥, and hence a countable set μ = ∪_{n}λ_{n} ⊂ I such that ∥g_{μ}∥ = α. It remains to prove that

Π_{i∈I}P_{0}(u_{i})f = Π_{i∈μ}P_{0}(u_{i})f.

For ϵ > 0, choose a finite set A_{0} ⊂ μ such that for all finite sets A with A_{0} ⊂ A ⊂ μ,

∥Π_{i∈A}P_{0}(u_{i})f − Π_{i∈μ}P_{0}(u_{i})f∥ < ϵ.

By the neutrality of P_{0}(u_{j}) and the definition of α, for any j ∉ μ,

P_{0}(u_{j})Π_{i∈μ}P_{0}(u_{i})f = Π_{i∈μ}P_{0}(u_{i})f.

Hence, for any finite subset B with A_{0} ⊂ B ⊂ I,

∥Π_{i∈B}P_{0}(u_{i})f − Π_{i∈μ}P_{0}(u_{i})f∥
= ∥Π_{i∈B−μ}P_{0}(u_{i})[Π_{i∈B∩μ}P_{0}(u_{i})f − Π_{i∈μ}P_{0}(u_{i})f]∥
≤ ∥Π_{i∈B∩μ}P_{0}(u_{i})f − Π_{i∈μ}P_{0}(u_{i})f∥ < ϵ.⊓ ⊔

2.2. Jordan decomposition

In this subsection we introduce the Jordan decomposition property. We use it in place of atomicity to obtain Proposition 2.2, which contains the analogs of [18, Prop. 2.9] and [19, Prop. 2.4]. Lemmas 2.3–2.6 are taken from an unpublished note of Yaakov Friedman and the second named author in 1990.

Lemma 2.3. Let F be a norm exposed face of the unit ball of a normed space Z, and let I denote the closed unit interval. The following are equivalent.

(a) (sp_{R}F)_{1} ⊂ co(IF ∪ −IF).

(b) For each non-zero f ∈ sp_{R}F, ∃ g, h ∈ R^{+}F with f = g − h and ∥f∥ = ∥g∥ + ∥h∥.

(c) ∂(sp_{R}F)_{1} ⊂ co(F ∪ −F).

(d) For each non-zero f ∈ sp_{R}F, ∃ g, h ∈ R^{+}F with f = g − h and g ⋄ h.

Proof. (a)⇒(c). If f ∈ sp_{R}F and ∥f∥ = 1, then f = λασ − (1 − λ)βτ, with α, β, λ ∈ I and σ, τ ∈ F. If λ = 0 or 1, then f ∈ ±F so assume that 0 < λ < 1. We have

1 = ∥f∥ = ∥λασ − (1 − λ)βτ∥ ≤ λα + (1 − λ)β ≤ α ∨ β ≤ 1.

Since λ < 1, α = β = 1.

(c)⇒(b). If 0 ≠ f ∈ sp_{R}F, then ∥f∥^{−1}f = λσ − (1 − λ)τ with λ ∈ I and σ, τ ∈ F. Since ∥λσ∥ + ∥(1 − λ)τ∥ = λ + (1 − λ) = 1, we have

∥f∥ = ∥f∥(∥λσ∥ + ∥(1 − λ)τ∥) = ∥(∥f∥λ)σ∥ + ∥(∥f∥(1 − λ))τ∥.

(b)⇒(a). Let f ∈ (sp_{R}F)_{1} and assume 0 < ∥f∥ ≤ 1. With f = g − h and ∥f∥ = ∥g∥ + ∥h∥ with g, h ∈ R^{+}F, we have

f = ∥g∥(∥g∥^{−1}g) + ∥h∥(−∥h∥^{−1}h) + (1 − ∥f∥) · 0 ∈ co(IF ∪ −IF).

(d)⇒(b). If g ⋄ h, then ∥f∥ = ∥g − h∥ = ∥g∥ + ∥h∥.

(b)⇒(d). If g, h ∈ R^{+}F and F = F_{x} for some x ∈ U of norm one, then ∥g + h∥ = g(x) + h(x) = ∥g∥ + ∥h∥. Therefore, ∥g ± h∥ = ∥g∥ + ∥h∥, i.e., g ⋄ h. ⊓ ⊔

Definition 2.1. A norm exposed face of the unit ball of a normed space Z satisfies the Jordan decomposition property if (one of) the conditions of Lemma 2.3 holds.

It is elementary that if F satisfies the Jordan decomposition property, then ext (sp_{R}F)_{1} = ext F ∪ ext (−F).

Lemmas 2.4-2.6 are not needed in the rest of this paper but are recorded here for their intrinsic interest and possible future use.

Lemma 2.4. Let Z be a neutral SFS space, let F be a norm exposed face of Z_{1}and let f ∈ F. Then S_{v(f)}(F) ⊂ F and P_{2}(f)F ⊂ R^{+}F, where P_{k}(f) denotes P_{k}(v(f)) for k = 0, 1, 2.

Proof. Let F = F_{u} for some u ∈ GT . Then by the minimality property of the polar decomposition ([16, Theorem 4.3(c)]), F_{v(f)} ⊂ F_{u} and by strong facial symmetry, S_{v(f)}^{∗}u = u. Thus if ρ ∈ F_{u}, ⟨S_{v(f)}ρ, u⟩ = ⟨ρ, u⟩ = 1 i.e., S_{v(f)}ρ ∈ F_{u}, which proves the first statement.

By what was just proved,

[P_{2}(f) + P_{0}(f)](F) = \frac{I + S_{v(f)}}{2} (F) ⊂ F.

Thus, if g ∈ F,

∥P_{2}(f)g∥ (\frac{P_{2}(f)g}{∥P_{2}(f)g∥}) + ∥P_{0}(f)g∥ (\frac{P_{0}(f)g}{∥P_{0}(f)g∥}) ∈ F.

Since F is a face, P_{2}(f)g/∥P_{2}(f)g∥ ∈ F. ⊓ ⊔

Lemma 2.5. In a neutral SFS space, the Jordan decomposition is unique whenever it exists, i.e., if u ∈ GT and for i = 1, 2, if f = σ_{1} −τ_{1} = σ_{2} −τ_{2} with τ_{i}, σ_{i} ∈ R^{+}F_{u}, 1 = ∥f∥ = ∥σ_{i}∥ + ∥τ_{i}∥, then σ_{1} = σ_{2} and τ_{1} = τ_{2}.

Proof. Apply P_{2}(σ_{1}) and P_{2}(τ_{1}) to f = σ_{1} − τ_{1} = σ_{2} − τ_{2} to obtain σ_{1} = P_{2}(σ_{1})σ_{2} − P_{2}(σ_{1})τ_{2} and −τ_{1} = P_{2}(τ_{1})σ_{2} − P_{2}(τ_{1})τ_{2}. Since σ_{i} ⋄ τ_{i},

1 = ∥σ_{1}∥ + ∥τ_{1}∥ ≤ ∥P_{2}(σ_{1})σ_{2}∥ + ∥P_{2}(σ_{1})τ_{2}∥ + ∥P_{2}(τ_{1})σ_{2}∥ + ∥P_{2}(τ_{1})τ_{2}∥
= ∥[P_{2}(σ_{1}) + P_{2}(τ_{1})]σ_{2}∥ + ∥[P_{2}(σ_{1}) + P_{2}(τ_{1})]τ_{2}∥
≤ ∥σ_{2}∥ + ∥τ_{2}∥ = 1.

Therefore ∥σ_{1}∥ = ∥P_{2}(σ_{1})σ_{2}∥ + ∥P_{2}(σ_{1})τ_{2}∥.

Case 1. P_{2}(σ_{1})τ_{2} ≠ 0. In this case, P_{2}(σ_{1})σ_{2} ≠ 0, otherwise we would have σ_{1} = 0 and hence σ_{1} = σ_{2} = 0. We then have

\frac{σ_{1}}{∥σ_{1}∥} = \frac{∥P_{2}(σ_{1})σ_{2}∥}{∥σ_{1}∥} \frac{P_{2}(σ_{1})σ_{2}}{∥P_{2}(σ_{1})σ_{2}∥} + \frac{∥P_{2}(σ_{1})τ_{2}∥}{∥σ_{1}∥} (\frac{−P_{2}(σ_{1})τ_{2}}{∥P_{2}(σ_{1})τ_{2}∥}) .

Since F_{u} is a face,

\frac{−P_{2}(σ_{1})τ_{2}}{∥P_{2}(σ_{1})τ_{2}∥} ∈ F_{u}.

On the other hand, by Lemma 2.4, P_{2}(σ_{1})τ_{2} ∈ R^{+}F_{u}, so that P_{2}(σ_{1})τ_{2} = 0, a contradiction, so this case does not occur.

Next, as above, apply P_{2}(σ_{2}) and P_{2}(τ_{2}) to f = σ_{1} − τ_{1} = σ_{2} − τ_{2} to obtain σ_{2} = P_{2}(σ_{2})σ_{1} − P_{2}(σ_{2})τ_{1} and −τ_{2} = P_{2}(τ_{2})σ_{1} − P_{2}(τ_{2})τ_{1}. Since σ_{i} ⋄ τ_{i}, as above we obtain ∥σ_{2}∥ = ∥P_{2}(σ_{2})σ_{1}∥ + ∥P_{2}(σ_{2})τ_{1}∥.

Case 2. P_{2}(σ_{2})τ_{1} ≠ 0. Exactly as in case 1, this implies that P_{2}(σ_{2})σ_{1} ≠ 0 and leads to a contradiction unless σ_{2} = 0. So this case does not occur.

Case 3. P_{2}(σ_{2})τ_{1} = P_{2}(σ_{1})τ_{2} = 0. In this case, σ_{1} = P_{2}(σ_{1})σ_{2} and σ_{2} = P_{2}(σ_{2})σ_{1}, so that ∥σ_{1}∥ = ∥σ_{2}∥. It follows that τ_{2}(v(σ_{1})) = ⟨P_{2}(σ_{1})τ_{2}, v(σ_{1})⟩ = 0 and

∥σ_{1}∥ = σ_{1}(v(σ_{1})) = f(v(σ_{1})) = σ_{2}(v(σ_{1})) − τ_{2}(v(σ_{1}))
= σ_{2}(v(σ_{1})) ≤ ∥σ_{2}∥ = ∥σ_{1}∥,

implying v(σ_{2}) ≤ v(σ_{1}). Similarly, using P_{2}(σ_{2})τ_{1} = 0 leads to ∥σ_{2}∥ = σ_{1}(v(σ_{2})) and v(σ_{1}) ≤ v(σ_{2}).

Thus v(σ_{2}) = v(σ_{1}), and we now have

σ_{1} = P_{2}(σ_{1})f = P_{2}(σ_{1})σ_{2} − P_{2}(σ_{1})τ_{2} = P_{2}(σ_{1})σ_{2} = P_{2}(σ_{2})σ_{2} = σ_{2}.⊓ ⊔

Lemma 2.6. Let F be a norm exposed face satisfying the Jordan decomposition property. Then

(a) sp_{R}F ∩ isp_{R}F = {0}.

(b) If Z is a neutral strongly symmetric space, then the projection of sp_{C}F = sp_{R}F + isp_{R}F onto sp_{R}F is contractive.

Proof. Let h ∈ sp_{R}F ∩isp_{R}F, and suppose that ∥h∥ = 1. By Lemma 2.3,

h = αif + β(−ig) = γf_{1} + δ(−g_{1})

for some f, g, f_{1}, g_{1} ∈ F and α, β, γ, δ ∈ R with α ≥ 0, β = 1 − α, γ ≥ 0, δ = 1 − γ, f ⋄ g, and f_{1} ⋄ g_{1}. With F = F_{u} for some u ∈ GT , we have i(α − β) = h(u) = γ − δ, so that

h = \frac{1}{2}i(f − g) = \frac{1}{2}(f_{1} − g_{1}).  (2.2)

Applying successively P_{2}(f) and P_{2}(g) to (2.2) we obtain

if = P_{2}(f)f_{1} − P_{2}(f)g_{1} and − ig = P_{2}(g)f_{1} − P_{2}(g)g_{1}.

Therefore

2 = ∥if∥ + ∥ig∥ ≤ ∥P_{2}(f)f_{1}∥ + ∥P_{2}(f)g_{1}∥ + ∥P_{2}(g)f_{1}∥ + ∥P_{2}(g)g_{1}∥
= (∥P_{2}(f)f_{1}∥ + ∥P_{2}(g)f_{1}∥) + (∥P_{2}(f)g_{1}∥ + ∥P_{2}(g)g_{1}∥)
≤ ∥f_{1}∥ + ∥g_{1}∥ = 2.

