CLASSIFICATION OF CONTRACTIVELY COMPLEMENTED HILBERTIAN OPERATOR SPACES

MATTHEW NEAL AND BERNARD RUSSO

Abstract. We construct some separable infinite dimensional homogeneous Hilbertian operator spaces H_{∞}^{m,R} and H_{∞}^{m,L}, which generalize the row and column spaces R and C (the case m = 0). We show that separable infinitedimensional Hilbertian JC^{∗}-triples are completely isometric to an element of the set of (infinite) intersections of these spaces . This set includes the operator spaces R, C, R ∩ C, and the space Φ spanned by creation operators on the full anti-symmetric Fock space. In fact, we show that H_{∞}^{m,R} (resp. H_{∞}^{m,L}) is completely isometric to the space of creation (resp. annihilation) operators on the m (resp. m + 1) anti-symmetric tensors of the Hilbert space. Together with the finite-dimensional case studied in [14], this gives a full operator space classification of all rank-one JC^{∗}-triples in terms of creation and annihilation operator spaces.

We use the above to show that all contractive projections on a C*-algebra A with infinite dimensional Hilbertian range are “expansions” (which we define precisely) of normal contractive projections from A^{∗∗} onto a Hilbertian space which is completely isometric to R, C, R ∩ C, or Φ. This generalizes the well known result, first proved for B(H) by Robertson in [17], that all Hilbertian operator spaces that are completely contractively complemented in a C*-algebra are completely isometric to R or C. We also compute various completely bounded Banach-Mazur distances between these spaces, or Φ.

1. Preliminaries

The goals of the present paper are to classify all infinite dimensional rank 1 JC*- triples up to complete isometry (Theorem 1 in section 2) and then use that result to give a suitable “classification” of all Hilbertian operator spaces which are contractively complemented in a C*-algebra or normally contractively complemented in a W*-algebra (Theorems 2 and 3 in section 3). In particular, we show that these spaces are “essentially” R, C, R ∩ C, or Φ modulo a “degenerate” piece.

In Theorem 4 in section 4 we compute various completely bounded Banach-Mazur distances between these JC*-triples. In Theorem 5 in section 5, we show that all of these JC*-triples in the separable infinite dimensional and finite dimensional cases can be represented completely isometrically as creation and annihilation operator spaces on pieces of the anti-symmetric Fock space.

In the rest of this section, we give some background on operator space theory and on JC^{∗}-triples.

Date: August 26, 2005. 1991 Mathematics Subject Classification. Primary 46L07. Key words and phrases. Hilbertian operator space, homogeneous operator space, contractive

projection, creation operator, anti-symmetric Fock space, completely bounded Banach-Mazur distance.

1.1. Operator spaces. Operator space theory is a non-commutative or quantized theory of Banach spaces. By definition, an operator space is a Banach space together with an isometric linear embedding into B(H), the bounded linear operators on a complex Hilbert space. While the objects are obviously the Banach spaces themselves, the more interesting aspects concern the morphisms, namely, the completely bounded maps. These are defined by considering an operator space as a subspace X of B(H). Its operator space structure is then given by the sequence of norms on the set of matrices M_{n}(X) with entries from X, determined by the identification M_{n}(X) ⊂ M_{n}(B(H)) = B(H ⊕ H ⊕ ⋯ ⊕ H). A linear mapping φ : X → Y between two operator spaces is completely bounded if the induced mappings φ_{n} : M_{n}(X) → M_{n}(Y) defined by φ_{n}([x_{ij}]) = [φ(x_{ij})] satisfy ∥φ∥cb := sup_{n} ∥φ_{n}∥ < ∞.

Operator space theory has its origins in the work of Stinespring in the 1950s, and Arveson in the 1960s. Many tools were developed in the 1970s and 1980s by a number of operator algebraists, and an abstract framework was developed in 1988 in the thesis of Ruan. All definitions, notation, and results used in this paper can be found in recent accounts of the subject, namely (in chronological order) [6],[15],[16],[3]. Let us just recall that a completely bounded map is a complete isomorphism if its inverse exists and is completely bounded. Two operator spaces are completely isometric if there is a linear isomorphism T between them with ∥T ∥cb = ∥T^{−1}∥cb = 1. We call T a complete isometry in this case. Other important types of morphisms in this category are complete contractions (∥φ∥cb ≤ 1) and complete semi-isometries (:= isometric complete contraction).

Examples of completely bounded maps are the restriction to a subspace of a C^{∗}- algebra of a *-homomorphism and multiplication by an fixed element. It is a fact that every completely bounded map is essentially a product of these two examples, [16, Th. 1.6]. The space CB(X, Y) of completely bounded maps between operator spaces X and Y is a Banach space with the completely bounded norm ∥ · ∥cb.

Analogous to the Banach-Mazur distance for Banach spaces, the class of all operator spaces can be made into a metric space by using the logarithm of the completely bounded Banach-Mazur distance:

dcb(E, F) = inf{∥u∥cb · ∥u^{−1}∥cb ; u : E → F complete isomorphism}.

Two important examples of Hilbertian operator spaces (:= operator spaces isometric to Hilbert space) are the row and column spaces R, C, and their finitedimensional versions R_{n}, C_{n}. These are defined as follows. In the matrix representation for B(ℓ_{2}), column Hilbert space C := \overline{sp}\{e_{i1} : i ≥ 1\} and row Hilbert space R := \overline{sp}\{e_{1j} : j ≥ 1\}. Their finite dimensional versions are C_{n} = sp\{e_{i1} : 1 ≤ i ≤ n\} and R_{n} = sp{e_{1j} : 1 ≤ j ≤ n}. Here of course e_{ij} is the operator defined by the matrix with a 1 in the (i, j)-entry and zeros elsewhere. Although R and C are Banach isometric, they are not completely isomorphic (dcb(R, C) = ∞); and R_{n} and C_{n}, while completely isomorphic, are not completely isometric. In fact, it is known that dcb(R_{n}, C_{n}) = n.

R, C, R_{n}, C_{n} are examples of homogeneous operator spaces, that is, operator spaces E for which ∀u : E → E, ∥u∥cb = ∥u∥. Another important example of an Hilbertian homogeneous operator space is Φ(I). The space Φ(I) is defined by Φ(I) = \overline{sp}\{V_{i} : i ∈ I\}, where the V_{i} are bounded operators on a Hilbert space satisfying the canonical anti-commutation relations. In some special cases, the notations Φ_{n} := Φ({1, 2, …, n}), and Φ = Φ({1, 2, …}) are used. For more properties of this space and related constructs, see [16, 9.3].

Two more examples of homogeneous operator spaces are min(E), max(E), where E is any Banach space. For any such E, the operator space structure of min(E) is defined by the embedding of E into the continuous functions on the unit ball of E^{∗} in the weak*-topology, namely, ∥(a_{ij})∥_{M_{n}(min(E))} = sup_{ξ∈B_{E^{∗}}} ∥(ξ(a_{ij}))∥_{M_{n}}. The operator space structure of max(E) is given by

∥(a_{ij})∥_{M_{n}(max(E))} = sup{∥(u(a_{ij}))∥_{M_{n}(B(H_{u}))} : u : E → B(H_{u}), ∥u∥ ≤ 1}.

More generally, if F and G are operator spaces, then in F ⟶^{u} min(E), ∥u∥cb = ∥u∥, and in max(E) ⟶^{v} G, ∥v∥cb = ∥v∥. The notations min(E) and max(E) are justified by the fact that for any Banach space E, the identity map on E is completely contractive in max(E) → E → min(E).

By analogy with the classical Banach spaces ℓ_{p}, c_{0}, L_{p}, C(K) (as well as their “second generation”, Orlicz, Sobolev, Hardy, Disc algebra, Schatten p-classes), we can consider the (Hilbertian) operator spaces R, C, min(ℓ_{2}), max(ℓ_{2}), OH, Φ, as well as their finite dimensional versions R_{n}, C_{n}, min(ℓ_{2}^{n}), max(ℓ_{2}^{n}), OH_{n}, Φ_{n}, as “classical operator spaces”. Among these spaces, only the spaces R, C, and Φ play important roles in this paper. (For the definition and properties of the space called OH, see [16, Chapter 7].) The classical operator spaces are mutually completely non-isomorphic. If E_{n}, F_{n} are n-dimensional versions, then dcb(E_{n}, F_{n}) → ∞, [16, Ch. 10].

We propose to add to this list of classical operator spaces the Hilbertian operator spaces H_{∞}^{m,R} and H_{∞}^{m,L} constructed here, as well as their finite-dimensional versions H_{n}^{k} studied in [13] and [14]. Like the space Φ, the spaces H_{∞}^{m,R}, H_{∞}^{m,L} and H_{n}^{k} can be represented up to complete isometry as spaces of creation operators or annihilation operators on anti-symmetric Fock spaces ([14, Lemma 2.1] and Theorem 5 below).

Let us recall from [13, Sections 6,7] the construction of the spaces H_{n}^{k}, 1 ≤ k ≤ n. Let I denote a subset of {1, 2, …, n} of cardinality |I| = k −1. The number of such I is q := \binom{n}{k−1}. Let J denote a subset of \{1, 2, …, n\} of cardinality |J| = n − k. The number of such J is p := \binom{n}{n−k}. We assume that each I = \{i_{1}, …, i_{k−1}\} is such that i_{1} < ⋯ < i_{k−1}, and that if J = {j_{1}, …, j_{n−k}}, then j_{1} < ⋯ < j_{n−k}.

The space H_{n}^{k} is the linear span of matrices b_{i}^{n,k}, 1 ≤ i ≤ n, given by

b_{i}^{n,k} = ∑_{I∩J=∅,(I∪J)^{c}={i}} ϵ(I, i, J)e_{J,I},

where e_{J,I} = e_{J} ⊗ e_{I} = e_{J}e_{I}^{t} ∈ M_{p,q}(C) = B(C^{q}, C^{p}), and ϵ(I, i, J) is the signature of the permutation taking (i_{1}, …, i_{k−1}, i, j_{1}, …, j_{n−k}) to (1, …, n). Since the b_{i}^{n,k} are the image under a triple isomorphism (actually ternary isomorphism) of a rectangular grid in a JW^{∗}-triple of rank one, they form an orthonormal basis for H_{n}^{k} (cf. [13, subsection 5.3 and section 7]).

The following definition from [16, 2.7] plays a key role in this paper. If E_{0} ⊂ B(H_{0}) and E_{1} ⊂ B(H_{1}) are operator spaces whose underlying Banach spaces form a compatible pair in the sense of interpolation theory, then the Banach space E_{0}∩E_{1} (with the norm ∥x∥_{E_{0}∩E_{1}} = max(∥x∥_{E_{0}}, ∥x∥_{E_{1}})) equipped with the operator space structure given by the embedding E_{0} ∩ E_{1} ∋ x ↦ (x, x) ∈ E_{0} ⊕ E_{1} ⊂ B(H_{0} ⊕ H_{1}) is called the intersection of E_{0} and E_{1} and is denoted by E_{0} ∩ E_{1}. We note, for examples, that ∩_{k=1}^{n}H_{n}^{k} = Φ_{n} ([14]) and the space R ∩ C is defined relative to the embedding of C into itself and R into C given by the transpose map ([16, p. 184]). The definition of intersection extends easily to arbitrary families of compatible operator spaces (cf. Theorem 1 below). Lemma 1.1. Let H be an Hilbertian operator space, and suppose that every finite dimensional subspace of H is homogeneous. Then H itself is homogeneous.

Proof. Let ϕ be any unitary operator on H. According to the first statement of [16, Prop.9.2.1], it suffices to prove that ϕ is a complete isometry.

Let F be any finite dimensional subspace of H and let G be the subspace spanned by F ∪ ϕ(F). By the second statement of [16, Prop.9.2.1], F and ϕ(F), being of the same dimension as subspaces of the homogeneous space G, are completely isometric, and ϕ|F is a complete isometry.

Now let [x_{ij}] ∈ M_{n}(H). Then {x_{ij}, ϕ(x_{ij}) : 1 ≤ i, j ≤ n} spans a finite dimensional subspace F of H, and

∥ϕ_{n}([x_{ij}])∥_{M_{n}(H)} = ∥ϕ_{n}([x_{ij}])∥_{M_{n}(F)} = ∥[x_{ij}]∥_{M_{n}(F)} = ∥[x_{ij}]∥_{M_{n}(H)}. □

1.2. Rank one JC^{∗}-triples. A JC^{∗}-triple is a norm closed complex linear subspace of B(H, K) (equivalently, of a C^{∗}-algebra) which is closed under the operation a ↦ aa^{∗}a. JC^{∗}-triples were defined and studied (using the name J^{∗}-algebra) as a generalization of C^{∗}-algebras by Harris [10] in connection with function theory on infinite dimensional bounded symmetric domains. By a polarization identity, any JC^{∗}-triple is closed under the triple product

(1)  (a, b, c) ↦ \{abc\} := \frac{1}{2}(ab^{∗}c + cb^{∗}a),

under which it becomes a Jordan triple system. A linear map which preserves the triple product (1) will be called a triple homomorphism. Cartan factors are examples of JC^{∗}-triples, as are C^{∗}-algebras, and Jordan C^{∗}-algebras. Cartan factors are defined for example in [13, Section 1]. We shall only make use of Cartan factors of type 1, that is, spaces of the form B(H, K) where H and K are complex Hilbert spaces.

