Operator space tensor norms
J.A. Chávez-Domínguez]Javier Alejandro Chávez-Domínguez
Department of Mathematics, University of Oklahoma, Norman, OK 73019-3103,
USA<EMAIL_ADDRESS>
V. Dimant]Verónica Dimant
Departamento de Matemática y Ciencias, Universidad de San
Andrés, Vito Dumas 284, (B1644BID) Victoria, Buenos Aires,
Argentina and CONICET<EMAIL_ADDRESS>
D. Galicer] Daniel Galicer
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires,
Argentina and CONICET<EMAIL_ADDRESS>
The first-named author was partially supported by NSF grant DMS-1900985. The second author was partially supported by CONICET PIP 11220200101609CO and ANPCyT PICT 2018-04104. The third author was partially supported by CONICET-PIP 11220200102366CO
and ANPCyT PICT 2018-4250.

namedefsubjclassname@2020 Mathematics Subject Classification

[2020]Primary: 47L25, 46M05, 46B28. Secondary: 46L06.

The use of a tensor product perspective has enriched functional analysis and other important areas of mathematics and physics. The context of operator spaces is clearly no exception.
The aim of this manuscript
is to kick off the development of a systematic theory of tensor products and tensor norms for operator spaces and its interplay with their associated  mapping ideals.
Based on the theory of tensor products in Banach spaces, we provide the corresponding natural definitions in the operator space framework.
The theory is not a mere translation of what is known in the classical setting and new insights, techniques, ideas or hypotheses are required in many cases.  As a consequence, notable differences in the theory appear when compared to the classical one.

§ INTRODUCTION

In the thirties von Neumann in collaboration with Murray  began a “mathematical quantization program”, aiming to formulate an operator version of integration theory.
They were successful, with the key idea being to replace function spaces by $*$-algebras of bounded operators on Hilbert spaces.
This was the seed that started a long road of “noncommutative analysis” which lead to what is now known as the theory of operator spaces.
Along this road and prior to the formalization of this theory, many concepts which are now typical in this framework (e.g., completely bounded mappings) were introduced in the context of the $C^*$-algebras.

An operator space is a Banach space $\textbf X$ with an extra “matricial norm structure”:
in addition to the usual norm on $\textbf X$, we have norms on all the spaces $M_n(\textbf X)$ of $n \times n$ matrices with entries from $\textbf X$ (often called the matrix levels), where these norms must satisfy certain consistency requirements.
The connection with operator algebras comes from the fact that when $\textbf X$ is a subspace of an algebra $\mathcal{B}(H)$ of bounded linear operators on a Hilbert space $H$, there is a canonical norm on $M_n(\textbf X)$ coming from the identification between $M_n(\mathcal{B}(H))$ and $\mathcal{B}(H^n)$.
The natural morphisms in this category are no longer just bounded maps, but the completely bounded ones: they are required to be uniformly bounded on all the matrix levels.
Operator spaces are thus a quantized or noncommutative version of Banach spaces, giving rise to a theory that not only is mathematically attractive but it is also naturally well-positioned to have applications to quantum physics.
The systematic study of operator spaces begun with Ruan's work, and
was deepened mainly by Effros and Ruan, Blecher and Paulsen, Pisier and Junge (see the monographs [27, 37, 5, 21, 38] and the references therein).
Operator spaces have often provided an appropriate framework to develop many areas of non-commutative analysis.

It is quite natural to investigate to what extent the classical theory of Banach spaces can be translated to the  noncommutative context, and this has been extensively studied throughout the years. Though some properties do carry over, many do not and these differences are one of the reasons making the new theory so interesting and rich. And even for those properties that do admit a generalization, the proofs are often different and require the development of new tools.
Along these lines, the main goal of this monograph is to take the first step towards the systematic development of a complete theory of tensor products and tensor norms in the operator space setting and its connections with the parallel theory of their associated mapping ideals.

It should be noted that both ideals of operators and tensor norms have inspired important developments in the operator space setting.
Noncommutative versions of nuclear, integral, summing, and other ideals of operators have played significant roles in, e.g.,
[21, 37, 27, 17, 26],
whereas various specific tensor norms for operator spaces (most notably the Haagerup one) have also played a significant role (see <cit.> for a good summary).

Blecher and Paulsen [6] have showed that part of elementary and basic theory of tensor norms of Banach spaces carries over to operator spaces, initiating a “tensor norm program” for operator spaces further developed in [4].
However, many of the abstract tools of operators ideals and tensor norms in the sense of [11] do not appear to have been treated yet.
Moreover, while it is clear to specialists that there are obstacles making it impossible to fully translate the classical tensor product theory into the operator space setting,
it has not been highlighted exactly which portions do translate and which ones do not.
Hence, we will focus on seeing to what extent the classical theory can be transferred. In many situations, this is not the case and additional assumptions about the structure of the operator spaces or the tensor norms involved are really necessary.
A good example is local reflexivity: since all Banach spaces have this property but not all operator spaces do, it is well known that operator space versions of classical results will often need local reflexivity as an additional assumption.
However, when it comes to tensor norms it turns out that in some cases we can still get operator space versions of some classical results without requiring local reflexivity, as long as the tensor norm satisfies an extra condition.

Although the theory of operator spaces has its own peculiarities, the classical theory will be our reference and roadmap to obtain and develop non-commutative versions of the theory of tensor norms.
Therefore, let us now recall briefly summarize the history of the classical theory's development .
Tensor products burst into the world of functional analysis in the late thirties with the works of Murray, von Neumann and Schatten,  but it was Grothendieck who first truly revealed the richness of properties for tensor products.
His famous article  “Résumé de la théorie métrique  des produits tensoriels
topologiques” [23] established the basis of what we know today as `local theory' — that is, the study of Banach spaces in terms of their finite-dimensional subspaces — and showed the relevance of the use of tensor products in the theory of normed spaces.
All this helped him to establish a very fruitful theory of duality.

While nowadays the “Résumé” is considered foundational work in the area, this masterpiece remained mostly unnoticed for many years.
It was not until the end of the sixties when it started to be better valued by the scientific community.  Lindenstrauss and Pełczyński [29]
presented some important connections and applications to the theory of absolutely $p$-summing operators, translating results written in terms of certain tensor norms by Grothendieck into properties of operator ideals.
On the other hand, almost simultaneously, a general theory of operator ideals on the class of Banach spaces was developed by  Pietsch (and his school in Jena) but without the use — and the language — of tensor norms. Pietsch's book “Operator Ideals” [32], which appeared at the end of the seventies, led this theory to  become one of the central themes of study in Banach space theory.

At the time functional analysts generally chose the language of operator ideals over that of tensor products, so the former received more attention.
The theory of tensor products in Banach spaces  became stronger and more ubiquitous during the eighties with the seminal work of Pisier [34, 35], which showed the usefulness of these tools.