If P_{2}(f)f_{1} = 0, then if = −P_{2}(f)f_{1} ∈ R^{+}F, a contradiction. Similarly, P_{2}(f)g_{1} ≠ 0. Since iF is a face, and

if = ∥P_{2}(f)f_{1}∥ (\frac{P_{2}(f)f_{1}}{∥P_{2}(f)f_{1}∥}) + ∥P_{2}(f)g_{1}∥ (\frac{−P_{2}(f)g_{1}}{∥P_{2}(f)g_{1}∥}) ,

\frac{P_{2}(f)f_{1}}{∥P_{2}(f)f_{1}∥} ∈ iF.

On the other hand, by Lemma 2.4,

\frac{P_{2}(f)f_{1}}{∥P_{2}(f)f_{1}∥} ∈ F

also. This is a contradiction which proves (a).

Now let g +ih ∈ sp_{R}F +isp_{R}F. Write g = aρ−bσ with ρ⋄σ, ρ, σ ∈ F, and ∥g∥ = a + b. Then ⟨g, v_{ρ} − v_{σ}⟩ = a + b, and

∥g + ih∥ ≥ |⟨g + ih, v_{ρ} − v_{σ}⟩| = |a + b + i⟨h, v_{ρ} − v_{σ}⟩|
= [(a + b)^{2} + ⟨h, v_{ρ} − v_{σ}⟩^{2}]^{1/2} ≥ a + b = ∥g∥,

proving (b). ⊓ ⊔

If Z is a dual space, so that each norm exposed face is weak*-compact, then (b) and the Jordan decomposition property imply that sp_{C}F is closed, so that Z_{2}(F) = sp_{C}F.

Definition 2.2. A WFS space satisfies property JD if every symmetric face satisfies the Jordan decomposition property. In this case, we say that Z is locally base normed.

It is important to note that this property is hereditary, that is, if Z satisfies JD, then so does any geometric Peirce space Z_{k}(u). Indeed, if F_{w} ∩ Z_{k}(u) is a local face corresponding to a geometric tripotent w ∈ U_{k}(u), and ρ ∈ sp_{R} [F_{w} ∩ Z_{k}(u)], then ρ = αg − βh, with g, h ∈ F_{w} and ∥ρ∥ = α + β. From this it follows that P_{k}(u)g, P_{k}(u)h ∈ F_{w} and ρ = αP_{k}(u)g − βP_{k}(u)h.

Proposition 2.2. Let Z be a locally base normed SFS space.

(a) I=M.

(b) Suppose furthermore that Z is neutral and satisfies JP. Let v ∈ M and suppose that w ∈ GT and w⊤v. Then w ∈ M.

Proof. Let v ∈ I and suppose F_{v} contains two distinct elements f_{1}, f_{2} and set f = f_{1} − f_{2}. Then f = αg − βh with α, β ∈ R^{+} and g, h ∈ F_{v}. By evaluating at v one sees that α = β = 1/2. Therefore F_{v} contains orthogonal elements g and h with orthogonal supports v_{g} and v_{h} such that v_{g} ≤ v, v_{h} ≤ v. Since v ∈ I, v_{g} = v = v_{h}, a contradiction. Thus F_{v} consists of a single point and v ∈ M. This proves (a).

To prove (b), we first show that F_{w} ⊂ Z_{1}(v). Let ψ = ψ_{2} + ψ_{1} + ψ_{0} be the Peirce decomposition of ψ ∈ F_{w} ⊂ Z_{2}(w) with respect to v. We shall show that ψ_{2} = ψ_{0} = 0. In the first place, by [18, Prop. 2.4], ψ_{2} = P_{2}(v)ψ = ψ(v)f_{v} and since v ∈ U_{1}(w), f_{v} ∈ Z_{1}(w). (To see this last step, note that for k = 0, 2, P_{k}(w)f_{v} = P_{k}(w)P_{2}(v)f_{v} = P_{2}(v)P_{k}(w)f_{v} = ⟨P_{k}(w)f_{v}, v⟩f_{v} = ⟨f_{v}, P_{k}(w)^{∗}v⟩f_{v} = 0.) On the other hand, since v and w are compatible, ψ_{2} = P_{2}(v)P_{2}(w)ψ = P_{2}(w)P_{2}(v)ψ ∈ Z_{2}(w), showing that ψ_{2} ∈ Z_{2}(w) ∩ Z_{1}(w) = {0}. Now ψ = ψ_{1} + ψ_{0} ∈ F_{w}, so S_{v}ψ = −ψ_{1} + ψ_{0} ∈ −F_{w} by [16, Theorem 2.5], so that ψ_{0} ∈sp_{R} F_{w}. Hence, if ψ_{0} ≠ 0,we can write ψ_{0}/∥ψ_{0}∥ = λσ − μτ with σ, τ ∈ F_{w}, λ, μ ≥ 0 and λ + μ = 1. Since σ, τ ∈ F_{w}, as shown above, σ_{2} = τ_{2} = 0 and ψ_{0}/∥ψ_{0}∥ = λ(σ_{1} + σ_{0}) − μ(τ_{1} + τ_{0}) so that λσ_{1} − μτ_{1} = 0, λ = μ = 1/2 (since σ_{1}(w) = τ_{1}(w) = 1) and ∥σ_{0} − τ_{0}∥ = ∥2ψ_{0}/∥ψ_{0}∥∥ = 2. Since

2 = ∥σ_{0} − τ_{0}∥ ≤ ∥σ_{0}∥ + ∥τ_{0}∥ ≤ 1 + 1 = 2,

∥σ_{0}∥ = 1 and by neutrality of P_{0}(v) ([16, Lemma 2.1], σ_{1} = 0, implying σ_{0} = σ ∈ F_{w}, a contradiction as σ_{0}(w) = ⟨P_{2}(w)P_{0}(v)σ, w⟩ = ⟨P_{2}(w)σ, P_{0}(v)^{∗}w⟩ = 0. Therefore ψ_{0} = 0 and F_{w} ⊂ Z_{1}(v).

Now that we know F_{w} ⊂ Z_{1}(v), we show that F_{w} is a single point. Suppose to the contrary that there exist g, h ∈ F_{w} with g ≠ h. Then f := g −h is a non-zero element of sp_{R}F_{w}, so f = σ −τ with σ, τ ∈ R^{+}F_{w} and ∥σ∥ + ∥τ∥ = ∥f∥. Since σ(w) = τ(w), σ ≠ 0 and τ ≠ 0, and since σ ⋄ τ, v_{σ} and v_{τ} are orthogonal geometric tripotents in U_{2}(w) and hence U_{2}(v_{σ} + v_{τ}) ⊂ U_{2}(w). Moreover, by [16, Theorems 2.3,2.5],

U_{2}(w) = \overline{sp}^{w^{∗}} \{v_{G} : G ∈ SF, G ⊂ Z_{2}(w)\}
⊂ \overline{sp}^{w^{∗}} \{v_{G} : G ∈ SF, G ⊂ Z_{1}(v)\}
⊂ \overline{sp}^{w^{∗}} \{v_{G} : G ∈ SF, S_{v}(G) = −G\} = U_{1}(v).

Then by [19, Cor. 2.3], v ∈ U_{1}(v_{σ}) ∩ U_{1}(v_{τ}) and by JP, v ∈ U_{2}(v_{σ} + v_{τ}) ⊂ U_{2}(w), that is, w ⊢ v, a contradiction. ⊓ ⊔

2.3. Rank 2 faces; spin factor

In this section we assume that Z is a neutral, strongly facially symmetric, locally base normed space satisfying JP.

Lemma 2.7. Let v ∈ M and φ ∈ Z_{1}(v), ∥φ∥ = 1, and suppose that w := v_{φ} is minimal in U_{1}(v). Then either φ is a global extreme point or the midpoint of two orthogonal global extreme points.

Proof. Since w is minimal in U_{1}(v), the face F_{w} ∩ Z_{1}(v) in Z_{1}(v) exposed by w, considered as a geometric tripotent of U_{1}(v), is the single point {φ}. For any element ψ ∈ F_{w}, P_{1}(v)ψ = φ, since for k = 0, 2, ψ_{k}(w) = ⟨P_{k}(v)ψ, w⟩ = ⟨ψ, P_{k}(v)^{∗}w⟩ = 0. Thus every ψ ∈ F_{w} has the form ψ = ψ_{2} + φ + ψ_{0} where ψ_{k} = P_{k}(v)ψ for k = 0, 2.

If F_{w} = {φ}, there is nothing more to prove. So assume otherwise in the rest of this proof. As in the proof of Proposition 2.2(a), F_{w} then contains two orthogonal elements σ = σ_{2} + φ + σ_{0} and τ = τ_{2} + φ + τ_{0}. Further

2 = ∥σ − τ∥ = ∥σ_{2} − τ_{2}∥ + ∥σ_{0} − τ_{0}∥ ≤ ∥σ_{2}∥ + ∥τ_{2}∥ + ∥σ_{0}∥ + ∥τ_{0}∥
= ∥σ_{2} + σ_{0}∥ + ∥τ_{2} + τ_{0}∥ ≤ ∥σ∥ + ∥τ∥ = 2.

This proves ∥σ_{2} + σ_{0}∥ = 1 = ∥τ_{2} + τ_{0}∥, and setting u := v_{σ_{2}} ∈ U_{2}(v) and ũ = v_{σ_{0}} ∈ U_{0}(v) one obtains σ(u + ũ) = ∥σ_{2}∥ + ∥σ_{0}∥ = ∥σ_{2} + σ_{0}∥ = 1 so σ ∈ F_{w} ∩ F_{u+ũ}.

We show next that F_{w} ∩ F_{u+ũ} is the single point {σ}. Suppose to the contrary that F_{w} ∩ F_{u+ũ} is not a singleton. Then, as above, it contains two orthogonal elements σ′ and τ′ with ∥σ′_{2} − τ′_{2}∥ + ∥σ′_{0} − τ′_{0}∥ = 2.

We next claim that

u = v_{σ′_{2}} = v_{τ′_{2}} , ũ(σ′_{0}) = ∥σ′_{0}∥ , ũ(τ′_{0}) = ∥τ′_{0}∥.

Indeed,

1 = ⟨u + ũ, σ′_{2} + σ′_{0}⟩ = σ′_{2}(u)+σ′_{0}(ũ) ≤ |σ′_{2}(u)|+|σ′_{0}(ũ)| ≤ ∥σ′_{2}∥+∥σ′_{0}∥ ≤ 1,

so that σ′_{0}(ũ) = ∥σ′_{0}∥ and σ′_{2}(u) = ∥σ′_{2}∥, and hence v_{σ′_{2}} ≤ u, and v_{σ′_{2}} = u, since u, being a multiple of v, is minimal, and hence indecomposable. The proofs for τ′_{2} and τ′_{0} are similar.

We next show that there are positive numbers λ, μ and an extreme point ρ such that σ′_{2} = λρ and τ′_{2} = μρ. Indeed, σ′_{2} = P_{2}(v)σ′ = σ′(v)f_{v} = σ′_{2}(v)f_{v}, where f_{v} is the extreme point corresponding to v ∈ M, and ∥σ′_{2}∥ = σ′_{2}(u) = σ′_{2}(v)f_{v}(u). Since u is a multiple of v, f_{v}(u) ≠ 0 and

σ′_{2} = \frac{∥σ′_{2}∥}{f_{v}(u)}f_{v} and similarly τ′_{2} = \frac{∥τ′_{2}∥}{f_{v}(u)}f_{v}.

Writing f_{v}(u) = re^{iθ}, we have σ′_{2} = \frac{∥σ′_{2}∥}{r}(e^{−iθ}f_{v}) and τ′_{2} = \frac{∥τ′_{2}∥}{r}(e^{−iθ}f_{v}). Finally, assuming without loss of generality that λ ≥ μ, we have

2 = ∥σ′_{2} − τ′_{2}∥ + ∥σ′_{0} − τ′_{0}∥ = ∥σ′_{2}∥ − ∥τ′_{2}∥ + ∥σ′_{0} − τ′_{0}∥ ≤ 1 + ∥τ′_{0}∥ − ∥τ′_{2}∥,

which implies that τ′_{2} = 0 and ∥τ′_{0}∥ = 1. By neutrality of P_{0}(v), τ′ = τ′_{0} which is a contradiction.