A special case of a JC^{∗}-triple is a ternary algebra, that is, a subspace of B(H, K) closed under the ternary product (a, b, c) ↦ ab^{∗}c. A ternary homomorphism is a linear map ϕ satisfying ϕ(ab^{∗}c) = ϕ(a)ϕ(b)^{∗}ϕ(c). These spaces are also called ternary rings of operators and abbreviated TRO. They have been studied both concretely in [11] and abstractly in [21]. Given a TRO M, its left (resp. right) linking C^{∗}-algebra is defined to be the norm closed span of the elements ab^{∗} (resp. a^{∗}b) with a, b ∈ M. Ternary isomorphic TROs have isomorphic left and right linking algebras.

TROs have come to play a key role in operator space theory, serving as the algebraic model in the category. Recall that the algebraic models for the categories of order-unit spaces, operator systems, and Banach spaces, are respectively Jordan C^{∗}-algebras, C^{∗}-algebras, and JB^{∗}-triples. Indeed, for TROs, a ternary isomorphism is the same as a complete isometry.

If v is a partial isometry in a JC^{∗}-triple M ⊂ B(H, K), then the projections l = vv^{∗} ∈ B(K) and r = v^{∗}v ∈ B(H) give rise to (Peirce) projections P_{k}(v) : M → M, k = 2, 1, 0 as follows; for x ∈ M,

P_{2}(v)x = lxr , P_{1}(v)x = lx(1 − r) + (1 − l)xr , P_{0}(v)x = (1 − l)x(1 − r).

The projections P_{k}(v) are contractive, and their ranges, called Peirce spaces and denoted by M_{k}(v), are JC^{∗}-subtriples of M satisfying M = M_{2}(v)⊕M_{1}(v)⊕M_{0}(v).

A partial isometry v is said to be minimal in M if M_{2}(v) = Cv. This is equivalent to v not being the sum of two non-zero orthogonal partial isometries. Recall that two partial isometries v and w (or any two Hilbert space operators) are orthogonal if v^{∗}w = vw^{∗} = 0. Orthogonality of partial isometries v and w is equivalent to v ∈ M_{0}(w) and will be denoted by v ⊥ w. Each finite dimensional JC^{∗}-triple is the linear span of its minimal partial isometries. More generally, a JC^{∗}-triple is defined to be atomic if it is the weak closure of the span of its minimal partial isometries. In this case, it has a predual and is called a JW^{∗}-triple. The rank of a JC^{∗}-triple is the maximum number of mutually orthogonal minimal partial isometries. For example, the rank of the Cartan factor B(H, K) of type 1 is the minimum of the dimensions of H and K; and the rank of the Cartan factor of type 4 (spin factor) is 2.

In a JC*-triple, there is a natural ordering on partial isometries. We write v ≤ w if vw^{∗}v = v; this is equivalent to vv^{∗} ≤ ww^{∗} and v^{∗}v ≤ w^{∗}w. Moreover, if v ≤ w, then there exists a partial isometry v′ orthogonal to v with w = v + v′.

Another relation between two partial isometries that we shall need is defined in terms of the Peirce spaces as follows. Two partial isometries v and w are said to be collinear if v ∈ M_{1}(w) and w ∈ M_{1}(v), notation v⊤w. Let u, v, w be partial isometries. The following is part of [13, Lemma 5.4], and is referred to as “hopping”: If v and w are each collinear with u, then uu^{∗}vw^{∗} = vw^{∗}uu^{∗} and u^{∗}uv^{∗}w = v^{∗}wu^{∗}u. If u, v, w are mutually collinear partial isometries, then {uvw} = 0.

JC^{∗}-triples of arbitrary dimension occur naturally in functional analysis and in holomorphy. A special case of a theorem of Friedman and Russo [8, Theorem 2] states that if P is a contractive projection on a C^{∗}-algebra A, then there is a linear isometry of the range P (A) of P onto a JC^{∗}-subtriple of A^{∗∗}. A special case of a theorem of Kaup [12] gives a bijective correspondence between Cartan factors and irreducible bounded symmetric domains in complex Banach spaces.

Contractive projections play a ubiquitous role in the structure theory of the abstract analog of JC^{∗}-triples (called JB^{∗}-triples). Of use to us will be both of the following two conditional expectation formulas for a contractive projection P on a JC*-triple M (which are valid for JB^{∗}-triples) ([7, Corollary 1]):

(2)  P {P x, P y, P z} = P {P x, P y, z} = P {P x, y, P z}, (x, y, z ∈ M).

By a special case of [4, Cor.,p.308], every JW^{∗}-triple of rank one is isometric to a Hilbert space and every maximal collinear family of partial isometries corresponds to an orthonormal basis. Conversely, every Hilbert space with the abstract triple product {xyz} := ((x|y)z + (z|y)x)/2 can be realized as a JC^{∗}-triple of rank one in which every orthonormal basis forms a maximal family of mutually collinear minimal partial isometries.

2. Operator space structure of Hilbertian JC*-triples

2.1. Hilbertian JC*-triples: The spaces H_{∞}^{m,R} and H_{∞}^{m,L}. The general setting for the next two sections will be the following: Y is a JC^{∗}-subtriple of B(H) which is Hilbertian in the operator space structure arising from B(H), and {u_{i} : i ∈ Ω} is an orthonormal basis consisting of a maximal family of mutually collinear partial isometries of Y. Note that the u_{i} are each minimal in Y, but not necessarily minimal in any JC^{∗}-triple containing Y.

We let T and A denote the TRO and the C^{∗}-algebra respectively generated by Y. For any subset G ⊂ Ω, (uu^{∗})_{G} := ∏_{i∈G} u_{i}u_{i}^{∗} and (u^{∗}u)_{G} := ∏_{i∈G} u_{i}^{∗}u_{i}. The elements (uu^{∗})_{G} and (u^{∗}u)_{G} lie in the weak closure of A and more generally in the left and right linking von Neumann algebras of T.

In the following lemma, parts (a) and (a′) justify the definitions of the integers m_{R} and m_{L} in parts (b) and (b′).

Lemma 2.1. Let Y be an Hilbertian operator space which is a JC^{∗}-subtriple of B(H) and let {u_{i} : i ∈ Ω} be an orthonormal basis consisting of a maximal family of mutually collinear partial isometries of Y.

(a): If (uu^{∗})_{Ω−F} = 0 for some finite set F ⊂ Ω, then, (uu^{∗})_{Ω−G} = 0 for every

finite set G with the same cardinality as F. (a′): If (u^{∗}u)_{Ω−F} = 0 for some finite set F ⊂ Ω, then, (u^{∗}u)_{Ω−G} = 0 for every

finite set G with the same cardinality as F. (b): Assume (uu^{∗})_{Ω−F} ≠ 0 for some finite set F. Let m_{R} be the smallest

nonnegative integer with (uu^{∗})_{Ω−F} ≠ 0 for every F with cardinality m_{R}. Define p_{R} = ∑_{|F |=m_{R}}(uu^{∗})_{Ω−F}. Then the maps y ↦ p_{R}y and y ↦ (1 − p_{R})y are completely contractive triple isomorphisms of Y onto rank one subtriples of the weak closure of T in B(H). Moreover, p_{R}Y ⊥ (1 − p_{R})Y.

(b′): Assume (u^{∗}u)_{Ω−F} ≠ 0 for some finite set F. Let m_{L} be the smallest nonnegative integer with (u^{∗}u)_{Ω−F} ≠ 0 for every F with cardinality m_{L}. Define p_{L} = ∑_{|F |=m_{L}}(u^{∗}u)_{Ω−F}. Then the maps y ↦ yp_{L} and y ↦ y(1 − p_{L}) are completely contractive triple isomorphisms of Y onto rank one subtriples of the weak closure of T in B(H). Moreover, Y p_{L} ⊥ Y (1 − p_{L}).

(c): In case (b), let w_{i} = p_{R}u_{i} and let m′_{R} be the smallest nonnegative integer with (ww^{∗})_{Ω−F} ≠ 0 for all F with cardinality m′_{R}. Then m′_{R} exists, and m′_{R} = m_{R}. Furthermore, (w^{∗}w)_{G} ≠ 0 for |G| = m_{R} + 1 and (w^{∗}w)_{G} = 0 for |G| = m_{R} + 2. Thus, if we define k_{R} to be the smallest integer k such that (w^{∗}w)_{G} ≠ 0 for |G| = k, then k_{R} = m_{R} + 1.

(c′): In case (b′), let w_{i} = u_{i}p_{L} and let m′_{L} be the smallest nonnegative integer with (w^{∗}w)_{Ω−F} ≠ 0 for all F with cardinality m′_{L}. Then m′_{L} exists, and m′_{L} = m_{L}. Furthermore, (ww^{∗})_{G} ≠ 0 for |G| = m_{L} + 1 and (ww^{∗})_{G} = 0 for |G| = m_{L} + 2. Thus, if we define k_{L} to be the smallest integer k such that (ww^{∗})_{G} ≠ 0 for |G| = k, then k_{L} = m_{L} + 1.

Proof. The proofs of (a) and (a′) are identical to the proof given in [13, Lemma 5.8]. The fact that the set Ω − F is infinite has no effect on the proof in [13].

The proofs of (b) and (b′) are identical to the proof given in [13, Lemma 5.9]. The facts that the set Ω − F is infinite and that the sums defining the projections p_{R} and p_{L} are infinite have no effect on the proof in [13].

We now prove (c), the proof of (c′) being entirely similar. For any finite set F ⊂ Ω,

(ww^{∗})_{Ω−F} = ∏_{i∈Ω−F} ( ∑_{|G|=m_{R}}(uu^{∗})_{Ω−G}) u_{i}u_{i}^{∗} ( ∑_{|H|=m_{R}}(uu^{∗})_{Ω−H} )

= ∏_{i∈Ω−F} (  ∑_{|G|=m_{R},i∈Ω−G}(uu^{∗})_{Ω−G}) = ∑_{G⊂F,|G|=m_{R}}(uu^{∗})_{Ω−G}.

From this it follows that (ww^{∗})_{Ω−F} = 0 if |F | < m_{R} and that (ww^{∗})_{Ω−F} = (uu^{∗})_{Ω−F} ≠ 0 if |F | = m_{R}. This proves that m′_{R} = m_{R}, that is (ww^{∗})_{Ω−F} = 0 ⇔ |F | < m_{R}.

Now let |F | = r and for convenience, suppose that F = {1, 2, …, r}. Then

(w^{∗}w)_{F} = (w^{∗}w)_{{1,2,...,r}} = ∑ u_{1}^{∗}(uu^{∗})_{Ω−F_{1}}u_{1}u_{2}^{∗}(uu^{∗})_{Ω−F_{2}}u_{2}u_{3}^{∗} ⋯ u_{r}^{∗}(uu^{∗})_{Ω−F_{r}}u_{r},

where the sum is over all |F_{j}| = m_{R}, j ∈ Ω − F_{j}, F − {j} ⊂ F_{j} (by “hopping”), and j = 1, 2, … r. Every term in this sum is zero if r − 1 > m_{R}, that is r ≥ m_{R} + 2. Further, if r = m_{R} + 1, there is only one term, namely x :=

(w^{∗}w)_{{1,2,...,m+1}} = u_{1}^{∗}(uu^{∗})_{Ω−{2,3,...,m+1}}u_{1}u_{2}^{∗}(uu^{∗})_{Ω−{1,3,4,...,m+1}}u_{2}u_{3}^{∗} × ⋯ ×
(uu^{∗})_{Ω−{1,2,...,m−1,m+1}}u_{m}u_{m+1}^{∗}(uu^{∗})_{Ω−{1,2,...,m}}u_{m+1}.

which by a sequence of “hoppings” becomes

x = (u^{∗}u)_{{1,2,...,m}}u_{m+1}^{∗}(uu^{∗})_{Ω−{1,2,...,m+1}}u_{m+1}.

In turn, using the collinearity of the u_{k}, this becomes

x = u_{m+1}^{∗}(uu^{∗})_{Ω−{1,2,...,m+1}}u_{m+1}.