With time, the theory of tensor norms became an interesting field in its own right.
Defant and Floret, in their famous monograph “Tensor Norms and Operator Ideals” [11], made it clear that the theory of tensor products and the  theory of operator ideals are two sides of the same coin. Their work had a tremendous legacy: nowadays authors use indistinctly both languages. This same approach has also been developed in various other books on tensor products of Banach spaces [44, 12].

As stated in [11], “both theories, the theory of tensor norms and of norm operator ideals, are more easily understood and also richer if one works with both simultaneously”.
With this in mind,  certain problems are more easily handled using either the categorical perspective due to Pietsch, or Grothendieck's cycle of ideas on tensor products.
With the present work, it is our intent to bring the benefits of the aforementioned parallel perspectives to the operator space setting.
As mentioned above both the approaches of ideals and tensor norms have been repeatedly used in the operator space literature, but there has not been a methodical exploration of the connections between the two.
It is worth mentioning that this work does not pretend to be an extensive monograph like [11]. Our purpose is much more modest: initiate a program that, over time,
can be enriched with the contributions of the operator space community. Thus, numerous topics covered in [11] have been left out.
Some expected results that do not appear in the literature can be found in this monograph, and also some new insights that do not have a counterpart or differ from the theory of Banach spaces.

The manuscript is organized as follows. Section <ref> is devoted to background material. We set some notation, give basic definitions that will be needed throughout the manuscript (in particular what we mean by an operator space tensor norm, or o.s. tensor norm), and present some known examples of operator space tensor norms.
Also, we introduce a new family of Haagerup-style tensor norms which are examples of
$\lambda$-tensor products (developed in [16, 46]).
Additionally, we emphasize the main relevant differences between the operator space and the Banach space frameworks which have an impact in the study of tensor norms in both settings.

In Section <ref> we deal with the notions of finitely and cofinitely generated o.s. tensor norms. The former essentially means that the tensor norm can be approximated using finite-dimensional subspaces, while the latter means we can approximate it using quotients over subspaces of cofinite dimension.
We also define the finite and cofinite hull of a given o.s. tensor norm.

In Section <ref> we present the “Five Basic Lemmas” (see Section 13 in [11]) for the operator space setting. These are the Approximation Lemma,
the Extension Lemma, the Embedding Lemma, the Density Lemma and the $\mathcal L_p$-Technique Lemma.
To prove some of these results, unlike the classical case, we additionally need certain strong hypotheses such as local reflexivity.
However, in the case of $\lambda$-o.s. tensor norms this extra assumption is often not necessary.

In Section <ref> we deal with dual tensor norms and develop an operator space version of the Duality Theorem. We also introduce right and left-accessible tensor norms and prove that any $\lambda$-o.s. tensor norm is accessible. Further, in order to prove
some duality relations we define the notion of local accessibility, a weaker version of accessibility which involves locally reflexive operator spaces.

In Section <ref> we discuss completely metric and completely bounded approximation properties and relate these properties with the weak density of a multiple of $ B_{E' \otimes_{\min} F}$ in $B_{\CB(E,F)}$. Also, we consider a weak completely bounded approximation property which provides an accurate characterization in terms of  the duality between the $\proj$ and  the $\min$ o.s. tensor norms.

In Section <ref>  we discuss the notion of mapping ideal (defined in <cit.>) and  present some examples already studied in the literature.  To maintain the core of the Banach space concept and allow the natural relationship  with tensor norms, we have to slightly strengthen the definition of mapping ideal from <cit.> and add an extra condition. Also, we introduce some typical procedures:  injective/surjective hulls of a given mapping ideal, the dual mapping ideal, and present some properties.

Sections <ref>  and <ref> deal with maximal and minimal mapping ideals and their Representation theorems.
As the names indicate, maximal and minimal ideals associated to a given tensor norms are the largest and smallest with respect to the inclusion which coincides over finite dimensional spaces.
The representation theorem for maximal mapping ideals shows how the ideal can be seen as a dual of a tensor product. Again, the hypothesis of local reflexivity is needed here in order to obtain a full characterization. As a consequence, we prove an Embedding Theorem (which gives a natural completely isometric inclusion from a tensor product into the mapping ideal) and show how dual and adjoint ideals behave in terms of their associated tensor norms.
For minimal ideals, the representation theorem provides a natural completely quotient mapping from a tensor product onto the mapping ideal.
Therefore, in a sense, both  theorems (for maximal and minimal ideals) relate a mapping ideal with a tensor product.

In Section <ref>, we give the definitions of right and left completely injective and completely projective hulls of an o.s. tensor
norm. We examine some of their interesting properties and see when they are finitely generated. Also, we define (both sides) completely injective and completely projective hulls showing that in both cases left and right procedures commute.

We establish in Section <ref> that, unlike what happens in the Banach space setting,  accessibility is a required property of a tensor norm in order to obtain the usual relations between duality and injective/projective hulls. Accessibility is also needed to associate a left (or right) injective hull of a norm to the surjective (respectively, injective) hull of a mapping ideal.
Based on the classical theory, we also provide a definition of accessibility for mapping ideals. As expected, right-accessible finitely-generated tensor norms are associated with right-accessible mapping ideals but, surprisingly, the left version of this result does not hold. We see that a left-accessible o.s. tensor norm might have an associated mapping ideal which is not left-accessible (although we show in Proposition <ref> that, in this case, the mapping ideal does have a weaker version of left-accessibility).

Section <ref> deals with the structure of natural norms, in the sense of Grothendieck.
We prove that some results from the Banach space theory about equivalences or inequalities between injective and projective hulls remain valid in this new context but the whole picture is undoubtedly substantially different. In the classical theory Grothendieck's inequality (which for instance implies that there is not possible for a tensor norm to be both injective and projective) has a strong impact in the description of natural tensor norms. The existence of an injective and projective Haagerup tensor norm shows us that the situation in the operator space setting could not be at all a translation from the classical case.
We use the term $\proj$ family for the set of norms obtained from the norm  $\proj$ after taking left or right injective/projective hulls finitely many times. Analogously, we define the $\min$ family. Unlike the classical setting, these families are not intertwined: each member of the $\proj$ family dominates any member of the $\min$ family. We present incomplete pictures of both of these families, proving some dominations and non-equivalences and leaving a large list of open questions about them.
On the other hand, we completely describe the list of all natural norms that come from applying to $\min$ or $\proj$ two-sided hull operations (injective or projective). Precisely, it consists of six o.s. tensor norms. Again, this differs from the Banach space case where there are actually four (see Section <ref>).

§ BASIC DEFINITIONS

We only assume familiarity with the basic theory of operator spaces, although we recall some elementary definitions. The books [38] and [21] are excellent references on this topic.
Our notation follows closely that from [37, 38].
For technical reasons, unlike the general literature in the subject, we will not assume that our operator spaces are complete. Thus, to avoid confusion, we  choose to call them
normed operator spaces (when completeness is not assumed) and   Banach operator spaces (when completeness is required).
The letters $E$, $F$ and $G$ will always denote normed operator spaces, that is, normed vector spaces with an additional structure at the matricial level that we now explain.