This proves that F_{w} ∩ F_{u+ũ} is a single point {σ} and hence σ = σ_{2}+φ+σ_{0} is a global extreme point. Then so is σ̃ := −S_{v}σ = −σ_{2}+φ−σ_{0} and φ = (σ + σ̃)/2, completing the proof. ⊓ ⊔

We can now prove versions of [18, Prop. 3.2,Lemma 3.6] without assuming our space is atomic. First, we need the following lemma, the conclusion of which is in the hypotheses of [18, Prop. 3.2,Lemma 3.6].

Lemma 2.8. Let v ∈ M, and let w ∈ GT ∩ U_{1}(v). Suppose that w ∉ M. Then F_{w} is a rank 2 face, that is, w = w_{1} + w_{2} where w_{1} and w_{2} are orthogonal minimal geometric tripotents.

Proof. There are two possibilities: (i) w is minimal in U_{1}(v); and (ii) w is not minimal in U_{1}(v).

In case (i), w is the support geometric tripotent for some extreme point φ of the unit ball of Z_{1}(v). Since by assumption φ is not a global extreme point, by Lemma 2.7, φ is the midpoint of two orthogonal global extreme points, and therefore w is the sum of two orthogonal minimal geometric tripotents.

In case (ii), w = w_{1} + w_{2} where w_{1}, w_{2} ∈ GT ∩ U_{1}(v), w_{1} ⋄ w_{2}, and by [19, Cor. 2.3], w_{1}⊤v, w_{2}⊤v, so w_{1}, w_{2} ∈ M by Proposition 2.2(b). ⊓ ⊔

Lemma 2.9. Assume that Z also satisfies FE and STP. Let v ∈ M, and let w ∈ GT ∩ U_{1}(v). Suppose that w ∉ M. Then

(a) If σ and τ are orthogonal elements of F_{w}, then σ and τ are extreme points, σ + τ = f_{w_{1}} + f_{w_{2}} and v_{σ} + v_{τ} = w, where w = w_{1} + w_{2} according to Lemma 2.8.

(b) Each norm exposed face of Z_{1}, properly contained in F_{w}, is a point.

(c) If ρ is an extreme point of F_{w}, then there is a unique extreme point ρ̃ of F_{w} orthogonal to ρ.

(d) With ξ = (f_{w_{1}} + f_{w_{2}})/2, F_{w} = ∪{[ξ, ρ] : ρ ∈ ext F_{w}}, where [ξ, ρ] is the line segment connecting ξ and ρ.

Proof. Case (i). w is minimal in U_{1}(v).

(a) In the proof of Lemma 2.7, it was shown that if F_{w} contains two orthogonal elements, then these elements are global extreme points. Once this is known, the equalities σ + τ = f_{w_{1}} + f_{w_{2}} and v_{σ} + v_{τ} = w follow exactly as in the proof of [18, Prop. 3.2].

(b) Suppose that F_{u} ⊂ F_{w} and F_{u} ≠ F_{w}. By [15, Lemma 2.7] if σ ∈ F_{u}, there exists τ ∈ F_{w} with τ ⋄ σ. Then σ and τ are extreme points. Thus F_{u} consists only of extreme points, and so it contains only one element.

(c) If ρ is an extreme point of F_{w}, then as in the proof of (b), there exists an extreme point ρ̃ ∈ F_{w} orthogonal to ρ. Since by (a), ρ + ρ̃ = f_{w_{1}} + f_{w_{2}}, ρ̃ is unique.

(d) The proof is exactly the same as in [18, Lemma 3.6].

Case (ii). w is not minimal in U_{1}(v).

In the first place, since Z_{1}(v) satisfies JD, and F_{w} ∩ Z_{1}(v) is not a point, it must contain two orthogonal elements g and h with orthogonal supports v_{g} and v_{h} in U_{1}(v). Then by [19, Cor. 2.3], v_{g}⊤v, v_{h}⊤v, so by Proposition 2.2(b), v_{g}, v_{h} ∈ M and g, h are global extreme points. After noting that Z_{1}(v) satisfies FE and STP (by [16, Lemma 2.8,Cor. 4.12]), it now follows exactly as in the proof of case (i) that (a)-(d) hold for the face F_{w} ∩ Z_{1}(v). In particular F_{w} ∩ Z_{1}(v) =

{λρ + (1 − λ)ρ̃: ρ, ρ̃ ∈ F_{w} ∩ Z_{1}(v) ∩ ext Z_{1}, ρ ⋄ ρ̃, v_{ρ} + vρ̃ = w, 0 ≤ λ ≤ 1}.

Now take two orthogonal elements σ, τ ∈ F_{w} and Peirce decompose each one with respect to v:

σ = σ_{2} + σ_{1} + σ_{0} , τ = τ_{2} + τ_{1} + τ_{0}.

Since σ_{1}, τ_{1} ∈ F_{w}, as noted above we may write

σ_{1} = λρ + (1 − λ)ρ̃ , τ_{1} = μϕ + (1 − μ)ϕ̃,

where ρ and ρ̃ are orthogonal global extreme points lying in F_{w} ∩ Z_{1}(v) with v_{ρ} + vρ̃ = w, and similarly for ϕ, ϕ̃.

We can partially eliminate ϕ and ϕ̃ as follows. Since τ_{1} = P_{1}(v)τ = P_{2}(w)P_{1}(v)τ ∈ Z_{2}(w) and w = v_{ρ} + vρ̃, by [18, Lemma 2.3]

τ_{1} = c_{1}ρ + c_{2}ρ̃ + ψ  (2.3)

for scalars c_{1}, c_{2} and ψ ∈ Z_{1}(ρ) ∩ Z_{1}(ρ̃). Since |c_{1}| + |c_{2}| = ∥c_{1}ρ + c_{2}ρ̃∥ = ∥P_{2}(v_{ρ})τ_{1} + P_{0}(vρ̃)τ_{1}∥ ≤ 1 and since 1 = τ_{1}(w) = c_{1} + c_{2} + ψ(w) = c_{1} + c_{2} we have c_{1} + c_{2} = 1 and 0 ≤ c_{1}, c_{2} ≤ 1. Denote c_{1} by c in what follows.

We shall now prove that

τ_{0}, σ_{0} ∈ Z_{1}(v_{ρ}) and v_{τ_{0}}v_{σ_{0}} ∈ M,  (2.4)

and

ψ in (2.3) is zero.  (2.5)

To prove (2.4), note that since v_{ρ} ∈ U_{1}(v), v_{ρ} is compatible with v , so P_{k}(v_{ρ})τ_{0} ∈ Z_{0}(v) for k = 2, 1, 0. Since τ_{2} = P_{2}(v)τ = ⟨τ, v⟩f_{v} and

f_{v} = P_{2}(ρ)f_{v} + P_{1}(ρ)f_{v} + P_{0}(ρ)f_{v}
= ⟨f_{v}, v_{ρ}⟩ρ + P_{1}(ρ)f_{v} + P_{0}(ρ)P_{2}(v)f_{v}
= \overline{⟨ρ, v⟩}ρ + P_{1}(ρ)f_{v} + ⟨P_{0}(ρ)f_{v}, v⟩f_{v}
= P_{1}(ρ)f_{v},

it follows that τ_{2} ∈ Z_{1}(ρ). Moreover, since S_{v_{ρ}}^{∗}w = w, we have S_{v_{ρ}}F_{w} ⊂ F_{w}. Hence

S_{v_{ρ}}τ = −τ_{2} + cρ + (1 − c)ρ̃ − ψ + S_{v_{ρ}}τ_{0} ∈ F_{w},

and therefore

\frac{τ + S_{v_{ρ}}τ}{2} = cρ + (1 − c)ρ̃ + (τ_{0} + S_{v_{ρ}}τ_{0})/2 ∈ F_{w}.

Let τ′ := (τ_{0} + S_{v_{ρ}}τ_{0})/2. We’ll show τ′ = 0. Recall that for any ϕ ∈ Z,

∥P_{1}(v)ϕ + P_{0}(v)ϕ∥ = ∥ − S_{v}[P_{1}(v)ϕ + P_{0}(v)ϕ]∥ = ∥P_{1}(v)ϕ − P_{0}(v)ϕ∥.

Hence, if τ′ ≠ 0, then cρ + (1 − c)ρ̃ − (τ_{0} + S_{v_{ρ}}τ_{0})/2 ∈ F_{w}, whence τ′ ∈ sp_{R} F_{w}, and by the property JD,

τ′/∥τ′∥ = α(ξ_{1} + ξ_{0}) − (1 − α)(η_{1} + η_{0})

with ξ, η ∈ F_{w} and α ∈ [0, 1]. Note here that

ξ_{2} = P_{2}(v)P_{2}(w)ξ = ⟨P_{2}(w)ξ, v⟩f_{v} ∈ Z_{1}(w),

so ξ_{2} = 0 and similarly η_{2} = 0. As in Proposition 2.2(b), this implies α = 1/2, ξ_{1} = η_{1}, ∥ξ_{0} − η_{0}∥ = 2 and ∥ξ_{0}∥ = 1 = ∥η_{0}∥. By neutrality, ξ_{1} = 0 = η_{1}, which contradicts the fact that ξ = ξ_{1} + ξ_{0} ∈ F_{w}. Thus, τ′ = 0, proving that τ_{0} ∈ Z_{1}(v_{ρ}). A similar proof shows that σ_{0} ∈ Z_{1}(v_{ρ}).

Now v_{τ_{0}} ∈ U_{0}(v) ∩ U_{1}(v_{ρ}), and if v_{τ_{0}} ⊢ v_{ρ}, then v_{ρ} ∈ U_{2}(v_{τ_{0}}) ⊂ U_{0}(v), by [16, Cor. 3.4], a contradiction. Now by the two case lemma ([19, Prop. 2.2]), v_{τ_{0}}⊤v_{ρ} and v_{τ_{0}} is a minimal geometric tripotent by Proposition 2.2(b). A similar proof shows that σ_{0} ∈ M. This proves (2.4).

We next prove (2.5). Recall that τ = τ_{2} + cρ + (1 − c)ρ̃ + ψ + τ_{0}, and note that

τ′ := −S_{v}S_{v_{ρ}}τ = τ_{2} + cρ + (1 − c)ρ̃ − ψ + τ_{0},

and −S_{v}S_{v_{ρ}}σ = σ. If we let τ′′ := (τ + τ′)/2 then τ, τ′, τ′′ ∈ F_{w} ∩ σ^{⋄}, in particular ψ = τ′′ − τ′ ∈ σ^{⋄}. Suppose that ψ is not a multiple of a global extreme point. Since ψ ∈ Z_{1}(v_{ρ}) ∩ Z_{1}(vρ̃), and v_{ψ} is not minimal, we have v_{ρ}, vρ̃ ∈ U_{2}(v_{ψ}) and w = v_{ρ} + vρ̃ ∈ U_{2}(v_{ψ}). But v_{σ} ≤ w ∈ U_{2}(v_{ψ}) and v_{σ} ⋄ v_{ψ}, implying v_{σ} ∈ U_{0}(v_{ψ}) ∩ U_{2}(v_{ψ}), a contradiction.

We conclude that ψ = αφ is a multiple of a global extreme point φ. From (2.3), if α ≠ 0, then φ is a difference of two elements of F_{w}, hence an extreme point of (sp_{R} F_{w})_{1}, which implies that φ ∈ F_{w} ∪ F_{−w}. This is a contradiction since ±α = αφ(w) = ψ(w) = ψ(v_{ρ} + vρ̃) = 0. Hence α = 0 proving (2.5).

We next show that F_{w} ∩ {σ}^{⋄} ∩ {τ}^{⋄} = ∅. Suppose there exists a point τ′ lying in F_{w} ∩ {σ}^{⋄} ∩ {τ}^{⋄}. By the above calculations, one member of the set {τ_{1}, (τ′)_{1}, σ_{1}} is a convex combination of the other two. From this it follows exactly as in the proof of Lemma 2.7 that the corresponding convex combination of two elements of the orthogonal set {τ, τ′, σ} is an extreme point, which is a contradiction. Thus F_{w} ∩ {σ}^{⋄} ∩ {τ}^{⋄} = ∅.

We can now complete the proof of (a), and (b)-(d) will follow as in case (i). If F_{v_{τ}} ≠ {τ}, then by JD, F_{v_{τ}} contains two orthogonal elements g, h. But we have proved that in this case F_{w} ∩ {g}^{⋄} ∩ {h}^{⋄} = ∅. However, this set contains σ and this contradiction shows that τ (and by symmetry σ) is an extreme point. This completes the proof of Lemma 2.9. ⊓ ⊔

Once we know the result of Lemma 2.9 above, the proof in [18] shows that the main result of [18] holds with atomic replaced by JD and JP. We formalize this in the next proposition.