Thus, if x = 0, then 0 = u_{m+1}xu_{m+1}^{∗} = (uu^{∗})_{Ω−{1,2,...,m}}, a contradiction. □

Our goal for the remainder of this section is to give a completely isometric representation for the spaces p_{R}Y and Y p_{L} in parts (b) and (b′) of Lemma 2.1. This will be achieved via a coordinatization procedure which we now describe.

In the following, let us restrict to the special case that Y is a Hilbertian JC*- triple which satisfies the properties of p_{R}Y in Lemma 2.1 part (c). For notational convenience, let m = m_{R}. Thus (u^{∗}u)_{G} ≠ 0 for |G| ≤ m + 1 and (u^{∗}u)_{G} = 0 for |G| ≥ m + 2.

Analogous to [13, Def. 6.1], we are going to define elements which are indexed by an arbitrary pair of subsets I, J of Ω satisfying

(3)  |Ω − I| = m + 1, |J| = m.

Here and throughout the rest of this paper, |F | denotes the cardinality of the finite set F.

The set I ∩ J is finite, and if |I ∩ J| = s ≥ 0, then |(I ∪ J)^{c}| = s + 1. Let us write I ∩ J = {d_{1}, …, d_{s}} and (I ∪ J)^{c} = {c_{1}, …, c_{s+1}}, and let us agree (for the moment) that there is a natural linear ordering on Ω such that c_{1} < c_{2} < ⋯ < c_{s+1} and d_{1} < d_{2} < ⋯ < d_{s}.

With the above notation, we define

(4)  u_{IJ} = u_{I,J} = (uu^{∗})_{I−J}u_{c_{1}}u_{d_{1}}^{∗}u_{c_{2}}u_{d_{2}}^{∗} ⋯ u_{c_{s}}u_{d_{s}}^{∗}u_{c_{s+1}}(u^{∗}u)_{J−I}.

Note that in general I − J is infinite and J − I is finite so that u_{I,J} lies in the weak closure of T.

In the special case of (4) where I ∩ J = ∅, we have s = 0 and u_{I,J} has the form

u_{I,J} = (uu^{∗})_{I}u_{c}(u^{∗}u)_{J},

where I ∪ J ∪ {c} = Ω is a partition of Ω. As in [13], we call such an element a “one”, and denote it also by u_{I,c,J}.

The proof of the following lemma, which is the analog of [13, Lemma 6.6], is complicated by the fact that the sets I are infinite if Ω is infinite. Lemma 2.2. Let Y be an Hilbertian operator space which is a JC^{∗}-subtriple of B(H) and let {u_{i} : i ∈ Ω} be an orthonormal basis consisting of a maximal family of mutually collinear partial isometries of Y. Assume that Y satisfies the properties of p_{R}Y in Lemma 2.1 part (c) with m = m_{R}, that is, (u^{∗}u)_{G} ≠ 0 for |G| ≤ m + 1 and (u^{∗}u)_{G} = 0 for |G| ≥ m + 2. For any c ∈ Ω,

(5)  u_{c} = ∑_{I,J} u_{I,J} = ∑_{I,J} u_{I,c,J}

where the sum is taken over all disjoint I, J satisfying (3) and not containing c, and converges weakly in the weak closure of T.

Proof. The proof of [13, Prop. 6.3] remains valid in our context insofar as {u_{I,J}} is a collection of pairwise orthogonal partial isometries in the weak closure of the ternary envelope T of Y. Since u_{c}^{∗}u_{c} commutes with (u^{∗}u)_{J}, u_{I,J}^{∗}u_{I,J}u_{c}^{∗}u_{c} = u_{I,J}^{∗}u_{I,J}, so that u_{I,J}^{∗}u_{I,J} ≤ u_{c}^{∗}u_{c} and similarly u_{I,J}u_{I,J}^{∗} ≤ u_{c}u_{c}^{∗} so that ∑ u_{I,J} ≤ u_{c}. To prove (5), we proceed as follows.

Let us write u_{c} = v_{c} + w_{c}, where w_{c} = ∑_{I,c,J} Iu_{c}J and for example, Iu_{c}J is shorthand for u_{I,c,J} = (uu^{∗})_{I}u_{c}(u^{∗}u)_{J}, and v_{c} is a partial isometry orthogonal to w_{c}. We shall show that v_{c} = 0. From the simple facts that Iu_{c}J = u_{c}J and w_{c}J = Iu_{c}J, it follows that v_{c}J = 0. Similarly, Iv_{c} = 0. From this, it follows that for each pair i ≠ j, v_{i} ∈ M_{1}(u_{j}). Indeed,

v_{i} + w_{i} = u_{i} = u_{i}u_{j}^{∗}u_{j} + u_{j}u_{j}^{∗}u_{i}
= v_{i}u_{j}^{∗}u_{j} + w_{i}u_{j}^{∗}u_{j} + u_{j}u_{j}^{∗}v_{i} + u_{j}u_{j}^{∗}w_{i}
= v_{i}u_{j}^{∗}u_{j} + ∑_{j∈J} u_{i}J + u_{j}u_{j}^{∗}v_{i} + ∑_{j∉J} u_{i}J
= v_{i}u_{j}^{∗}u_{j} + u_{j}u_{j}^{∗}v_{i} + w_{i}.

Hence v_{i} = v_{i}u_{j}^{∗}u_{j} + u_{j}u_{j}^{∗}v_{i}, so v_{i} ∈ M_{1}(u_{j}).

Next, we observe that v_{i} is orthogonal to w_{j} for every i and j. For i = j this is clear by definition. For i ≠ j, we have v_{i}(Ju_{j}^{∗}I) = (v_{i}J)u_{j}^{∗}I = 0 and similarly Ju_{j}^{∗}Iv_{i} = 0 so that v_{i}w_{j}^{∗} = w_{j}^{∗}v_{i} = 0.

It now follows that v_{i}⊤v_{j} for i ≠ j. Indeed,

v_{j} = v_{j}u_{i}^{∗}u_{i} + u_{i}u_{i}^{∗}v_{j} = v_{j}v_{i}^{∗}v_{i} + v_{j}w_{i}^{∗}w_{i} + v_{i}v_{i}^{∗}v_{j} + w_{i}w_{i}^{∗}v_{j} = v_{j}v_{i}^{∗}v_{i} + v_{i}v_{i}^{∗}v_{j}.

Let us adopt the notation J_{v} for (v^{∗}v)_{J} = ∏_{j∈J} v_{j}^{∗}v_{j}. (What we previously denoted by J would now be denoted by J_{u} = (u^{∗}u)_{J}.) We know that v_{i}J_{v} = v_{i}J_{u} = 0. Suppose that v_{i}J′_{v} ≠ 0 for some J′ ⊂ J. We will show that J′ = ∅. In the first place, v_{i}J′_{v} = I′_{v}v_{i}J′_{v}, since letting I′ = Ω − (J′ ∪ {i}) = {i_{α} : α ∈ Λ} say,

v_{i}J′_{v} = v_{i}(v^{∗}v)_{J′} = (v_{i_{α}}v_{i_{α}}^{∗}v_{i} + v_{i}v_{i_{α}}^{∗}v_{i_{α}})(v^{∗}v)_{J′} = v_{i_{α}}v_{i_{α}}^{∗}v_{i}(v^{∗}v)_{J′} = ⋯ = I′_{v}v_{i}J′_{v}.

In the second place, by the orthogonality of v_{j} and w_{j},

I′_{u}u_{j} = (uu^{∗})_{I′}u_{j} = [∏_{i∈I′}(v_{i} + w_{i})(v_{i}^{∗} + w_{i}^{∗})](v_{j} + w_{j}) = I′_{v}v_{j} + I′_{w}w_{j}.

Hence, I′_{v}v_{j} = I′_{u}u_{j} − I′_{w}w_{j} and each term is zero because, as with the {u_{j}}, the {w_{j}} satisfy (ww^{∗})_{I′} = 0 if |Ω − I′| < m. This contradiction shows that J′ = ∅, and therefore either v_{j}v_{i}^{∗} = 0 for all i, j or v_{j} = 0 for all j. In the latter case, there is nothing to prove. In the former case, since v_{j}⊤v_{i},

v_{j} = v_{i}v_{i}^{∗}v_{j} + v_{j}v_{i}^{∗}v_{i} = ( ∏_{i∈Ω−{j}} v_{i}v_{i}^{∗})v_{j} = ( ∏_{i∈Ω−{j}} u_{i}u_{i}^{∗})u_{j} + ( ∏_{i∈Ω−{j}} w_{i}w_{i}^{∗})w_{j} = 0

as required. □

We shall now assume that our set Ω is countable and for convenience set Ω = N = {1, 2, 3, …} with its natural order. Note that in this case, the number of possible sets I in (3) is ℵ_{0} and the number of such J is also ℵ_{0}.

Again as in [13], we assign a signature to each “one” u_{I,k,J} as follows: Let the elements of I be i_{1} < i_{2} < ⋯ < i_{p} < ⋯ and the elements of J be j_{1} < j_{2} < ⋯ < j_{m}, where p is chosen such that max{k, j_{m}} < i_{p}. Then ϵ(I, k, J) is defined to be the signature of the permutation taking the (p + m + 1)-tuple (i_{1}, …, i_{p}, k, j_{1}, …, j_{m}) onto (1, 2, …, p+m+1). This is clearly independent of p as long as max{k, j_{m}} < i_{p}.

The proof of [13, Lemma 6.7] shows that every element u_{I,J} decomposes uniquely into a product of “ones.” The signature ϵ(I, J) (also denoted by ϵ(IJ)) of u_{I,J} is defined to be the product of the signatures of the factors in this decomposition. Then the proof of [13, Prop. 6.10] shows that the family {ϵ(IJ)u_{I,J}} forms a rectangular grid which satisfies the extra property

(6)  ϵ(IJ)u_{IJ}[ϵ(IJ′)u_{IJ′}]^{∗}ϵ(I′J′)u_{I′J′} = ϵ(I′J)u_{I′J}.

It follows as in [13] that the map ϵ(IJ)u_{IJ} → E_{JI} is a ternary isomorphism (and hence complete isometry) from the norm closure of sp_{C} u_{IJ} to the norm closure of sp_{C} {E_{JI}}, where E_{JI} denotes an elementary matrix, whose rows and columns are indexed by the sets J and I, with a 1 in the (J, I)-position. By [4, Lemma 1.14], this map can be extended to a ternary isomorphism from the w*-closure of sp_{C} u_{IJ} onto the Cartan factor of type I consisting of all ℵ_{0} by ℵ_{0} complex matrices which act as bounded operators on ℓ_{2}. By restriction to Y and (5), Y is completely isometric to a subtriple Ỹ, of this Cartan factor of type 1. Definition 1. We shall denote the space Ỹ above by H_{∞}^{m,R}. An entirely symmetric argument (with J infinite and I finite) under the assumption that Y satisfies the conditions of Y p_{L} in Lemma 2.1 part (c) with m = m_{L} defines the space H_{∞}^{m,L}.

Explicitly,

H_{∞}^{m,R} = \overline{sp}_{C} \{b_{i}^{m} = ∑_{I∩J=∅,(I∪J)^{c}=\{i\},|J|=m} ϵ(I, i, J)e_{J,I} : i ∈ N\}.

and

H_{∞}^{m,L} = \overline{sp}_{C} \{ ∑_{I∩J=∅,(I∪J)^{c}=\{i\},|I|=m} ϵ(I, i, J)e_{J,I} : i ∈ N\},

with ϵ(I, i, J) defined in the obvious analogous way with I finite instead of J.

Corollary 4.1 below shows that these spaces are all distinct from each other and from Φ. This discussion has proved the following lemma.

Lemma 2.3. The spaces p_{R}Y and Y p_{L} in Lemma 2.1 parts (c) and (c′) are completely isometric to H_{∞}^{m_{R},R} and H_{∞}^{m_{L},L}, respectively. Remark 2.4. It is immediate from [13, Cor. 5.3] that H_{∞}^{0,R} = C and H_{∞}^{0,L} = R. Also note that H_{∞}^{m,R} and H_{∞}^{m,L} are homogeneous Hilbertian operator spaces by Lemma 1.1 and [14, Theorem 1].

2.2. The coordinatization of Hilbertian JC*-triples. Let Y satisfy the hypothesis of Lemma 2.1. Our analysis will consider the following three mutually exhaustive and (by Corollary 4.1(b)) mutually exclusive possibilities (in each case, the set F is allowed to be empty):

Case 1: (uu^{∗})_{Ω−F} ≠ 0 for some finite set F ⊂ Ω; Case 2: (u^{∗}u)_{Ω−F} ≠ 0 for some finite set F ⊂ Ω; Case 3: (uu^{∗})_{Ω−F} = (u^{∗}u)_{Ω−F} = 0 for all finite subsets F of Ω.

