OPERATOR ANALOGUES OF LOCALLY CONVEX SPACES

EDWARD G. EFFROS AND CORRAN WEBSTER

Dedicated to the memory of Lajos Pukanszky.

Abstract. Local operator spaces are defined to be projective limits of operator spaces. These limits arise when one considers linear spaces of unbounded operators, and they may be regarded as the “quantized” or “operator” analogues of locally convex spaces. It is shown that for nuclear spaces, the maximal and minimal quantizations coincide. Thus in a striking contrast to normed spaces, a nuclear space has precisely one quantization. Furthermore, it is shown that a local operator space is nuclear in the operator sense if and only if its underlying locally convex space is nuclear. Operator versions of bornology and duality are also considered.

1. Introduction

Quantum physics has provided functional analysts with some of their most challenging problems. In order to justify the physical theory, it is necessary to consider linear spaces of (generally unbounded) Hilbert space operators. Although in some cases, such as the theory of “functional integrals” on nuclear spaces, functional analytic methods have proved to be remarkably effective (see [14], [26]), a generally accepted framework for the Feynman integral remains elusive.

In this paper we consider an aspect of the functional analytic theory that may have a bearing on these problems. The transition from functions to operators introduces a new kind of linear structure into the systems. A linear space of operators is automatically equipped with natural orderings or norms on its matrices, which is usually not determined by those structures on the space itself. This phenomenon was first observed by operator algebraists, who found it necessary to replace the positive and bounded mappings of classical analysis with those having these properties on the matrices as well (see, e.g., [1]).

In recent years, several models have been introduced in order to better understand these notions. M.-D. Choi and the first author axiomatized self-adjoint unital linear spaces of bounded operators [3] by considering the underlying matrix orderings. A more inclusive theory was formulated by Z.-J. Ruan, who succeeded in characterizing arbitrary linear spaces of bounded operators by using the intrinsic matrix norms [24]. These spaces are known as operator spaces, and they may be thought of as “quantized normed spaces”. There is now a rapidly growing literature on the theory and applications of these objects.

Date: April 15, 1996.

1991 Mathematics Subject Classification. Primary 46A32,47D15; Secondary 46L89, 46M05, 47D20.

Key words and phrases. operator spaces, locally convex spaces, nuclearity.

Supported in part by NSF.

R. Powers was the first to consider the matrix structure underlying spaces of unbounded operators. Focussing his attention on the enveloping algebra A(𝔤) of a Lie algebra 𝔤, he showed that one can identify the integrable representations of 𝔤 by using suitable matrix orderings on A(𝔤) [22]. In this paper we consider a non-ordered theory that is applicable to more general linear spaces. Since locally convex spaces are just the projective limits of normed spaces (see [23]), it is natural to define a local operator space to be a projective limit of operator spaces. We wish to show that this simple idea leads to a satisfactory theory.

We have followed the general program proposed some forty years ago by Grothendieck for the study of locally convex vector spaces (see [16], [17], [15], and especially [20]). In that pioneering work, Grothendieck carefully examined the relationship between various tensor products and mapping spaces. His approach has thus far proved to be the most effective way of generalizing classical Banach space techniques to operator space theory (see, e.g., [2], [10], [7], [19], [21]).

The theory of locally convex spaces is purely topological, i.e., it is concerned with the topology on a vector space, rather than any specific seminorms. To illustrate this, we recall that any two locally convex spaces of the same finite dimension are linearly homeomorphic, and thus may be identified as locally convex spaces. Similarly, a local operator space V is characterized by a matrix topology on the linear space 𝔪(V) of infinite matrices [v_{i,j}]_{i,j∈ℕ} with v_{i,j} ∈ V zero for all but finitely many indices. Once again, a linear isomorphism between two finite dimensional local operator spaces induces a linear homeomorphism of their matrix spaces (see §6). On the other hand, an infinite dimensional Hilbert space H supports noncompletely isomorphic operator space structures, and the corresponding topologies on 𝔪(H) are therefore distinct (see §5 for more details).

It should also be noted that the matrix topology on 𝔪(V) for V a local operator space is asymptotic, i.e., it cannot be detected on matrices of bounded size. Letting V and the n × n matrices M_{n}(V) have the relative topologies in 𝔪(V), we have the natural linear homeomorphism

M_{n}(V) ≅ V^{n^{2}}, (1)

where we have used the product topology on the right (see §5).

We begin in §2 and §3 with a discussion of the notions of absolute matrix convexity for sets and their gauges. In §4 we prove a version of the bipolar theorem. The local operator spaces are introduced in §5. In §6 we define the local operator space versions of the projective and injective tensor products by using projective limits of the corresponding operator space products.

We use the operator projective tensor product to define the notion of matrix nuclearity for local operator spaces in §7. We prove that a local operator space is nuclear if and only if the underlying locally convex space is nuclear. It will be recalled that the nuclear spaces behave in many respects like finite dimensional spaces. This notion is reinforced by the surprising fact that a nuclear locally convex space has precisely one quantization. We consider the matrix analogues of boundedness and “bornology” in §8, and we then discuss local matrix topologies for mapping spaces and duals in §9. We conclude in §10 with a simple example of a nuclear operator space that is relevant to elementary quantum mechanics.

Local operator spaces display unusual non-classical phenomena that require additional care. The appropriate bipolar theorem involves a few subtleties, which have already been considered in [13]. Owing to the fact that operator spaces need not be locally reflexive in the appropriate sense (this is in contrast to the situation for normed spaces) [5], the theory of operator integral operators is also more difficult (see §7).

We have chosen not to be encyclopedic. Instead, our goal has been to prove a selection of results that show how one can also use Grothendieck’s theory in the setting of local operator spaces. We will explore other aspects of the theory, such as the Haagerup tensor product (see §6), in a subsequent paper.

We are indebted to Marc Rieffel, who for some time has encouraged us to develop the local theory. His studies of quantum deformations had led him as well to conclude that projective limits of operator spaces will play an important role in noncommutative analysis. We also wish to thank Barry Johnson, who has advocated a Banach module approach to operator space theory. His influence may be found in our frequent use of matrix modules, and in particular, in his elegant characterization of 𝔪-convex hulls in Lemma 3.2.

Turning to notation, we define a graded set S_{⋇} = (S_{n}) to be a sequence of sets S_{n} (n ∈ ℕ). Given a sequence of points x_{⋇} = (x_{n}), we write x_{⋇} ∈ S_{⋇} if x_{n} ∈ S_{n} for all n. Given two graded sets S_{⋇} and T_{⋇}, we write S_{⋇} ⊆ T_{⋇} if S_{n} ⊆ T_{n} for all n ∈ ℕ. When convenient, we omit the symbol ⋇.

We let l_{∞}, l_{2}, l_{1}, and c_{0} denote the usual sequence spaces, and M_{∞}, HS_{∞,}T_{∞}, and K_{∞} denote the operator analogues, i.e., the Banach spaces of bounded operators, Hilbert-Schmidt, trace class, and compact operators, respectively.

In this paper all vector spaces are assumed complex. Given vector spaces V and W, we let L(V, W) denote the vector space of linear mappings φ : V → W, and we let V^{d} = L(V, ℂ). We define a pairing of complex vector spaces V and W to be a bilinear function F : V × W → ℂ for which the corresponding functions V → W^{d} and W → V^{d} are injective. If we have fixed F, we will often use the notation ⟨v, w⟩ = F (v, w), whereas we will use the notation ⟨·|·⟩ for sesquilinear forms.

All locally convex spaces in this paper are assumed to be Hausdorff. We have found it convenient to use slightly different notations for normed and locally convex spaces. Given normed spaces, we let B(V, W) denote the normed space of bounded linear mappings from V to W, and V^{∗} = B(V, ℂ). For locally convex spaces we use the notation C(V, W) for the vector space of continuous linear mappings φ : V → W, and V′ = C(V, ℂ). On the other hand we let B(V, W) denote the bounded linear mappings φ : V → W (these need not be continuous – see §9).

The reader may read about operator spaces in [2], [11], and [12]. We use the following slightly modified notation. If V and W are operator spaces, we let B_{op}(V, W) denote the completely bounded mappings from V to W with the completely bounded operator norm ∥·∥_{cb} . This may be regarded as an operator space structure provided one uses the usual identification

M_{n}(B_{op}(V, W)) ≅ B_{op}(V, M_{n}(W)).

On the other hand, we may identify the normed spaces B_{op}(V, ℂ) and V^{∗}, and the operator space structure on the latter space is determined by the linear isomorphism

M_{n}(V^{∗}) ≅ B_{op}(V, M_{n}). (2)

Finally we recall that there are two especially important operator structures that can be placed on a Hilbert space H. We may use the linear isomorphisms

H ≅ B(ℂ, H)

and

H ≅ B(H^{∗}, ℂ)

of H with “column” and “row” operators to determine the column and row matrix norms ρ_{c} and ρ_{r} on H, and we denote the corresponding column and row Hilbert operator spaces by H_{c} and H_{r}, respectively (see, e.g., [8]). To be more explicit, the corresponding matrix norms are determined by the linear identifications

M_{m,n}(H_{c}) = B(ℂ^{n}, H^{m}),

and

M_{m,n}(H_{r}) = B(H^{∗n}, ℂ^{m}).

We have natural complete isometries (H_{c})^{∗} ≅ (H^{∗})_{r}.

There is a basic terminological difficulty with matrix structures, since the terms “completely”, “operator”, and “matrix” have all been used to signal that one is considering a property of the order or topology on 𝔪(V) ratherthan on V. Unfortunately none of these terms would be satisfactory in all situations since, for example, the term “completely continuous” would conflict with an unrelated notion. We have therefore chosen to use these terms interchangeably. It should also be noted that the “operator” or “matrix” nuclear spaces considered below have nothing to do with nuclear C^{∗}-algebras. Since the theory of nuclear locally convex spaces preceded that of nuclear C^{∗}-algebras by several decades, we propose that the latter be called C^{∗}-nuclear if there is a possibility of confusion.

2. Matrix conventions

We let M_{m,n} denote the vector space of complex m by n matrices α = [α_{i,j}], and M_{n} = M_{n,n}. We write ε_{i,j}^{m,n} (1 ≤ i ≤ m, 1 ≤ j ≤ n) for the usual basis of matrix units in M_{m,n}, and we let ε_{i,j}^{n} = ε_{i,j}^{n,n}. We identify M_{m,n} with the normed space B(ℂ^{n}, ℂ^{m}) and we regard the conjugate transpose operation α ↦ α^{∗} as an isometric conjugate linear mapping from M_{m,n} to M_{n,m}. Identifying M_{n} with the C^{∗}-algebra B(ℂ^{n}), M_{n} has the multiplicative identity ε^{n} = ∑ ε_{i,i}^{n}.

Given a (complex) vector space V, we let M_{m,n}(V) be the vector space of all m by n matrices v = [v_{i,j}] with v_{i,j} ∈ V, and we let M_{n}(V) = M_{n,n}(V), and M_{m,n} = M_{m,n}(ℂ). We identify a matrix of matrices over V with a single large matrix by deleting the inner brackets. Given v ∈ M_{m,n}(V) and w ∈ M_{p,q}(V), and scalar matrices α ∈ M_{p,m} and β ∈ M_{n,q}, we define

v ⊕ w ∈ M_{m+p,n+q}(V)

and

αvβ ∈ M_{p,q}(V)

in the obvious manner. Given v = [v_{i,j}] ∈ M_{p,q}(V), we have that

v_{i,j} = E_{i}^{(p)}vE_{j}^{(q)∗}, (3)

where in general we let

E_{k}^{(n)} = [0 … 1_{k} … 0_{n}] ∈ M_{1,n}.

On the other hand we have that

v = ∑_{i,j} E_{i}^{(p)∗}v_{i,j}E_{j}^{(q)} (4)

In many situations it is advantageous to replace the finite matrix spaces M_{m,n}(V) with the space 𝔪(V) of infinite matrices [v_{i,j}] (v_{i,j} ∈ V, i, j ∈ ℕ), where all but finitely many of the v_{i,j} are zero. Unless we indicate otherwise, we use the mapping

[ \begin{matrix}v_{1,1} & … & v_{1,n} \\ v_{m,1} & … & v_{m,n} \end{matrix} ] ↦ [ \begin{matrix}v_{1,1} & … & v_{1,n} & 0 & … \\ v_{m,1} & … & v_{m,n} & 0 & … \\ 0 & … & 0 & 0 \\ … & & … & & … \end{matrix} ]

to identify M_{m,n}(V) with a subspace of 𝔪(V). Letting 𝔪 = 𝔪(ℂ), we have that 𝔪(V) is an 𝔪-bimodule in the obvious manner. It will also be useful to consider more general index sets. If 𝔰 and 𝔱 are finite sets, we let M_{𝔰,𝔱}(V) be the vector space of matrices [v_{i,j}] (i ∈ 𝔰, j ∈ 𝔱), and given arbitrary sets 𝔰 and 𝔱, we define

𝔪_{𝔰,t}(V) = lim_{→} M_{𝔰_{0},𝔱_{0}}(V),

where the limit is taken over finite subsets 𝔰_{0} ⊆ 𝔰, and 𝔱_{0} ⊆ 𝔱. In this more general context it is useful to let n ∈ ℕ also stand for the set {1, …, n}, and ∞ for the set ℕ. Our previous conventions have obvious formulations in this more general setting. Letting ε_{i,j} ⊗ ε_{k,l} correspond to ε_{(i,k),(j,l)}, we have a natural identification

𝔪 ⊗ 𝔪 ≅ 𝔪_{∞×∞}.