Proposition 2.3. Let Z be a neutral strongly facially symmetric locally base normed space which satisfies FE, STP, and JP. If v ∈ M and u ∈ GT ∩ U_{1}(v), then Z_{2}(u) is isometric to the dual of a complex spin factor.

Proof. The argument in [18], from [18, Corollary 3.7] to [18, Theorem 4.16] uses only the following results from [18] and does not otherwise invoke the atomic assumption made there: [18, Prop. 2.9,Cor. 2.11,Prop. 3.2,Lemma 3.6].

On the one hand, [18, Prop. 2.9] and [18, Cor. 2.11] remain true if atomic is replaced there by JD and JP, as shown in our Proposition 2.2(a). On the other hand, [18, Prop 3.2] remains true if atomic is replaced by JD and JP, as shown in our Lemma 2.9(a),(b),(c); and [18, Lemma 3.6] remains true if atomic is replaced by JD and JP, as shown in our Lemma 2.9(d). Thus Proposition 2.3 is proved. ⊓ ⊔

2.4. Atomic decomposition

The following is the main result of this section.

Theorem 2.1. Let Z be a locally base normed neutral strongly facially symmetric space satisfying FE, JP, and the pure state properties STP and ERP. Then Z = Z_{a} ⊕^{ℓ^{1}} N, where Z_{a} and N are strongly facially symmetric spaces satisfying the same properties as Z, N has no extreme points in its unit ball, and Z_{a} is the norm closed complex span of the extreme points of its unit ball.

Proof. If Z has no extreme points in its unit ball, there is nothing to prove. If it has an extreme point, then there exists a maximal family {u_{i}}_{i∈I} of mutually orthogonal minimal geometric tripotents. Let Q := Π_{i∈I}P_{0}(u_{i}) be the contractive projection on Z with Q(Z) = ∩_{i∈I}Z_{0}(u_{i}) guaranteed by Proposition 2.1. We shall show that N := Q(Z) and Z_{a} := (I − Q)(Z) have the required properties. By maximality, N has no extreme points in its unit ball.

For a finite subset A of I and Q_{A} := Π_{i∈A}P_{0}(u_{i}), by JP,

(I − Q_{A})(Z) = Z_{2}(Σ_{A}u_{i}) ⊕ Z_{1}(Σ_{A}u_{i})  (2.6)
= (⊕_{A}Z_{2}(u_{i})) ⊕ (⊕_{i≠j}[Z_{1}(u_{i}) ∩ Z_{1}(u_{j})]) ⊕ (⊕_{A}[Z_{1}(u_{i}) ∩ Z_{0}(Σ_{j≠i}u_{j})]) .

Since I − Q_{A} → I − Q strongly, it follows that every element of (I − Q)(Z) is the norm limit of elements from ∪_{A}(I −Q_{A})(Z). Since obviously Z_{2}(u_{i}) ⋄ Q(Z), in order to prove Z_{a} ⋄ N, it suffices to prove that for every i ∈ I,

Z_{1}(u_{i}) ⋄ Q(Z).  (2.7)

For each i, let Q_{i} = Π_{j∈I−{i}}P_{0}(u_{j}) and for φ ∈ Z_{1}(u_{i}), write φ = Q_{i}φ + (I − Q_{i})φ. Note that

Q_{i}(Z_{1}(u_{i})) = Z_{1}(u_{i}) ∩ [∩_{j∈I−{i}}Z_{0}(u_{j})]

and that

(I − Q_{i})(Z_{1}(u_{i})) is the norm closure of ⊕finite_{j∈I−{i}} [Z_{1}(u_{i}) ∩ Z_{1}(u_{j})].

For the latter, note that for a finite subset A ⊂ I − {i}, if Q_{i,A} denotes the partial product for Q_{i}, then

(I − Q_{i,A})P_{1}(u_{i}) = ∑_{A} P_{2}(u_{j})P_{1}(u_{i}) + ∑_{k≠l} P_{1}(u_{k})P_{1}(u_{l})P_{1}(u_{i})
+ ∑_{A} P_{1}(u_{j})P_{0}(∑_{k≠j} u_{k})P_{1}(u_{i})
= 0 + 0 + ∑_{A} P_{1}(u_{j})P_{1}(u_{i}).

Thus, (I−Q_{i})φ can be approximated in the norm by elements from spaces of the form ⊕_{j∈A}[Z_{1}(u_{i}) ∩ Z_{1}(u_{j})], where A is a finite subset of I − {i}.

Now (2.7) is reduced to proving that Q_{i}(Z_{1}(u_{i})) ⋄ Q(Z) and (I − Q_{i})(Z_{1}(u_{i})) ⋄ Q(Z). Since Z_{1}(u_{i}) ∩ Z_{1}(u_{j}) ⊂ Z_{2}(u_{i} + u_{j}) and Q(Z) ⊂ Z_{0}(u_{i} + u_{j}), it is clear that [Z_{1}(u_{i}) ∩ Z_{1}(u_{j})] ⋄ Q(Z). It remains to show that

(Z_{1}(u_{i}) ∩ [∩_{j∈I−{i}}Z_{0}(u_{j})]) ⋄ Q(Z).  (2.8)

Suppose g ∈ Z_{1}(u_{i})∩[∩_{j∈I−{i}}Z_{0}(u_{j})] and h ∈ Q(Z). Then either v_{g} ⊢ u_{i} or v_{g}⊤u_{i}. In the first case, since by Proposition 2.3, U_{2}(v_{g}) is isometric to a spin factor, there is a minimal geometric tripotent ũ_{i} with ũ_{i} ⋄u_{i} and ũ_{i} ∈ U_{0}(Σ_{j∈I−{i}}u_{j}). This contradicts the maximality. Therefore v_{g} is a minimal geometric tripotent and g is a multiple of an extreme point ψ. If h = h_{2} + h_{1} + h_{0} is the geometric Peirce decomposition of h with respect to v_{g}, then since v_{g} is compatible with all the u_{k}, h_{j} ∈ Q(Z). Now h_{2} is also a multiple of ψ and ψ ∈ Z_{1}(u_{i}); hence h_{2} ∈ Z_{0}(u_{i}) ∩ Z_{1}(u_{i}) = {0}.

Since v_{h_{1}} ∈ U_{1}(v_{g}), either v_{h_{1}} ⊢ v_{g} or v_{h_{1}}⊤v_{g}. In the first case we would have v_{g} ∈ U_{2}(v_{h_{1}}) ⊂ Q(Z), a contradiction. In the second case, h_{1} would be a multiple of ψ, again a contradiction. We conclude that h_{1} = 0 and therefore h = h_{0} ∈ Z_{0}(v_{g}) so that g ⋄ h as required, proving (2.8) and thus the decomposition Z = Z_{a} ⊕^{ℓ^{1}} N.

It is elementary that all the properties of Z transfer to any L-summand. Finally, the set of extreme points of the unit ball of Z which lie in (I − Q)(Z) are norm total in (I − Q)(Z), since every element from the right side of (2.6) is a linear combination of at most two extreme points by Lemmas 2.7 and 2.9(d). ⊓ ⊔

3. Characterization of one-sided ideals in C^{∗}-algebras

3.1. Contractive projections on Banach spaces

An interesting question about general Banach spaces, which is relevant to this paper, is to determine under what conditions the intersection of 1-complemented subspaces is itself 1-complemented. Although this may be true if the contractive projections onto the subspaces form a commuting family, we have been unable to prove it or find it in the literature, without adding some other assumptions. The hypothesis of weak sequential completeness used in Corollary 3.2 and Proposition 3.1 is satisfied in L-embedded spaces, as noted in subsection 1.1.

Lemma 3.1. Let X be a Banach space and let {P_{i}}_{i∈I} be a family of commuting contractive projections on X. Then W := ∩_{i∈I}P_{i}^{∗}(X^{∗}) is the range of a contractive projection on X^{∗}.

Proof. Let F denote the collection of finite subsets of I. For each A ∈ F, let Q_{A} = Π_{i∈A}P_{i}. Since the unit ball B(X^{∗})_{1} is compact in the weak*- operator topology (= point-weak*-topology), there is a subnet {R_{δ}}_{δ∈D} of the net {Q_{A}^{∗}}_{A∈F} converging in this topology to an element R ∈ B(X^{∗})_{1}. Thus R_{δ} = Q_{u(δ)}^{∗}, where u : D → F is a finalizing map (∀A ∈ F, ∃δ_{0} ∈ D, u(δ) ≥ A, ∀δ ≥ δ_{0}), and for every x ∈ X^{∗} and f ∈ X,

⟨Rx, f⟩ = lim_{δ}⟨R_{δ}x, f⟩.

It is now elementary to show that R^{2} = R and Rx = x if and only if x ∈ W. For completeness, we include the details.

For x ∈ X^{∗}, f ∈ X,

⟨R^{2}x, f⟩ = lim_{δ}⟨R_{δ}Rx, f⟩ = lim_{δ}⟨Rx, Q_{u(δ)}f⟩
= lim_{δ} lim_{δ′}⟨R_{δ′}x, Q_{u(δ)}f⟩ = lim_{δ} lim_{δ′}⟨x, Q_{u(δ′)}Q_{u(δ)}f⟩
= lim_{δ′}⟨x, Q_{u(δ′)}f⟩ = lim_{δ′}⟨R_{δ′}x, f⟩ = ⟨Rx, f⟩.

Thus R^{2} = R.

If x ∈ W, then Q_{A}^{∗}x = x for every A ∈ F, so that ⟨Rx, f⟩ = lim_{δ}⟨R_{δ}x, f⟩ = lim_{δ}⟨Q_{u(δ)}^{∗}x, f⟩ = ⟨x, f⟩, so that Rx = x.

Conversely, if Rx = x, then

⟨P_{i}^{∗}x, f⟩ = ⟨P_{i}^{∗}Rx, f⟩ = lim_{δ}⟨R_{δ}x, P_{i}f⟩
= lim_{δ}⟨P_{i}^{∗}Q_{u(δ)}^{∗}x, f⟩ = lim_{δ}⟨R_{δ}x, f⟩ = ⟨Rx, f⟩ = ⟨x, f⟩,

so that x ∈ W. ⊓ ⊔

We cannot conclude from the above proof that ∩_{i∈I}P_{i}(X) is the range of a contractive projection on X. On the other hand, we have the following two immediate consequences.

Corollary 3.1. Let X be a reflexive Banach space, and let {P_{i}}_{i∈I} be a family of commuting contractive projections on X with ranges X_{i} = P_{i}(X). Then Y := ∩_{i∈I}X_{i} is the range of a contractive projection on X.

Corollary 3.2. Let X be a weakly sequentially complete Banach space, and let {P_{i}}_{i∈N} be a sequence of commuting contractive projections on X with ranges X_{i} = P_{i}(X). Then Y := ∩_{i∈N}X_{i} is the range of a contractive projection on X.

Proof. With Q_{n} = P_{1} ⋯ P_{n}, there is a subsequence Q_{n_{k}}^{∗} converging to an element R ∈ B(X^{∗})_{1} in the weak*-operator topology, that is, for x ∈ X^{∗} and f ∈ X, ⟨x, Q_{n_{k}}f⟩ → ⟨Rx, f⟩, so that {Q_{n_{k}}f} is a weakly Cauchy sequence. By assumption, Q_{n_{k}}f converges weakly to an element Sf, and it is elementary to show that R = S^{∗}, and S is a contractive projection on X with range Y. ⊓ ⊔

Proposition 3.1. Let X be a weakly sequentially complete Banach space, and let {P_{i}}_{i∈I} be a family of neutral commuting contractive projections on X with ranges X_{i} = P_{i}(X). Then Y := ∩_{i∈I}X_{i} is the range of a contractive projection on X.

Proof. We note first that for any countable subset λ ⊂ I, by Corollary 3.2, there is a contractive projection Q_{λ} (not necessarily unique). with range ∩_{i∈λ}X_{i} Now, for f ∈ X, define

α_{f} = inf_{λ} inf_{Q_{λ}} ∥Q_{λ}f∥.

There exists a sequence λ^{(n)} and a choice of contractive projection Q_{λ^{(n)}} with α_{f} ≤ ∥Q_{λ^{(n)}}f∥ ≤ α_{f} + 1/n. Set μ = ∪_{n}λ^{(n)} and let Q_{μ} be a contractive projection on X with range ∩_{i∈μ}X_{i}. Since Q_{μ}(X) ⊂ Q_{λ^{(n)}}(X), we have ∥Q_{μ}f∥ = ∥Q_{μ}Q_{λ^{(n)}}f∥ ≤ ∥Q_{λ^{(n)}}f∥ implying α_{f} = ∥Q_{μ}f∥, and so ∥Q_{μ}f∥ ≤ ∥Q_{λ}f∥ for all countable subsets λ of I.