We will first address cases 1 and 2. Proposition 2.5. Let Y be a separable infinite dimensional Hilbertian operator space which is a JC^{∗}-subtriple of B(H) and let {u_{i} : i ∈ Ω} (Ω = N) be an orthonormal basis consisting of a maximal family of mutually collinear partial isometries of Y.

(a): Suppose there exists a finite subset F of Ω such that (uu^{∗})_{Ω−F} ≠ 0. Then Y is completely isometric to an intersection Y_{1} ∩ Y_{2} such that Y_{1} is completely isometric to a space H_{∞}^{m,R} (that is, in the notation of Lemma 2.1, m′_{R} = m ≥ 1 and k′_{R} = m + 1), or C, and Y_{2} is a Hilbertian JC^{∗}-triple. (b): Suppose there exists a finite subset F such that (u^{∗}u)_{Ω−F} ≠ 0. Then Y is completely isometric to an intersection Y_{1} ∩Y_{2} such that Y_{1} is completely isometric to a space H_{∞}^{m,L} (that is, m′_{L} = m ≥ 1 and k′_{L} = m + 1), or R, and Y_{2} is an Hilbertian JC^{∗}-triple.

Proof. (a) follows from Lemma 2.1 (b) and (c), the coordinatization procedure outlined in subsection 2.1, and Lemma 2.3. (b) follows by symmetry using Lemma 2.1 (b′) and (c′), and Lemma 2.3. □

It is worth emphasizing that the space H_{∞}^{m,R} (resp. H_{∞}^{m,L}) is determined up to complete isometry among Hilbertian JC*-triples by the condition

(7)  (uu^{∗})_{Ω−F} = 0 ⇔ |F | < m, (u^{∗}u)_{G} = 0 ⇔ |G| > m + 1

(resp.

(u^{∗}u)_{Ω−F} = 0 ⇔ |F | < m, (uu^{∗})_{G} = 0 ⇔ |G| > m + 1),

and that H_{∞}^{0,R} = C and H_{∞}^{0,L} = R. Remark 2.6. Recall from [13, p. 2245] that i_{R} (resp. i_{L}) is the largest i such that (uu^{∗})_{J} ≠ 0 (resp. (u^{∗}u)J ≠ 0) for any J with |J| = i. For the spaces H_{n}^{k} from [13], we have i_{R} = k and i_{L} = n − k + 1 so that i_{R} + i_{L} = n + 1. We may therefore think of the condition (7) as “i_{R} = ∞ − m, i_{L} = m + 1”, so that “i_{R} + i_{L} = ∞ + 1”.

To handle the remaining case 3, we shall need the following lemma. Lemma 2.7. Let Y be a separable infinite dimensional Hilbertian operator space which is a JC^{∗}-subtriple of a C^{∗}-algebra A and let {u_{i} : i ∈ Ω} be an orthonormal basis consisting of a maximal family of mutually collinear partial isometries of Y.

Let S and T be finite subsets of Ω and let k ∈ Ω−(S∪T). If (uu^{∗})_{S}u_{k}(u^{∗}u)_{T} = 0, then (uu^{∗})_{S′}u_{k′}(u^{∗}u)_{T′} = 0 for all sets S′, T′ with |S′| = |S|, |T′| = |T | and for all k′ ∈ Ω − (S′ ∪ T′).

Proof. It suffices to prove this with (S, k, T) replaced in turn by (S∪{l}−{j}, k, T) (with l ∉ S and j ∈ S); by (S, l, T) (with l ≠ k); and by (S, k, T ∪ {l} − {i}) (with l ∉ T and i ∈ T).

In the first case,

u_{l}u_{l}^{∗}(uu^{∗})_{S−j}u_{k}(u^{∗}u)_{T} = (u_{j}u_{j}^{∗}u_{l} + u_{l}u_{j}^{∗}u_{j})u_{l}^{∗}(uu^{∗})_{S−j}u_{k}(u^{∗}u)_{T}
= 0 + u_{l}u_{j}^{∗}u_{j}u_{l}^{∗}(uu^{∗})_{S−j}u_{k}(u^{∗}u)_{T}
= u_{l}u_{j}^{∗}(uu^{∗})_{S−j}u_{j}u_{l}^{∗}u_{k}(u^{∗}u)_{T} (by hopping)
= −u_{l}u_{j}^{∗}(uu^{∗})_{S−j}u_{k}u_{l}^{∗}u_{j}(u^{∗}u)_{T}
= −u_{l}u_{j}^{∗}(uu^{∗})_{S−j}u_{k}(u^{∗}u)_{T} u_{l}^{∗}u_{j} = 0.

By symmetry, (uu^{∗})_{S}u_{k}(u^{∗}u)_{T −i}u_{l}^{∗}u_{l} = 0, proving the second case.

Finally,

(uu^{∗})_{S}u_{l}(u^{∗}u)_{T} = (uu^{∗})_{S}(u_{l}u_{k}^{∗}u_{k} + u_{k}u_{k}^{∗}u_{l})(u^{∗}u)_{T}
= (uu^{∗})_{S}u_{l}u_{k}^{∗}u_{k}(u^{∗}u)_{T} + (uu^{∗})_{S}u_{k}u_{k}^{∗}u_{l}(u^{∗}u)_{T}
= u_{l}u_{k}^{∗}(uu^{∗})_{S}u_{k}(u^{∗}u)_{T} + (uu^{∗})_{S}u_{k}(u^{∗}u)_{T} u_{k}^{∗}u_{l} = 0. □

We can now handle the final case 3. Proposition 2.8. Let Y be a separable infinite dimensional Hilbertian JC*-triple and let {u_{i} : i ∈ Ω} be an orthonormal basis consisting of a maximal family of mutually collinear partial isometries of Y. Suppose that for all finite subsets G ⊂ Ω, (uu^{∗})_{Ω−G} = 0 and (u^{∗}u)_{Ω−G} = 0. Then Y is completely isometric to Φ.

Proof. We show first that all finite products (uu^{∗})_{F} u_{i}(u^{∗}u)_{G} with F, G, {i} pairwise disjoint (and F, G not both empty), are not zero. Suppose, on the contrary, that (uu^{∗})_{F} u_{i}(u^{∗}u)_{G} = 0 for some F, G, i. If F and G are both non-empty, pick a subset F′ ⊂ F of maximal cardinality such that (uu^{∗})_{F′}u_{i}(u^{∗}u)_{G} ≠ 0 (F′ could be empty). Then by repeated use of collinearity and passing to the limit, we arrive at (uu^{∗})_{F′}u_{i}(u^{∗}u)_{G} = (uu^{∗})_{F′}u_{i}(u^{∗}u)_{Ω−({i}∪F′)} = 0, a contradiction. So either F = ∅ and u_{i}(u^{∗}u)_{G} = 0, or G = ∅ and (u^{∗}u)_{F} u_{i} = 0. In the first case, picking a subset G′ ⊂ G of maximal cardinality such that u_{i}(u^{∗}u)_{G′} ≠ 0, then by collinearity u_{i}(u^{∗}u)_{G′} = (uu^{∗})_{Ω−({i}∪G′)}u_{i}(u^{∗}u)_{G′} = 0, a contradiction, and similarly in the second case. We have now shown that all finite products (uu^{∗})_{F} u_{i}(u^{∗}u)_{G} with F, G, {i} pairwise disjoint, are not zero.

Now consider the space Y_{n} := sp{u_{1}, …, u_{n}}. By [13, Th. 3(b)], Y_{n} is completely isometric to a space H_{n}^{k_{1}} ∩ ⋯ ∩ H_{n}^{k_{m}}, where n ≥ k_{1} > ⋯ > k_{m} ≥ 1. We claim that m = n and k_{j} = n − j + 1 for j = 1, …, n. By way of contradiction, suppose that there is a k, 1 ≤ k ≤ n such that the space H_{n}^{k} is not among the spaces H_{n}^{k_{j}}, 1 ≤ j ≤ m. Let ψ : x ↦ (x^{(k_{1})}, …, x^{(k_{m})}) denote the ternary isomorphism of the ternary envelope of Y_{n} whose restriction to Y_{n} implements the complete isometry of Y_{n} with H_{n}^{k_{1}} ∩ ⋯ ∩ H_{n}^{k_{m}}, and consider the element x := (uu^{∗})_{{1,...,k−1}}u_{k}(u^{∗}u)_{{k+1,...,n}}. As shown above, x ≠ 0. However, x^{(k_{j})} = 0 for each j, a contradiction. To see that x^{(k_{j})} = 0, suppose first that k_{j} < k. Since ψ is a ternary isomorphism, x^{(k_{j})} = (u^{(k_{j})}u^{(k_{j})∗})_{{1,...,k−1}}u_{k}^{(k_{j})}(u^{(k_{j})∗}u^{(k_{j})})_{{k+1,...,n}} = 0 since (u^{(k_{j})}u^{(k_{j})∗})_{{1,...,k−1}}u_{k}^{(k_{j})} is zero in H_{n}^{k_{j}}. Similarly, if k_{j} > k, then n−k_{j} +1 < n − k + 1, u_{k}^{(k_{j})}(u^{(k_{j})∗}u^{(k_{j})})_{{k+1,...,n}} = 0 so that x^{(k_{j})} = 0 in this case as well.

We now have for each n that, completely isometrically, Y_{n} = ∩_{k=1}^{n}H_{n}^{k} and the latter space is completely isometric to Φ_{n} by [14, Lemma 2.1]. Since Y = \overline{∪Y_{n}} and Φ = \overline{∪Φ_{n}}, it follows that Y = Φ completely isometrically. □

We come now to the first main result of this paper. Theorem 1. Let Y be a separable infinite dimensional Hilbertian operator space which is concretely represented as a JC^{∗}-triple. Then Y is completely isometric to one of the following spaces, where each Y_{i} is completely isometric to one of the spaces H_{∞}^{m_{i},R} or H_{∞}^{m_{i},L}.

• Φ.

• Z ∩ Y_{1} ∩ ⋯ ∩ Y_{n}, where n ≥ 1 and Z is Φ or is absent.

• Z ∩ Y_{1} ∩ ⋯ ∩ Y_{n} ∩ ⋯, where Z is Φ or is absent. Proof. Let {u_{i}} be an orthonormal basis for Y consisting of a maximal family

of mutually collinear minimal partial isometries. By Proposition 2.8 and Proposition 2.5, either Y is completely isometric to Φ, in which case the theorem is proved, or Y is completely isometric to an intersection Y_{1} ∩ Z, where Y_{1} is completely isometric to either H_{∞}^{m,R} or H_{∞}^{k,L}, with m, k ≥ 0. It follows by induction, in the case that Y is not completely isometric to Φ, that Y = Y_{1} ∩ Y_{2} ∩ ⋯ ∩ Y_{n} ∩ Z_{n}, where n ≥ 1, each Y_{i} is completely isometric to one of the spaces H_{∞}^{m_{i},R} or H_{∞}^{k_{i},L}, and either Z_{n} is completely isometric to the space Φ, in which case the theorem is proved, or Z_{n} can be further decomposed as Y_{n+1} ∩ W, where Y_{n+1} is completely isometric to either H_{∞}^{m_{n+1},R} or H_{∞}^{k_{n+1},L}.

It remains to consider the case in which no Z_{n} is completely isometric to Φ. By the constructions in the proof of Proposition 2.5, Z_{n} is obtained by multiplying Y on the left and right by a sequence of projections of the form 1 − p_{i,R} or 1 − p_{j,L} (see Lemma 2.1). The resulting products of projections converge strongly and it follows that Y is the intersection of an infinite sequence Y_{i} and a space Z, which has the property that all products (ww^{∗})_{Ω−F} and (w^{∗}w)_{Ω−F} are zero, for all finite sets F. An appeal to Proposition 2.8 now shows that Z is completely isometric to Φ, completing the proof. □ Remark 2.9. By the argument in [13, p. 2259], the sequences m_{i} and k_{i} are strictly increasing. This fact is not needed in the preceding proof. Remark 2.10. In section 5, we show that the spaces H_{∞}^{m,R} and H_{∞}^{k,L} are completely isometric to spaces of creation and annihilation operators on pieces of the antisymmetric Fock space. Hence all rank 1 JC*-triples are really spaces of creation and annihilation operators.