Each linear mapping φ : V → W determines linear mappings

φ_{n} : M_{n}(V) → M_{n}(W) : [v_{i,j}] ↦ [φ(v_{i,j})]

and

φ_{∞} : 𝔪(V) → 𝔪(W) : [v_{i,j}] → [φ(v_{i,j})].

Given a matrix

φ = [φ_{i,j}] ∈ M_{n}(L(V, W))

we define a linear mapping φ : V → M_{n}(W) by letting φ(v) = [φ_{i,j}(v)]. We use this to make the identification

M_{n}(L(V, W)) ≅ L(V, M_{n}(W)), (5)

and similarly we have the identification

𝔪(L(V, W)) ≅ L(V, 𝔪(W)). (6)

Given a non-degenerate pairing of vector spaces

⟨·, ·⟩ : V × W → ℂ, (7)

each element v ∈ V (respectively, w ∈ W) determines a linear functional v : W → ℂ (respectively, w : V → ℂ) by

v(w) = w(v) = ⟨v, w⟩.

The pairing (7) determines the scalar pairings

⟨·, ·⟩ : M_{n}(V) × M_{n}(W) → ℂ (8)

6 EDWARD G. EFFROS AND CORRAN WEBSTER

and ⟨·, ·⟩ : 𝔪(V) × 𝔪(W) → ℂ (9) where

⟨v, w⟩ = ∑_{i,j}⟨v_{i,j}, w_{i,j}⟩

On the other hand it also determines the matrix pairings ⟨⟨·, ·⟩⟩ : M_{m}(V) × M_{n}(W) → M_{m×n} (10) and ⟨⟨·, ·⟩⟩ : 𝔪(V) × 𝔪(W) → 𝔪_{∞×∞} (11) where ⟨⟨v, w⟩⟩ = [⟨v_{i,j}, w_{k,l}⟩] = w_{m}(v) = v_{n}(w) (12) for m, n = 1, …, ∞. In the latter formulae we regard v and w as linear mappings v:W → M_{m} and w:V → M_{n}, or mappings into 𝔪, respectively.

3. Matrix gauges and matrix convexity

A gauge on a vector space V is a function

γ : V → [0, ∞]

such that • G1. γ(v + w) ≤ γ(v) + γ(w), and

• G2. γ(αv) ≤ |α| γ(v) for all v, w ∈ V and α ∈ ℂ.We say that a gauge p is faithful if p(v) = 0 implies v = 0. A gauge ρ is a seminorm on V if ρ(v) < ∞ for all v ∈ V. A faithful gauge p determines a norm on the space pV = {v ∈ V : p(v) < ∞} , (13) whereas a seminorm ρ determines a norm ∥·∥_{ρ} on the quotient vector space V_{ρ} = V/N_{ρ}, (14) where

N_{ρ} = {v ∈ V : ρ(v) = 0} .

Letting π_{ρ} be the quotient mapping of V onto V_{ρ}, we have that

ρ(v) = ∥π_{ρ}(v)∥_{ρ} .

If γ is a gauge on V, then the unit set

B^{γ} = {v : γ(v) ≤ 1}

is absolutely convex, i.e., if we are given v_{i} ∈ B^{γ}, and α_{i} ∈ ℂ (1 ≤ i ≤ n) for which ∑ |α_{i}| ≤ 1, then ∑ α_{i}v_{i} ∈ B^{γ}. Conversely, given an absolutely convex set B ⊆ V, the corresponding Minkowski functional γ^{B} is a gauge on V. We recall that

γ^{B}(v) = inf {λ > 0 : v ∈ λB} ,

where we let γ^{B}(v) = ∞ if v ∉ λB for any λ > 0.

Given an arbitrary vector space V, a matrix (or operator) gauge γ_{⋇} on V is a sequence of gauges

γ_{n} : M_{n}(V) → [0, ∞]

such that for any v ∈ M_{m}(V), w ∈ M_{n}(V), α ∈ M_{n,m}, and β ∈ M_{m,n}, • MG1. γ_{m+n}(v ⊕ w) = max {γ_{m}(v), γ_{n}(w)} , and

• MG2. γ_{n}(αvβ) ≤ ∥α∥ γ_{m}(v) ∥β∥ .

Owing to MG1, we have a well-defined gauge γ_{∞} on 𝔪(V) determined by the relation

γ_{∞}(v) = γ_{n}(v),

when v ∈ M_{n}(V). We say that elements v_{i} ∈ 𝔪(V) (i = 1 …, n) are orthogonal if there exist orthogonal projections e_{i} ∈ 𝔪 for which e_{i}v_{i}e_{i} = v_{i}.

Lemma 3.1. If γ_{⋇} is a matrix gauge on V, then the corresponding gauge γ_{∞} satisfies the following properties: 𝔪G1. If v and w are orthogonal in 𝔪(V), then γ_{∞}(v + w) = max{γ_{∞}(v), γ_{∞}(w)},

and 𝔪G2. γ_{∞}(αvβ) ≤ ∥α∥ γ_{∞}(v) ∥β∥ for all α, β ∈ 𝔪, v ∈ M_{m}(V), and m ∈ ℕ.

Proof. Given v, w ∈ 𝔪(V) and orthogonal projections e, f ∈ 𝔪 with eve = v and fwf = w, we fix an integer m with v, w ∈ M_{m}(V) and e, f ∈ M_{m}. We may choose partial isometries α and β ∈ M_{m} which are equivalences of e and f with the projections ε^{(k)} and ε^{(l)}, where k and l are the dimensions of these projections. It follows that αvα^{∗} ∈ M_{k}(V), βwβ^{∗} ∈ M_{l}(V), and

v + w = [ α^{∗} β^{∗} ] (αvα^{∗} ⊕ βwβ^{∗}) [ \begin{matrix}α \\ β \end{matrix} ] .

Noting that

∥[ \begin{matrix}α \\ β \end{matrix} ]∥^{2} = ∥[ α^{∗} β^{∗} ] [ \begin{matrix}α \\ β \end{matrix} ]∥ = ∥e + f∥ ≤ 1,

it follows that

γ_{∞}(v + w) ≤ max{γ_{∞}(v), γ_{∞}(w)}.

The remaining verifications are trivial.

We say that a gauge γ on 𝔪(V) that satisfies 𝔪G1 and 𝔪G2 is an 𝔪-module gauge. It is evident that given such a gauge, the restrictions γ_{n} = γ|_{M_{n}(V)} determine a matrix gauge γ_{⋇} on V, and that we have a one-to-one relation between matrix gauges γ_{⋇} on V and 𝔪-module gauges γ on 𝔪(V).

We say that a graded set B_{⋇} in M_{⋇}(V), is absolutely matrix convex if for all m, n ∈ ℕ,

• MC1. B_{m} ⊕ B_{n} ⊆ B_{m+n}, and

• MC2. αB_{m}β ⊆ B_{n} for any n by m and m by n contractions α and β.

Given a matrix gauge γ_{⋇} on V, the sequence of unit sets B_{⋇}^{γ} = (B^{γ_{n}}) is an absolutely matrix convex set. Conversely, given an absolutely matrix convex graded set B_{⋇} = (B_{n}), the corresponding Minkowski norms γ^{B_{n}} associated with the convex sets B_{n} ⊆ M_{n}(V) determine an operator seminorm γ_{⋇}^{B}. It is a simple matter to verify that B_{∞} = ∪B_{n} is the unit ball of the 𝔪-module gauge γ_{∞}.

We say that a set B ⊆ 𝔪(V) is absolutely 𝔪-convexif • 𝔪C1. for any orthogonal elements v, w ∈ B, we have that v + w ∈ B, and • 𝔪C2. αBβ ⊆ B for any contractions α,β ∈ 𝔪.

The argument used in Lemma 3.1 shows that the mapping B_{⋇} ↦ B_{∞} = ∪B_{n} provides a one-to-one correspondence between matrix convex graded sets for V and 𝔪-convex sets in 𝔪(V).

It is evident that an intersection of absolutely 𝔪-convex sets is an 𝔪-convex set, and thus if we are given a set S ⊆ 𝔪(V), we may define the 𝔪-convex hull h_{op}(S) of S to be the smallest absolutely 𝔪-convex set containing S. We are indebted to B. E. Johnson for the following result.

Lemma 3.2. Given an arbitrary subset S ⊆ 𝔪(V),

h_{op}(S) = {∑ α_{i}v_{i}β_{i} : v_{i} ∈ S}

where α_{1}, …, α_{n}, and β_{1}, …, β_{n} ∈ 𝔪 satisfy

∑ α_{i}α_{i}^{∗}, ∑ β_{i}^{∗}β_{i} ≤ 1.

Proof. We have that

∑ α_{i}v_{i}β_{i} = [α_{1} … α_{n}] (v_{1} ⊕ … ⊕ v_{n}) [ β_{1} ; ⋮ ; β_{n} ] ,

and thus it is evident that if these elements satisfy the indicated relations, we have that ∑ α_{i}v_{i}β_{i} ∈ h_{op}(S). Conversely, given contractions γ and δ, we have that

γ(∑ α_{i}v_{i}β_{i})δ = [γα_{1} … γα_{n}] (v_{1} ⊕ … ⊕ v_{n}) [ β_{1}δ ; ⋮ ; β_{n}δ ]

where

∑ γα_{i}α_{i}^{∗}γ^{∗} = γ^{∗} (∑ α_{i}α_{i}^{∗}) γ

is again a contraction, and the corresponding relation holds for the β_{i}δ terms. If we are given orthogonal projections e, f and we have that v = ∑ α_{i}v_{i}β_{i} and w = ∑ γ_{j}w_{j}δ_{j} satisfy eue = u and fvf = v, then we have that

∑ eα_{i}α_{i}^{∗}e + ∑ fγ_{j}γ_{j}^{∗}f ≤ 1

and

∑ eβ_{i}^{∗}β_{i}e + ∑ fδ_{j}^{∗}δ_{j}f ≤ 1,

and thus

u + v = ∑ eα_{i}v_{i}γ_{i}e + ∑ fβ_{j}w_{j}δ_{j}f

is a decomposition of the same type.

Let us suppose that γ is a matrix gauge on V. Given a finite set 𝔰_{0} with n elements, we may use a bijection of 𝔰_{0} onto {1, …, n} to identify M_{𝔰_{0}}(V) with M_{n}(V), and thus provide M_{𝔰_{0}}(V) with a gauge γ_{𝔰_{0}}. This does not depend on the particular bijection since a permutation of {1, …, n} determines a unitary α : ℂ^{n} → ℂ^{n}, and from 𝔪G2 the mapping

v ↦ α^{∗}vα

leaves the gauge γ on M_{n}(V) invariant. If 𝔰 is an arbitrary set, we may define a gauge on 𝔪_{𝔰}(V) by letting

γ_{𝔰}(v) = sup{γ_{𝔰_{0}}(v_{𝔰_{0}})},

where the supremum is taken over all finite subsets 𝔰_{0} of 𝔰. In particular we may use the norms on the spaces M_{𝔰_{0}} to define a norm on 𝔪_{𝔰}.

Given a matrix gauge γ on a vector space V and a matrix v = [v_{i,j}] ∈ M_{n}(V), we have from (3) and (4) and MC2 the important constraints

γ_{1}(v_{i,j}) = γ_{1}(E_{i}^{(n)}vE_{j}^{(n)∗})
≤ γ_{n}(v)
= γ_{n}(∑ E_{i}^{(n)∗}v_{i,j}E_{j}^{(n)})
(15)  ≤ n^{2} max{γ_{1}(v_{i,j})}.

It follows that if p_{⋇} is a matrix gauge for which p_{1} is faithful, then the same is true for each of the p_{n} and thus for p_{∞}, and similarly if ρ_{⋇} is a matrix gauge for which ρ_{1} is a seminorm, i.e., it is finite, then that is also the case for each ρ_{n} (n ∈ ℕ∪{∞}). In the latter case we say that ρ is a matrix or operator seminorm. If ρ is a matrix seminorm and N_{ρ} = {v ∈ V : ρ_{1}(v) = 0}, we have that ρ determines an operator norm ∥·∥_{⋇}^{ρ} on V_{ρ} = V/N_{ρ}, and thus V_{ρ} is an operator space with

∥(π_{ρ})_{n}(v)∥_{n}^{ρ} = ρ_{n}(v).

4. The bipolar theorem.

If V is a locally convex topological vector space, then the canonical topology on M_{n}(V) is that determined by the linear isomorphism (1), i.e., we have that a net v_{γ} ∈ M_{n}(V) converges to an element v ∈ M_{n}(V) if and only if (v_{γ})_{i,j} → v_{i,j}.