As usual, $M_n(E)$ will stand for the set of $n\times n$-matrices of elements in $E$ and for $a\in M_{n, m}$ ($n\times m$-matrices of complex scalars), $\|a\|$ denotes its norm as an operator from $\ell_2^m$ to $\ell_2^n$.

Given $x=(x_{i,j})\in M_n(E)$ and $y=(y_{k,l})\in M_m(E)$,  we consider $x\oplus y\in M_{n+m}(E)$ to be the matrix
x\oplus y=\left(
\begin{array}{cc}
(x_{i,j}) & 0 \\
0 & (y_{k,l}) \\
\end{array}
\right).

$E$ is a  normed operator space if, for each $n$, there is  a norm $\|\cdot\|_n$ on $M_n(E)$ satisfying:

M1 $\|x\oplus y\|_{M_{n+m}(E)}=\max\big\{\|x\|_{M_n(E)},\|y\|_{M_m(E)}\big\}$, for all $x\in M_n(E)$ and $y\in M_m(E)$.

M2 $\|a x b\|_{M_n(E)}\leq \|a\|\cdot \|x\|_{M_m(E)} \cdot \|b\|$, for all $x\in M_m(E)$, $a\in M_{n, m}$ and $b\in M_{m, n}$.

If $H$ is a Hilbert space and $E\subset \mathcal B(H)$ is a subspace, there is a natural norm in $M_n(E)$ given through the identification
$M_n(\mathcal B(H))=\mathcal B(H^n)$, which endows
$E$ with a normed operator space structure.
On  the other hand, by Ruan's representation theorem for any  operator space $E$ there exist a Hilbert space $H$ and an inclusion  $E\subset\mathcal B(H)$ which is an isometry for all the matrix levels.

Every  linear mapping $T:E\to F$ induces, for each $n\in\mathbb{N}$,  a linear mapping  $T_n: M_n(E)\to M_n(F)$ (called the amplification of $T$) given by

T_n(x)=\left(T(x_{ij})\right), \textrm{ for all }x=(x_{ij})\in M_n(E).

The completely bounded norm of $T$ is defined by

\|T\|_{\cb}=\sup_{n\in\mathbb{N}}\|T_n\|=\sup_{n\in\mathbb{N}} \big\{\|T_n(x)\|_{M_n(F)}:\, \|x\|_{M_n(E)}\le 1\big\}.

We say that $T$ is completely bounded if $\|T\|_{\cb}$ is finite and we denote by $\CB(E,F)$ the space of completely bounded mappings from $E$ to $F$. This is an operator space with the structure given by the identification $M_n(\CB(E,F))=\CB(E,M_n(F))$.

We say that $T$ is a complete isometry if each $T_n:M_n(E)\to M_n(F)$ is an isometry
and we write $E = F$ to indicate that $E$ and $F$ are completely isometrically isomorphic.

The operator space structure of a dual space $E'$ comes from the identification $M_n(E')=\CB(E,M_n)$. The norm of a matrix in $M_n(E')$ can also be computed through the matrix pairing $\langle\langle \cdot, \cdot\rangle\rangle: M_n(E)\times M_n(E')\to M_{n^2}$,  where $\langle\langle (x_{ij}),(\varphi_{kl})\rangle\rangle =(\varphi_{kl}(x_{ij}))$. Precisely,

\|(\varphi_{kl})\|_{M_n(E')}=\sup\big\{\|\langle\langle (x_{ij}),(\varphi_{kl})\rangle\rangle\|_{M_{n^2}}: \|(x_{ij})\|_{M_n(E)}\le 1\big\}.

Given a bilinear mapping $\phi:E\times F\to G$, we denote by $\phi_n:M_n(E)\times M_n(F)\to M_{n^2}(G)$ the mapping defined, for each $n\in \mathbb{N}$, as follows:

\phi_n(v,w)=\left(\phi(v_{ij},w_{kl})\right), \textrm{ for all }v=(v_{ij})\in M_n(E), w=(w_{kl})\in M_n(F).

We say that $\phi$ is jointly completely bounded when

\|\phi\|_{\jcb}\equiv\sup_{n\in\mathbb{N}}\|\phi_n\|<\infty.

The space $\JCB(E\times F,G)$ of all  jointly completely bounded bilinear mappings from $E\times F$ to $G$ has an operator space structure given by the identification

$$M_n\left(\JCB(E\times F,G)\right)= \JCB\left(E\times F,M_n(G)\right).$$

Another natural associate to $\phi$ is the bilinear mapping  $\phi_{(n)}:M_n(E)\times M_n(F)\to M_n(G)$ related with the matrix product and given by

\phi_{(n)}(v,w)=\left(\sum_{k=1}^n\phi(v_{ik},w_{kl})\right), \textrm{ for all }v=(v_{ij})\in M_n(E), w=(w_{kl})\in M_n(F).

The bilinear mapping $\phi$ is multiplicatively bounded if

\|\phi\|_{\mb}=\sup_{n\in\mathbb{N}}\|\phi_{(n)}\|<\infty.

Again, for the space $\MB(E\times F,G)$ of  all  multiplicatively bounded bilinear mappings from $E\times F$ to $G$ an operator space structure is provided via the identification

M_n\left(\MB(E\times F,G)\right) = \MB\left(E\times F,M_n(G)\right).

§.§ Definition of o.s. tensor norm

We begin by recalling the definitions of the three most common tensor norms in operator space theory: the minimal, projective, and Haagerup tensor norms.

The operator space injective tensor norm (also known as minimal tensor norm) of $u \in M_n(E \otimes F)$ is defined as

\begin{equation}\label{eqn-injective-tensor-norm}
\min(u) = \sup \{ \|(f \otimes g)_n(u)\| : f\in M_p(E'), g \in M_q(F'), \Vert f \Vert \leq 1, \Vert g \Vert \leq 1 \}
\end{equation}

The operator space projective tensor norm of $u \in M_n(E \otimes F)$ is defined as

\begin{equation}\label{eqn-projective-tensor-norm}
\proj(u) = \inf \{ \Vert a \Vert \cdot \Vert x\Vert \cdot \Vert y\Vert \cdot \Vert b\Vert \},
\end{equation}

where the infimum is taken over all the representations of $u= a(x \otimes y)b$ with $x \in M_p(E)$, $y \in M_q(F)$, $a \in M_{n,p \cdot q}$, $b \in M_{p\cdot q,n}$ for some $p,q\in \mathbb{N}.$

The Haagerup tensor norm of $u \in M_n(E \otimes F)$ is defined as

\begin{equation}\label{eqn-Haagerup-tensor}
h(u) = \inf \{ \Vert x \Vert \cdot \Vert y\Vert : u = x \odot y, x \in M_{n,r}(E), y \in M_{r,n}(F), r\in \mathbb{N} \},
\end{equation}

where $\odot$ denotes the standard  matrix  product. Precisely,

x \odot y = \left(\sum_{k=1}^r x_{ik} \otimes y_{kj}\right)_{i,j}.

The three aforementioned well-known tensor norms can be framed into the following concept.