We close this section with a well known lemma about Hilbertian TRO’s. Recall that TROs are operator subspaces of a C*-algebra which are closed under the product xy^{∗}z, and are fundamental in operator space theory. Indeed, every operator space has both a canonical injective envelope [18] and a canonical “Shilov boundary”

[2] which are TROs. A proof of the following lemma can be found in [19], which classifies all W*-TRO’s up to complete isometry. We include a quick alternate proof from the point of view of this section. Lemma 2.11. If X is a Hilbertian TRO, then X is completely isometric to R or C

Proof. Let {u_{j}} be an orthonormal basis consisting of mutually collinear minimal partial isometries in X. For a fixed i ≠ j, since u_{i}u_{i}^{∗}u_{j} is a partial isometry in X, u_{i}u_{i}^{∗}u_{j} = P_{2}(u_{j})(u_{i}u_{i}^{∗}u_{j}) is either equal to e^{iθ}u_{j} or 0. If the latter case holds, then by the calculation in [13, Lemma 5.1], u_{i}u_{i}^{∗}u_{j} = 0 for all i ≠ j, and X is ternary isomorphic and thus completely isometric to C. On the other hand, if u_{i}u_{i}^{∗}u_{j} = e^{iθ}u_{j}, then by collinearity, e^{iθ} = 1, u_{j}u_{i}^{∗}u_{i} = 0, and again by [13, Lemma 5.1], X is completely isometric to R. □

3. Contractively complemented Hilbertian operator spaces

Suppose that a Hilbert space H is complemented in a C*-algebra A via a contractive projection P. Let L be a contractive linear map from H into A with the properties that L(H) ⊥ H and P (L(H)) = 0. Then the space K = {h + L(h) : h ∈ H} is clearly contractively complemented by P + LP. From this it follows that a classification of contractively complemented Hilbertian operator spaces is hopeless without some qualifications.

3.1. Expansions of contractive projections. The following definitions are crucial. Definition 2. Consider a triple {K, A, P } consisting of a Hilbertian operator space K, a C*-algebra A, and a contractive projection P from A onto K. If there exists a Hilbertian subspace H of A which is contractively complemented by a projection Q and a contractive linear map L from H into A such that P = Q + LQ, L(H) ⊥ H and Q(L(H)) = 0, we say that{K, A, P } is an expansion of {H, A, Q}. (Note that this implies that K = {h + L(h) : h ∈ H}.)

The following is immediate. Lemma 3.1. If {K, A, P } is an expansion of {H, A, Q} then Q|_{K} is a completely contractive isometry from K onto H

Suppose X ⊂ A is a contractively complemented Hilbertian operator subspace by a projection Q. Further suppose that Y is a Hilbertian operator subspace of A which is isometric to X and which is orthogonal to X and lies in ker(Q). Then {x+Lx : x ∈ X} is contractively complemented in A by the projection P = Q+LQ, where L is any isometry from X onto Y. It is clear that {x + Lx : x ∈ X} is an expansion of X. Thus one cannot hope to classify contractively complemented Hilbertian operator spaces up to complete isometry. However, we will show in this section that all contractively complemented Hilbert spaces are expansions of a“minimal” 1-complemented Hilbert space which is a JC*-triple. Definition 3. The support partial isometry of a non-zero element ψ of the predual A_{∗} of a JW*-triple A is the smallest element of the set of partial isometries v such that ψ(v) = ∥ψ∥, and is denoted by v_{ψ}. For each non-empty subset G of A_{∗}, the support space s(G) of G is the smallest weak*-closed subspace of A containing the support partial isometries of all elements of G.

The existence and uniqueness of the support partial isometry was proved and exploited in the more general case of a JB^{∗}-triple (in which case the partial isometries are replaced by their abstract analog, the tripotents) in [9]. One of its important properties is that of “faithfulness”: if a non-zero partial isometry w satisfies w ≤ v_{ψ}, then ψ(w) > 0.

We now give two examples of expansions which naturally occur and are relevant to our work. Example 1. From [14, Theorem 2], if P is a contractive projection on a C^{∗}-algebra A, with X := P (A) which is isometric to a Hilbert space, then there are projections p, q ∈ A^{∗∗}, such that, X = P^{∗∗}A^{∗∗} = {pxq+(1−p)x(1−q) : x ∈ X}. The space pXq is exactly the norm closed span of the support partial isometries of the elements of P^{∗}A^{∗} (see [8] for the construction). The map E_{0} : x ↦ pxq is an isometry of X onto a JC^{∗}-subtriple E_{0}X of A^{∗∗}, E_{0}P^{∗∗} is a normal contractive projection on A^{∗∗} with range E_{0}X and clearly pXq ⊥ (1 − p)X(1 − q). It follows that

{X, A^{∗∗}, P^{∗∗}} is an expansion of {E_{0}X, A^{∗∗}, E_{0}P^{∗∗}}.

Specifically, let L : E_{0}X → A^{∗∗} be the map pxq ↦ (1 − p)x(1 − q). Then P (A) = P^{∗∗}A^{∗∗} = {pxq + (1 − p)x(1 − q) : x ∈ P (A)} and P^{∗∗} = E_{0}P^{∗∗} + LE_{0}P^{∗∗}, since if a ∈ A^{∗∗}, there is x ∈ A with a = P x = p(P x)q + (1 − p)P x(1 − q) and E_{0}P^{∗∗}a+LE_{0}P^{∗∗}a = E_{0}P x+LE_{0}P x = p(P x)q+(1−p)P x(1−q). Finally, if x ∈ A, then LE_{0}P x = (1−p)P x(1−q) and E_{0}P^{∗∗}LE_{0}P x = E_{0}P^{∗∗}((1−p)P x(1−q)) = 0 by [14, Theorem 2(e)]. Definition 4. The triple {E_{0}X, A^{∗∗}, E_{0}P^{∗∗}} (or simply E_{0}X) will be called the support of {X, A^{∗∗}, P^{∗∗}}. It is also called the enveloping support of {X, A, P }. Example 2. It follows from [5], that, for a normal contractive projection P from a von Neumann algebra (or JW^{∗}-triple) A onto a Hilbert space X, there is a similar projection E on A such that

{X, A, P } is an expansion of {EA, A, E}

and EA is the norm closure of the span of support partial isometries of elements of P_{∗}A_{∗}.

Indeed, as set forth in [5, Lemma 3.2], P (A) ⊂ s(P_{∗}(A_{∗})) ⊕ s(P_{∗}(A_{∗}))^{⊥} ⊂ A, and E : A → A is a normal contractive projection onto s(P_{∗}(A_{∗})) given by E = ϕ ∘ P where ϕ : P (A) → s(P_{∗}(A_{∗})) is the restriction of the M-projection of s(P_{∗}(A_{∗})) ⊕ s(P_{∗}(A_{∗}))^{⊥} onto s(P_{∗}(A_{∗})). (Although we will not use these facts, ϕ is a triple isomorphism from P (A) with the triple product {xyz}_{P (A)} := P {xyz} onto the JW^{∗}-subtriple s(P_{∗}(A_{∗})) of A, and ϕ^{−1} coincides with P on s(P_{∗}(A_{∗})).

The map L : E(A) → E(A)^{⊥} in this case is given by L = ϕ^{⊥} ∘ ϕ^{−1} = ϕ^{⊥} ∘ P, where ϕ^{⊥} : P (A) → s(P_{∗}(A_{∗}))^{⊥} is the restriction of the M-projection of s(P_{∗}(A_{∗}))⊕ s(P_{∗}(A_{∗}))^{⊥} onto s(P_{∗}(A_{∗}))^{⊥}. Then for h ∈ s(P_{∗}(A_{∗})), say h = ϕ ∘ P (x) for some x ∈ A, h + Lh = ϕ(P x) + ϕ^{⊥}(P x) = P x so that P (A) = {h + Lh : h ∈ s(P_{∗}(A_{∗}))}. Furthermore, for x ∈ A, Ex + LEx = ϕ(P x) + ϕ^{⊥}(P x) = P x. It is obvious that LE(A) ⊥ E(A). Finally, for x ∈ A, E(LEx) = ϕ ∘ P (ϕ^{⊥}(P x)) = 0 since P x = P P x = P ϕP x + P ϕ^{⊥}P x and ϕ ∘ P x = (ϕ ∘ P)^{2}x + ϕ ∘ P (ϕ^{⊥}(P x)). Definition 5. By analogy with Example 1, we will call {EA, A, E} (or simply EA) the support of {X, A, P } in this case. If {X, A, P } is not the expansion of any tuple other than itself, we say that {X, A, P } is essential and that X is essentially normally complemented in A.

A concrete instance of Example 2 is the projection of B(H) onto R (or C). It is easy to see that R and C are essentially normally complemented in B(H), as is R ∩ C in B(H ⊕ H). (See the paragraph preceding Theorem 3 belowd.) Remark 3.2. If {P (A), A, P } is as in Example 1, then {P^{∗∗}(A^{∗∗}), A^{∗∗}, P^{∗∗}} is as in Example 2, and the enveloping support of P is the same as the support of P^{∗∗}, since both E_{0}P (A) and E(A^{∗∗}) coincide with the norm closed linear span of A^{∗∗} generated by s(P^{∗}(A^{∗})). Proposition 3.3. Suppose X is Hilbertian and complemented in a von Neumann algebra A by a normal contractive projection P. Then {X, A, P } is essential if and only if it equals its support.

Proof. Suppose {X, A, P } equals its support and is the expansion of {Y, A, Q} given by a contractive map L. For each partial isometry v ∈ X, v = w + z where w and z are orthogonal partial isometries, w = Qv, QL = 0, z = L(w)and P = Q + LQ. Suppose v is the support partial isometry of ψ ∈ P_{∗}A_{∗}. Then

ψ(v) = ψ(P v) = ψ((Q + LQ)(v)) = ψ((Q + QLQ)(v)) = ψ(Qv) = ψ(w),

and hence w = v, L = 0 and {X, A, P }={Y, A, Q}. The converse is immediate. □

3.2. Operator space structure of 1-complemented Hilbert spaces. As noted at the beginning of the previous subsection, we cannot classify 1-complemented Hilbert spaces up to complete isometry. However, in Theorem 2 below, we are able to give a classification up to “trivial” expansion.

We assume in what follows that P is a normal contractive projection on a von Neumann algebra A, whose range Y = P (A) is a JC^{∗}-subtriple of A of rank one, and {u_{i}} is an orthonormal basis for Y consisting of a maximal family of minimal (in Y) collinear partial isometries. We shall assume for convenience that Y is infinite dimensional and separable. In Theorem 2 below, we shall also be able to handle the case of a contractive projection on a C^{∗}-algebra.

We know from Theorem 1 that Y is completely isometric to an intersection of operator spaces Ỹ = R ∩ L ∩ Φ, where R = ∩_{i}H_{∞}^{r_{i}} and L = ∩_{k}H_{∞}^{l_{k}}. Some of these spaces may be missing, and for short we have written H_{∞}^{r_{i}} = H_{∞}^{r_{i},R} and H_{∞}^{l_{k}} = H_{∞}^{l_{k},L}.

As shown in section 2, the weak*-ternary envelope of Y in A is generated by the partial isometries {u_{I,J}} and is ternary isomorphic, hence completely isometric, to a Cartan factor M of type I which is generated by the matrix units {E_{J,I}}. We may therefore assume that P is defined on M and has range Ỹ = R ∩ L ∩ Φ, which is a JC^{∗}-subtriple of M. We shall identify Y with Ỹ = R ∩ L ∩ Φ, and the weak*-ternary envelope of Y with M.

Note that by the definition of intersection, if the operator space structures of H_{∞}^{r_{i}}, H_{∞}^{l_{k}}, Φ come from embeddings H_{∞}^{r_{i}} ⊂ B(H^{r_{i}}), H_{∞}^{l_{k}} ⊂ B(H^{l_{k}}), Φ ⊂ B(H^{Φ}), then

[∩_{i}H_{∞}^{r_{i}}] ∩ [∩_{k}H_{∞}^{l_{k}}] ∩ Φ ⊂ M ⊂ B([⊕H^{r_{i}}] ⊕ [⊕H^{l_{k}}] ⊕ H^{Φ}).

Lemma 3.4. If u_{j} = ∑_{i} u_{j}^{r_{i}} +∑_{k} u_{j}^{l_{k}} +u_{j}^{Φ} is the decomposition of u_{j} into orthogonal partial isometries of [∩_{i}H_{∞}^{r_{i}}] ∩ [∩_{k}H_{∞}^{l_{k}}] ∩ Φ, and if P (u_{j}^{r_{i}}) = 0 for some i (resp. P (u_{j}^{l_{k}}) = 0 for some k), then u_{j}^{r_{i}} = 0 (resp. u_{j}^{l_{k}} = 0).