If V is a locally convex vector space, a gauge γ on V is lower semicontinuous if the corresponding unit set B^{γ} is closed. The mappings γ ↦ B^{γ} and B ↦ γ^{B} determine one-to-one correspondences between the lower semi-continuous gauges γ and the weakly closed absolutely convex subsets B ⊆ V. On the other hand, we have that B^{γ} is a neighborhood of 0 if and only if γ is a continuous seminorm (see [23], p. 14).

Two locally convex spaces V and W are in duality (or a dual pair) if one is given a distinguished pairing

⟨·, ·⟩ : V × W → ℂ

such that all the continuous functionals on V are given by elements of W, and vice versa. If this is the case, we have that M_{n}(V) and M_{n}(W) are in duality under the scalar pairing (8). Both the scalar pairing and the matrix pairing (10) determine the same topologies on M_{n}(V) and M_{n}(W), which we refer to as the weak topologies. We define the weak topology on 𝔪 to be that determined by the scalar pairing

𝔪 × 𝔪 → ℂ : (α, β) ↦ ⟨α, β⟩. (16)

We may similarly use the scalar pairing (9), or letting 𝔪 have the weak topology, the matrix pairing (11) to define the weak topologies on 𝔪(V) and 𝔪(W). Using (12), it is easy to see that the bilinear mapping

⟨⟨·, ·⟩⟩ : V × M_{n}(W) → M_{n}

determines all the weakly continuous mappings φ : V → M_{n}, i.e., we have

M_{n}(W) ≅ C(V, M_{n}). (17)

Given a subset D ⊆ V, the absolute polar of D is defined by

D^{∘} = {w ∈ W : |⟨v, w⟩| ≤ 1 for all v ∈ D}.

The classical bipolar theorem implies that D^{∘∘} is the smallest weakly closed absolutely convex set containing D, and thus D ↦ D^{∘} is a one-to-one correspondence between the weakly closed absolutely convex sets of V and W.

Given a graded set D_{⋇} ⊆ M_{⋇}(V), the (absolute) operator polar D_{⋇}^{⊚} ⊆ M_{⋇}(W), is defined by

D_{n}^{⊚} = {w ∈ M_{n}(W) : ∥⟨⟨v, w⟩⟩∥ ≤ 1 for all v ∈ D_{r}, and r ∈ ℕ}.

We note that

D_{1}^{⊚} = D_{1}^{∘} (18)

since if f ∈ V′ satisfies |⟨f, x⟩| ≤ 1 for all x ∈ D_{1}, then ∥⟨⟨f, x⟩⟩∥ ≤ 1 for all x ∈ D_{n}. To see this let us suppose that ξ, η ∈ ℂ^{n} are unit vectors. Then regarding these as unit column matrices, we have that

⟨ ⟨⟨f, x⟩⟩ η|ξ⟩ = ⟨f|ξ^{∗}xη⟩ < 1

(see Lemma 5.2 for a more general result).

The following bipolar theorem is variation of [11]. Owing to the fact that we must allow infinite gauges in our applications, it is necessary to modify that argument, using a technique introduced in [13]. Unfortunately, we are not aware of a simple argument that would reduce this result to the bipolar theorem for non-absolutely matrix convex sets proved in [13]. The reader may prefer to initially skip over the rather lengthy proof.

We say that D_{⋇} is weakly closed if that is the case for each set D_{n}.

Proposition 4.1. Suppose that V and W are dual locally convex spaces. Given any graded set D_{⋇} ⊆ M_{⋇}(V), we have that D_{⋇}^{⊚⊚} is the smallest weakly closed absolutely matrix convex set containing D_{⋇}.

Proof. It suffices to prove that if D_{⋇} is absolutely matrix convex and weakly closed, then D_{⋇}^{⊚⊚} = D_{⋇}. Thus we must prove that if v_{0} ∉ D_{n}, then v_{0} ∉ D_{n}^{⊚⊚}. To do this we will show that there exists a w_{0} ∈ M_{n}(W) for which w_{0} ∈ (D_{n}^{⊚}), i.e.,

∥⟨⟨v, w_{0}⟩⟩∥ ≤ 1,

for all v ∈ D_{r} and r ∈ ℕ, but

∥⟨⟨v_{0}, w_{0}⟩⟩∥ > 1

Letting γ be the matrix gauge of D_{⋇} and using the identification (17), it suffices to find a mapping φ : V → M_{n} for which

∥φ_{r}(v)∥ ≤ 1 (19)

for all v ∈ D_{r} and r ∈ ℕ, but

∥φ_{n}(v_{0})∥ > 1. (20)

Using the classical bipolar theorem, we have that v_{0} ∉ D_{n}^{∘∘}, i.e., we may find a functional F ∈ M_{n}(V)′ for which

|F |_{D_{n}}| ≤ 1 < |F (v_{0})| . (21)

As in [13] we will extract the desired matrix valued function φ from a perturbation of F.

Our argument depends on a simple convexity result. Let us suppose that E is a cone of real continuous affine functions on a compact convex subset K of a topological vector space E, and that for each e ∈ E, there is a corresponding point k_{e} ∈ K with e(k_{e}) ≥ 0. Then there is a point k_{0} ∈ K for which e(k_{0}) ≥ 0 for all e ∈ E. A simple proof of this result may be found in [13], Lemma 5.2.

Following [11], we claim that there exist states p_{0} and q_{0} on M_{n} such that

|F (αvβ)| ≤ p_{0}(αα^{∗})^{1/2}γ_{r}(v)q_{0}(β^{∗}β)^{1/2} (22)

for all α ∈ M_{n,r}, β ∈ M_{r,n}, and v ∈ M_{r}(V) for which γ_{r}(v) < ∞, with r ∈ ℕ arbitrary. If γ_{r}(v) = 0, then

γ_{n}(αvβ) ≤ ∥α∥ γ_{r}(v) ∥β∥ = 0,

i.e., λ(αvβ) ∈ D_{n} for all scalars λ > 0. Since we chose F with |F |_{D_{n}}| ≤ 1, we have λ|F (αvβ)| ≤ 1 for all scalars λ > 0, hence F (αvβ) = 0 and (22) is trivial. Dividing by a constant, we may assume that γ_{r}(v) = 1. Summarizing, our task is to find states p_{0} and q_{0} on M_{n} such that for all v ∈ M_{r}(V) with γ_{r}(v) = 1, we havethat

|F (αvβ)| ≤ p_{0}(αα^{∗})^{1/2}q_{0}(β^{∗}β)^{1/2}. (23)

It suffices to find states p_{0}, q_{0} such that

Re F (αvβ) ≤ p_{0}(αα^{∗})^{1/2}q_{0}(β^{∗}β)^{1/2}

since we can then replace α by e^{iθ}α for a suitable θ ∈ [0, 2π]. In turn, it is enough to prove that

Re F (αvβ) ≤ (1/2)[p_{0}(αα^{∗})+q_{0}(β^{∗}β)]. (24)

To see this, we replace α by t^{1/2}α and β by t^{−1/2}β for t > 0. It follows that

Re F (αvβ) ≤ (1/2)[tp_{0}(αα^{∗}) +t^{−1}q_{0}(β^{∗}β)],

and minimizing for t > 0 (or simply letting t = p_{0}(αα^{∗})^{−1/2}q_{0}(β^{∗}β)^{1/2} for α ≠ 0), we obtain (23).

Letting S_{n} be the set of states on M_{n}, S = S_{n} × S_{n} is a compact and convex subset of (M_{n} ⊕ M_{n})^{∗}. We write A(S) for the continuous affine functions on S. Given α ∈ M_{n,r}, β ∈ M_{r,n} and v ∈ M_{r}(V) with γ_{r}(v) = 1, we may define a corresponding function e_{α,v,β} ∈ A(K) by

e_{α,v,β}(p, q) = p(αα^{∗}) + q(β^{∗}β) − 2ReF (αvβ).

Letting E denote the collection of all such functions, we must show that there is a point (p_{0}, q_{0}) ∈ S for which e(p_{0}, q_{0}) ≥ 0 for all e ∈ E. But we have that

a) Each function e ∈ E is non-negative at some point (p_{e}, q_{e}) ∈ S. To see this suppose that e = e_{α,v,β}, and select states p_{e} and q_{e} with p_{e}(αα^{∗}) = ∥αα^{∗}∥ = ∥α∥^{2}, and q_{e}(β^{∗}β) = ∥β∥^{2}. Then we have

e_{α,v,β}(p_{e}, q_{e}) = ∥α∥^{2} + ∥β∥^{2} − 2ReF (αvβ) ≥ 0

since

Re F (αvβ) ≤ |F (αvβ)| ≤ ∥αvβ∥ ≤ ∥α∥∥β∥ ≤ (1/2)[∥α∥^{2} + ∥β∥^{2}].

b) The collection E is a cone of functions, i.e., E + E ⊆ E, and λE ⊆ E for λ ≥ 0. The second assertion is trivial. For the first we note that

e_{α,v,β} + e_{α′,v′,β′} = e_{α′′,v′′,β′′}

where α′′ = [ α α′ ] , β′′ = [ \begin{matrix}β \\ β′ \end{matrix} ] , and v′′ = v ⊕ v′ ∈ M_{r+r′} satisfies γ_{r+r′}(v ⊕

v′) = 1.

¿From the convexity result, there exists a point (p_{0}, q_{0}) at which all of the functions in E are positive. Thus we have proved (22). We claim that we may perturb these states so that they are faithful. We recall that the normalized trace τ on M_{n} is faithful, i.e., we have that τ(α^{∗}α) = 0 implies that α = 0. Given 0 < ε < 1, it follows that p = (1 − ε)p_{0} + ετ and q = (1 − ε)q_{0} + ετ are faithful. Letting G = (1 − ε)F, we have from (22) that if γ_{r}(v) < ∞,

|G(αvβ)| ≤ (1 − ε)p_{0}(αα^{∗})^{1/2}q_{0}(β^{∗}β)^{1/2}γ_{r}(v)
≤ \frac{(1 − ε)}{2}[p_{0}(αα^{∗}) + q_{0}(β^{∗}β)]γ_{r}(v)
≤ \frac{1}{2}[p(αα^{∗}) + q(β^{∗}β)]γ_{r}(v).

Replacing α by t^{1/2}α and β by t^{−1/2}α, where t is a positive scalar, and then minimizing, we conclude that

|G(αvβ)| ≤ p(αα^{∗})^{1/2}γ_{r}(v)q(β^{∗}β)^{1/2}. (25)

On the other hand, if we let ε be sufficiently small, we have from (21) that

|G|_{D_{n}}| ≤ 1 < |G(v_{0})| .

Applying the GNS theorem, we have corresponding faithful representations π and θ of M_{n} on finite dimensional Hilbert spaces H and K, respectively, with separating and cyclic vectors ξ_{0} ∈ H and η_{0} ∈ K satisfying p(α) = ⟨π(α)ξ_{0}|ξ_{0}⟩ and q(α) = ⟨θ(α)η_{0}|η_{0}⟩, respectively, for all α ∈ M_{n}.

Given a row matrix α = [α_{1}, α_{2}, …, α_{n}] ∈ M_{1,n}, we define α̃ ∈ M_{n} by

α̃ = [ \begin{matrix}α_{1} & α_{2} & … \\ 0 & 0 & … \\ … & … & … \end{matrix} ] .

We let M̃_{1,n} be the linear space of all such n by n matrices, and we let H_{0} = π(M̃_{1,n})ξ_{0} ⊆ H and K_{0} = θ(M̃_{1,n})η_{0} ⊆ K. Owing to the fact that ξ_{0} and η_{0} are separating, H_{0} and K_{0} are n-dimensional spaces.

Fixing an element v ∈ V, the sesquilinear form B_{v} defined on K_{0} × H_{0} by

B_{v}(θ(β̃)η_{0}, π(α̃)ξ_{0}) = G(α^{∗}vβ)

is well-defined since if, for example, θ(β̃)η_{0} = 0, then since η_{0} is separating and θ is a faithful representation, we have that β̃ = 0 and G(α^{∗}vβ) = 0. Thus there exists a unique linear map φ(v) : K_{0} → H_{0} for which

G(α^{∗}vβ) = ⟨φ(v)θ(β̃)η_{0} | π(α̃)ξ_{0}⟩.

It is a simple matter to verify that the corresponding map φ : V → B(K_{0}, H_{0}) is linear. Since H_{0} and K_{0} are n-dimensional, we may identify each of these spaces with ℂ^{n}, and φ with a mapping φ : V → M_{n}.

Given a matrix v ∈ M_{n}(V), we have from (4)

G(v) = Σ⟨φ(v_{ij})θ(Ẽ_{j}^{(n)})η_{0} | π(Ẽ_{i}^{(n)})ξ_{0}⟩
(26)  = ⟨φ_{n}(v)η | ξ⟩,

where

η = ( θ(Ẽ_{1}^{(n)})η_{0} ; ⋮ ; θ(Ẽ_{n}^{(n)})η_{0} ) , ξ = ( π(Ẽ_{1}^{(n)})ξ_{0} ; ⋮ ; π(Ẽ_{n}^{(n)})ξ_{0} ) ∈ ℂ^{n}

satisfy

∥ξ∥^{2} = Σ∥π(Ẽ_{j}^{(n)})ξ_{0}∥^{2} = Σp(E_{j}^{(n)∗}E_{j}^{(n)}) = p(ε^{(n)}) = 1,

and similarly, ∥η∥^{2} = 1.