If Q′_{μ} is any other contractive projection with range Q_{μ}(X), then Q_{μ}f = Q′_{μ}Q_{μ}f = Q_{μ}Q′_{μ}f = Q′_{μ}f so that we may unambiguously define an element Qf ∈ ∩_{i∈μ}X_{i} by Qf := Q_{μ}f. By the neutrality of the projections, it follows that Qf ∈ ∩_{i∈I}X_{i}. Indeed, if j ∈ I − μ, then

∥Q_{μ∪{j}}f∥ = ∥P_{j}Q_{μ}f∥ ≤ ∥Q_{μ}f∥ ≤ ∥Q_{μ∪{j}}f∥,

and by the neutrality of P_{j}, P_{j}Q_{μ}f = Q_{μ}f. Hence Qf ∈ ∩_{i∈I}X_{i}. Conversely, if f ∈ ∩_{i∈I}X_{i}, then in particular, f ∈ Q_{μ}(X), so Qf = Q_{μ}f = f.

We have shown that Q is a nonlinear nonexpansive projection of X onto Y. It remains to show that Q is actually linear. For this it suffices to observe that, by neutrality, if Qf = Q_{μ}f, then Qf = Q_{λ}f for any countable set λ ⊃ μ. Then, if f, g ∈ X and Qf = Q_{μ}f, Qg = Q_{ν}g, and Q(f + g) = Q_{σ}(f + g) for suitable countable sets μ, ν, σ of I, then with τ = μ ∪ ν ∪ σ,

Q(f + g) = Q_{τ} (f + g) = Q_{τ} f + Q_{τ} g = Qf + Qg.⊓ ⊔

3.2. Characterization of predual of Cartan factor

In this subsection we show that the entire machinery of [19] can be repeated with appropriate modifications to yield a variation of the main result of [19] to non-atomic facially symmetric spaces satisfying JD, and stated in Proposition 3.2 below. As noted below, the assumption that Z is L-embedded in its second dual needs to be added to the assumptions in [19]. As was done in the proof of Propositiion 2.3, we shall explicitly indicate the modifications needed in [19], section by section, to prove Proposition 3.2.

In the proof of [19, Lemma 1.2] it was stated that the intersection of a certain family of 1-complemented subspaces, is itself 1-complemented. As noted in Section 3.1, this is problematical in general. However, [19, Lemma 1.2] is used in [19] only in the context of a reflexive Banach space, hence it is covered by Corollary 3.1. The role of the assumption of atomic in [19, Proposition 1.5] is to obtain the property expressed in Proposition 1.1(b). But as shown in Theorem 2.1, this property will be available to us. Thus [19, Section 1] is valid with atomic replaced by JD.

By Proposition 2.2(b) and Lemma 2.8 respectively, [19, Proposition 2.4] and [19, Proposition 2.5] remain true with atomic replaced by JD. [19, Corollary 2.7] depends only on [19, Proposition 2.5] and Proposition 2.3, while the part of [19, Lemma 2.8] concerned with the property FE is immediate from [19, Corollary 2.7] and [19, Proposition 2.4]. Finally, [19, Corollary 2.9] is immediate from [18, Proposition 2.9] which, as already remarked in the proof of Proposition 2.3, remains true with atomic replaced by JD (Proposition 2.2(a)). Thus [19, Section 2] is valid with atomic replaced by JD.

The only reliance on atomicity in [19, Section 3] occurs in [19, Lemma 3.2] and [19, Proposition 3.7]. The former depends only on [19, Corollaries 2.7 and 2.9] and the latter on [19, Proposition 1.5], which as just noted, are both valid with atomic replaced by JD. In the proof of [19, Proposition 3.12] it was stated that the intersection of a family of Peirce-0 subspaces of an orthogonal family of minimal geometric tripotents is 1-complemented, and in fact the net of partial products converges strongly to the projection on the intersection. As no proof was provided for this in [19], we provided a proof in Proposition 2.1. Recall that Proposition 2.1 was also the key engredient of the proof of the atomic decomposition in Theorem 2.1 above.

With these remarks we can now assert the following modification of [19, Theorem 3.14], the main result of [19, Section 3].

Lemma 3.2. Let Z be a neutral locally base normed SFS space and assume the pure state properties ERP and STP, and the properties FE and JP. Assume that there exists a minimal geometric tripotent v with U_{1}(v) of rank 1 and a geometric tripotent u with u ⊢ v. Then U has an Msummand which is linearly isometric with the complex JBW^{∗}-triple of all symmetric “matrices” on a complex Hilbert space (Cartan factor of type 3). In particular, if Z is irreducible, then Z^{∗} is isometric to a Cartan factor of type 3.

The only possible reliance on atomicity in [19, Section 4] occurs in [19, Lemma 4.9] and [19, Proposition 4.11]. The former depends only on [18, Lemma 3.6], which is valid in the presence of JD by Lemma 2.9(d), and the latter on [19, Lemma 1.2], which as noted above is needed only for reflexive Banach spaces. However, [19, Lemma 4.9] states explicitly that reflexivity. Note that the “classification scheme”, embodied in [19, Proposition 4.20] does not involve atomic so is valid in the presence of JD.

The only reliance on atomicity in [19, Section 5] occurs in [19, Lemma 5.2], which depends on [18, Corollary 2.11]. As already noted, the latter is valid in the presence of JD. In the proof of [19, Lemma 5.5] it was stated that the intersection of a family of Peirce-0 subspaces of a family of geometric tripotents which are either orthogonal or collinear is 1complemented, and in fact the net of partial products converges strongly to the projection on the intersection. As no proof was provided for this in [19], we provided a proof of the 1-complementedness of the intersection in Proposition 3.1. This is the only place in this paper and one of two places in [19] where the assumption of L-embeddedness is used. Although it is problematical whether the strong convergence of the partial products exists, nevertheless, it is sufficient to take a subnet of the net of partial sums in the proof of [19, Lemma 5.5]. The same remark applies to [19, Lemma 6.6].

With these remarks we can now assert the following modification of [19, Theorem 5.10], the main result of [19, Section 5].

Lemma 3.3. Let Z be a neutral locally base normed SFS space of spin degree 4, which is L-embedded and which satisfies FE,STP,ERP, and JP. Then Z has an L-summand which is linearly isometric to the predual of a Cartan factor of type 1. In particular, if Z is irreducible, then Z^{∗} is isometric to a Cartan factor of type 1.

The only reliance on atomicity in [19, Section 6] occurs in [19, Lemma 6.2]. However, this dependence is on earlier results which have been established in the presence of JD. As noted above for [19, Lemma 5.5], [19, Lemma 6.6] holds under the assumption of L-embeddedness.

With these remarks we can now assert the following modification of [19, Theorem 6.8], the main result of [19, Section 6].

Lemma 3.4. Let Z be a neutral locally base normed SFS space of spin degree 6, which is L-embedded and which satisfies FE,STP,ERP, and JP. Then Z has an L-summand which is linearly isometric to the predual of a Cartan factor of type 2. In particular, if Z is irreducible, then Z^{∗} is isometric to a Cartan factor of type 2.

The results of [19, Sections 7 and 8] carry over verbatim in the presence of JD. The proof of [19, Theorem 7.1] on pages 75–79 of [19] yields the following modification.

Lemma 3.5. Let Z be a neutral locally base normed SFS space which satisfies FE, STP, ERP, and JP, and let v, ṽ be orthogonal minimal geometric tripotents in U := Z^{∗} such that the dimension of U_{2}(v + ṽ) is 8 and U_{1}(v + ṽ) ≠ {0}. Then there is an L-summand of Z which is isometric to the predual of a Cartan factor of type 5, i.e., the 16 dimensional JBW^{∗}-triple of 1 by 2 matrices over the Octonions. In particular, if Z is irreducible, then Z^{∗} is isometric to the Cartan factor of type 5.

Similarly, the proof of [19, Theorem 7.8] appearing on pages 79–82 of [19] yields the following modification.

Lemma 3.6. Let Z be a neutral locally base normed SFS space of spin degree 10 which satisfies FE,STP,ERP, and JP, and has no L-summand of type I_{2}. Then Z contains an L-summand which is isometric to the predual of a Cartan factor of type 6, i.e., the 27 dimensional JBW^{∗}- triple of all 3 by 3 hermitian matrices over the Octonions. In particular, if Z is irreducible, then Z^{∗} is isometric to the Cartan factor of type 6.

Finally, the proof of [19, Theorem 8.2] on pages 83–84 of [19] yields the following modification.

Proposition 3.2. Let Z be a neutral locally base normed strongly facially symmetric space satisfying FE, STP, ERP, which is L-embedded and which satisfies JP. For any minimal geometric tripotent v in U, there is an L-summand J(v) of Z isometric to the predual of a Cartan factor of one of the types 1-6 such that v̂ ∈ J(v). If Z is the norm closure of the complex linear span of its extreme points, then it is isometric to the predual of an atomic JBW^{∗}-triple.

3.3. Spectral duality and Characterization of dual ball of JB^{∗}-triple

If Z is an L-embedded, locally base normed, neutral strongly facially symmetric space satisfying FE, JP and the pure state properties, then by Proposition 3.2 and Theorem 2.1, its dual Z^{∗} is a direct sum Z^{∗} = (Z_{a})^{∗} ⊕^{ℓ^{∞}} N^{∗} where (Z_{a})^{∗} is isometric to an atomic JBW^{∗}-triple. We shall identify (Z_{a})^{∗} with this JBW^{∗}-triple in what follows.

Lemma 3.7. Suppose that Z is as above and assume that Z is the dual of a Banach space B. For a ∈ B, if â denotes the canonical image of a in Z^{∗}, and Q is the projection of Z^{∗} onto (Z_{a})^{∗}, then ∥Qâ∥ = ∥a∥.

Proof. For a ∈ B with ∥a∥ = 1, let g be an extreme point of the nonempty convex w*-compact set {f ∈ Z : ∥f∥ = 1 = f(a)}. Then g ∈ ext Z_{1}, so g vanishes on N^{∗}. Thus

1 = ∥a∥ = ∥â∥ ≥ ∥Qâ∥ ≥ |⟨Qâ, g⟩| = |⟨â, g⟩| = |⟨g, a⟩| = 1.⊓ ⊔

In order to show that the space B is isometric to a JB^{∗}-triple, it suffices to show that the image of the map a ↦ Qâ is closed under the cubing operation in (Z_{a})^{∗}, and is hence a subtriple of (Z_{a})^{∗}. To show this we need a spectral assumption on the elements of B. To make this definition, we need a lemma.

Lemma 3.8. Let Z be a neutral WFS space satisfying PE. Let {F_{B} : B ∈ B} be a family of norm closed faces of Z_{1}, where B denotes the set of non-empty Borel subsets of the closed interval [a, b].

(a) Suppose that (i) if B_{1} ∩ B_{2} = ∅, then F_{B_{1}} ⋄ F_{B_{2}} and v_{B_{1}∪B_{2}} = v_{B_{1}} + v_{B_{2}}. For f ∈ C[a, b], if P = {s_{0}, …, s_{n}} is a partition of [a, b] and T = {t_{1}, …, t_{n}} are points with s_{i−1} ≤ t_{i} ≤ s_{i}, the Riemann sums S(P, T, f) = ∑_{1}^{n} f(t_{j})v_{(s_{j−1},s_{j}]} converge in norm to an element ∫ f dv_{B} = ∫ f(t) dv_{B}(t) of Z^{∗} as the mesh |P | = min{s_{j} − s_{j−1}} → 0.

(b) Assume that Z is the dual of a Banach space Z_{∗} , that (i) holds, with [a, b] = [0, ∥x∥] for some x ∈ Z_{∗}, and suppose that x satisfies the further conditions: (ii) ⟨x, F_{B}⟩ ⊂ B for each interval B ∈ B; (iii) S_{F_{B}}^{∗}x = x for B ∈ B; (iv) ⟨x, F_{(0,∥x∥]}^{⋄}⟩ = 0. Then x = ∫ t dv_{B}(t).