Proof. u_{j} is the support partial isometry of some functional ψ_{j} ∈ Y_{∗} = P_{∗}(A_{∗}). By the faithfulness of ψ_{j} on its support, ψ_{j}(u_{j}^{r_{i}}) = ψ_{j}(P (u_{j}^{r_{i}})) = 0 implies, since u_{j}^{r_{i}} ≤ u_{j}, that u_{j}^{r_{i}} = 0. Similarly for u_{j}^{l_{k}}. □

We again adopt the more compact notation Iu_{i}J, used in the proof of Lemma 2.2, for the “one” (uu^{∗})_{I}u_{i}(u^{∗}u)_{J}. We note next that for j ≠ i, {u_{j}, Iu_{i}J, u_{j}} = u_{j}(Iu_{i}J)^{∗}u_{j} = u_{j}Ju_{i}^{∗}Iu_{j} = (J ∪ {j}u_{i}^{∗}(I ∪ {j}) = 0 since either j ∉ I or j ∉ J. By a conditional expectation formula in (2),

0 = P ({u_{j}, Iu_{i}J, u_{j}}) = {u_{j}, P (Iu_{i}J), u_{j}}.

Since every element of Y is in the closed linear span of the u_{j}, we may write P (Iu_{i}J) = ∑_{k} λ_{k}^{i,J}u_{k} and thus 0 = ∑_{k} \overline{λ_{k}^{i,J}}\{u_{j}u_{k}u_{j}\} = \overline{λ_{j}^{i,J}}u_{j}. We conclude that λ_{j}^{i,J} = 0 for j ≠ i and hence P (Iu_{i}J) = λ_{i,J}u_{i} for each “one” Iu_{i}J, where we have written λ_{i,J} for λ_{i}^{i,J}.

Now suppose that i is fixed and k ≠ i, say k ∈ J. Then

2{u_{k}, u_{i}, Iu_{i}J} = u_{k}u_{i}^{∗}Iu_{i}J + Iu_{i}Ju_{i}^{∗}u_{k} = u_{k}u_{i}^{∗}Iu_{i}J (as k ∈ J)
= Iu_{k}u_{i}^{∗}u_{i}J (by “hopping”)
= Iu_{k}((J − {k}) ∪ {i}).

Thus by another conditional expectation formula in (2),

λ_{k,(J−{k})∪{i}}u_{k} = P (Iu_{k}((J − {k}) ∪ {i}))
= P (2{u_{k}, u_{i}, Iu_{i}L})
= 2{u_{k}, u_{i}, P (Iu_{i}J)}
= 2λ_{i,J}{u_{k}u_{i}u_{i}} = λ_{i,J}u_{k}.

Thus λ_{i,J} = λ_{k,(J−{k})∪{i}} and so λ_{i,J} = λ is independent of i, J such that i ∉ J and |J| = m. Similarly, it can be shown that λ_{i,J} = λ_{k,J} for any i ∉ J, k ∉ J.

We have now shown that there is a complex number λ = λ_{m} such that P (Iu_{j}J) = λu_{j}, for all partitions I ∪ {j} ∪ J of Ω with |J| = m.

Now, since P (∑_{|J|=m} Iu_{i}J) = ∑_{|J|=m} P (Iu_{i}J) = ∑_{|J|=m} λ_{m}u_{i}, we must have λ_{m} = 0 and P (∑_{|J|=m} Iu_{i}J)) = 0 unless r_{i} = 0. Thus P (u_{i}^{r_{i}}) = 0 unless m = 0. Similarly, P (u_{i}^{l_{k}}) = 0 unless l_{k} = 0. By Lemma 3.4, u_{j}^{r_{i}} = u_{j}^{l_{k}} = 0 for r_{i} ≠ 0, and l_{k} ≠ 0. By Theorem 1, P (A) is an intersection of at most the three spaces R, C, Φ. Together with Examples 1 and 2, Remark 3.2, and Proposition 3.3 in subsection 3.1, this proves the second main theorem of this paper. Theorem 2. Suppose Y is a separable infinite dimensional Hilbertian operator space which is contractively complemented (resp. normally contractively complemented) in a C^{∗}-algebra A (resp. W*-algebra A) by a projection P. Then,

(a): {Y, A^{∗∗}, P^{∗∗}} (resp. {Y, A, P }) is an expansion of its support {H, A^{∗∗}, Q}

(resp. {H, A^{∗∗}, Q}, which is essential) (b): H is contractively complemented in A^{∗∗} (resp. A) by Q and is completely

isometric to either R, C, R ∩ C, or Φ.

This theorem says that, in A^{∗∗}, Y is the diagonal of a contractively complemented space H which is completely isometric to R, C, R ∩ C or Φ and an orthogonal degenerate space K which is in the kernel of P. As pointed out at the beginning of section 3.1, this is the best possible classification.

By [20], the range Y of a completely contractive projection on a C*-algebra is a TRO. By Lemma 2.11 it follows that, if Y is Hilbertian, Y is completely isometric to R or C. This gives an alternate proof of the result of Robertson [17], stated here for completely contractive projections on a C^{∗}-algebra.

Although Theorem 2 is only a classification modulo expansions, the following Lemma shows that it is the correct analogue for contractively complemented Hilbert spaces. Lemma 3.5. Suppose that {Y, A, P } is an expansion of {H, A, Q} and that P is a completely contractive projection. Then Y is completely isometric to H.

Proof. By definition of expansion, in A^{∗∗}, Y coincides with {h + L(h) : h ∈ H}, Q + LQ = P, L(H) ⊥ H and Q(L(H)) = 0. Thus, P^{∗∗}|_{H} is a complete contraction from H onto Y with completely contractive inverse Q|_{Y}. Hence, Y is completely isometric to H. □

3.3. An essential contractive projection onto Φ. As noted earlier, the spaces R, C and R ∩ C are each essentially normally contractively complemented in a von Neumann algebra. We now proceed to show that the same holds for Φ.

We begin by taking a closer look at the contractive projection P = P_{n}^{k} of the ternary envelope T = T (H_{n}^{k}) = M_{p_{k},q_{k}}(C) of H_{n}^{k}, onto H_{n}^{k}. This projection and the space H_{n}^{k} were first constructed in [1] and rediscovered in [13]. By [13, Cor. 7.3],

P_{n}^{k}x = \frac{1}{\binom{n−1}{k−1}} ∑_{i=1}^{n} tr (xu_{i}^{∗})u_{i}.

Consistent with the identification of Y with Ỹ in the previous subsection, we let u_{i} denote the image of the orthonormal basis u_{i}, of a finite dimensional JC^{∗}-triple of rank one. Thus, u_{i} = ∑ ϵ(I, i, J)E_{J,I}. Lemma 3.6. The action of P = P_{n}^{k} is as follows: if x ∈ T is not a “one”, then

P x = 0. If x = ϵ(I, i, J)E_{J,I} is a “one”, then P (ϵ(I, i, J)E_{J,I}) = \frac{1}{\binom{n−1}{k−1}}u_{i}.

Proof. Suppose first that x = ϵ(I, J)E_{J,I} ∈ T is not a “one”, that is, I ∩ J ≠ ∅. Then

xu_{i}^{∗} = ϵ(I, J)E_{J,I} ∑_{I′,J′} ϵ(I′, i, J′)E_{I′,J′}
= ϵ(I, J) ∑_{I′,J′} ϵ(I′, i, J′)E_{J,I}E_{I′,J′}
= ϵ(I, J) ∑_{J′} ϵ(I, i, J′)E_{J,J′}.

Since J′ ∩ I = ∅ and J ∩ I ≠ ∅, J′ is never equal to J and so tr (xu_{i}^{∗}) = 0 and P x = 0.

Suppose now that x = ϵ(I, i, J)E_{J,I} ∈ T is a “one”. Then for 1 ≤ j ≤ n, u_{j} = ∑_{I′∩J′=∅} ϵ(I′, j, J′)E_{J′,I′}, and as above xu_{j}^{∗} = ϵ(I, i, J) ∑_{J′} ϵ(I, i, J′)E_{J,J′}. Thus, tr (xu_{j}^{∗}) = 1 if j = i and tr (xu_{j}^{∗}) = 0 if j ≠ i. It follows that

P (ϵ(I, i, J)E_{J,I}) = \frac{1}{\binom{n−1}{k−1}} ∑_{j} tr (xu_{j}^{∗})u_{j} = \frac{1}{\binom{n−1}{k−1}}u_{i}. □

We proceed to construct a contractive projection defined on a TRO A which has range Φ. Since every TRO is the corner of a C^{∗}-algebra, we will have constructed a projection on a C^{∗}-algebra with range Φ. Now, let u_{i} be an orthonormal basis for the Hilbertian operator space Φ and let H_{n} = sp{u_{1}, …, u_{n}}. As noted in the proof of Proposition 2.8, H_{n} = Φ_{n} is completely isometric to the intersection ∩_{i=1}^{n}H_{n}^{i} ⊂ ⊕_{k=1}^{n}T (H_{n}^{i}) = the ternary envelope T (H_{n}) of H_{n} in A.

We construct a contractive projection P^{n} on T (H_{n}) with range H_{n} as follows. For x = ⊕_{i=1}^{n}x_{i} ∈ T (H_{n}), write x = ∑_{i=1}^{n}(0 ⊕ ⋯ ⊕ x_{i} ⊕ ⋯ ⊕ 0), (x_{i} is in the ith-position). Then define

P^{n}(x) = ∑_{i=1}^{n} P^{n}(0 ⊕ ⋯ ⊕ x_{i} ⊕ ⋯ ⊕ 0) := \frac{1}{n} ∑_{i=1}^{n}(P_{n}^{i}(x_{i}), …, P_{n}^{i}(x_{i})).

Note that since (P_{n}^{i}(x_{i}), …, P_{n}^{i}(x_{i})) belongs to H_{n} = ∩_{i=1}^{n}H_{n}^{i}, we shall sometimes write it as ((P_{n}^{i}(x_{i}))^{1}, …, (P_{n}^{i}(x_{i}))^{n}) and view (P_{n}^{i}(x_{i}))^{j} as an element of H_{n}^{j}.

With u_{k} = (u_{k}, …, u_{k}) = (u_{k}^{1}, …, u_{k}^{n}) = ∑_{i}(0, …, u_{k}^{i}, …, 0), we have

P^{n}(u_{k}) = ∑_{i} P^{n}((0, …, u_{k}^{i}, …, 0))

= \frac{1}{n} ∑_{i}(P_{n}^{i}(u_{k}^{i}), …, P_{n}^{i}(u_{k}^{i}))
= ∑_{i}(u_{k}^{i}, …, u_{k}^{i})/n
= ∑_{i}(u_{k}^{1}, …, u_{k}^{n})/n
= (u_{k}^{1}, …, u_{k}^{n}) = u_{k}.

By Lemma 3.6, P^{n} is zero an any non-“one”, so the range of P^{n} is H_{n}. To calculate the action of P^{n} on “ones”, let x = Iu_{k}J be such and write x = ⊕x_{i} = ⊕I^{i}u_{k}^{i}J^{i}, where x_{i} ∈ H_{n}^{i}. We claim that

(8)  P^{n}(x) = \frac{u_{k}}{n\binom{n−1}{i−1}},

where |I| = i − 1. Let us illustrate this first in a specific example: Let n = 3, x = u_{2}u_{2}^{∗}u_{1}u_{3}^{∗}u_{3} = x_{1} ⊕ x_{2} ⊕ x_{3} ∈ H_{3}^{1} ∩ H_{3}^{2} ∩ H_{3}s, so that x_{1} = 0, x_{3} = 0, and i = 2. By Lemma 3.6 again,

P^{3}(x) = P^{3}(x_{1} ⊕ 0 ⊕ 0) + P^{3}(0 ⊕ x_{2} ⊕ 0) + P^{3}(0 ⊕ 0 ⊕ x_{3})
= \frac{1}{3}[(P_{1}^{3}(x_{1}), P_{1}^{3}(x_{1}), P_{1}^{3}(x_{1})) + (P_{2}^{3}(x_{2}), P_{2}^{3}(x_{2}), P_{2}^{3}(x_{2}))
+ (P_{3}^{3}(x_{3}), P_{3}^{3}(x_{3}), P_{3}^{3}(x_{3}))]
= \frac{1}{3}[(0, 0, 0) + (\frac{1}{2}u_{1}^{2}, \frac{1}{2}u_{1}^{2}, \frac{1}{2}u_{1}^{2}) + (0, 0, 0)]
= \frac{1}{3} \frac{1}{2}(u_{1}^{2}, u_{1}^{2}, u_{1}^{2}) = \frac{1}{6}u_{1}.