To show that φ satisfies (19), we must prove that if v ∈ M_{r}(V) and γ_{r}(v) < ∞, then

|⟨φ_{r}(v)η_{1} | ξ_{1}⟩| ≤ γ_{r}(v)∥η_{1}∥∥ξ_{1}∥

for unit vectors ξ_{1} and η_{1} in (ℂ^{n})^{r}. Letting

ξ_{1} = ( π(α̃_{1})ξ_{0} ; ⋮ ; π(α̃_{r})ξ_{0} ) , η_{1} = ( θ(β̃_{1})η_{0} ; ⋮ ; θ(β̃_{r})η_{0} )

where α_{i}, β_{j} ∈ M_{1,n}, we have that

∥ξ_{1}∥^{2} = Σ∥π(α̃_{i})ξ_{0}∥^{2} = Σp(α_{i}^{∗}α_{i}) = p(α^{∗}α),

and similarly ∥η_{1}∥^{2} = q(β^{∗}β) , where

α = [ α_{1} ; ⋮ ; α_{r} ] , β = [ β_{1} ; ⋮ ; β_{r} ] ∈ M_{r,n}.

It follows that

⟨φ_{r}(v)η_{1} | ξ_{1}⟩ = Σ⟨φ_{0}(v_{i,j})θ(β̃_{j})η_{0} | π(α̃_{i})ξ_{0}⟩
= ΣG(α_{i}^{∗}v_{ij}β_{j})
= G(α^{∗}vβ),

and thus if v ∈ D_{r},

|⟨φ_{r}(v)η_{1} | ξ_{1}⟩| ≤ |G(α^{∗}vβ)|
≤ p(αα^{∗})^{1/2}γ_{r}(v)q(β^{∗}β)^{1/2}
= γ_{r}(v)∥ξ_{1}∥∥η_{1}∥
≤ 1.

On the other hand we have that v_{0} ∈ M_{n}(V) satisfies

1 < G(v_{0}) = ⟨φ_{n}(v_{0})η | ξ⟩,

and since that η and ξ are unit vectors, we have proved (20).

Given a subset S ⊆ 𝔪(V), we define its 𝔪-polar S^{⊚} ⊆ 𝔪(W) by

S^{⊚} = {w ∈ 𝔪(W) : ∥⟨⟨v, w⟩⟩∥ ≤ 1 for all v ∈ S} .

We claim that S^{⊚} is 𝔪-convex. If w, w′ ∈ S^{⊚} and e, f ∈ 𝔪 are orthogonal projections with ewe = w and fw′f = w′, then given v ∈ S_{n} = S ∩ M_{n}, we have that ẽ = ε^{(n)} ⊗ e and f̃ = ε^{(n)} ⊗ f are orthogonal projections in 𝔪 ⊗ 𝔪, satisfying ẽ⟨⟨v, w⟩⟩ẽ = ⟨⟨v, w⟩⟩ and f̃⟨⟨v, w′⟩⟩f̃ = ⟨⟨v, w′⟩⟩. It follows that

∥⟨⟨v, ewe + fw′f⟩⟩∥ = ∥ẽ⟨⟨v, w⟩⟩ẽ + f̃⟨⟨v, w′⟩⟩f̃∥
= max {∥⟨⟨v, w⟩∥ , ∥⟨⟨v, w′⟩⟩∥} ≤ 1.

On the other hand if α, β ∈ 𝔪 are contractions, then that is also the case for α̃ = ε^{(n)} ⊗ α and β^{(n)} ⊗ β, and thus

∥⟨⟨v, αvβ⟩⟩∥ = ∥α̃⟨⟨v, w⟩⟩β̃∥ ≤ ∥⟨⟨v, w⟩⟩∥ ≤ 1.

Corollary 4.1. Given a subset S ⊆ 𝔪(V), S^{⊚⊚} is the smallest weakly closed 𝔪convex set containing S.

Proof. If we let B be the smallest 𝔪-convex set containing S, the weak closure B̄ is obviously 𝔪-convex, and thus our task is to show that B^{⊚⊚} = B̄. The sets

B_{n} = ε^{(n)}Bε^{(n)} ⊆ M_{n}(V)

determine an absolutely matrix convex set B_{⋇}. The weak closure B̄ is equal to ∪B̄_{n} since if v_{λ} is a net in ∪B_{n} converging to v ∈ 𝔪(V), then assuming that v ∈ M_{n}(V),

v = lim_{λ} ε^{(n)}v_{λ}ε^{(n)} ∈ B̄_{n}.

Thus if v_{0} ∉ B̄, then we may assume that for some n ∈ ℕ,

v_{0} ∈ M_{n}(V)\B̄_{n}.

From the bipolar theorem there exists a w_{0} ∈ M_{n}(W) such that

∥⟨⟨v_{0}, w_{0}⟩⟩∥ > 1 ≥ ∥⟨⟨v, w_{0}⟩⟩∥

for all v ∈ B_{r}, with r ∈ ℕ arbitrary. Regarding w_{0} as an element of 𝔪(W), we have that w_{0} ∈ B_{n}^{⊚}, and thus v_{0} ∉ B^{⊚⊚}.

Given a gauge γ on V, we define the dual gauge γ^{∘} to be the gauge of the polar of the unit set of γ, or equivalently,

γ^{∘}(f) = sup {|⟨v, f⟩| : γ(v) ≤ 1} .

If γ_{⋇} is a matrix gaugeon V, the corresponding dual matrix gauge γ_{⋇}^{⊚} is determined for f ∈ M_{n}(V′) by

γ_{n}^{⊚}(f) = sup {∥⟨⟨v, f⟩⟩∥ : γ_{r}(v) ≤ 1, r ∈ ℕ}

whereas, if γ is an 𝔪-module gauge on 𝔪(V), then the dual 𝔪-module gauge γ^{⊚} on 𝔪(V′) is defined by

γ^{⊚}(f) = sup {∥⟨⟨v, f⟩⟩∥ : γ(v) ≤ 1, v ∈ 𝔪(V)} .

Given a locally convex space V, we have that V and W = V′ = C(V, ℂ) form a dual pair. An absolutely convex set K ⊆ V determines a graded set K_{⋇}, where

K_{n} = \{ \begin{matrix}K & if n = 1 \\ \{0\} & if n > 1 \end{matrix} . (27)

We define the minimal envelope K̂ ⊆ M_{⋇}(V) to be (K_{⋇})^{⊚⊚}. It follows from the preceding theorem that K̂ is the weak closure of the matrix convex set

(28)  K̃ = {α(v_{1} ⊕ … ⊕ v_{n})β : v_{i} ∈ K_{n_{i}}, α, β contractions}

We define the maximal envelope Ǩ to be the matrix polar of the classical polar K^{∘} ⊆ V′. To be more precise, Ǩ = (K_{⋇}^{∘})^{⊚}, where

K_{n}^{∘} = \{ \begin{matrix}K^{∘} & if n = 1 \\ \{0\} & if n > 1 \end{matrix} . (29)

We claim that if L_{⋇} is a weakly closed absolutely matrix convex set with L_{1} = K, then

K̂ ⊆ L ⊆ Ǩ.

We have L_{⋇} ⊇ K_{⋇} implies that L^{⊚} ⊆ K^{⊚} and thus

L = L^{⊚⊚} ⊇ K^{⊚⊚} = K̂.

On the other hand from (18), L_{1}^{⊚} = L_{1}^{∘} = K_{1}^{∘} implies that L_{⋇}^{⊚} ⊇ K_{⋇}^{∘} and

L = L^{⊚⊚} ⊆ (K_{⋇}^{∘})^{⊚} = Ǩ.

If ρ is the gauge of an absolutely convex weakly closed set K in V, we let ρ̂ and ρ̌ be the corresponding matrix gauges of K̂ and Ǩ, respectively. It follows from above that if ρ is any matrix gauge with ρ_{1} = ρ, then

ρ̌ ≤ ρ ≤ ρ̂. (30)

In particular, if V is a normed space with the norm ρ(v) = ∥v∥ , the two extremal matrix norms on V are defined for v ∈ M_{n}(V) by

∥v∥_{min} = sup {∥⟨⟨v, f⟩⟩∥ : f ∈ V^{∗}, ∥f∥ ≤ 1} = ρ̌_{n}(v) (31)

(see [6], [2]) and

∥v∥_{max} = sup {∥⟨⟨v, f⟩⟩∥ : f ∈ B(V, M_{p}), ∥f∥ ≤ 1, p ∈ ℕ} = ρ̂_{n}(v). (32)

(see [2]). The corresponding operator spaces are denoted by min V and max V, respectively.

5. Local operator spaces

If V is a locally convex space, then its topology is determined by the family 𝔑(V) of continuous seminorms ρ : V → [0, ∞). In the remainder of the paper, we regard a locally convex space V as in duality with its continuous dual V′, and we refer to the topology determined by V′ as the weak topology on V. Similarly, we define the weak topology on 𝔪(V) to be that determined by either the scalar pairing 𝔪(V) × 𝔪(V′) → ℂ, or equivalently, by the matrix pairing

𝔪(V) × 𝔪(V′) → 𝔪 ⊗ 𝔪,

where we let 𝔪 ⊗ 𝔪 = 𝔪_{∞×∞} have the weak topology determined by (16).

We recall that an operator space is a vector space V together with a distinguished operator norm ρ. Given a locally convex space V, we define a local operator structure on V to be a family ℜ of continuous operator seminorms ρ on V, for which the seminorms ρ_{1} generate the topology on V. We define the matrix topology on 𝔪(V) determined (or generated) by ℜ to be the locally convex topology determined by the seminorms ρ_{∞} with ρ ∈ ℜ. We define a local operator space to be a locally convex space V together with a matrix topology on 𝔪(V) which arises in the above manner. We shall say that a given local operator structure on a locally convex space is a quantization of that space. On the other hand when we shall on occassion refer to results in locally convex theory as being classical.

Let us suppose that V is a local operator space generated by a family of continuous operator seminorms ℜ. We say that an operator seminorm ν on V is matrix continuous if the corresponding 𝔪-module seminorm

ν_{∞} : 𝔪(V) → [0, ∞)

is continuous. Equivalently, we have that there exists an operator seminorm ρ ∈ ℜ and a constant k > 0 with ν_{n} ≤ kρ_{n} for all n. Letting 𝔑_{op}(V) be the collection of all matrix continuous operator seminorms, it is evident that 𝔑_{op}(V) and ℜ determine the same topology on 𝔪(V). We say that a local operator space V is countably generated if the matrix topology is generated by a countable family ℜ of operator seminorms. The collection 𝔑_{op}(V) is partially ordered by the relation ρ ≤ σ if for each v ∈ 𝔪(V) we have that ρ_{∞}(v) ≤ σ_{∞}(v).

Given two local operator spaces V and W, we say that a linear mapping φ : V → W is matrix or operator continuous if the corresponding mapping φ_{∞} : 𝔪(V) → 𝔪(W) is continuous in the matrix topologies. Thus we have that φ is operator continuous if and only if for any operator seminorm σ ∈ 𝔑_{op}(W), there is an operator seminorm ρ ∈ 𝔑_{op}(V) and a constant k > 0 such that

σ_{n}(φ_{n}(v)) ≤ kρ_{n}(v).

for all n ∈ ℕ. Similarly we say that φ is a matrix or operator homeomorphism if φ_{∞} is a homeomorphism in the matrix topologies. We let C_{op}(V, W) denote the vector space of all matrix continuous linear mappings φ : V → W. If V and W are operator spaces, these notions coincide with those of completely bounded and completely isomorphic linear mappings φ : V → W, respectively, i.e., in this case we have the linear space equality C_{op}(V, W) = B_{op}(V, W).

It is apparent from (15) that the seminorms ρ_{n} with ρ ∈ 𝔑_{op}(V) determine the canonical topology on M_{n}(V). Thus since the seminorm ρ_{∞} restricts to ρ_{n} on M_{n}(V), we conclude that the injection M_{n}(V) ↪ 𝔪(V) is a homeomorphism onto its image. On the other hand, it is already apparent from the theory of operator spaces that the norm topology on V does not determine that on 𝔪(V). This is evident if one considers the row and column Hilbert spaces H_{c} and H_{r}. Any completely bounded linear mapping φ : H_{c} → H_{r} is compact [8], Cor.4.5, hence if H is infinite dimensional, H_{c} and H_{r} are not completely isomorphic.

We next show that any matrix topology on 𝔪(V) contains the weak topology determined by the pairings (9) and (11), and thus 𝔪-polars of sets in 𝔪(V′) are automatically closed in 𝔪(V). For this purpose we use several lemmas due to Roger Smith [27].

Lemma 5.1. Given integers m, n ∈ ℕ with m ≥ n and a vector η ∈ ℂ^{m} ⊗ℂ^{n}, there exists an isometry β : ℂ^{n} ↪ ℂ^{m} and a vector η̄ ∈ ℂ^{n}⊗ℂ^{n} for which (β⊗I_{n})(η̄) = η.

Proof. There exist unique vectors η_{j} ∈ ℂ^{m}, (j = 1, …, n) with

η = ∑_{j=1}^{n} η_{j} ⊗ ε_{j}^{(n)}.