Proof. For the proof of (a), it suffices to show that for every ϵ > 0, there is a δ > 0, such that

∥S(P, T, f) − S(P′, T′, f)∥ < ϵ if |P |, |P′| < δ.  (3.1)

By the uniform continuity of f, let δ > 0 correspondence to a tolerance of ϵ/2. If |P |, |P′| < δ, then S(P, T, f) − S(P ∪ P′, T′′, f), where T′′ is any selection of points, is of the form ∑_{1}^{m} α_{j}v_{j}, where |α_{j}| < ϵ/2 and v_{1}, …, v_{m} are orthogonal geometric tripotents. Thus

∥S(P, T, f) − S(P ∪ P′, T′′, f)∥ = max_{j} |α_{j}| < ϵ/2

and (3.1) follows.

For the proof of (b), it suffices to prove that x is the weak*-limit of the Riemann sums corresponding to f_{0}(t) := t, for by (a), x will also be the norm limit. In what follows, F_{(0,∥x∥]} will be denoted by F. By (iii) and (iv)

⟨x, Z_{1}(F) + Z_{0}(F)⟩ = 0.

Also, each Riemann sum ∑ t_{j}v_{(s_{j−1},s_{j}]} ∈ U_{2}(F), so

⟨∑ t_{j}v_{(s_{j−1},s_{j}]}, Z_{1}(F) + Z_{0}(F)⟩ = 0.

Since Z_{2}(F) = sp_{C}F, it suffices to prove that for every ψ ∈ F,

⟨x − S(P, T, f_{0}), ψ⟩ → 0 as |P | → 0.

Since v_{F} = ∑ v_{i} where v_{i} = v_{(s_{i−1},s_{i}]}, if ψ ∈ F ⊂ ⊕_{i}Z_{2}(v_{i})⊕⊕_{i≠j}[Z_{1}(v_{i})∩ Z_{1}(v_{j})], then

1 = ⟨v_{F} , ψ⟩ = ⟨v_{F} , ∑ P_{2}(v_{i})ψ + ∑_{i≠j} P_{1}(v_{i})P_{1}(v_{j})ψ⟩
= ∑⟨v_{i}, P_{2}(v_{i})ψ⟩ ≤ ∑ ∥P_{2}(v_{i})ψ∥ = ∥ ∑ P_{2}(v_{i})ψ∥ ≤ ∥ψ∥ = 1.

Therefore

ψ = ∑ ∥P_{2}(v_{i})∥\frac{P_{2}(v_{i})}{∥P_{2}(v_{i})∥} + ∑_{i≠j} P_{1}(v_{i})P_{1}(v_{j})ψ
∈ co (F_{v_{1}} ∪ ⋯ ∪ F_{v_{n}}) + ⊕_{i≠j}[Z_{1}(v_{i}) ∩ Z_{1}(v_{j})].

By (iii), ⟨x, Z_{1}(F_{B})⟩ = 0 for every B ∈ B. Therefore ⟨x, ψ⟩ = ⟨x, ∑ λ_{i}ψ_{i}⟩, where ψ_{i} ∈ F_{v_{i}}, λ_{i} ≥ 0, ∑ λ_{i} = 1. Also, ⟨S(P, T, f_{0}), ψ⟩ = ⟨∑ t_{i}v_{i}, ∑ λ_{j}ψ_{j}⟩ = ∑ t_{i}λ_{i}.

By (ii), ⟨x, ψ_{i}⟩ ∈ (s_{i−1}, s_{i}], so

|⟨x − S(P, T, f_{0}), ψ⟩| = | ∑ λ_{i}(⟨x, ψ_{i}⟩ − t_{i})| ≤ |P |.

The lemma is proved. ⊓ ⊔

Let us observe that if Z is the dual of a JB^{∗}-triple A, then each element x ∈ A satisfies the conditions (i)-(iv) of Lemma 3.8. Indeed, if C denotes the JB*-subtriple of A generated by x, then C is isometric to a commutative C*-algebra and consists of norm limits of elements p(x) where p is an odd polynomial on (0, ∥x∥], cf. [22, 1.15] and [5, p. 438]; and if W denotes the JBW^{∗}-triple generated by x in A^{∗∗}, then W is a commutative von Neumann algebra. Thus, if x = w|x| is the polar decomposition of x in W, and |x| = ∫ λ de_{λ} is the spectral decomposition of |x| in W, and the face F_{B} is defined as the face exposed by the tripotent we(B) ∈ A^{∗∗}, then the family {F_{B} : B ∈ B} satisfies (i), as shown in [21, Theorem 3.2]. It also follows from [21, Theorem 3.2] that for every ϵ > 0, there is a partition of [0, ∥x∥] such that ∥x − ∑ t_{j}v_{(s_{j−1},s_{j}]}∥ < ϵ. If B is a subinterval of [0, ∥x∥], and ρ ∈ F_{B}, then with v_{j} = v_{(s_{j−1},s_{j}]}, B_{j} = B ∩ (s_{j−1}, s_{j}], there exist ρ_{k} ∈ F_{B_{k}} (if B_{k} ≠ ∅) and λ_{k} ≥ 0 with ∑ λ_{k} = 1 such that ⟨x, ρ⟩ is approximated by

⟨∑ t_{j}v_{j}, ∑_{B_{k}≠∅} λ_{k}ρ_{k}⟩ = ∑_{B_{j}≠∅} t_{j}λ_{j} ∈ co (∪_{B_{j}≠∅}B_{j}),

proving (ii). Again, using [21, Theorem 3.2] we shall show that (iii) and (iv) hold. Since x is approximated in norm by ∑ t_{j}v_{j}, where v_{j} = v_{(s_{j−1},s_{j}]}, to prove (iii), it suffices to prove that v_{B}v_{B}^{∗}v_{j}v_{B}^{∗}v_{B} = v_{B}v_{B}^{∗}v_{j} = v_{j}v_{B}^{∗}v_{B}. Since v_{B} = ∑ v_{B_{j}} where B_{j} = B ∩ (s_{j−1}, s_{j}], it is trivial to check that each of the terms v_{B}v_{B}^{∗}v_{j}v_{B}^{∗}v_{B}, v_{B}v_{B}^{∗}v_{j}, v_{j}v_{B}^{∗}v_{B} collapses to v_{B_{j}}. Since the support of the spectral measure of |x| lies in [0, ∥x∥], (iv) also holds.

There is another property of elements of a JB^{∗}-triple that we need to incorporate into our definition. It is based on the following observation. If x is an element of a JB^{∗}-triple A, let f(x) denote the element of C which is the norm limit of odd polynomials p_{n} which converge uniformly to f ∈ C_{0}([0, ∥x∥), and let f̃(x) = ∫ f(λ) de_{λ}. Since p_{n}(x) = p̃_{n}(x),

f(x) − f̃(x) = f(x) − p_{n}(x) + p̃_{n}(x) − ∑ p_{n}(t_{k})v_{k}
+ ∑ p_{n}(t_{k})v_{k} − ∑ f(t_{k})v_{k} + ∑ f(t_{k})v_{k} − f̃(x),

which shows that f̃(x) = f(x) ∈ A.

Definition 3.1. A strongly facially symmetric space Z with a predual Z_{∗} is strongly spectral if, for every element x ∈ Z_{∗}, there exists a family {F_{B} : B ∈ B} of norm closed faces of the closed unit ball Z_{1}, where B is the set of nonempty Borel subsets of (0, ∥x∥], satisfying (i)-(iv) in Lemma 3.8 and which also satisfies

(v) For every f ∈ C_{0}(0, ∥x∥), the element ∫ f dv_{B} is weak*-continuous, that is, lies in Z_{∗}.

Although somewhat complicated, this condition is precisely the analogue of a strongly spectral compact base K of a base normed space V given by Alfsen and Shultz in [1]. There it is given simply as the condition that in the order unit space V_{∗} each element a decomposes as an orthogonal difference a_{+} − a_{−} of two positive elements. Here orthogonal means that a_{+} and a_{−} are supported on real spans of orthogonal faces of K. Since V_{∗} is unital, the unit may be used together with a and this property to carve out an orthogonal collection of faces similar to the one above, and a lattice of orthogonal elements of V_{∗} which generate a space which is isometric to a full space of continuous functions, and hence closed under the continuous functional calculus. Since there is no unit in our space Z_{∗}, we must assume that elements x ∈ Z_{∗} may be decomposed in the above fashion, and that the resulting continuous functional calculus operates in Z_{∗}. Note that this is entirely a linear property, and has obvious quantum mechanical significance. The faces F_{B} are the states corresponding to observations of some value in B for the observable x. The probability of this happening for a state ψ is |ψ(v_{B})|.

We now have the following characterizations of JB*-triples. In this characterization, the property JP must hold for all orthogonal faces, not just extreme points. Thus it simply says that the (necessarily commutative) product of the symmetries S_{F} and S_{G} corresponding to orthogonal faces F and G is S_{F ∨G}.

Theorem 3.1. A Banach space X is isometric to a JB*-triple if and only if X^{∗} is an L-embedded, locally base normed, strongly spectral, strongly facially symmetric space which satisfies FE, JP, and the pure state properties STP and ERP.

Before proving this theorem, we require one more lemma.

Lemma 3.9. Let X be as in Theorem 3.1, Z := X^{∗} and let Ψ (resp. Ψ^{⋄}) denote the projection of Z onto its atomic part Z_{a} (resp. nonatomic part Z_{n}) given by Theorem 2.1. For any norm exposed face G ⊂ Z_{1}, G_{a} := Ψ(G) ∩ ∂Z_{1} and G_{n} := Ψ^{⋄}(G) ∩ ∂Z_{1} are faces in Z_{a} and Z_{n} respectively, and

G = co (G_{a} ∪ G_{n}).  (3.2)

Moreover, writing G = F_{w} for some geometric tripotent w, then Ψ^{∗}w is a geometric tripotent, and

F_{Ψ^{∗}w} = G_{a}.  (3.3)

Proof. To show that G_{a} is a face in (Z_{a})_{1}, let λρ + (1 − λ)σ ∈ G_{a} where ρ, σ ∈ (Z_{a})_{1}. Then λρ + (1 − λ)σ = Ψf for some f ∈ G, and f = λρ + (1 − λ)σ + f_{n}. Since ∥f∥ = 1 = ∥λρ + (1 − λ)σ∥, f_{n} = 0 and ρ, σ ∈ G, ∥ρ∥ = 1 = ∥σ∥, and ρ ∈ G ∩ Z_{a}, proving that G_{a} is a face. Similarly for G_{n}.

If f ∈ G has decomposition f = f_{a} + f_{n} = ∥f_{a}∥\frac{f_{a}}{∥f_{a}∥} + ∥f_{n}∥\frac{f_{n}}{∥f_{n}∥}, then

since G is a face, \frac{f_{a}}{∥f_{a}∥}, \frac{f_{n}}{∥f_{n}∥} ∈ G. This proves ⊂ in (3.2). If g_{a} := Ψg ∈ ΨG ∩ ∂Z_{1} for some g ∈ G, then ∥g_{a}∥ = 1 so g = g_{a} = Ψg ∈ G. A similar argument for Ψ^{⋄}(G) ∩ ∂Z_{1} proves ⊃ in (3.2).

To prove (3.3), let g ∈ Ψ(G)∩∂Z_{1}. Then ⟨g, Ψ^{∗}w⟩ = ⟨g, w⟩ = 1 so that g ∈ F_{Ψ^{∗}w}. On the other hand, if g ∈ F_{Ψ^{∗}w}, then 1 = ∥g∥ = ⟨g, Ψ^{∗}w⟩ = ⟨Ψg, w⟩ so that Ψg ∈ F_{w}. Since g = Ψg + Ψ^{⋄}g and ∥g∥ = ∥Ψg∥, Ψ^{⋄}g = 0, Ψg = g and g ∈ Ψ(G) ∩ ∂Z_{1}.

It remains to show that Ψ^{∗}w is a geometric tripotent, that is,

⟨Ψ^{∗}w, (G_{a})^{⋄}⟩ = 0.