In general, for x = ⊕x_{i} as above,

P^{n}(x) = (1/n)[∑(P_{n}^{i}(x_{i}), P_{n}^{i}(x_{i}), P_{n}^{i}(x_{i}))] = (1/n)[\frac{1}{\binom{n−1}{i−1}}(u_{k}1, …, u_{k}^{n})],

as required for (8). Lemma 3.7. Under the embedding T (H_{n}) = ⊕_{i=1}^{n}T (H_{n}^{i}) ⊂ ⊕_{i=1}^{n+1}T (H_{n}^{i}) = T (H_{n+1}), given by x_{1} ⊕ ⋯ ⊕ x_{n} ↦ x_{1} ⊕ ⋯ ⊕ x_{n} ⊕ 0, we have P^{n+1}|T (H_{n}) = P^{n}.

Proof. This is obviously true for generators u_{I,J} of T (H_{n}) which are not “ones” since all of the P_{n}^{k} and P_{n+1}^{k} vanish on them. On the other hand, if Iu_{k}J is a “one” in T (H_{n}), then by collinearity u_{k} = u_{k}u_{n+1}^{∗}u_{n+1} + u_{n+1}u_{n+1}^{∗}u_{k}, and by (8),

P^{n+1}(Iu_{k}J) = P^{n+1}((I ∪ {n + 1})u_{k}J + Iu_{k}(J ∪ {n + 1}))
= \frac{1}{n + 1} \frac{1}{\binom{n}{i}}u_{k} + \frac{1}{n + 1} \frac{1}{\binom{n}{i−1}}u_{k}
= \frac{1}{n} \frac{1}{\binom{n−1}{i−1}}u_{k} = P^{n}(Iu_{k}J). □

Lemma 3.7 enables the definition of a contractive projection P on a TRO A which is the norm closure in ⊕_{i=1}^{∞}M_{p_{i},q_{i}}(C) of ∪_{n=1}^{∞}T (H_{n}) with P (A) = Φ. As noted earlier, we can assume that A is a C^{∗}-algebra. By Example 1, {Φ, A^{∗∗}, P^{∗∗}} is an expansion of {E_{0}Φ, A^{∗∗}, E_{0}P^{∗∗}}, so E_{0}P^{∗∗}(A^{∗∗}) = E_{0}Φ. Thus E_{0}Φ is a normally contractively complemented JC^{∗}-subtriple of A^{∗∗}. By Theorem 2, E_{0}Φ is completely isometric to one of R, C, R ∩ C, Φ, which we shall write as R ∩ C ∩ Φ, with the understanding that one or two terms in this intersection may be missing. We claim in fact that R and C are both missing. Lemma 3.8. The support E_{0}Φ of {Φ, A^{∗∗}, P^{∗∗}} for the above construction is completely isometric to Φ.

Proof. Because of (8), for any partition {i_{1}, i_{2}, …} ∪ {k} ∪ {j_{1}, j_{2}, …, j_{m}} of {1, 2, 3, …},

P^{∗∗}(Iu_{k}J) = lim_{n→∞} P^{n+m+1}(\{i_{1}, …, i_{n}\}u_{k}J) = lim_{n→∞} \frac{1}{n + m + 1} \frac{1}{\binom{n+m}{n}}u_{k} = 0.

Thus, writing E_{0}u_{j} = E_{0}u_{j}^{C} +E_{0}u_{j}^{R} +E_{0}u_{j}^{Φ} as in the notation of Lemma 3.4 and using Lemma 2.2, P^{∗∗}E_{0}(u_{j}^{C}) = P^{∗∗}E_{0}(u_{j}^{R}) = 0 and thus P^{∗∗}(E_{0}u_{j}) = P^{∗∗}(E_{0}u_{j}^{Φ}). Recall that E_{0}(u_{j}) is the support partial isometry of a norm 1 element ψ in P^{∗}A^{∗}. Since ψ(E_{0}u_{j}) = ψ(P^{∗∗}E_{0}u_{j}) = ψ(P^{∗∗}E_{0}u_{j}^{Φ}) = ψ(E_{0}u_{j}^{Φ}), it follows that E_{0}(u_{j}) = E_{0}u_{j}^{Φ}, so that E_{0}(u_{j}^{C}) = E_{0}(u_{j}^{R}) = 0, proving that E_{0}Φ is completely isometric to Φ. □

Since R, C and R ∩ C are trivially contractively complemented in B(H) as spans of finite rank operators in such a way that they clearly equal their support spaces, this proves that each of the spaces occurring in (b) of Theorem 2 are essentially contractively complemented. Theorem 3. The operator spaces R, C, R ∩ C , and Φ are each essentially normally contractively complemented in a von Neumann algebra.

4. Completely bounded Banach-Mazur distance

Since all of the Hilbertian operator spaces under consideration in this paper are homogeneous (by Lemma 1.1 and [14, Theorem 1]), the completely bounded distances can be computed by simply computing ∥ψ∥cb∥ψ^{−1}∥cb for any fixed unitary operator between the two Hilbert spaces, [22, Theorem 3.1]. Theorem 4. For any m, k ≥ 1,

(a): dcb(C, H_{∞}^{m,R}) = \sqrt{m + 1}. (b): dcb(H_{∞}^{k,L}, H_{∞}^{m,R}) = ∞. (c): dcb(Φ, H_{∞}^{m,R}) = ∞.

Proof. We first prove (a). Let {u_{i}} (resp. {v_{i}}) be any orthonormal basis for C (resp. H_{∞}^{m,R}), and let ψ be the isometry that takes u_{i} to v_{i}. For each n > m + 1, let H̃_{n}^{1} = sp {u_{1}, …, u_{n}}, H̃_{n,R} = sp {v_{1}, …, v_{n}}, and ψ^{(n)} = ψ|H̃_{n}^{1}. Note that for H̃_{n}^{1}, we have i_{R} = 1 and i_{L} = n (see Remark 2.6) so that by [14, Cor. 5.3], H̃_{n}^{1} is completely isometric to column space C_{n} = H_{n}^{1}. Because of this, in what follows, we will write H_{n}^{1} for H̃_{n}^{1}. The space H̃_{n,R} has i_{R} = m + 1 < n and i_{L} = n, and by [13, Th. 3(b)], is completely isometric to an intersection H_{n}^{k_{1}} ∩ ⋯ ∩ H_{n}^{k_{r}}, where m + 1 = k_{1} > k_{2} > ⋯ > k_{r}. Now, for any p,

∥(ψ^{(n)})_{p}∥ = sup_{0≠x∈M_{p}(H_{n}^{1})} \frac{∥ψ_{p}(x)∥_{M_{p}(H̃_{n})}}{∥x∥_{M_{p}(H_{n}^{1})}} .

Let us write x = [x_{ij}] with x_{ij} ∈ H_{n}^{1} and y = [y_{ij}] = ψ_{p}(x), with y_{ij} = ψ(x_{ij}) = (y_{ij}^{k_{1}}, …, y_{ij}^{k_{r}}) ∈ H̃_{n} where y_{ij}^{k_{l}} ∈ H_{n}^{k_{l}}.

Now M_{p}(H̃_{n}) ⊂ M_{p}(H_{n}^{k_{1}}) ⊕ ⋯ ⊕ M_{p}(H_{n}^{k_{r})}, and M_{p}(H_{n}^{1}) ∋ [x_{ij}] ↦ [y_{ij}^{k_{l}}] ∈ M_{p}(H_{n}^{k_{l}}) has norm \sqrt{k_{l}} by [14, Lemma 3.1]. Thus y = [y_{ij}^{k_{1}}] ⊕ ⋯ ⊕ [y_{ij}^{k_{r}}] and

∥y∥_{M_{p}(H̃_{n})} = max_{1≤l≤r} ∥[y_{ij}^{k_{l}}]∥_{M_{p}(H_{n}^{(k_{l})})}
≤ max_{1≤l≤r} \sqrt{k_{l}}∥x∥_{M_{p}(H_{n}^{1})}
= \sqrt{m + 1}∥x∥_{M_{p}(H_{n}^{1})}.

Thus ∥ψ : H_{n}^{1} → H̃_{n}∥cb ≤ \sqrt{m + 1}, and by a simple approximation argument based on the fact that H_{∞}^{m,R} (resp. C) is the norm closure of the increasing union of the H̃_{n,R} (resp. H_{n}^{1}), it follows that ∥ψ∥cb ≤ \sqrt{m + 1}. Moreover, equality holds. Indeed, by the proof of [14, Lemma 3.1], there exists, for each n ≥ 1, an element (h_{n1}^{m}, …, h_{nn}^{m}) ∈ M_{1,n}(H_{n}^{1}), such that ∥(h_{n1}^{m}, …, h_{nn}^{m})∥_{M_{1,n}(H_{n}^{1})} = 1 and

∥(ψ(h_{n1}^{m}), …, ψ(h_{nn}^{m}))∥_{M_{1,n}(H_{n}^{m})} = \sqrt{m + 1}. Then with x_{n} := [ \begin{matrix}h_{n1}^{m}, …, h_{nn}^{m} & 0 \\ 0 & 0 \end{matrix} ] ∈

M_{p}(C) and y_{n} = ψ_{p}(x_{n}), we have ∥x_{n}∥_{M_{p}(C)} = 1 and ∥y_{n}∥_{M_{p}(H_{∞}^{m,R})} = \sqrt{m + 1}, so that ∥ψ_{p}∥ = \sqrt{m + 1} and ∥ψ∥cb = \sqrt{m + 1}.

We next show that ∥ψ^{−1}∥cb = 1, which will complete the proof of (a). Let y = [y_{ij}] ∈ M_{p}(H̃_{n}) and x = [x_{ij}] = (ψ^{−1})_{p}(y) ∈ M_{p}(H_{n}^{1}) so that x_{ij} = ψ^{−1}(y_{ij}). Then for any 1 ≤ l ≤ r, by [14, Lemma 3.1] and for sufficiently large p,

∥ψ^{−1} : H̃_{n} → H_{n}^{1}∥cb = ∥(ψ^{−1})_{p} : M_{p}(H̃_{n}) → M_{p}(H_{n}^{1})∥
= sup_{y≠0} \frac{∥(ψ^{−1})_{p}y∥_{M_{p}(H_{n}^{1})}}{∥y∥_{M_{p}(H̃_{n})}}

≤ sup_{y≠0} \frac{\sqrt{\frac{n}{n−k_{l}+1}}∥y∥_{M_{p}(H_{n}^{k_{l}})}}{max_{1≤q≤r} ∥[y_{ij}^{k_{q}}]∥_{M_{p}(H_{n}^{kq})}}

≤ \sqrt{\frac{n}{n − k_{l} + 1}} ≤ \sqrt{\frac{n}{n − m + 1}} ≤ 1.

Again, by the proof of [14, Lemma 3.1], for each n ≥ 1, there exists an element (h_{n1}^{m}, …, h_{nn}^{m})^{t} ∈ M_{n,1}(H_{n}^{m}), such that ∥(h_{n1}^{m}, …, h_{nn}^{m})^{t}∥_{M_{n,1}(H_{n}^{m})} = 1 and

∥(ψ^{−1}(h_{n1}^{m}), …, ψ^{−1}(h_{nn}^{m}))^{t}∥_{M_{n,1}(H_{n}^{1})} = \sqrt{\frac{n}{n−m+1}}. Then with

y_{n} := [ \begin{matrix}(h_{n1}^{m}, …, h_{nn}^{m})^{t} & 0 \\ 0 & 0 \end{matrix} ] ∈ M_{p}(H_{∞}^{m,R})

and x_{n} = (ψ_{p})^{−1}(y_{n}), we have ∥y_{n}∥_{M_{p}(H_{∞}^{m,R})} = 1 and ∥x_{n}∥_{M_{p}(C)} = \sqrt{\frac{n}{n−m+1}}.

Hence ∥ψ^{−1}∥cb = 1 and this proves (a).

We now prove (b). Let {u_{i}} (resp. {v_{i}}) be any orthonormal basis for H_{∞}^{k,L} (resp. H_{∞}^{m,R}), and let ψ be the isometry that takes u_{i} to v_{i}. For each n > max(k + 1, m), let H̃_{n,L} = sp {u_{1}, …, u_{n}}, H̃_{n,R} = sp {v_{1}, …, v_{n}}, and ψ^{(n)} = ψ|H̃_{n,L}. Note that for H̃_{n,L}, we have i_{R} = n and i_{L} = k + 1 so that by [13, Th. 3(b)], H̃_{n,L} is completely isometric to an intersection H_{n}^{j_{1}} ∩ ⋯ ∩ H_{n}^{j_{s}}, where n = j_{1} > j_{2} > ⋯ > j_{s}. Similarly for H̃_{n,R}, we have i_{R} = m + 1 and i_{L} = n so that by [13, Th. 3(b)], H̃_{n,R} is completely isometric to an intersection H_{n}^{k_{1}} ∩ ⋯ ∩ H_{n}^{k_{r}}, where m + 1 = k_{1} > k_{2} > ⋯ > k_{r}.