An operator space cross-norm $\alpha$, on the class $\ONORM$ of all  normed operator spaces,
is an assignment of a normed operator space $E \otimes_\alpha F$ to each pair $(E,F)$ of normed operator spaces, in such a way that $E \otimes_\alpha F$ is the algebraic tensor product $E \otimes F$ together with a matricial norm structure on $E \otimes F$, that we write as $\alpha_n$ or $\n{\cdot}_{\alpha_n}$, and such that

\begin{equation}  \label{norma producto matrices}
\alpha_{nm}(x \otimes y) = \n{x}_{M_n(E)} \cdot \n{y}_{M_m(F)} \text{ for every } x \in M_n(E), y \in M_m(F).
\end{equation}

This implies that the identity map $E \otimes_{\proj} F \to E \otimes_{\alpha} F$ is completely contractive <cit.>.
If in addition the identity map $E \otimes_\alpha F \to E \otimes_{\min} F$ is also completely contractive, we say that $\alpha$ is  reasonable. Note that if both mappings $E \otimes_{\proj} F \to E \otimes_{\alpha} F$ and $E \otimes_\alpha F \to E \otimes_{\min} F$ are completely contractive then we obviously obtain (<ref>).

Moreover, an operator space cross-norm $\alpha$ is called uniform if additionally
it satisfies the complete metric mapping property:
if $S \in \CB(E_1,E_2)$ and $T \in \CB(F_1,F_2)$, then

\left\| S \otimes T \colon  E_1\otimes_\alpha F_1 \to E_2\otimes_\alpha F_2 \right\|_{\cb} \le \n{S}_{\cb}\n{T}_{\cb}.

An operator space cross-norm on $\ONORM$ that is both reasonable and uniform will be called an operator space tensor norm, or o.s. tensor norm for simplicity.

An operator tensor norm $\alpha$ is an assignment  to each  $E,F \in \ONORM$  of a matricial norm in $E \otimes F$  such that the following two conditions are satisfied:

* $\alpha$ is reasonable: The mappings $E\otimes_{\proj} F\to E \otimes_\alpha F \to E \otimes_{\min} F$ are completely contractive.

* $\alpha$ is uniform: If $S \in \CB(E_1,E_2)$ and $T \in \CB(F_1,F_2)$, then

\left\| S \otimes T \colon  E_1\otimes_\alpha F_1 \to E_2\otimes_\alpha F_2 \right\|_{\cb} \le \n{S}_{\cb}\n{T}_{\cb}.

A degree of caution is required when consulting different works dealing with operator space tensor products, since the term “tensor norm” is not always taken to have the exact same meaning.
We are using the same definition as  [21, 13],
which is slightly different than the one from [6, 4];
we refer to <cit.> for  an explicit comparison.
It is clear that analogous definitions for operator space tensor norms can also be made on a subclass of operator spaces,
for example the class $\OFIN$ of all finite-dimensional operator spaces,
or the class of dual operator spaces.

§.§ Basic properties of tensor norms

An o.s. tensor norm $\alpha$ is symmetric if $E \otimes_{\alpha}F$ and $F \otimes_{\alpha}E$ are canonically completely isometric (via the transposition map). A norm $\alpha$ is associative if $(E \otimes_{\alpha}F)\otimes_{\alpha}G$ and $E \otimes_{\alpha}(F\otimes_{\alpha}G)$ are canonically completely isometric (via the identity map).
It is known that $\min$ and $\proj$ are both symmetric and associative, while the Haagerup o.s. tensor norm $h$ is associative but not symmetric <cit.>.

For operator space tensor norms $\alpha$ and $\beta$ and a constant $c$, we write “$\alpha \le c \beta$ on $E\otimes F$” to indicate that the identity map $E \otimes_\beta F \to E \otimes_\alpha F$ has $\cb$-norm at most $c$.
If no reference to spaces is made, we mean that the inequality holds for any pair of normed operator spaces.

§.§ The Schatten spaces and their vector-valued versions

Recall that for a Hilbert space $H$ the Schatten class $S_p(H)$ is defined for $1 \le p < \infty$ as the space of all compact operators $T$ on $H$ such that $\tr\big( |T|^p\big) < \infty$ equipped with the norm $\n{T}_{S_p} = \big( \tr\big( |T|^p \big) \big)^{1/p}$;
in the case $p=\infty$, we denote by $S_\infty(H)$ the space of all compact operators on $H$ (also denoted by $\mathcal K(H)$) endowed with the operator norm.
To endow the spaces $S_p(H)$ with an operator space structure, we follow Pisier's approach [37]:
since $S_\infty(H)$ is a $C^*$-algebra it has a canonical operator space structure <cit.>, by the duality $S_1(H)'= B(H)$ we also get a natural operator space structure on $S_1(H)$,
and by complex interpolation for operator spaces <cit.>
we get an operator space structure for each of the intermediate $S_p(H)$ spaces.
In the special case $H=\ell_2$, we simply write $S_p$ instead of $S_p(\ell_2)$.
For an operator space $E$ and $1\le p \le \infty$ we can similarly obtain an operator space which is an $E$-valued version of $S_p$ and will be denoted by $S_p[E]$: we define
$S_\infty[E] = S_\infty \otimes_{\min} E$ and
$S_1[E] = S_1 \otimes_{\proj} E$,
and once again in the case $1 < p < \infty$ we define $S_p[E]$ via complex interpolation between $S_\infty[E]$ and $S_1[E]$.

Replacing $S_1$ by the space $S_1^n$ of $n\times n$ matrices with the trace norm, and $S_\infty$ by the space $M_n$, the same procedure analogously yields operator spaces $S_p^n$ and $S_p^n[E]$.
We refer the reader to [37] for a detailed study of these spaces.

§.§ Direct sums
Given a family $\{E_\gamma\}_{\gamma \in \Gamma}$ of operator spaces, its $\ell_\infty$ direct sum
which we will denote by $\ell_\infty( \{E_\gamma \;:\; \gamma \in \Gamma\} )$ is the direct sum $E = \bigoplus_{\gamma \in \Gamma} E_\gamma$ with the operator space structure given as follows:
for $u = (u_\gamma)_{\gamma \in \Gamma} \in  M_n(E)$,

\|u\|_{\ell_\infty( \{E_\gamma \;:\; \gamma \in \Gamma\}) } = \sup \big\{ \|u_\gamma\|_{M_n(E_\gamma)} \;:\; \gamma \in \Gamma \}.

When we have only two summands, we will often use the notation $E_1 \oplus_\infty E_2$ instead.

The $\ell_1$-sum of $\{E_\gamma\}_{\gamma \in \Gamma}$ will be similarly denoted by $\ell_1( \{E_\gamma \;:\; \gamma \in \Gamma\} )$; for simplicity, below we only give the full details in the case
$E_1 \oplus_1 E_2$ where we have two summands. For $u  \in M_n(E_1 \oplus E_2)$,

\|u\|_{M_n(E_1 \oplus_1 E_2)} = \sup\big\{ \n{(T_1 \oplus T_2)_n(u)}_{M_n(\mathcal{B}(H))} \;:\; \n{T_j : E_j \to \mathcal{B}(H)}_{\cb} \le 1 \text{ for } j=1,2\big\}

where $T_1 \oplus T_2 : E_1 \oplus E_2 \to \mathcal{B}(H)$ is the mapping $(T_1\oplus T_2)(x,y) = x+y$.