Letting F ⊆ ℂ^{m} be the subspace spanned by the vectors η_{j}, we have that dim F ≤ n ≤ m. Thus we may find an isometry β : ℂ^{n} → ℂ^{m} with image containing F. For each j we have a unique vector η̄_{j} ∈ ℂ^{n} for which β(η̄_{j}) = η_{j}. Letting η̄ = ∑_{j} η̄_{j} ⊗ ε_{j}^{(n)}, we have that β ⊗ I_{n}(η̄) = η.

Lemma 5.2. Suppose that ρ is a matrix gauge on a vector space V, and that φ : V → M_{n} is a linear map. If

∥φ_{n}(v)∥ ≤ kρ_{n}(v)

for all v ∈ M_{n}(V), we have that

∥φ_{r}(v)∥ ≤ kρ_{r}(v)

for all v ∈ M_{r}(V) with r ∈ ℕ arbitrary.

Proof. If r ≤ n, then we have that

∥φ_{r}(v)∥ = ∥φ_{n}(v ⊕ 0_{n−r})∥ ≤ kρ_{n}(v ⊕ 0_{n−r}) = kρ_{r}(v).

Let us suppose that r > n. We may assume that ρ_{r}(v) < ∞. Given arbitrary unit vectors η, ξ ∈ (ℂ^{n})^{m} = ℂ^{m} ⊗ ℂ^{n} we have from the above lemma that, there exist isometries α, β : ℂ^{n} → ℂ^{m} and unit vectors ξ̄, η̄ ∈ ℂ^{n}⊗ℂ^{n} for which ξ = (α⊗I_{n})(ξ̄) and η = (β ⊗ I_{n})(η̄). We have that

|⟨φ_{r}(v)η|ξ⟩| = |⟨ φ_{r}(v)(β ⊗ I_{n})(η̄)|(α ⊗ I_{n})(ξ̄) ⟩|
= |⟨ φ_{n}(α^{∗}vβ)η̄ | ξ̄⟩|
≤ ∥ φ_{n}(α^{∗}vβ)∥
≤ kρ_{n}(α^{∗}vβ)
≤ kρ_{r}(v).

Since η and ξ are arbitrary unit vectors, we conclude that

∥φ_{r}(v)∥ ≤ kρ_{r}(v).

Corollary 5.1. If V is a local operator space, then the matrix topology contains the weak topology on 𝔪(V).

Proof. Given f ∈ 𝔪(V′), we may assume that f ∈ M_{n}(V′) for some n ∈ ℕ. By assumption the seminorms ρ_{1} with ρ ∈ 𝔑_{op}(V) determine the topology on V. It follows that there exists such a ρ and a constant k > 0 with

∥f(v)∥ ≤ kρ_{1}(v)

for all v ∈ V. It follows that if v ∈ M_{n}(V) then

∥f_{n}(v)∥ ≤ n^{2} max{|f(v_{i,j})|}
≤ n^{2}kρ_{1}(v_{i,j})
≤ n^{2}kρ_{n}(v).

Thus from Lemma 5.2, we have that

∥f_{r}(v)∥ ≤ n^{2}kρ_{r}(v)

for all v ∈ M_{r}(V) with r ∈ ℕ arbitrary. We conclude that f is continuous as a linear functional on 𝔪(V).

We may associate extremal quantizations of a locally convex space V in the following manner. Given a continuous seminorm ρ ∈ 𝔑(V), we have that the unit ball B^{ρ} is closed and convex, and thus it is closed in the weak topology determined by V′. Following the discussion of §4, we may associate with ρ the minimal and maximal envelopes ρ̂ and ρ̌, respectively, for each continuous seminorm ρ ∈ 𝔑(V). We let max V and min V denote the local operator spaces determined by the ρ̂ and ρ̌ for ρ ∈ 𝔑(V) (the correspondence between sets and gauges is order inverting). If V is a local operator space, then given ρ ∈ 𝔑_{op}(V), we have from (30) a diagram of matrix continuous mappings

max V → V → min V.

Given a subspace W of a locally convex operator space V, it is evident that the restrictions of operator seminorms in 𝔑_{op}(V) determine a matrix topology on 𝔪(W). If W is closed in V, it follows from the inequalities (15) that M_{n}(W) is closed in M_{n}(V). We also have that if V is complete, then it also follows from (15) that each of the spaces M_{n}(V) is complete. It is often convenient to refer to a linear matrix homeomorphism of a local operator space W into another local operator space V as an “inclusion W ⊆ V of local operator spaces”.

If ρ ∈ 𝔑_{op}(V), we let ρ̃_{n} denote the corresponding quotient seminorm on

M_{n}(V/W) = M_{n}(V)/M_{n}(W). (33)

These determine the quotient operator space structure on V/W. In general the quotient of a complete locally convex vector space need not be complete. However, this is the case for a Frechet space, i.e., a countably seminormed complete locally convex vector space (see [25]). If V is an operator space, we say that it is Frechet if it is Frechet as a locally convex space. It follows that each space M_{n}(V) is Frechet, and if W is a closed subspace of V, M_{n}(W) is closed in M_{n}(V), and (33) determines a Frechet operator space V/W.

6. Projective limits and tensor products

The development of projective limits and tensor products for operator spaces closely parallels that for locally convex spaces. After recalling the classical definition, we continue with the development for local operator spaces. We will need the classical tensor product theory as well, but this can be deduced from the operator case, or found in [23] and [25] .

Let us suppose that we are given a vector space V, a family of locally convex spaces V_{γ}, and linear mappings π_{γ} : V → V_{γ}. We associate with each γ ∈ Γ and ρ ∈ 𝔑(V_{γ}) a seminorm σ on V by letting σ_{n} = ρ_{n} ∘ (π_{γ})_{n}. Letting γ vary, the resulting collection 𝔑(V) of seminorms determines the projective locally convex topology on V determined by the π_{γ}, and we write

V = lim_{←}{V_{γ}, π_{γ}} = lim_{←} V_{γ}. (34)

If V_{γ} is a locally convex subspace of a locally convex space W_{γ}, then it is trivial that V = lim_{←} W_{γ}. In this paper we will be only interested in projective limits of Banach spaces. If V is a locally convex space, then we have

V = lim_{←} V̄_{ρ}

(see (14)), i.e., V is a projective limit of Banach spaces [23]. It is customary to take the completions in order to simplify the discussion of nuclearity (see the following section). It should be noted that even if V is complete, this need not be the case for V_{ρ}. Given another locally convex space W = lim_{←} W̄_{σ}, we have that a linear mapping φ : V → W is continuous if and only if for each σ ∈ 𝔑(W), there is a ρ ∈ 𝔑(V) and a constant k > 0 for which

∥θ_{σ}(φ(v))∥ ≤ k ∥π_{ρ}(v)∥ ,

or equivalently, we have a commutative diagram

\begin{matrix}V & ⟶^{φ} & W \\ ↓ & & ↓ \\ V̄_{ρ} & ⟶_{φ_{σ,ρ}} & W̄_{σ} \end{matrix} (35)

where φ_{σ,ρ} is a contraction.

We may similarly define the projective operator space topology on V determined by a family of local operator spaces V_{γ}, (γ ∈ Γ), and linear mappings π_{γ} : V → V_{γ} by using the collecton of operator seminorms σ_{n} = ρ_{n} ∘ (π_{γ})_{n} and we write

(36) V = oplim_{←}{V_{γ}, π_{γ}} = oplim_{←} V_{γ}.

The corresponding matrix topology on 𝔪(V) is just the usual projective limit

𝔪(V) = lim_{←} {𝔪(V_{γ}), (π_{γ})_{∞}} .

Once again we have that if V is an arbitrary local operator space, then

V = oplim_{←}{V̄_{ρ}, π_{ρ} : ρ ∈ 𝔑_{op}(V)}. (37)

where V̄_{ρ} is the completion of V_{ρ}. We recall that the completion of an operator space may be obtained by completing the underlying normed space, and then observing that each matrix space is then automatically complete (see (15)). Given ρ, σ ∈ 𝔑_{op}(V) with ρ ≤ σ, we have a corresponding complete contraction

π_{ρ,σ} : V̄_{σ} → V̄_{ρ}

determined by

π_{ρ,σ}(π_{σ}(v)) = π_{ρ}(v)

for v ∈ V.

Given V = oplim_{←}{V_{γ}, π_{γ}} and W = oplim_{←}{W_{δ}, θ_{δ}} with V_{γ} and W_{δ} operator

spaces, we have that a linear mapping φ : V → W is matrix continuous if and only if for each δ there exists a γ and a constant k > 0 for which

∥(θ_{δ})_{n}(φ_{n}(v))∥ ≤ k ∥(π_{γ})_{n}(v)∥ ,

or equivalently, we have (35), where φ_{σ,ρ} is a completely bounded mapping of complete operator spaces.

It is important to note that we are considering the projective limit of topologies on 𝔪(V), and not the projective limit of the underlying vector spaces 𝔪(V_{γ}). To illustrate how one can use this fact, let us suppose that V is a finite dimensional locally convex space. It is easy to see that we may choose a generating collection of continuous norms ρ_{μ}. We have from ([5], Lemma 2.3) that if V is a finite dimensional operator space, then any linear mapping of V into another operator space is completely bounded. It follows that all operator norms determine the same matrix topology on 𝔪(V). Thus the limit topology is determined by any one of the corresponding norms (we are indebted to S. Winkler for this observation), and we conclude that there is only one local operator structure on a finite dimensional vector space.

If we are given a subspace W of V, we have that the relative matrix topology on 𝔪(V) is determined by the restriction of the seminorms ∥π_{ν}(·)∥, i.e., we have that

W = oplim_{←}{W̄_{ν}, π̃_{ν}}

where W_{ν} = π_{ν}(W) and π̃_{ν} = π_{ν}|_{W} .

The following will play an important role in the next section.

Proposition 6.1. If V is a locally convex space, then

min V = oplim_{←} {min V̄_{ρ} : ρ ∈ 𝔑(V)} ,
max V = oplim_{←} {max V̄_{ρ} : ρ ∈ 𝔑(V)} .

Proof. Given a continuous seminorm ρ on V, it suffices to prove that V̄ρ̂ = max V̄_{ρ} and V̄ρ̌ = min V̄_{ρ}. For the second we observe that the adjoint of the quotient mapping V → V̄_{ρ} maps the unit ball of V_{ρ}^{∗} onto the polar (B^{ρ})^{∘} of the unit set for ρ. Thus for any v ∈ M_{n}(V) we have that

ρ̌_{n}(v) = sup {∥⟨⟨v, g⟩⟩∥ : g ∈ (B^{ρ})^{∘}}
= sup {∥⟨⟨π_{ρ,n}(v), f⟩⟩∥ : f ∈ V^{∗}, ∥f∥ ≤ 1}
= ∥π_{ρ,n}(v)∥_{min} .

On the other hand, the quotient mapping also determines a mapping of the unit ball of B(V_{ρ}, M_{p}) onto (B^{ρ})_{p}^{⊚}, from which we conclude that

ρ̂_{n}(v) = sup {∥⟨⟨v, g⟩⟩∥ : g ∈ (B^{ρ})_{p}^{⊚}, p ∈ ℕ}
= sup {∥⟨⟨π_{ρ,n}(v), f⟩⟩∥ : f ∈ B(V_{ρ}, M_{p}), ∥f∥ ≤ 1}
= ∥π_{ρ,n}(v)∥_{max}

(see (31) and (32)).

Turning to tensor products, we assume that the reader is familiar with the properties of the (complete) projective and injective tensor products V ˆ⊗W and V ˇ⊗W of normed spaces V and W. We denote the analogous operator space tensor products for operator spaces V and W by V ˆ⊗_{op}W and V ˇ⊗_{op}W. Motivated by the situation for locally convex spaces, we may use projective limits to define these tensor products for local operator spaces.

Given local operator spaces V and W, and continuous operator seminorms μ ∈ 𝔑_{op}(V) and ν ∈ 𝔑_{op}(W) we have corresponding linear mappings

π_{μ} ⊗ π_{ν} : V ⊗ W → V̄_{μ} ˆ⊗_{op}W̄_{ν}

and

π_{μ} ⊗ π_{ν} : V ⊗ W → V̄_{μ} ˇ⊗_{op}W̄_{ν}.

We define the (incomplete) projective and injective tensor products V ⊗_{∧}^{op} W, and V ⊗_{∨}^{op}W, respectively, to be the vector space V ⊗W together with the corresponding limit matrix topologies, i.e., we let

V ⊗_{∧}^{op} W = oplim_{←} V̄_{μ} ˆ⊗_{op}W̄_{ν}

and

V ⊗_{∨}^{op} W = oplim_{←} V̄_{μ} ˇ⊗_{op}W̄_{ν}.

where the limits are taken over μ ∈ 𝔑_{op}(V) and ν ∈ 𝔑_{op}(W). We define the (complete) projective and injective operator tensor products V ˆ⊗_{op}W and V ˇ⊗_{op}W to be the completion of these spaces. It is a simple matter to verify that one obtains the same spaces if one uses any generating families of operator norms ℜ and 𝔖 for V and W, respectively.