Note first that G^{⋄} = G_{a}^{⋄} ∩ G_{n}^{⋄} by (3.2). If ρ ∈ G_{a}^{⋄}, ⟨Ψ^{∗}w, ρ⟩ = ⟨w, Ψ(ρ)⟩ and this will be zero if Ψ(ρ) ∈ G^{⋄}. To prove this, first let σ ∈ G_{a}. Then ρ ⋄ σ, hence Ψ(ρ) ⋄ Ψ(σ) and since Ψ(σ) = σ, Ψ(ρ) ∈ G_{a}^{⋄}. Then Ψ(ρ) ∈ G_{a}^{⋄} ∩ G_{n}^{⋄} = G^{⋄} as required. ⊓ ⊔

Proof of Theorem 3.1. Assume that Z = X^{∗} is a strongly facially symmetric space satisfying the hypotheses of the theorem. Suppose x is an element of X = Z_{∗}. By the spectral axiom and Lemma 3.8, there is an element y ∈ X such that for ϵ > 0 there exists δ > 0 such that, with f_{0}(t) = t and f_{1}(t) = t^{3},

∥x − S(P, T, f_{0})∥ < ϵ and ∥y − S(P′, T′, f_{1})∥ < ϵ

for all partitions P, P′ with mesh less than δ. Fix a common partition P = {s_{0}, …, s_{n}} with |P | < δ, and write v_{i} = v_{(s_{i−1},s_{i}]} and (v_{i})_{a} = Ψ^{∗}(v_{i}). Then by (3.3),

∥Ψ^{∗}(x̂) − ∑ t_{i}(v_{i})_{a}∥ < ϵ and ∥Ψ^{∗}(ŷ) − ∑ t_{i}^{3}(v_{i})_{a}∥ < ϵ.

Since in a JB^{∗}-triple, ∥{aaa}−{bbb}∥ ≤ ∥a−b∥(∥a∥^{2} +∥a∥∥b∥+∥b∥^{2}), and since the (v_{i})_{a} are orthogonal tripotents in the JBW^{∗}-triple (Z_{a})^{∗}, we have

∥{Ψ^{∗}(x̂), Ψ^{∗}(x̂), Ψ^{∗}(x̂)} − ∑ t_{i}^{3}(v_{i})_{a}∥ < 3ϵ∥x∥^{2},

and therefore ∥{Ψ^{∗}(x̂), Ψ^{∗}(x̂), Ψ^{∗}(x̂)} − Ψ^{∗}(y)∥ < ϵ(3∥x∥^{2} + 1). It follows that Ψ(X̂) is a norm closed subspace of the JBW*-triple (Z_{a})^{∗} that is closed under the cubing operation. Hence Ψ(X) is a subtriple of (Z_{a})^{∗} as required.

The converse, that the dual Z of a JB*-triple is a strongly facially symmetric space satisfying the conditions of the theorem, has already been mentioned above. That the spectral axiom is satisfied was shown preceding Definition 3.1. The proofs that it is a strongly facially symmetric locally base normed space can be found in [17], the proofs that it satisfies the pure state properties can be found in [13], the proof of the L-embeddedness can be found in [6], and the proof of FE can be found in [10]. ⊓ ⊔

We can restate Theorem 3.1 from another viewpoint as follows: for a Banach space X, its open unit ball is a bounded symmetric domain if and only if X^{∗} is an L-embedded, locally base normed, strongly spectral, neutral strongly facially symmetric space which satisfies FE, JP, and the pure state properties STP and ERP.

3.4. One-sided ideals in C^{∗}-algebras

Proposition 3.4 and Theorem 3.2 below, together with Theorem 1.2, give facial and linear operator space characterizations of C*-algebras and left ideals of C*-algebras. This work was inspired by [7], in which Theorem 1.2 is used to characterize left ideals as TRO’s which are simultaneously abstract operator algebras with right contractive approximate unit.

We start by motivating the main result of this subsection. Recall that a TRO is made into a JB^{∗}-triple by symmetrizing the ternary product.

Remark 3.1. If J is a closed left ideal in a C*-algebra and J possesses a

right identity e of norm 1, then J is a TRO and E := [ \begin{matrix}0 \\ e \end{matrix} ] is a maximal

partial isometry in M_{2,1}(J), that is, P_{0}(E) = 0.

Proof. By a remark of Blecher (see [7, Lemma 2.9]), xe^{∗} = x for all x ∈ J, so that x = xe^{∗}e and in particular, e is a partial isometry, and so is E.

For [ \begin{matrix}x \\ y \end{matrix} ] ∈ M_{2,1}(J),

P_{0}(E) [ \begin{matrix}x \\ y \end{matrix} ] = (I − EE^{∗}) [ \begin{matrix}x \\ y \end{matrix} ] (I − E^{∗}E)
= [ \begin{matrix}1 & 0 \\ 0 & 1 − ee^{∗} \end{matrix} ] [ \begin{matrix}x \\ y \end{matrix} ] (I − e^{∗}e)
= [ \begin{matrix}x(1 − e^{∗}e) \\ (1 − ee^{∗})y(1 − e^{∗}e) \end{matrix} ] = 0.⊓ ⊔

Conversely, we have the following.

Proposition 3.3. Let A be a TRO. Suppose there is a norm one element x in A such that the face in (M_{2,1}(A)^{∗})_{1} exposed by

X := [ \begin{matrix}0 \\ x \end{matrix} ] ∈ M_{2,1}(A)

is maximal. Then A is completely isometric to a left ideal in a C*-algebra, which ideal contains a right identity element.

Proof. Let B = M_{2,1}(A). If V is the partial isometry in B^{∗∗} such that F_{X} = F_{V}, then X = V + P_{0}(V)^{∗}X = V, so that x is a partial isometry in A, which we denote by v.

We next prove that v is a right unitary in A; that is, x = xv^{∗}v, for all x ∈ A. Indeed, for x ∈ A,

D(V) [ \begin{matrix}x \\ 0 \end{matrix} ] = \frac{1}{2} ([ \begin{matrix}0 \\ v \end{matrix} ] [ \begin{matrix}0 \\ v \end{matrix} ]^{∗} [ \begin{matrix}x \\ 0 \end{matrix} ] + [ \begin{matrix}x \\ 0 \end{matrix} ] [ \begin{matrix}0 \\ v \end{matrix} ]^{∗} [ \begin{matrix}0 \\ v \end{matrix} ])
= [ \begin{matrix}xv^{∗}v/2 \\ 0 \end{matrix} ] ,

and

P_{2}(V) [ \begin{matrix}x \\ 0 \end{matrix} ] = [ \begin{matrix}0 \\ v \end{matrix} ] [ \begin{matrix}0 \\ v \end{matrix} ]^{∗} [ \begin{matrix}x \\ 0 \end{matrix} ] [ \begin{matrix}0 \\ v \end{matrix} ]^{∗} [ \begin{matrix}0 \\ v \end{matrix} ] = 0.

Since P_{1}(V) [ \begin{matrix}x \\ 0 \end{matrix} ] = [ \begin{matrix}x \\ 0 \end{matrix} ],

[ \begin{matrix}xv^{∗}v/2 \\ 0 \end{matrix} ] = D(V) [ \begin{matrix}x \\ 0 \end{matrix} ] = P_{2}(V) [ \begin{matrix}x \\ 0 \end{matrix} ] + \frac{1}{2}P_{1}(V) [ \begin{matrix}x \\ 0 \end{matrix} ] = \frac{1}{2} [ \begin{matrix}x \\ 0 \end{matrix} ] .

We next show that the map ψ : a ↦ av^{∗} is a complete isometry of A onto a closed left ideal J of the C*-algebra \overline{AA^{∗}} and vv^{∗} is a right identity of J. In the first place, since ∥ψ(x)∥^{2} = ∥xv^{∗}∥^{2} = ∥(xv^{∗})(xv^{∗})^{∗}∥ = ∥xv^{∗}vx^{∗}∥ = ∥xx^{∗}∥, ψ is an isometry. By the same argument, with W = diag (v, v, …, v), for X ∈ M_{n}(A), ∥XW^{∗}∥ = ∥X∥, so that ψ is a complete isometry.

If c ∈ \overline{AA^{∗}} is of the form c = ab^{∗} with a, b ∈ A, and y ∈ J := ψ(A), say y = xv^{∗}, then cy = ab^{∗}xv^{∗} ∈ Av^{∗} = J. By taking finite sums and then limits, J is a left ideal in C. Finally, with e = vv^{∗} and y = xv^{∗} ∈ J, ye = xv^{∗}vv^{∗} = xv^{∗} = y. ⊓ ⊔

For the general case we have the following result.

Theorem 3.2. Let A be a TRO. Then A is completely isometric to a left ideal in a C*-algebra if and only if there exists a convex set C = {x_{λ} : λ ∈ Λ} ⊂ A_{1} such that the collection of faces

F_{λ} := F_{[ \begin{matrix}0 \\ x_{λ}/∥x_{λ}∥ \end{matrix} ]} ⊂ M_{2,1}(A)^{∗},

form a directed set with respect to containment, F := sup_{λ} F_{λ} exists, and

(a) The set \{[ \begin{matrix}0 \\ x_{λ} \end{matrix} ] : λ ∈ Λ\} separates the points of F;

(b) F^{⊥} = 0 (that is, the partial isometry V ∈ (M_{2,1}(A))^{∗∗} with F = F_{V} is maximal);

(c) ⟨F, [ \begin{matrix}0 \\ x_{λ} \end{matrix} ]⟩ ≥ 0 for all λ ∈ Λ;

(d) S_{F}^{∗} ([ \begin{matrix}0 \\ x_{λ} \end{matrix} ]) = [ \begin{matrix}0 \\ x_{λ} \end{matrix} ] for all λ ∈ Λ.

Proof. We first assume that we have a closed left ideal L in a C*-algebra B. In this part of the proof, to avoid confusion with dual spaces, we denote the involution in B by x^{♯}. The set of positive elements of the open unit ball of the C*-algebra L ∩ L^{♯}, which we will denote by (u_{λ})_{λ∈Λ}, is a contractive right approximate unit for L. Let u = w^{∗}-lim u_{λ} ∈ B^{∗∗}. Identifying L^{∗∗} with B^{∗∗}u, we now verify the properties (a)-(d).

For each λ, u_{λ}/∥u_{λ}∥ = v_{λ} + v_{λ}^{0} where v_{λ} = w^{∗}- lim(u_{λ}/∥u_{λ}∥)^{n} is the support projection of u_{λ}/∥u_{λ}∥, that is, F_{u_{λ}/∥u_{λ}∥} = F_{v_{λ}} ⊂ B^{∗}, and v_{λ}^{0} is an element orthogonal to v_{λ}. Since u_{λ} ↑ u, u = sup_{λ} r(u_{λ}/∥u_{λ}∥), where r(u_{λ}/∥u_{λ}∥) is the range projection of u_{λ}/∥u_{λ}∥. For each fixed μ ∈ Λ, we apply the functional calculus to u_{μ}/∥u_{μ}∥ as follows. Let f_{n}(0) = 0, f_{n}(t) = 1 on [1/n, 1] and linear on [0, 1/n]. Then f_{n}(u_{μ}/∥u_{μ}∥) ∈ (L ∩ L^{♯})_{1}^{+} and so as above f_{n}(u_{μ}/∥u_{μ}∥) = v_{λ(μ,n)} + v_{λ(μ,n)}^{0} and sup_{n} v_{λ(μ,n)} = r(u_{μ}/∥u_{μ}∥). Therefore

u = sup_{μ} r(u_{μ}/∥u_{μ}∥) = sup_{μ} sup_{n} v_{λ(μ,n)} ≤ sup_{λ} v_{λ} = v say.

On the other hand, since v_{λ} ≤ (1 + \frac{1−∥u_{λ}∥}{∥u_{λ}∥})u, it follows that v ≤ u and therefore u = v.

It is clear that

F_{λ} = F_{[ \begin{matrix}0 \\ u_{λ}/∥u_{λ}∥ \end{matrix} ]} = F_{[ \begin{matrix}0 \\ v_{λ} \end{matrix} ]} ⊂ F_{[ \begin{matrix}0 \\ u \end{matrix} ]},

and therefore that sup_{λ} F_{λ} exists. We show that it equals F_{[ \begin{matrix}0 \\ u \end{matrix} ]}. Suppose

that for some a, b ∈ B^{∗∗}, F_{λ} ⊂ F_{[ \begin{matrix}a \\ b \end{matrix} ]} for every λ. This is equivalent to

[ \begin{matrix}0 \\ v_{λ} \end{matrix} ] = Q([ \begin{matrix}0 \\ v_{λ} \end{matrix} ]) [ \begin{matrix}a \\ b \end{matrix} ] = [ \begin{matrix}0 \\ v_{λ}b^{∗}v_{λ} \end{matrix} ] ,

or v_{λ}b^{∗}v_{λ} = v_{λ}. On the other hand, since v_{λ}^{0} = u_{λ}/∥u_{λ}∥ − v_{λ} → 0,

u_{λ}b^{∗}u_{λ} = ∥u_{λ}∥^{2}(v_{λ} + v_{λ}^{0})b^{∗}(v_{λ} + v_{λ}^{0}) → u,

so that ub^{∗}u = u and as above, F_{[ \begin{matrix}0 \\ u \end{matrix} ]} ⊂ F_{[ \begin{matrix}a \\ b \end{matrix} ]}, proving that sup_{λ} F_{λ} =

F_{[ \begin{matrix}0 \\ u \end{matrix} ]}.