§.§ Injections and projections

A linear map $q : E \to F$ between normed operator spaces is called a complete quotient or complete metric surjection or complete projection  if it is onto and the associated map from $E/\ker(q)$ to $F$ is a completely isometric isomorphism.
In <cit.>,  it is proved that
a linear map $q:E \to F$ is a complete quotient if and only if its adjoint $q' : F' \to E'$ is a completely isometric embedding.
Note that if a linear map $q : E \to F$ between normed operator spaces is a complete quotient, then for every $x \in M_n(F)$ we have

\n{x}_{M_n(F)} = \inf \left\{ \n{y}_{M_n(E)} \; : \; y \in M_n(E), \;  q_n(y) = x   \right\}.

A completely isometric embedding $T : E \to F$ will be called a complete injection for short.

An o.s. tensor norm $\alpha$ is called completely left projective (resp.  completely left injective) if for any normed operator space $G$ and any complete projection (resp. complete injection) $T : E \to F$, the map $T \otimes id_G : E \otimes_\alpha G \to F \otimes_\alpha G$ is a complete projection (resp. complete injection) as well.
Completely right projective and completely right injective operator space tensor norms are defined analogously, and an o.s. tensor norm will be called completely projective (resp. completely injective) when it is both completely left and right projective (resp. injective).

A normed operator space $E$ is said to be completely injective if whenever $F \subseteq G$ are operator spaces and $T : F \to E$ is a completely bounded linear map, there exists an extension $\widetilde{T} : G \to E$ with $\n{T}_{\cb} = \|\widetilde{T}\|_{\cb}$.
By the Arveson extension theorem, $\mathcal{B}(H)$ is completely injective for any Hilbert space $H$.
Let us now observe that any completely injective normed operator space $E$ has to be complete. Indeed, consider the diagram

\xymatrix{
\widetilde{E} \ar[dr]^T &\\
E \ar[u] \ar[r]_{id_E} & E}

where the vertical arrow is simply the inclusion of $E$ in its completion $\widetilde{E}$, and the operator $T$ is given by the injectivity of $E$.
If we take a Cauchy sequence $(v_n)_n$ in $E$, and we take its limit $v \in \widetilde{E}$, then $T(v) \in E$ is the limit of $(v_n)_n$ since

T(v) = T\Big( \lim_{n\to\infty} v_n \Big) = \lim_{n\to\infty} T(v_n) = \lim_{n\to\infty}  v_n.

A normed operator space $E$ is said to be completely projective if for any completely bounded linear map $T : E \to G/F$ into a quotient space and any $\varepsilon>0$ there exists a lifting $\widetilde{T} : E \to G$ of $T$ with $\|\widetilde{T}\|_{\cb} \le (1+\varepsilon) \n{T}_{\cb}$.
The basic examples of projective operator spaces are the spaces $S_1^n$ for any $n\in\N$, and we will also use the fact that an $\ell_1$-sum of projective operator spaces is again projective.

Every Banach operator space $E$ can be seen as a quotient of a completely projective space. Indeed, there is a set $I$ and a family $(n_i)_{i \in I} \subset \mathbb{N}$ such that $E$ is the quotient of $\ell_1(\{S_1^{n_i} : i \in I \})$ (see e.g., <cit.>). We denote the latter space by $Z_E$ and $q_E : Z_E \twoheadrightarrow E$ the corresponding complete quotient mapping. The fact that $Z_E$ is completely projective can be tracked in <cit.>.

See <cit.> for more on completely injective and completely projective operator spaces.

§.§ Examples of o.s. tensor norms

In addition to the three fundamental o.s. tensor norms defined above (injective, projective, Haagerup), a number of other examples have appeared in the literature and we list some of them below.
The first three are discussed in [22], where it is shown that they satisfy the complete metric mapping property.
To conclude that they are o.s. tensor norms, we then just need to check that they are between $\min$ and $\proj$.

§.§.§ The nuclear tensor product

$E \otimes_{\nuc} F$ is defined as the quotient of $E \otimes_{\proj} F$ by the kernel of the canonical identity map $E \otimes_{\proj} F \to E \otimes_{\min} F$.
From the definition it is clear that $\min \le \nuc \le \proj$, so $\nuc$ is reasonable.

§.§.§ The extended Haagerup tensor product [18]

$E \otimes_{\eh} F$ is the space of maps $u : E' \times F' \to \C$ which are normal (i.e. weak$^*$ continuous in each variable) and multiplicatively bounded.
If we denote the space of such maps by $\MB^\sigma(E'\times F', \C)$,
the matrix norms on $E \otimes_{\eh} F$ are given by the identification ${M_n\big( \MB^\sigma(E'\times F', \C) \big)} = \MB^\sigma(E'\times F', M_n)$.
It then follows from (<ref>) that $\min \le \eh$.

The operator space structure on $E \otimes_{\eh} F$ can be described in the same way as (<ref>) but replacing $r\in\N$ with an arbitrary set <cit.>, hence the extended in the name.
This in particular shows $\eh \le h$, and thus $\eh \le \proj$.
The extended Haagerup tensor product is associative <cit.>, and completely injective but not completely projective <cit.>.

§.§.§ The normal Haagerup tensor product [18]

This is only defined for dual operator spaces, by $E' \otimes_{\nh} F' = (E \otimes_{\eh} F)'$.
Since the identity $E \otimes_{\proj} F \to E\otimes_{\eh} F$ is a complete contraction, dualizing yields a contraction

E' \otimes_{\nh} F' = (E \otimes_{\eh} F)' \to (E \otimes_{\proj} F)' = \CB(E,F'),

which implies that the identity $E' \otimes_{\nh} F' \to E' \otimes_{\min} F'$ is a complete contraction.
The fact that $\nh \le \proj$ is a complete contraction, i.e. the identity $E' \otimes_{\proj} F' \to (E \otimes_{\eh} F)'$ is a complete contraction, follows easily from (<ref>) and the description of $E \otimes_{\eh} F$ as normal multiplicatively bounded mappings (see   <ref>).

Recall from above that the
extended Haagerup tensor product is not completely projective, that is, there exist complete projections $\pi_j : E_j \to F_j$, $j=1,2$ such that
$\pi_1 \otimes \pi_2 : E_1 \otimes_{\eh} E_2 \to F_1 \otimes_{\eh} F_2$ is not a complete projection.
By dualizing, we have complete injections $\pi_j' : F_j' \to E_j'$, $j=1,2$
such that $\pi_1' \otimes \pi_2' : F_1' \otimes_{\nh} F_2' \to E_1' \otimes_{\nh} E_2'$
is not a complete injection, so
the normal Haagerup tensor
product is not completely injective.