We may also use projective limits to define the Haagerup tensor product V ⊗^{h} W for local operator spaces. Alternatively we can appeal to a result of B. Johnson to the effect that the Haagerup tensor product has a natural interpretation in terms of module tensor products (see [6]). Since we shall not be using this tensor product below, we have put off further discussion to a subsequent paper.

Given local operator spaces V,W, and X, we say that a bilinear mapping φ : V × W → X is jointly matrix continuous, if the corresponding mapping

φ_{∞,∞} : 𝔪(V) × 𝔪(W) → 𝔪_{∞×∞}(X)

is jointly continuous in the matrix topologies. Equivalently, for each ρ ∈ 𝔑(X) we have that there exist μ ∈ 𝔑_{op}(V) and ν ∈ 𝔑_{op}(W) and a constant k > 0 such that

ρ_{m×n}(φ_{m,n}(v, w)) ≤ kμ_{m}(v)ν_{n}(w)

for all v ∈ M_{m}(V) and w ∈ M_{n}(W). Letting φ̃ : V ⊗W → X be the linear mapping determined by φ, it follows that we have a commutative diagram

V ⊗ W ⟶^{φ̃} X
↓  ↓
V̄_{μ} ˆ⊗_{op}W̄_{ν} ⟶_{φ̃_{μ,ν;ρ}} X̄_{ρ}

where the completely bounded linear mappings φ̃_{μ,ν;ρ} are determined by the corresponding bilinear functions

V̄_{μ} × W̄_{ν} → X̄_{ρ}.

It follows that φ̃ : V ⊗_{∧} W → X is matrix continuous. Letting C_{op}(V × W, X) be the vector space of all matrix continuous bilinear mappings φ : V × W → X, it follows that we have a natural vector space isomorphism

C_{op}(V × W, X) ≅ C_{op}(V ⊗_{∧}^{op} W, X).

If W is an operator space, and we let W^{∗} have the dual operator space structure (see (2)) we have the linear isomorphism

(V ⊗_{∧}^{op} W)′ ≅ C_{op}(V, W^{∗}). (38)

To see this, let us suppose that we are given a bilinear mapping F : V × W → ℂ, and that we define

φ_{F} : V → W^{∗}

by letting φ_{F} (v)(w) = F (v, w). We have that φ_{F} is matrix continuous if and only if for each ρ ∈ 𝔑_{op}(V) there is a k_{ρ} > 0 with

∥F_{m,n}(v, w)∥ = ∥(φ_{F})_{m}(v)_{n}(w)∥ ≤ k_{ρ}ρ_{m}(v) ∥w∥

for all v ∈ M_{m}(V), w ∈ M_{n}(W), m, n ∈ ℕ. Since the operator norms determine the operator space topology for V, these inequalities correspond exactly to the condition that F ∈ C_{op}(V × W, ℂ).

Given local operator space inclusions V_{0} ⊆ V and W_{0} ⊆ W (see §5), we claim that we a have natural inclusion

V_{0} ˇ⊗_{op}W_{0} ⊆ V ˇ⊗_{op}W. (39)

To see this, we observe that these inclusions determine a continuous bijection of V_{0} ⊗_{∨}^{op} W_{0} onto a subspace E of V ⊗_{∨}^{op} W. Given μ ∈ 𝔑_{op}(V) and ν ∈ 𝔑_{op}(W) with restrictions μ_{0} ∈ 𝔑_{op}(V_{0}) and ν_{0} ∈ 𝔑(W_{0}), the completely isometric inclusions (V_{0})_{μ_{0}} ↪ V_{μ} and (W_{0})_{ν_{0}} ↪ W_{ν}determine a complete isometry of V̄_{0μ} ˇ⊗_{op}W̄_{0ν} onto a subspace E_{μ,ν} of V̄_{0μ} ˇ⊗_{op}W̄_{0ν}. Thus the relation (39) is apparent from the diagram

\begin{aligned}V_{0} ⊗_{∨}^{op} W_{0} & ⇄ & E & ⊆ & V ⊗_{∨}^{op} W \\ ↓ & & ↓ & & ↓ \\ V̄_{0μ} ˇ⊗_{op}W̄_{0ν} & ⇄ & E_{μ,ν} & ⊆ & V̄_{μ} ˇ⊗_{op}W̄_{ν} \end{aligned} .

If V is a Banach space, we have that

M_{n}(min V) = M_{n} ˇ⊗V = M_{n} ⊗_{∨} V,

and thus

𝔪(min V) = 𝔪 ⊗_{∨} V.

On the other hand, we have that the vector space identification

M_{n}(max V) = M_{n} ⊗ V

determines a cross norm on M_{n} ⊗ V since if we are given α ∈ M_{n} and v ∈ V, we have that

∥α ⊗ v∥_{max} = sup {∥⟨⟨α ⊗ v, f⟩⟩∥ : f ∈ B(V, M_{p}), ∥f∥ ≤ 1, p ∈ ℕ}
= sup {∥(id ⊗ f)(α ⊗ v)∥ : f ∈ B(V, M_{p}), ∥f∥ ≤ 1, p ∈ ℕ}
= sup {∥α ⊗ f(v)∥ : f ∈ B(V, M_{p}), ∥f∥ ≤ 1, p ∈ ℕ}
= sup {∥α∥ ∥f(v)∥ : f ∈ B(V, M_{p}), ∥f∥ ≤ 1, p ∈ ℕ}
= ∥α∥ ∥v∥ .

It follows that we have a contraction

M_{n} ⊗_{∧} V → M_{n}(max V).

Owing to the fact that we have a contraction θ : 𝔪 → M_{n} for which

M_{n} ↪ 𝔪 →^{θ} M_{n}

is the identity mapping, we have that 𝔪 ⊗_{∧} V is just the union of the normed spaces M_{n} ⊗_{∧} V. Since 𝔪(max V) is the union of the normed spaces M_{n}(max V), we conclude from the diagram

M_{n} ⊗_{∧} V → M_{n}(max V)
↓  ↓
𝔪 ⊗_{∧} V → 𝔪(max V)

that the natural mapping

𝔪 ⊗_{∧} V → 𝔪(max V)

is a contraction.

If V is any locally convex space, we conclude that we have the homeomorphic identifications

(40) 𝔪(min V) = lim_{←} 𝔪(min V_{ρ}) = lim_{←} 𝔪 ⊗_{∨} V_{ρ} = 𝔪 ⊗_{∨} V_{.}

On the other hand, we have from the diagrams

\begin{aligned}𝔪 ⊗_{∧} V & → & 𝔪(max V) \\ ↓ & & ↓ \\ 𝔪 ⊗_{∧} V̄_{ρ} & → & 𝔪(max V̄_{ρ}) \end{aligned} (41)

that the top row is a continuous mapping.

The bottom row of the diagram

V ⊗_{∧}^{op} W ⟶^{id} W ⊗_{∨}^{op} W
↓  ↓
V̄_{μ} ˆ⊗_{op}W̄_{ν} ⟶ V̄_{μ} ˇ⊗_{op}W̄_{ν}

is completely contractive and thus we have the canonical operator continuous mapping

Φ_{op} : V ˆ⊗_{op}W → V ˇ⊗_{op}W. (42)

7. Nuclear mappings and spaces

Given Banach spaces V and W, a linear mapping φ : V → W is said to be nuclear if it is in the image of the canonical mapping

V^{∗} ˆ⊗W →^{Φ} V^{∗} ˇ⊗W ⊆ B(V, W).

The space N(V, W) of nuclear mappings φ : V → W is defined to be the image of Φ together with the quotient norm ∥φ∥_{nuc} determined by

N(V, W) ≅ \frac{V^{∗} ˆ⊗W}{ker Φ} .

We have that a mapping φ : V → W satisfies ∥φ∥_{nuc} < 1 if and only if we have a commutative diagram

l_{∞} ⟶^{θ_{λ}} l_{1}
σ↑  ↓^{τ}
V ⟶_{φ} W

where the column mappings are contractions, and θ_{λ} is defined for a λ ∈ l_{1} of norm < 1, by

θ_{λ}(a_{i}) = (λ_{i}a_{i})

Perturbing these mappings, we may replace l_{∞} in this diagram by c_{0}. A locally convex space V is said to be nuclear if for each continuous seminorm ρ on V, there exists a continuous seminorm σ ≥ ρ on V for which the corresponding mapping V̄_{σ} → V̄_{ρ} is nuclear. We have (see [25]):

Theorem 7.1. Suppose that V is a locally convex space. Then the following are equivalent:

a) V is nuclear

b) For all locally convex spaces W, we have that the mapping V ⊗_{∧}W → V ⊗_{∨}W is a homeomorphism, and thus V ˆ⊗W = V ˇ⊗W.

Given operator spaces V and W we have a corresponding diagram

V^{∗} ˆ⊗_{op}W ⟶^{Φ_{op}} V^{∗} ˇ⊗_{op}W ⊆ CB(V, W).

We define the space N_{op}(V, W) of matrix nuclear mappings φ : V → W to be the range of Φ_{op} together with the quotient norm ∥φ∥_{nuc}^{op} determined by

N_{op}(V, W) ≅ \frac{V^{∗} ˆ⊗_{op}W}{ker Φ_{op}}

(see [9] for the properties of these mappings). We say that a mapping φ : V → W is a proper matrix nuclear contraction if it satisfies ∥φ∥_{nuc}^{op} < 1. This will be the case if and only if we have a commutative diagram

\begin{matrix}M_{∞} & ⟶^{θ_{a,b}} & T_{∞} \\ ^{σ}↑ & & ↓^{τ} \\ V & ⟶_{φ} & W \end{matrix} (43)

where σ and τ are complete contractions, and the mapping θ_{a,b} is defined for properly contractive Hilbert-Schmidt operators a, b by

θ_{a,b}(x) = axb.

It is equivalent to assume that we have commutative diagrams (43) with M_{∞} replaced by K_{∞}.

Given two operator spaces V and W, we have a natural diagram of complete contractions interrelating the scalar and operator spaces:

V^{∗} ˆ⊗W ⟶ V^{∗} ˆ⊗_{op}W ⟶ V^{∗} ˇ⊗_{op}W ⟶ V^{∗} ˇ⊗W
↓  ↓  ↓  ↓
N(V, W) ⟶ N_{op}(V, W) ⟶ B_{op}(V, W) ⟶ B(V, W)

In particular, we conclude that if φ : V → W is nuclear in the classical sense, then it is operator nuclear.

Proposition 7.1. Suppose that V and W are operator spaces. If φ : V → W is a proper matrix nuclear mapping with ∥φ∥_{nuc}^{op} < 1, then for any operator space X the mapping

φ ⊗ id : V ⊗_{∨}^{op} X → W ⊗_{∧}^{op} X

is a complete contraction, and thus determines a complete contraction

φ ⊗ id : V ˇ⊗_{op}X → W ˆ⊗_{op}X.

Proof. To see this we recall that if G ∈ (W ˆ⊗_{op}X)^{∗} and ∥G∥ ≤ 1, then G determines a complete contraction

θ_{G} : W → X^{∗}

by

θ_{G}(w)(x) = G(w ⊗ x)

(see (38)). It is evident from (43) that the composition

V ⟶^{φ} W ⟶^{θ_{G}} X^{∗}

is again a proper matrix nuclear contraction, and thus has the form π(F) for some F ∈ V^{∗} ˆ⊗_{op}X^{∗} with ∥F ∥ < 1. We have that

⟨G, φ(v) ⊗ w⟩ = π(F)(v)(w) = ⟨F, v ⊗ w⟩,

and by linearity,

⟨G, (φ ⊗ id)(u)⟩ = ⟨F, u⟩

for all u ∈ V ⊗ X. Since the mapping

V ˇ⊗_{op}W ↪ (V^{∗} ˆ⊗_{op}W^{∗})^{∗}

is completely isometric (see [2]), it follows that

∥φ ⊗ id(u)∥ = sup {|⟨G, (φ ⊗ id)(u)⟩| : ∥G∥ ≤ 1}
≤ sup {|⟨F, u⟩| : ∥F ∥ ≤ 1}
≤ ∥u∥_{∨} .

Given operator spaces V and W and matrix nuclear mappings φ : V → W and ψ : X → Y, we have that the mapping

φ ⊗ ψ : V ˆ⊗_{op}X → W ˆ⊗_{op}Y

is matrix nuclear. To see this let us assume that we have the commutative diagram (43) for φ as well as a commuative diagram

K_{∞} ⟶^{θ_{c,d}} T_{∞}
σ′↑  ↓_{τ′}
X ⟶_{ψ} Y

Owing to the universal property of the norms ˆ⊗_{op}, we have that the first column in the commutative diagram

K_{∞} ˇ⊗_{op}K_{∞} = K_{∞×∞} ⟶^{θ_{a⊗c,b⊗d}} T_{∞×∞} = T_{∞} ˆ⊗_{op}T_{∞}
σ⊗σ′↑  ↓^{τ⊗τ′}
V ˆ⊗_{op}X  ⟶_{φ⊗ψ}  W ˆ⊗_{op}Y

is completely contractive. Since it is evident that a⊗c and b⊗d are Hilbert-Schmidt, we are done.