Let us now prove (a). Since u_{λ} ↑ u, the convergence is strong convergence. We claim first that B ∩ B_{2}^{∗∗}(u) is weak*-dense in B_{2}^{∗∗}(u). Indeed with x ∈ B_{2}^{∗∗}(u) of norm 1, there is a net b_{α} ∈ B with b_{α} → x strongly. Then u_{λ}b_{α}u_{μ} → uxu = x, and since u_{μ}u = u_{μ}r(u_{μ})u = u_{μ}r(u_{μ}) = u_{μ}, u_{λ}b_{α}u_{μ} ∈ B ∩ B_{2}^{∗∗}(u), proving the claim. We claim next that L ∩ L^{♯} = B_{2}^{∗∗}(u) ∩ B. If y ∈ L ∩ L^{♯}, then y = bu = uc^{♯} for some b, c ∈ B^{∗∗}, hence y = uyu ∈ B_{2}^{∗∗} ∩ B. Since B_{2}^{∗∗}(u) = uB^{∗∗}u ⊂ B^{∗∗}u = L^{∗∗}, we have B_{2}^{∗∗}(u) ∩ B ⊂ L^{∗∗} ∩ B = L. If x ∈ B_{2}^{∗∗}(u) ∩ B, then x^{♯} ∈ B_{2}^{∗∗}(u) ∩ B, proving that x ∈ L ∩ L^{♯}.

Let M denote the TRO [ \begin{matrix}L \\ L \end{matrix} ]. Let f, g be two elements of F_{[ \begin{matrix}0 \\ u \end{matrix} ]} which

are not separated by [ \begin{matrix}0 \\ C \end{matrix} ]. It follows that [ \begin{matrix}0 \\ C \end{matrix} ] annihilates f − g ∈ M_{2}^{∗}([ \begin{matrix}0 \\ u \end{matrix} ]) = sp_{C}F_{[ \begin{matrix}0 \\ u \end{matrix} ]}. This contradicts the fact, implicit in the preced-

ing paragraph, that the linear span of C is w*-dense in L_{2}^{∗∗}(u) = B_{2}^{∗∗}(u). This proves (a).

To prove (b), it suffices to show that for a, b ∈ L^{∗∗},

(1 − [ \begin{matrix}0 \\ u \end{matrix} ] [ \begin{matrix}0 \\ u \end{matrix} ]^{∗}) [ \begin{matrix}a \\ b \end{matrix} ] (1 − [ \begin{matrix}0 \\ u \end{matrix} ]^{∗} [ \begin{matrix}0 \\ u \end{matrix} ]) = 0.

This reduces to

[ \begin{matrix}1 & 0 \\ 0 & 1 − u \end{matrix} ] [ \begin{matrix}a \\ b \end{matrix} ] (1 − u) = [ \begin{matrix}a(1 − u) \\ (1 − u)b(1 − u) \end{matrix} ] = 0,

which is true since u is a right identity for L^{∗∗}.

To prove (c), let N denote the TRO [ \begin{matrix}B \\ B \end{matrix} ]. Note that F_{[ \begin{matrix}0 \\ u \end{matrix} ]} is the

normal state space of the von Neumann algebra N_{2}^{∗∗} ([ \begin{matrix}0 \\ u \end{matrix} ]) and that

[ \begin{matrix}0 \\ u_{λ} \end{matrix} ] = [ \begin{matrix}0 \\ √ & \overline{u_{λ}} \end{matrix} ] [0, u] [ \begin{matrix}0 \\ √ & \overline{u_{λ}} \end{matrix} ]

is the square of the self-adjoint element

[ \begin{matrix}0 \\ √ & \overline{u_{λ}} \end{matrix} ]^{♯} = [ \begin{matrix}0 \\ u \end{matrix} ] [0, \sqrt{u_{λ}}] [ \begin{matrix}0 \\ u \end{matrix} ] = [ \begin{matrix}0 & 0 \\ 0 & \sqrt{u_{λ}} \end{matrix} ] [ \begin{matrix}0 \\ u \end{matrix} ] = [ \begin{matrix}0 \\ √ & \overline{u_{λ}} \end{matrix} ] .

Hence (c) follows.

From the proof of (c), [ \begin{matrix}0 \\ u_{λ} \end{matrix} ] ∈ N_{2}^{∗∗} ([ \begin{matrix}0 \\ u \end{matrix} ]), so it is fixed by S_{F}^{∗}.

To prove the converse, assume that A is a TRO satisfying the conditions of the theorem. Let B denote the TRO [ \begin{matrix}A \\ A \end{matrix} ]. As in the first part of the proof, for each λ, there exists a partial isometry v_{λ} ∈ A^{∗∗}

and an element v_{λ}^{0} ∈ A_{0}^{∗∗}(v_{λ}) such that [ \begin{matrix}o \\ x_{λ}/∥x_{λ}∥ \end{matrix} ] = [ \begin{matrix}0 \\ v_{λ} \end{matrix} ] + [ \begin{matrix}0 \\ v_{λ}^{0} \end{matrix} ] and

F_{[ \begin{matrix}0 \\ x_{λ}/∥x_{λ}∥ \end{matrix} ]} = F_{[ \begin{matrix}0 \\ v_{λ} \end{matrix} ]}. Since sup_{λ} F_{[ \begin{matrix}0 \\ v_{λ} \end{matrix} ]} = F exists, let F = F_{[ \begin{matrix}u \\ v \end{matrix} ]} with

[ \begin{matrix}u \\ v \end{matrix} ] a partial isometry in (M_{2,1}(A))^{∗∗}. We shall show that u = 0 and

State spaces of JB^{∗}-triples  39

hence that v is a partial isometry. In the first place, P_{2}([ \begin{matrix}0 \\ v_{λ} \end{matrix} ])([ \begin{matrix}u \\ v \end{matrix} ]) = [ \begin{matrix}0 \\ v_{λ} \end{matrix} ], which reduces to P_{2}(v_{λ})v = v_{λ}. Since [ \begin{matrix}0 \\ v \end{matrix} ] is the image of [ \begin{matrix}u \\ v \end{matrix} ] under a contractive projection, ∥v∥ ≤ 1, and therefore P_{1}(v_{λ})^{∗}v = 0 (by [13, Lemma 1.5]). Thus v = v_{λ} + v_{λ}^{0} with v_{λ}^{0} orthogonal to v_{λ}, and it follows that the support partial isometry u(v) of the element v ∈ A^{∗∗} satisfies u(v) ≥ v_{λ}. It follows that [ \begin{matrix}0 \\ v_{λ} \end{matrix} ] ≤ [ \begin{matrix}0 \\ u(v) \end{matrix} ] and since [ \begin{matrix}u \\ v \end{matrix} ] is the least upper bound, we have [ \begin{matrix}u \\ v \end{matrix} ] ≤ [ \begin{matrix}0 \\ u(v) \end{matrix} ]. Thus

[ \begin{matrix}u \\ v \end{matrix} ] = P_{2}([ \begin{matrix}0 \\ u(v) \end{matrix} ]) [ \begin{matrix}u \\ v \end{matrix} ] = [ \begin{matrix}0 \\ P_{2}(u(v))v \end{matrix} ] = [ \begin{matrix}0 \\ v \end{matrix} ] ,

showing that u = 0.

Conditions (b) and (d) imply that [ \begin{matrix}0 \\ x_{λ} \end{matrix} ] lies in the von Neumann algebra B_{2}^{∗∗}([ \begin{matrix}0 \\ v \end{matrix} ]) while condition (c) implies that [ \begin{matrix}0 \\ x_{λ} \end{matrix} ] ≥ 0 in that von Neumann algebra. In particular, [ \begin{matrix}0 \\ x_{λ} \end{matrix} ] is self-adjoint; vx_{λ}^{∗}v = x_{λ}. We claim that condition (a) implies that [ \begin{matrix}0 \\ C \end{matrix} ] cannot annihilate any non-zero element of B_{2}^{∗}([ \begin{matrix}0 \\ v \end{matrix} ]). Indeed, suppose [ \begin{matrix}0 \\ C \end{matrix} ] (ψ_{1} − ψ_{2}) = 0 where ψ_{1} − ψ_{2} is the Jordan decomposition of a functional ψ in the self adjoint part of B_{2}^{∗}([ \begin{matrix}0 \\ v \end{matrix} ]). Note that since \{v_{λ}\} is directed, and v_{λ} ≤ x_{λ} ≤ v, it follows that

∥ψ_{1}∥ = [ \begin{matrix}0 \\ v \end{matrix} ] (ψ_{1}) = sup [ \begin{matrix}0 \\ C \end{matrix} ] (ψ_{1}) = sup [ \begin{matrix}0 \\ C \end{matrix} ] (ψ_{2}) = [ \begin{matrix}0 \\ v \end{matrix} ] (ψ_{2}) = ∥ψ_{2}∥

and this contradicts (a), as ψ_{1}/α, ψ_{2}/α ∈ F, where α is the common norm of ψ_{1} and ψ_{2}. It follows that the bipolar ([ \begin{matrix}0 \\ C \end{matrix} ]_{0})^{0} = B_{2}^{∗∗}([ \begin{matrix}0 \\ v \end{matrix} ]). Consequently, the w* closure of sp_{C}C is A_{2}^{∗∗}(v) and since the norm closure of a convex set is the same as its weak closure

A ∩ A_{2}^{∗∗}(v) = A ∩ \overline{sp C}w* = A ∩ \overline{sp C}^{∥·∥} = \overline{sp C}^{∥·∥}

is a C*-subalgebra of A_{2}^{∗∗}(v).

We are now in a position to show that A is completely isometric to a left ideal of a C*-algebra. Exactly as in the proof of the right unital case we have A ⊂ Av^{∗}v. We define a map Ψ : A → AA^{∗} by Ψ(a) = av^{∗}. The crux of the matter is to show that the range of Ψ lies in AA^{∗}. If that is the case, then since for X, Y, Z ∈ M_{n}(A), with D = diag(v^{∗}, …, v^{∗}),

XY^{∗}ZD = XD(Y D)^{∗}ZD,

ψ is a complete isometry. Moreover, if b, c ∈ A then (bc^{∗})av^{∗} = (bc^{∗}a)v^{∗} shows that the range of ψ is a left ideal. It remains to show that Av^{∗} ⊂ AA^{∗}.

Note first that, for a ∈ A, av^{∗}x_{λ} ∈ A, since

av^{∗}x_{λ} = av^{∗}x_{λ}^{1/2} · x_{λ}^{1/2} = av^{∗}(v(x_{λ}^{1/2})^{∗}v)v^{∗}x_{λ}^{1/2}
= (av^{∗}v)(x_{λ}^{1/2})^{∗}(vv^{∗}x_{λ}^{1/2}) = a(x_{λ}^{1/2})^{∗}x_{λ}^{1/2} ∈ A.

Next, since v belongs to the w*-closure of sp_{R} C, and for each a ∈ A, {av^{∗}y : y ∈ sp_{R} C} is a convex subset of A (since av^{∗}y = ∑ α_{i}av^{∗}x_{λ_{i}} = ∑ α_{i}a(x_{λ_{i}}^{1/2})^{∗}x_{λ_{i}}^{1/2} ∈ A), it follows that a belongs to the norm closure of {av^{∗}y : y ∈ sp_{R} C}. Now va^{∗}av^{∗}y = va^{∗}av^{∗}vy^{∗}v = va^{∗}ay^{∗}v = (ya^{∗}a)^{♯} ∈ A ∩ A_{2}^{∗∗}(v) and therefore va^{∗}a belongs to the norm closure of the set {va^{∗}av^{∗}y : y ∈ sp_{R} C} and hence va^{∗}a ∈ A. Using the triple functional calculus in the TRO A (see [23]) , we have

av^{∗} = a^{1/3}(a^{1/3})^{∗}a^{1/3}v^{∗} = a^{1/3}(v(a^{1/3})^{∗}a^{1/3})^{∗} ∈ AA^{∗}.⊓ ⊔

In Theorem 3.2, the elements x_{λ} represent a right approximate unit cast in purely linear terms. Similar language can be used to characterize C*-algebras.

Proposition 3.4. Let A be a TRO. Then A is completely isometric to a unital C*-algebra if and only if there is a norm one element x in A such that the complex linear span sp_{C}(F)) of the face F in A^{∗} exposed by x coincides with A^{∗}.

Note that a characterization of non-unital C*-algebras can also be given with obvious modifications as in Theorem 3.2.

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