§.§.§ The symmetrized Haagerup tensor norms

Given normed operator spaces $E$ and $F$ and $u \in M_n(E \otimes F)$, we define

(h \cap h^t)_n(u;E,F) = \max\big\{ h_n(u; E,F), h^t_n(u; E,F) \big\}.

It is easy to see that $h \cap h^t$ is a symmetric o.s. tensor norm, and from the definition it is obvious that there is a completely isometric embedding

E \otimes_{ h \cap h^t} F \hookrightarrow (E \otimes_h F) \oplus_\infty (E \otimes_{h_t} F)

given by $u \mapsto (u,u)$.
Since $h$ and $h^t$ are completely injective, it is clear that $h \cap h^t$ is completely injective as well.
Notice that if $\alpha$ is a symmetric o.s. tensor norm such that $h \le \alpha$, by transposing we have $ h^t \le \alpha$ and therefore $h \cap h^t \le \alpha$.
Thus, we call $h \cap h^t$ the minimal symmetrized Haagerup tensor norm.
This tensor norm has appeared in [42, 24, 14].

On the other hand, for normed operator spaces $E$ and $F$ and $u \in M_n(E \otimes F)$, we define

(h+h^t)_n(u;E,F) = \inf\big\{ \|(v,w)\|_{M_n\big( (E \otimes_h F)\oplus_1(E \otimes_{h^t} F) \big)} \;:\; u = v+w \big\}.

That is, by definition the mapping

\[
q : (E \otimes_h F) \oplus_1 (E \otimes_{h^t} E) \to E \otimes_{h+h^t} F
\]

given by $q(u,v) = u + v$ is a complete quotient.
Once again, it is easy to check that this defines a symmetric o.s. tensor norm which is completely projective because so is $h$.
The o.s. tensor norm $h+h^t$ was introduced in [30], were it was denoted by $\mu$ and it was shown that it is neither associative nor completely injective.
We will call $h+h^t$ the maximal symmetrized Haagerup tensor norm because of the following property. Suppose that $\beta$ is a symmetric tensor norm with $\beta \le h$. Then $\beta \le h^t$, so that the formal identity maps

E \otimes_h F \to E \otimes_\beta F, \quad E \otimes_{h^t} F \to E \otimes_\beta F

are complete contractions. Therefore, the map $(v,w) \mapsto v+w$ is a complete contraction

(E \otimes_h F) \oplus_1 (E \otimes_{h^t} E) \to E \otimes_\beta F

and by the standard properties of quotients <cit.>, the identity map $E \otimes_{h+h^t} F \to E \otimes_\beta F$ is a complete contraction as well, that is, $\beta \le {h+h^t}$.

§.§.§ The Chevet-Saphar tensor products

Inspired by the definition of the Banach space case,  for $1 \leq p \leq \infty$, the right and left $p$-Chevet-Saphar o.s. tensor norms, $d_p $ and $g_p $, are defined in  [8]   in following way: given normed operator spaces $E$ and $F$, for $u \in E \otimes F$,

\begin{align}\label{dp-os}
d_p (u) = \inf \left\{\Vert (x_{ij}) \Vert_{S^n_{p'}\otimes_{\min} E} \; \Vert(y_{ij})\Vert_{S^n_p[F]} : u = \sum_{i,j=1}^n x_{ij} \otimes y_{ij} \right\}, \\
g_p (u) = \inf \left\{ \Vert(x_{ij})\Vert_{S^n_p[E]} \; \Vert (y_{ij}) \Vert_{S^n_{p'}\otimes_{\min} F}   : u = \sum_{i,j=1}^n x_{ij} \otimes y_{ij} \right\}
\end{align}

where the infimum runs over  all the possible ways in which the tensor $u$ can be written as a finite sum as above.

Note that the above expressions only define norms on $E \otimes F$, the full operator space structure of the tensor products $E \widehat \otimes_{d_p } F$ and  $E \widehat \otimes_{g_p } F$ are given by the following quotient mappings  (see <cit.>)

\begin{align}
q_{d_p} : (S_{p'} \widehat{\otimes}_{\min} E) \widehat{\otimes}_{\proj} S_p[F] \to E \widehat{\otimes}_{d _p} F, \\
q_{g_p} : S_p[E] \widehat{\otimes}_{\proj} (S_{p'} \widehat{\otimes}_{\min} F) \to E \widehat{\otimes}_{g _p} F,
\end{align}

where $\widehat \otimes $ means the completion of the corresponding tensor product.
In <cit.> it is shown that both $d_p $ and $g_p $ are o.s. tensor norms.

§.§.§ $\lambda$-tensor products [46, 16]

For each $k \in \N$, let
$$\bl_k: M_k\times M_k \to M_{\tau(k)}$$
(where $\tau(k) \in \N$ is a natural number only depending on $k$) be a bilinear map;
we will denote the sequence $(\bl_k)_k$ by $\lambda$.
Two basic examples to keep in mind are the tensor product $\otimes$ (i.e. when $\bl_k(x,y) = x \otimes y$ for all $k$) and the matrix product $\odot$ (i.e. when $\bl_k(x,y) = x \odot y$ for all $k$).
We point out that the notation above is slightly different from that of [46, 16], in order to keep consistency in the notation of the present work.
For $u \in M_k(E \otimes F)$, define

\begin{equation}\label{lnorm}
\lambda_k(u)=\inf\{\|a\|\|v_1\|\|v_2\|\|b\|\}
\end{equation}

where  the infimum is taken over arbitrary decompositions $u = a \otimes_{\bl_j} (v_1,v_2) b$ with $a\in M_{k, \tau(j)}$, $b\in M_{\tau(j), k}$, $v_1\in M_j(E)$, $v_2\in M_j(F)$, where $\otimes_{\bl_j} : M_j(E) \times M_j(F) \to M_{\tau(j)}(E \otimes F)$ is the bilinear map given by $(a_1 \otimes x, a_2 \otimes y) \mapsto \bl_j(a_1,a_2) \otimes x \otimes y$.
Observe that the case $\lambda=\otimes$ corresponds to the projective tensor norm, and $\lambda=\odot$ yields the Haagerup tensor norm.

In order to guarantee that the norms defined above give an o.s. tensor norm, we will need for $\lambda$ to satisfy certain technical conditions. First, some notation.
We let $\varepsilon_{i,j}:=\varepsilon^{[k,l]}_{i,j}\in M_{k,l}$ denote the matrix which is 1 in the $(i,j)$-th entry and zero elsewhere, $\varepsilon^{[k]}_{i,j}:=\varepsilon^{[k,k]}_{i,j},\; \varepsilon_i:=\varepsilon_{i,i}, \;\varepsilon^{[k]}_i:=\varepsilon^{[k,k]}_{i,i}$ and $\varepsilon^{[k,l]}_{i,j}=0$ if $(i,j) \notin \{1,\ldots,k\} \times \{1,\ldots,l\}$. As usual we will denote by $I_n$ the identity $n\times n$-matrix.