Given local operator spaces V and W, we say that a linear mapping φ : V → W is matrix or operator nuclear if there exists a commutative diagram

E ⟶^{ψ} F
σ↑  ↓^{τ}
V ⟶^{φ} W

where E and F are complete operator spaces, σ and τ are continuous, and ψ is nuclear.

We say that a local operator space V is matrix or operator nuclear if for any continuous operator seminorm ρ ∈ 𝔑_{op}(V), we have that there is a σ ∈ 𝔑_{op}(V) with σ ≥ ρ and for which the corresponding mapping

π_{ρ,σ} : V̄_{σ} → V̄_{ρ} (44)

is matrix nuclear.

Theorem 7.2. If V is local operator space, then the following are equivalent:

a) V is a matrix nuclear local operator space,

b) V is a nuclear locally convex space.

Proof. Let us suppose that V is matrix nuclear. Then given ρ ∈ 𝔑_{op}(V) we may select σ, τ ∈ 𝔑_{op}(V) with τ ≥ σ ≥ ρ for which the mappings

V̄_{τ} → V̄_{σ} → V̄_{ρ}

are matrix nuclear. But it was shown in [10], Th. 4.3, that any matrix nuclear mapping has factorizations through both row and column Hilbert spaces (see §1). Thus we obtain a diagram of completely contractive mappings

H_{c}  ⟶  K_{r}
↗  ↘  ↗  ↘
V̄_{τ}  ⟶  V̄_{σ}  ⟶  V̄_{ρ}

with H and K Hilbert spaces, and where the top row is just the composition H_{c} → V̄_{σ} → K_{r}. But any such completely contractive mapping from H_{c} → K_{r} must be Hilbert Schmidt (see [8], Cor. 4.5), and the same therefore applies to the composition V̄_{τ} → V̄_{ρ}. We may next use the fact that a composition of Hilbert-Schmidt operators is a trace class operator, and the trace class operators coincide with the nuclear operators between Hilbert spaces. Thus if we select a larger seminorm λ, we obtain the Hilbert spaces F and G, and a diagram of mappings

F → G  ⟶  H → K
↑  ↘  ↗  ↓
V̄_{λ}  ⟶  V̄_{τ}  ⟶  V̄_{ρ}

where the mapping G → H is just the obvious composition. It follows that we obtain diagrams

F ⟶^{θ} K
↑  ↓
V̄_{λ} ⟶ V̄_{ρ}

where the mappings θ are nuclear, and thus V is nuclear as a locally convex space.

Conversely suppose that V is a nuclear locally convex space. Then given ρ ∈ 𝔑_{op}(V), we may find a σ ∈ 𝔑(V) such that σ ≥ ρ, and the corresponding mapping V̄_{σ} → V̄_{ρ} is nuclear. But the operator seminorms are assumed to define the topology on V, hence we may choose τ ∈ 𝔑_{op}(V) such that τ ≥ σ. It follows that the composition V̄_{τ} → V̄_{ρ} is nuclear and from our earlier remarks, it must also be operator nuclear.

In this regard, we note that 𝔪(V) = 𝔪 ⊗_{∨} V is never nuclear as a locally convex space. To see this we simply observe that if we fix an element v_{0} ∈ V, we have that we have the homeomorphic inclusion

𝔪 ≅ 𝔪 ⊗_{∨} ℂv_{0} ↪ 𝔪 ⊗_{∨} V,

and we have that a subspace of a nuclear space is nuclear. Thus if 𝔪(V) is nuclear, then so is the infinite dimensional normed space 𝔪, which is a contradiction (see [25], pp. 100-103).

Theorem 7.3. If V is a nuclear locally convex space, then it has precisely one quantization.

Proof. It suffices to prove that max V = min V. But we have a diagram of topological identifications and continuous mappings

𝔪(min V) = 𝔪 ⊗_{∨} V = 𝔪 ⊗_{∧} V → 𝔪(max V) → 𝔪(min V).

for which the composition is the identity mapping. The first equality follows from (40), the second from the fact that V is a nuclear locally convex space. The first mapping is explained in (41). The final mapping is continuous since the corresponding result is trivial for operator spaces, and we have the commutative diagrams

max V → min V
↓  ↓
max V̄_{ρ} → min V̄_{ρ}

Thus we have the desired equality.

Theorem 7.4. If V is a matrix nuclear local operator space, then for any local operator space W we have that

V ˆ⊗_{op}W = V ˇ⊗_{op}W. (45)

Proof. We have already seen from (42) that we have a matrix continuous mapping from the first space to the second. On the other hand given ρ ∈ 𝔑_{op}(V), let us select ρ_{1} ≥ ρ for which the mapping V̄_{ρ_{1}} → V̄_{ρ} is matrix nuclear. Then for any σ ∈ 𝔑_{op}(W), we have from Proposition 7.1, that the bottom row of the diagram

V ⊗_{∨}^{op} W ⟶ V ⊗_{∧}^{op} W
↓  ↓
V̄_{ρ_{1}} ˇ⊗_{op}W̄_{σ} ⟶ V̄_{ρ} ˆ⊗_{op}W̄_{σ}

is completely bounded, and thus the top row is matrix continuous.

In contrast to the situation for locally convex spaces, it is not evident that the property (45) characterizes matrix nuclearity for V. The difficulty stems from the fact that operator spaces need not satisfy the analogue of local reflexivity [5]. However, if we take the latter condition into consideration, we obtain a completely satisfactory result.

Given operator space V and W, we say that a mapping φ : V → W is operator integral, if we have approximately commutative diagrams

M_{n} ⟶^{θ_{c,d}} T_{n}
σ↑  ↓^{τ}
V ⟶_{φ} W

(see [9]). As in the classical case, any matrix nuclear mapping is operator integral. On the other hand, the composition of two operator integral mappings is matrix nuclear. It follows from the latter fact that a local operator space V is nuclear if and only if we have diagrams (44) with π_{ρ,σ} integral. We let I_{op}(V, W) denote the linear space of integral mappings between operator spaces V to W. We will have no need to define the corresponding notion for local operator spaces.

An operator space W is locally operator reflexive if for each finite dimensional operator space E we have that any completely contractive mapping φ : E → W^{∗∗} can be approximated in the point-norm topology by completely contractive mappings φ : E → W. Equivalently, we have that for any operator space V, the natural mapping

I_{op}(V, W^{∗}) → (V ˇ⊗_{op}W)^{∗} (46)

is a completely isometric bijection (see [9], Th. 3.6).

Proposition 7.2. Given a local operator space V, we have that V is matrix nuclear if and only if we have

a) V_{ρ}^{∗} is locally reflexive for a generating family of continuous operator seminorms ρ ∈ 𝔑_{op}(V), and

b) for all operator spaces W, V ˆ⊗_{op}W = V ˇ⊗_{op}W.

Proof. Let us assume that we are given ρ ∈ 𝔑_{op}(V). From (38) the mapping π_{ρ} : V → V̄_{ρ} determines a continuous linear functional

F ∈ (V ˆ⊗_{op}V_{ρ}^{∗})′,

where F (v ⊗ f) = f(π_{ρ}(v)). From b), we have F ∈ (V ˇ⊗_{op}V_{ρ}^{∗})′. Since by definition,

V ˇ⊗_{op}V_{ρ}^{∗} = lim_{←}_{σ}V_{σ} ˇ⊗_{op}V_{ρ}^{∗},

there is a σ ∈ 𝔑_{op}(V) and an element

F̄ ∈ (V_{σ} ˇ⊗_{op}V_{ρ}^{∗})^{∗}

with

F (v ⊗ f) = F̄(π_{σ}(v) ⊗ f).

for f ∈ V_{ρ}^{∗}. From a) and (46)we may assume that

(V_{σ} ˇ⊗_{op}V_{ρ}^{∗})^{∗} = I_{op}(V_{σ}, V_{ρ}^{∗∗}).

We may also assume that σ ≥ ρ, and thus that we have a corresponding mapping

π_{ρ,σ} : V̄_{σ} → V̄_{ρ}

for which π_{ρ} = π_{ρ,σ} ∘ π_{σ}. If v ∈ V, then

F̄(π_{σ}(v) ⊗ f) = F (v ⊗ f)
= f(π_{ρ}(v))
= f(π_{ρ,σ}(π_{σ}(v)),

and thus F̄ is determined by the mapping π_{ρ,σ} : V̄_{σ} → V̄_{ρ}. Owing to the fact that the natural inclusion

I_{op}(V_{σ}, V_{ρ}) ↪ I_{op}(V_{σ}, V_{ρ}^{∗∗})

is completely isometric (see [9]) we conclude that the connecting mapping π_{ρ,σ} is operator integral. From our earlier remarks, we conclude that V is matrix nuclear.

Conversely if V is matrix nuclear, then it is evident from the proof of Theorem 7.2 we may assume that V is a projective limit of column Hilbertspaces. Since reflexive operator spaces are obviously locally reflexive, we have condition a). Condition b) follows from Theorem 7.4

Proposition 7.3. If V and W are matrix nuclear spaces, then the same is true for V ˆ⊗_{op}W.

Proof. This is immediate from the fact that if we choose ρ_{1} ≥ ρ and σ_{1} ≥ σ with the corresponding mappings V̄_{ρ_{1}} → V̄_{ρ} and W̄_{σ_{1}} → W̄_{σ} matrix nuclear, it follows that

V̄_{ρ_{1}} ˆ⊗_{op}W̄_{σ_{1}} → V̄_{ρ} ˆ⊗_{op}W̄_{σ}

is matrix nuclear.

8. Direct limits and operator bounded sets

Given a vector space V and a family of normed spaces E_{γ} together with mappings ψ_{γ} : E_{γ} → V, the direct limit topology on V is defined to be the finest locally convex topology on V for which these mappings are all continuous. We write

V = lim_{→}{E_{γ,}ψ_{γ}}

to indicate V with this topology. Given a locally convex topology on V, this relation holds if and only if it has the following property: a linear mapping φ:V → W, with W locally convex, is continuous if and only if each of the compositions

E_{γ} → V → W

is continuous (see [23], p. 79).

If the E_{γ} are operator spaces, then we have corresponding mappings

(ψ_{γ})_{∞} : 𝔪(E_{γ}) → 𝔪(V),

and we define the direct limit matrix topology on V to be the corresponding direct limit topology on 𝔪(V), i.e.,

𝔪(V) = lim_{→} 𝔪(E_{γ}).

We use the notation

V = oplim_{→}{E_{γ}, ψ_{γ}}

to indicate V with this matrix topology. Again it is evident that we have that if V is a local operator space, this relation will be true if and only if V has the following property: a linear mapping φ:V → W, with W a local operator space, is matrix continuous if and only if each of the compositions

E_{γ} → V → W

is matrix continuous.

A subset B of a locally convex space is said to be bounded if it is absorbed by each neighborhood N of 0, i.e., we have that cB ⊆ N for some constant c > 0. Equivalently, each continuous seminorm ν ∈ 𝔑(V) is bounded on B. If B is a bounded closed absolutely convex set with matrix gauge p = p^{B}, we have that for any ρ ∈ 𝔑(V) there is a constant k > 0 such that p ≤ kρ . It follows that p is faithful, and it defines a norm p is a norm on _{p}V (see (13)) for which the inclusion mapping

ι_{p} : _{p}V ↪ V

is continuous. We let 𝔅(V) denote the set of all gauges of bounded closed absolutely convex sets in V. Given locally convex spaces V and W, we say that a linear mapping φ : V → W is bounded if it maps bounded sets into bounded sets. We let B(V, W) denote the linear space of bounded linear mappings. Any continuous linear mapping φ : V → W is bounded, hence we have the linear injection

C(V, W) ↪ B(V, W).

A subset P of a locally convex space V is said to be bornivorous (see [18]) if it absorbs arbitrary bounded sets, i.e., for any bounded set B there is a k > 0 with kB ⊆ P. Any neighborhood of the origin is obviously bornivorous. A locally convex space V is said to bornological if conversely, any bornivorous set is a neighborhood of 0. Any countably generated locally convex space is bornological [23], p.82. The following result may be found in [18], p. 220 and [25].

Proposition 8.1. Suppose that V is a locally convex space. Then the following are equivalent:

a) V is bornological. b) If W is locally convex and φ : V → W is bounded, then it is continuous, i.e.,

we have that C(V, W) = B(V, W).

c) V = lim_{→} {_{p}V : p ∈ 𝔅(V)} .

We let 𝔅_{op}(V) denote the gauges of bounded 𝔪-convex sets in 𝔪(V). Any bounded set in 𝔪(V) is contained in a bounded 𝔪-convex set. To see this let us suppose that ρ ∈ 𝔑_{op}(V), and that ρ_{∞}(B) ≤ α. Then it is immediate from MG1, MG2, and Lemma 3.2 that if ρ ∈ 𝔑_{op}(V), and ρ_{∞}(B) ≤ α then ρ_{∞}(h_{op}(B)) ≤ α. It follows that

lim_{→} {_{p_{∞}}𝔪(V) : p ∈ 𝔅_{op}(V)} = lim_{→} {_{q}𝔪(V) : q ∈ 𝔅(𝔪(V))} (47)

We say that a mapping φ : V → W is matrix (or operator) bounded if φ_{∞} : 𝔪(V) → 𝔪(W) is bounded, and we let B_{op}(V, W) be the linear space of matrix bounded mappings.