Now, for any p, with x = [x_{ij}] = (ψ)_{p}^{−1}(y),

∥((ψ^{(n)})^{−1})_{p}∥ = sup_{0≠y∈M_{p}(H̃_{n,R})} \frac{∥ψ_{p}^{−1}(y)∥_{M_{p}(H̃_{n,L})}}{∥y∥_{M_{p}(H̃_{n,R})}}

= sup_{0≠y∈M_{p}(H̃_{n,R})} \frac{max_{1≤q≤s} |[x_{ij}^{j_{q}}]∥_{M_{p}(H_{n}^{jq})}}{max_{1≤l≤r} ∥[y_{ij}^{k_{l}}]∥_{M_{p}(H_{n}^{k_{l}})}} ,

which, for suitable choices of y, as above, is greater than

\frac{max(\sqrt{j_{1}}, …, \sqrt{j_{s}})}{max(\sqrt{k_{1}}, …, \sqrt{k_{r}})} = \begin{matrix}√ & \overline{n} \\ √ & \overline{m + 1} \end{matrix}.

Thus, ∥ψ^{−1}∥cb ≥ ∥(ψ^{(n)})^{−1}∥cb ≥ ∥((ψ^{(n)})^{−1})_{p}∥ ≥ \begin{matrix}√ & \overline{n} \\ √ & \overline{m+1} \end{matrix} → ∞. This proves (b).

Finally, we prove (c). Let {u_{i}} (resp. {v_{i}}) be any orthonormal basis for Φ (resp. H_{∞}^{m,R}, and let ψ be the isometry that takes u_{i} to v_{i}. For each n > m, let H̃_{n,R} = sp {u_{1}, …, u_{n}}, H̃_{n,Φ} = sp {v_{1}, …, v_{n}}, and ψ^{(n)} = ψ|H̃_{n,Φ}. Note that for H̃_{n,R}, we have i_{R} = m + 1 and i_{L} = n so that by [13, Th. 3(b)], H̃_{n,R} is completely isometric to an intersection H_{n}^{j_{1}} ∩ ⋯ ∩ H_{n}^{j_{s}}, where m + 1 = j_{1} > j_{2} > ⋯ > j_{s}. For H̃_{n,Φ}, we have i_{R} = n and i_{L} = n so that by [13, Th. 3(b)], H̃_{n,Φ} is completely isometric to an intersection H_{n}^{k_{1}} ∩ ⋯ ∩ H_{n}^{k_{r}}, where n = k_{1} > k_{2} > ⋯ > k_{r} (in fact, as shown in the proof of Proposition 2.8, r = n and k_{j} = n − j + 1 but we do not need this fact).

Now, for any p, with x = [x_{ij}] ∈ M_{p}(H̃_{n,Φ}), y = [y_{ij}] = ψ_{p}(x),

∥(ψ^{(n)})_{p}∥ = sup_{0≠x∈M_{p}(H̃_{n,R})} \frac{∥ψ_{p}(x)∥_{M_{p}(H̃_{n,R})}}{∥x∥_{M_{p}(H̃_{n,Φ})}}

= sup_{0≠x∈M_{p}(H̃_{n,R})} \frac{max_{1≤l≤r} |[y_{ij}^{k_{l}}]∥_{M_{p}(H_{n}^{k_{l}})}}{max_{1≤q≤s} ∥[x_{ij}^{j_{q}}]∥_{M_{p}(H_{n}^{js})}} ,

which, for suitable choices of x, as above, is ≥ \frac{max(\sqrt{k_{1}},...,\sqrt{k_{r}})}{max(\sqrt{j_{1}},...,\sqrt{j_{s}})} = \begin{matrix}√ & \overline{n} \\ √ & \overline{m+1} \end{matrix}, showing ∥ψ∥cb = ∞ and proving (c). □ Corollary 4.1. For m, k ≥ 0,

(a): H_{∞}^{m,R} and H_{∞}^{k,R} are completely isomorphic but not completely isometric

if m ≠ k. (b): H_{∞}^{k,L} and H_{∞}^{m,R} are not completely isomorphic. (c): Φ and H_{∞}^{m,R} are not completely isomorphic.

Similar arguments yield the following distances as well as their corresponding consequences:

• dcb(R, H_{∞}^{k,L}) = \sqrt{k + 1}.

• dcb(R, H_{∞}^{m,R}) = dcb(C, H_{∞}^{m,L}) = dcb(Φ, H_{∞}^{m,L}) = ∞

5. Representation on the Fock space

We begin by recalling the construction of the spaces H_{∞}^{m,R}; see subsection 2.1. Let I denote a subset of Ω with |Ω − I| = m + 1. and let J denote a subset of Ω of cardinality |J| = m. We assume that each I = {i_{1}, i_{2}, …} is such that i_{1} < i_{2} < ⋯, and that the collection of all such subsets I is ordered lexicographically. Similarly, if J = {j_{1}, …, j_{m}}, then j_{1} < ⋯ < j_{m} and the collection of all such subsets J is ordered lexicographically.

We shall use the notation e_{i} to denote the column vector with a 1 in the ith position and zeros elsewhere. Thus e_{1}, e_{2}, … denotes the canonical basis of column vectors for separable column space C. More generally, e_{I} denotes the basis vector for ℓ_{2} consisting of a 1 in the “Ith” position.

The space H_{∞}^{m,R} is the closed linear span of matrices b_{i}^{m}, i ∈ Ω, given by

b_{i}^{m} = ∑_{I∩J=∅,(I∪J)^{c}={i},|J|=m} ϵ(I, i, J)e_{J,I},

where e_{J,I} = e_{J} ⊗ e_{I} = e_{J}e_{I}^{t} ∈ M_{ℵ_{0},ℵ_{0}}(C) = B(ℓ_{2}), and ϵ(I, i, J) is the signature of the permutation defined for disjoint I, J in subsection 2.1.

Let H be any separable infinite dimensional Hilbert space. For any h ∈ H, let C_{h}^{m} denote the wedge (or creation) operator from ∧^{m}H to ∧^{m+1}H given by

C_{h}^{m}(h_{1} ∧ ⋯ ∧ h_{m}) = h ∧ h_{1} ∧ ⋯ ∧ h_{m}.

The space of creation operators \overline{sp}\{C_{e_{i}}^{m}\} will be denoted by C^{m}. Its operator space structure is given by its embedding in B(∧^{m}H, ∧^{m+1}H).

It will be convenient to identify the space ∧^{k}H with ℓ_{2}({J ⊂ Ω : |J| = k}) or with ℓ_{2}({I ⊂ Ω : |Ω − I| = k}).

Define the unitary operators V_{k} and W_{k} on ∧^{k}H by

V_{k} : ℓ_{2}({I ⊂ Ω : |Ω − I| = k}) → ℓ_{2}({K ⊂ Ω : |K| = k})

and

W_{k} : ℓ_{2}({J ⊂ Ω : |J| = k}) → ℓ_{2}({J ⊂ Ω : |J| = k})

as follows:

• V_{k}(e_{I}) = e_{N−I}; More specifically, V_{k}(e_{I}) = e_{j_{1}} ∧ ⋯ ∧ e_{j_{k}} where N − I =

{j_{1} < ⋯ < j_{k}}.
• W_{k}(e_{I}) = ϵ(i, I)ϵ(I, i, J)e_{I} where I ∪ {i} ∪ J = N is a disjoint union.

It is easy to see, as in [14, section 2], that the definition of W_{k} is independent of the choice of i. Indeed, if p is chosen so that i_{p} > max{i, j_{k−1}}, then ϵ(I, i, J) = (−1)^{p}ϵ(i, i_{1}, …, i_{p}, j_{1}, …, j_{k−1}) = (−1)^{p}ϵ(i, i_{1}, …, i_{p})ϵ({i}∪{i_{1}, …, i_{p}}, j_{1}, …, j_{k−1}) and therefore for any i′ ≠ i, ϵ(i, I)ϵ(I, i, J) = ϵ(i′, I′)ϵ(I′, i′, J). Lemma 5.1. H_{∞}^{m,R} is completely isometric to C^{m}.

Proof. With b_{i}^{m} = ∑ ϵ(I, i, J)e_{J,I} we have

b_{i}^{m}W_{m}(e_{I_{0}}) = b_{i}^{m}((ϵ(i, I_{0})ϵ(I_{0}, i, J_{0})e_{I_{0}}) = ϵ(i, I_{0})e_{J_{0}},

and

V_{m+1}C_{e_{i}}^{m}(e_{I_{0}}) = V_{m+1}(ϵ(i, I_{0})e_{{i}∪I_{0}}) = ϵ(i, I_{0})e_{J_{0}}. □

Since the anti-creation operator space A^{m} is simply the adjoint of the creation operator space C^{m}, by construction it is clear that Lemma 5.2. H_{∞}^{m,L} is completely isometric to the space of anti-creation operators A^{m}.

By Lemma 5.1, Lemma 5.2, Theorem 1, and [14, Lemma 2.1] we now have

Theorem 5. Every n-dimensional or infinite dimensional separable Hilbertian JC*- triple is completely isometric to an intersection over a set of values of k of the spaces of creation and annihilation operators on k-fold antisymmetric tensors of the Hilbert space.

References

1. Jonathan Arazy and Yaakov Friedman, Contractive projections in C_{1} and C_{∞}, Mem. Amer. Math. Soc. 13 (1978), no 200.

2. David Blecher, The Shilov boundary of an operator space and the characterization theorems., J. Funct. Anal. 182 (2001), no. 2, 280–343.

3. David P. Blecher and Christian Le Merdy, Operator algebras and their modules—an operator space approach, Clarendon Press, Oxford 2004.

4. Truong Dang and Yaakov Friedman, Classification of JBW^{∗}-triple factors and applications, Math. Scand. 61 (1987), 292–330.

5. C. Martin Edwards, Remo V. Hügli, and Gottfried T. Rüttimann, A geometric characterization of structural projections on a JBW^{∗}-triple, J. Funct. Anal. 202 (2003), 174–194.

6. Edward Effros and Zhong-Jin Ruan, Operator Spaces, Oxford University Press, 2000.

7. Yaakov Friedman and Bernard Russo, Conditonal expectation without order, Pac. J. Math. 115 (1984), 351–360.

8. Yaakov Friedman and Bernard Russo, Solution of the Contractive Projection Problem, J. Funct. Anal. 60 (1985), 56-79.

9. Yaakov Friedman and Bernard Russo, Structure of the predual of a JBW^{∗}-triple, J. Reine Angew. Math. 356 (1985), 67–89.

10. Lawrence A. Harris, Bounded symmetric domains in infinite dimensional spaces, In: Hayden, T. L. ,Suffridge,T. J. (eds.) Infinite dimensional holomorphy. Proceedings, 1973 (Lecture Notes in Mathematics 364,pp. 13–40), Springer 1974.

11. Magnus R. Hestenes, A ternary algebra with applications to matrices and linear transformations, Arch. Rational Mech. Anal. 11 (1962), 138–194.

12. Wilhelm Kaup, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Zeit. 183 (1983), 503–529.

13. Matthew Neal and Bernard Russo, Contractive projections and operator spaces, Trans. Amer. Math. Soc. 355 (2003), 2223–2362.

14. Matthew Neal and Bernard Russo, Representation of contractively complemented Hilbertian operator spaces on the Fock space, Proc. Amer. Math. Soc. (to appear).

15. Vern Paulson, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics 78, Cambridge University Press, Cambridge, 2002.

16. Gilles Pisier, Introduction to Operator Space Theory, Cambridge University Press 2003.

17. A. Guyan Robertson, Injective matricial Hilbert spaces, Math. Proc.Cambridge Philos. Soc. 110 (1991), no.1, 183-190.

18. Zhong-Jin Ruan, Injectivity of operator spaces., Trans. Amer. Math. Soc. 315 (1989), no. 1, 89–104.

19. Zhong-Jin Ruan, Type decomposition and the rectangular AFD property for W^{∗}-TRO’s., Canad. J. Math. 56 (2004), no. 4, 843–870.

20. M. A. Youngson, Completely contractive projections on C*-algebras, Quart. J. Oxford Ser. (2) 34 (1983), 507-511.

21. Heinrich Zettl, A characterization of ternary rings of operators, Adv. Math. 48 (1983), 117– 143.

22. Chun Zhang, Completely bounded Banach-Mazur distance, Proc. Edinburgh Math. Soc. 40 (1997),247–260.

Department of Mathematics, Denison University, Granville, Ohio 43023

E-mail address: nealm@denison.edu

Department of Mathematics, University of California, Irvine, California 92697-3875

E-mail address: brusso@math.uci.edu