For any operator spaces $E$ and $F$ the sequence $\lambda_k(\cdot)$ defined above gives an operator space structure on $E \otimes F$, whenever $\lambda$ satisfies the following conditions:

(E1) For all $k \in \mathbb{N}$ there exist $p \in \N$ and matrices $S \in M_{k,\tau (p)}$, $T \in M_{\tau (p),k}$, $a_1 , \cdots , a_k \in M_p$ such that for all $ j_1 , j_2
\in \{1, \cdots, k\}$:

S \bl_p(a_{j_1},a_{j_2}) T= \left\{
\begin{array}{ll}
\varepsilon_j^{[k]} & \qquad \text{if}\;\; j_1=j_2=j, \\
0 & \qquad \text{otherwise.}
\end{array}
\right.

(E2) For all $ r, s \in  \mathbb{N}$ there exist matrices  $P \in M_{\tau (r)+\tau (s), \tau(r+s)}$, with $\|P\|\leq 1$ such that for all $ (i_k , j_k ) \in
\{1, \cdots , r\}^2 \cup \{r + 1, \cdots , r + s\}^2$ with $k=1,2$:

\begin{align}
\begin{split}
& P \bl_{r+s}( \varepsilon_{i_1, j_1}^{[r+s]} , \varepsilon_{i_2 ,j_2 }^{[r+s]})P^* \\
&= \mathrm{diag}\Big( \bl_r (\varepsilon_{i_1 ,j_1}^{[r]} ,  \varepsilon_{i_2,j_2}^{[r]}), \bl_{s}(\varepsilon_{i_1-r ,j_1-r}^{[s]} , \varepsilon_{i_2-r,j_2-r}^{[s]})\Big).\notag
\end{split}
\end{align}

(E3) $\bl_1(1,1)=1$ and $\sup_{k\in \N}\|\bl_k\| <\infty$.

In this case, we denote by $E \otimes_\lambda F$ the corresponding operator space.
Moreover, the complete metric mapping property is satisfied.

Note that, in an abuse of notation, we are using the symbol $\lambda$ to denote both the sequence of bilinear maps $M_k \times M_k \to M_{\tau(k)}$ and the operator space structure induced on the tensor product.

Just as for the projective and Haagerup tensor products, the dual of a $\lambda$-tensor product can be identified with a certain space of bilinear forms.
For any bilinear map $\phi : E \times F \to W$ where $E$, $F$, $W$ are operator spaces, define the bilinear maps
$\phi_{\bl_k} : M_k(E) \times M_k(F) \to M_{\tau(k)}(W)$ given on elementary tensors by

\phi_{\bl_k}\big( a_1 \otimes v_1, a_2 \otimes v_2 \big) = \bl_k(a_1,a_2) \otimes \phi(v_1,v_2), \qquad a_1, a_2 \in M_k, v_1 \in E, v_2 \in F.

We also define

\n{\phi}_{\cb,\lambda} = \sup_{k\in\N} \left\{\n{\phi_{\bl_k}(x,y)}_{M_{\tau(k)}(W)} \;:\; \n{x}_{M_k(E)} \le 1, \n{y}_{M_k(F)} \le 1 \right\}

\CB_\lambda(E \times F; W) = \left\{ \phi : E \times F \to W \text{ bilinear} \;:\; \n{\phi}_{\cb,\lambda}<\infty \right\}.

This space is an operator space with the identification

M_k\big(\CB_\lambda(E \times F; W)\big) = \CB_\lambda\big(E \times F; M_k(W)\big).

Note that for the cases  $\lambda=\otimes$ (projective tensor product) and $\lambda=\odot $ (Haagerup tensor product) we recover the usual dual spaces $\CB_\otimes=\JCB$ and $\CB_\odot=\MB$.

If $\lambda$ satisfies (E1), (E2), and (E3), then
the natural identification yields a canonical complete isometry

\CB( E\otimes_{\lambda} F, W ) = \CB_\lambda(E \times F;W)

so in particular

(E\otimes_{\lambda} F)' = \CB_\lambda(E \times F;\C) =: \CB_\lambda(E \times F).

Let us denote by $\odot^t : M_k \times M_k \to M_k$ the transposition of the usual matrix product, that is, $A \odot^t B = B \odot A$ for $A, B \in M_k$.
It is clear that taking $\lambda=\odot^t$, the associated operator space structure on $E \otimes F$ is precisely $E \otimes_{h^t} F$, the transposition of the Haagerup tensor product.
We now introduce a new family of Haagerup-style o.s. tensor products which are $\lambda$-tensor products, and which can be understood as a sort of  “interpolation” between $h$ and $h^t$.
For any $\theta \in [0,1]$ and $k\in\N$, we let
$\odot^\theta_k : M_k \times M_k \to M_k$ be $(1-\theta) \odot + \theta \odot^t$, that is, $\odot^\theta_k(A,B) = (1-\theta)AB+\theta BA$ for $A,B\in M_k$. It is clear that each $\odot^\theta_k$ is bilinear, and let us now show that they all induce operator space structures on the tensor product.

For each $\theta\in[0,1]$, $\odot^\theta$ satisfies (E1), (E2) and (E3).

Clearly $1 \odot 1 = 1 \odot^t 1 = 1$, and $\n{\odot_k} = \n{\odot^t_k} = 1$ (since $\n{A\odot B} \le \n{A} \n{B}$ for $A,B\in M_k$). Taking a convex combination yields (E3) for $\odot^\theta$.

As in the proof of <cit.>, (E1) is satisfied for $\odot$ by taking $p=k$, $S=T=I_k$, and $a_{j_i} = \varepsilon^{[k]}_{j_i}$. Since all the matrices involved are diagonal all the products commute, and it follows that the same choices yield (E1) for $\odot^t$. Taking convex combinations, the same choices once again yield (E1) for $\odot^\theta$.
The exact same type of argument works for (E2): this is proved for $\odot$ in <cit.> using $P=I_{r+s}$.

For each $\theta\in[0,1]$, $\odot^\theta$ is a cross-norm.

Let $E$, $F$ be operator spaces, $x \in M_m(E)$, $y \in M_n(F)$.
It is well-known that $x\otimes y = (x\otimes I_n)\odot (I_m \otimes y)$, see e.g. <cit.>. The exact same calculation shows $x\otimes y = (x\otimes I_n)\odot^t (I_m \otimes y)$, and therefore $x\otimes y = \odot^\theta_{mn}(x\otimes I_n, I_m \otimes y)$. Thus, by the definition of $\odot^\theta_{mn}$,

\odot^\theta_{mn}( x\otimes y ) \le \n{x\otimes I_n}_{M_{mn}(E)} \n{ I_m \otimes y }_{M_{mn}(F)} =  \n{x}_{M_m(E)} \cdot \n{y}_{M_n(F)}.

In order to conclude that $\odot^\theta$ is an o.s. tensor norm, the only missing ingredient is to check that $\min \le \odot^\theta$.
We prove a more general result.

Suppose that $\lambda$ satisfies (E1), (E2), (E3), and $\n{\bl_k}_{\jcb} \le 1$ for each $k\in\N$.
Then the identity mapping $E\otimes_\lambda F \to E \otimes_{\min} F$ is a complete contraction.