Corollary 8.1. Suppose that V is a local operator space. Then the following are equivalent:

a) 𝔪(V) is bornological

b) V = oplim_{→} {_{p}V : p ∈ 𝔅_{op}(V)} .

If V satisfies either of these properties, then for any operator space W, we have that C_{op}(V, W) = B_{op}(V, W). Furthermore, any countably generated local operator space is matrix bornological.

Proof. If V is matrix bornological, then we have from the proposition and (47) that

V = lim_{→} {_{q}𝔪(V) : q ∈ 𝔅(𝔪(V))} = lim_{→} {_{p_{∞}}𝔪(V) : p ∈ 𝔅_{op}(V)}

and thus we have b). The converse relation is immediate, as are the remaining two assertions.

If B is a closed bounded convex set in a locally convex space V, then its polar B^{∘} is absorbing in V′, since any continuous linear functional f on V must be bounded on B. Thus we have that if p is the gauge of a bounded set in V, it is faithful, and its polar ρ = p^{∘} is a seminorm on V′. We have the obvious norm isomorphism

(_{p}V)^{∗} ≅ (V′)_{ρ} (48)

The same arguments apply to the gauges of bounded 𝔪-convex sets in 𝔪(V). We have that if p ∈ 𝔅_{op}(V), then ρ = p^{⊚} is an 𝔪-module seminorm on 𝔪(V′), and in this context we have that (48) is a complete isometry of operator spaces. We shall use such seminorms in the following section to define topologies for mapping spaces.

9. Mapping spaces and duality

Let us suppose that we are given local operator spaces V and W. For each operator bounded gauge p ∈ 𝔅_{op}(V) and continuous operator seminorm ρ ∈ 𝔑_{op}(W), we define

π_{ρ,p} : C_{op}(V, W) → B_{op}(V_{p}, W_{ρ})

by taking a linear mapping φ ∈ C_{op}(V, W) into the composition

pV ⟶ V ⟶^{φ} W ⟶ W_{ρ}.

Given a collection of operator bounded gauges 𝔖 ⊆ 𝔅(V). we define the 𝔖 matrix topology on C_{op}(V, W) by the relation

C_{op}(V, W)_{𝔖} = oplim_{←}{B_{op}(_{p}V, W_{ρ}), π_{p,ρ} : p ∈ 𝔖, ρ ∈ 𝔑_{op}(W)}.

In particular, we let

V′_{𝔖} = C_{op}(V, ℂ)_{𝔖}.

Given a net

φ_{ν} ∈ 𝔪(C_{op}(V, W)) = C_{op}(V, 𝔪(W)),

we have that φ_{ν} → φ if and only if for each ρ ∈ 𝔑_{op}(V) we have that the compositions

pV ↪ V ⟶^{φ_{ν}} 𝔪_{∞}(W) → 𝔪_{∞}(W̄_{ρ})

converge in B_{op}(_{p}V, 𝔪(W_{ρ})) to

pV ↪ V ⟶^{φ} 𝔪_{∞}(W) → 𝔪_{∞}(W̄_{ρ})

In particular, we have that f_{ν} ∈ 𝔪(V′) converges to f ∈ 𝔪(V′) if and only if

⟨⟨f_{ν}, v⟩⟩ → ⟨⟨f, ν⟩⟩

in the norm topology on 𝔪_{∞×∞}, uniformly on the bounded unit sets B^{p} ⊆ 𝔪(V) for p ∈ 𝔖.

We define the strong matrix topology on

V′ = C_{op}(V, ℂ)

by using the collection β = 𝔅_{op}(V) of all bounded operator norms on V. The matrix topology on 𝔪(V′) is determined by the 𝔪-module seminorms ρ = p^{⊚} with p ∈ β. We have that a net f_{ν} ∈ 𝔪(V′) converges to a function f ∈ 𝔪(V′) in this topology if the functions v ↦ ⟨⟨f_{ν}, v⟩⟩ converge uniformly to the function v ↦ ⟨⟨f, v⟩⟩ on the operator bounded sets in 𝔪(V). We note that since the seminorms ρ_{1} with ρ ∈ 𝔑_{op}(V) determine the topology on V, if B is a subset in V, then it is bounded in V if and only if it is bounded in 𝔪(V) (we are using the obvious inclusion V ↪ 𝔪(V)). Thus the strong matrix topology on 𝔪(V′) restricts to the usual strong topology on V′.

We let V′′ denote the strong bidual of a local operator space V, i.e., V′′ = (V′_{β})′_{β}.

Proposition 9.1. If V is an operator bornological space, then the mapping V → V′′ is a matrix homeomorphic injection.

Proof. The matrix topology on V′′ is generated by the matrix seminorms σ = p^{⊚} with p the matrix gauge of a bounded 𝔪-convex set D ⊆ 𝔪(V′). In particular, if N is an open set in 𝔪(V), it absorbs bounded sets K in 𝔪(V), and thus N^{⊚} is absorbed by sets of the form K^{⊚} in 𝔪(V′). Since the sets K^{⊚} with K bounded in 𝔪(V) generate the strong topology on 𝔪(V′), it follows that N^{⊚} is bounded. On the other hand, if D is a bounded set in 𝔪(V′), then it is absorbed by sets of the form K^{⊚} with K bounded in V. It follows that D_{⊚} = D^{⊚} ∩ 𝔪(V) absorbs the bounded 𝔪-convex sets in 𝔪_{∞}(V). Thus if we assume that V is operator bornological, D_{⊚} contains a neighborhood N of 0 in 𝔪(V), and (D_{⊚})^{⊚} is contained in a set of the form N^{⊚}. Thus the strong topology on V′′ is determined by sets of the form N^{⊚⊚}. Since we have that

N^{⊚⊚} ∩ V = N ∩ V,

it follows that the injection V ↪ V′′ is a matrix homeomorphism.

10. A nuclear quantization arising in quantum mechanics

It follows from §7 that there is a one-to-one correspondence between nuclear locally convex spaces and nuclear operator spaces. It perhaps useful to consider a very simple example that arises in mathematical physics [4], in which we have explicit operator nuclear mappings.

For any Hilbert spaces H and K we have the natural isometry

H_{r} ˆ⊗K_{c} = H_{r} ˆ⊗_{op}K_{c} (49)

(see [8], Cor. 4.4c). Replacing H by H^{∗}, it follows that we have a commutative diagram of isometries

H^{∗} ˆ⊗K ≅ (H_{c})^{∗} ˆ⊗_{op}K_{c}
↓  ↓
T (H, K) = N(H, K) ≅ N_{op}(H_{c}, K_{c})

i.e., each trace class contraction φ : H → K determines a matrix nuclear mapping φ : H_{c} → K_{c}. Given an arbitrary nuclear Frechet space V, we may find a diagram of Hilbert spaces and nuclear mappings

H_{1} ⟵^{φ_{1}} H_{2} ⟵^{φ_{2}} …

with

V = lim_{←} H_{k}.

We then have corresponding nuclear operator spaces V_{c} and V_{r} with the same underlying locally convex topologies, and the matrix topologies

V_{c} = oplim_{←}(H_{k})_{c}, and

V_{r} = oplim_{←}(H_{k})_{r}.

We let s be the space of sequences a = (a_{n})_{n∈ℕ} which are rapidly vanishing in the sense that

sup {n^{k} |a_{n}|} < ∞

for all k ∈ ℕ, and we define a mapping

π_{k} : s → l_{2}

by letting

π_{k}(a)_{n} = (1 + n)^{2k}a_{n}.

We then have a diagram of spaces

\begin{aligned}& & & & s \\ & & & ↙ π_{2} & ↓ π_{1} \\ … & →^{θ} & l_{2} & →^{θ} & l_{2} \end{aligned} ,

where θ : l_{2} → l_{2} is defined by

θ(a)_{n} = (1 + n)^{−2}a_{n}.

We may write θ = σ ∘ σ where the diagonal operator

σ(a)_{n} = (1 + n)^{−1}a_{n}

is a Hilbert-Schmidt mapping since

∑_{i,j} |σ_{i,j}|^{2} = ∑_{j}(1 + j)^{−2} < ∞.

It follows that θ is of trace class, and thus nuclear. Letting s have the topology defined by the mappings π_{k}, we see that s = lim_{←}{l_{2}, π_{k}}is a nuclear space. Using the terminology of [4], the (one parameter) space of physical states T is defined to be the locally convex tensor product

T =s ˆ⊗s.

It is the dual of the algebra A of observables.

In this context we may define a corresponding local operator structure on T as follows. We let s_{c} = lim_{←}(l_{2})_{c} and s_{r} = lim_{←}(l_{2})_{r} where (l_{2})_{c} and (l_{2})_{r} are the column and row Hilbert spaces determined by l_{2}. From above we have that the mappings θ : (l_{2})_{c} → (l_{2})_{c} and θ : (l_{2})_{r} → (l_{2})_{r} are matrix nuclear, and thus

T_{op} = s_{r} ˆ⊗_{op}s_{c} = lim_{←}(l_{2})_{r} ˆ⊗_{op}(l_{2})_{c}

is a matrix nuclear local operator space.

References

[1] W. Arveson, Subalgebras of C^{∗}-algebras, Acta Math. 123 (1969), 141-224.

[2] D. Blecher and V. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1991), 262-292.

[3] M.-D. Choi and E. Effros, Injectivity and operator spaces, J. Funct. Anal. 24 (1977), 156-209.

[4] D. Dubin and M. Hennings, Quantum Mechanics, Algebras, and Distributions, Pitman Research Notes in Mathematics Series, No. 238, Longman Scientific and Technical, Essex, 1990.

[5] E. Effros and U. Haagerup, Lifting problems and local reflexivity for C^{∗}-algebras, Duke Math. J. 52 (1985)103-128.

[6] E. Effros and Z.-J. Ruan, On matricially normed spaces, Pacific J. Math. 132 (1988), 243-264.

[7] E. Effros and Z.-J. Ruan, On approximation properties for operator spaces, Int. J. Math.1 (1990), 163-187.

[8] E. Effros and Z.-J. Ruan, Self-duality for the Haagerup tensor product, and Hilbert space factorizations, J. Funct. Anal. 200 (1991), 257-284.

[9] E. Effros and Z.-J. Ruan, Mapping spaces and liftings for operator spaces, Proc. London Math. Soc.,(3)69 (1994), 171-194.

[10] E. Effros and Z.-J. Ruan, The Grothendieck Pietsch and Dvoretzky-Rogers theorems for operator spaces, J. Funct. Anal. 122 (1994), 428-450.

[11] E. Effros and Z.-J. Ruan, On the abstract characterization of operator spaces, Proc. Amer. Math. Soc. 119 (1993), 579-584.

[12] E. Effros and Z.-J. Ruan, Operator Spaces, Oxford University Press, to appear.

[13] E. Effros and S. Winkler, Matrix convexity: operator analogues of the bipolar and Hahn-Banach theorems, J. Fnal. Anal., to appear.

[14] M. Gel’fand and N. Vilenkin, Generalized Functions IV, Applications of Harmonic Analysis, Academic Press, New York, 1964.

[15] J. Gilbert and T. Leib, Factorization, tensor products, and bilinear functions in Banach space theory, Notes in Banach Spaces, ed. H. Lacey, Univ. of Texas Press, Austin, 1980.

[16] A. Grothendieck, Résumé de la théorie métriques des produits tensoriels topologiques, Bol. Soc. Mat. São Paolo 8 (1956), 1-79.

[17] A. Grothendieck, Produits tensoriels topologiques et espace nucléaires, Mem. Amer. Math. Soc. no. 16, 1955.

[18] J. Horváth, Topological Vector Spaces and Distributions, vol. 1, Addison-Wesley, Reading, Mass. 1966.

[19] M. Junge and G. Pisier, Bilinear forms on exact operator spaces and B(H)⊗B(H), to appear.

[20] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conference Series in Math., No. 69, Washington, DC, 1984.

[21] G. Pisier, Espaces de Banach quantiques: une introduction à la théorie des espaces d’opérateurs, Journée Annuelle, Soc. Math. France, 1994.

[22] R. Powers, Self-adjoint algebras of unbounded operators. II, Trans. Amer. Math. Soc. 187 (1974), 261-293.

[23] A. Robertson and W. Robertson, Topological Vector Spaces, Cambridge Tracts in Math. and Math. Phys. Cambridge Univ. Press, Cambridge, 1964.

[24] Z.-J. Ruan, Subspaces of C^{∗}-algebras, J. Funct. Anal. 29 (1988), 217-230.

[25] H. H. Schaefer, Toplogical Vector Spaces, Springer-Verlag, New York, 1970.

[26] B. Simon, The P (ϕ)_{2} Euclidean (Quantum) Field Theory, Princeton Series in Physics, Princeton University Press, 1974.

[27] R. Smith, Completely bounded maps between C^{∗}-algebras, J. London Math. Soc. (2) 27 (1983), 157-166.

[28] S. Winkler, Dissertation, UCLA, to appear.

Mathematics Dept., UCLA, Los Angeles, CA 90024

   E-mail address: ege@@math.ucla.edu
   E-mail address: cwebster@@math.ucla.edu.