By <cit.>, it suffices to show that for every $U \in M_k(E \otimes F)$, $\phi \in M_m(E')$, $\psi \in M_n(F')$ we have

\n{\mpair{\phi \otimes \psi}{U}}_{M_{mnk}} \le \n{\phi}_{M_m(E')} \n{\psi}_{M_n(F')}\lambda_k(U).

Now, by the identification $(E\otimes_{\lambda} F)' = \CB_\lambda(E \times F)$ we already know that

\n{\mpair{\phi \otimes \psi}{U}}_{M_{mnk}} \le \n{\phi \otimes \psi}_{M_{mn}(\CB_\lambda(E \times F))}\lambda_k(U).

Recalling that $M_{mn}\big(\CB_\lambda(E \times F)\big) = \CB_\lambda\big(E \times F; M_{mn}\big)$, all we need is to show that

\n{\varphi}_{\lambda,\cb} \le \n{\phi}_{M_m(E')} \n{\psi}_{M_n(F')}

where $\varphi =\phi \otimes \psi : E \times F \to M_{mn}$; note that here we are interpreting $\phi \in M_m(E') = \CB(E,M_m)$ and $\psi \in M_n(F') = \CB(F,M_n)$.

To calculate the norm of $\varphi$ in $\CB_\lambda(E \times F ; M_{mn})$, take $u \in M_k(E)$ and $v \in M_k(F)$ with $\n{u}_{M_k(E)}, \n{v}_{M_k(F)}\le 1$.
Represent $u = \sum_i a_i \otimes u_i$, $v = \sum_j b_j \otimes v_j$ where $a_i,b_j \in M_k$, $u_i \in E$, $v_j \in F$.
Then, by the definition of $\varphi_{\bl_k}$,

\begin{multline*}
\varphi_{\bl_k}(u,v) = \sum_{i,j} \bl_k(a_i,b_j) \otimes \varphi(u_i,v_j) =  \sum_{i,j} \bl_k(a_i,b_j) \otimes \phi(u_i) \otimes \psi(v_j) \\
= \otimes_{\bl_k} \Big(\sum_i a_i \otimes \phi(u_i),   \sum_j b_j \otimes \psi(v_j) \Big)
= \otimes_{\bl_k} \big( (I_k \otimes \phi)u, (I_k\otimes \phi)v \big).
\end{multline*}

Note that the assumption $\n{\bl_k}_{\jcb} \le 1$ precisely means that for any $A \in M_k(M_m)$ and $B \in M_k(M_n)$ we have

\n{ \otimes_{\bl_k}(A,B) }_{M_{\tau(k)mn}} \le \n{A}_{M_k(M_m)} \n{B}_{M_k(M_n)},

\begin{multline*}
\n{\varphi_{\bl_k}(u,v)} \le \n{(I_k \otimes \phi)u}_{M_k(M_m)} \n{(I_k\otimes \phi)v}_{M_k(M_n)} \\
\le \n{I_k \otimes \phi} \n{u} \n{I_k\otimes \phi} \n{v} \le \n{\phi} \n{\phi}
\end{multline*}

which yields the desired conclusion.

For each $\theta \in [0,1]$, $\odot^\theta$ is an o.s. tensor norm.

This is an immediate consequence of Propositions <ref>, <ref> and <ref>, where for the latter we use the fact that $\n{\odot}_{\jcb} \le 1$ and therefore $\n{\odot^\theta_k}_{\jcb} \le 1$.

Throughout the rest of this paper, whenever we consider any $\lambda$-tensor norm
we will always be assuming that it satisfies (E1), (E2), (E3) and additionally it is an o.s. tensor norm. To emphasize this point, we will call them $\lambda$-o.s. tensor norms.

We have previously pointed out that our notion of uniformity in the definition of o.s. tensor norm is weaker than that of [6].
This has the advantage of allowing the $\odot^\theta$ to be covered by the theory: the only assumptions we are aware of which imply that a $\lambda$-tensor norm satisfies the uniformity condition in [6] are conditions (W1) and (W2) in <cit.>, but $\odot^\theta$ only satisfies (W2) in the extreme cases $\theta=0,1$.

§.§ Some usual notation

For a normed operator space $E$, we denote by $\OFIN(E)$ (resp. $\OCOFIN(E)$) the set of all finite-dimensional (resp. finite-codimensional) subspaces of $E$.
Given $L \in \OCOFIN(E)$, let $q^E_L \colon E \to E/L$ be the canonical projection
and given a subspace $F$ of $E$ let $i^E_F \colon  F \to E$ be the canonical injection. To avoid an overload of notation, when the inclusion is the typical complete isometry for operator spaces $E\subset \mathcal{B}(H)$ we just denote the canonical injection by $i_E:E\to \mathcal B(H)$.
Also, following the usual Banach space notation, for a normed operator space $E$, $\kappa_E \colon  E \to E''$ denotes the canonical injection into the bidual.

Given vector spaces $E$ and $F$, we identify a linear map $T : E \to F'$ with a bilinear map $\beta_T : E \times F \to \C$ via $\beta_T(x,y) = (Tx)(y)$; note that $\beta_T$ can also be identified with an element of $(E \otimes F)'$.

A normed operator space $E$ is an $\mathscr{OS}_{p,C}$ space <cit.>
if there is a family $(F_i)_{i\in I}$ of finite-dimensional subspaces of $E$ whose union is dense in $E$ and such that for every index $i$ there is a natural number $n_i$ such that $d_{\cb}(S_p^{n_i}, F_i) \leq C$ where $d_{\cb}$ stands for the completely bounded Banach-Mazur distance, i.e.,

d_{\cb}(E, F):= \inf\big\{\Vert T \Vert_{cb} \Vert T^{-1} \Vert_{cb} : T \in \CB(E,F) \mbox{ is a complete isomorphism} \big\}.

If $E$ is an $\mathscr{OS}_{p,C'}$ space for every $C'>C$, we say that $E$ is an $\mathscr{OS}_{p,C+}$ space.

If $W$ is a Banach space, its maximal operator space structure $\Max(W)$ is defined by, for $A \in M_n(W)$

\n{A}_{M_n(\Max(W))} = \big\{ \n{  u_n(A) }_{M_n(B(H))} \;:\; \n{ u : W \to B(H) } \le 1 \big\}.

See <cit.> for more details.

§.§ Operator approximation property

A normed operator space $E$ is said to have the operator approximation property (OAP)
<cit.> if there exists a net of finite rank mappings $T_\eta \in\CB(E,E)$ such that the net $id_\mathcal{K} \otimes T_\eta$ converges pointwise to the identity in $K_\infty(E)=\mathcal{K} \otimes_{\min} E$, where $\mathcal{K}$ denotes the compact operators on $\ell_2$.
Similarly, $E$ is said to have the
$C$-completely bounded approximation property ($C$-CBAP) <cit.> if there exists a net of finite rank mappings $T_\eta \in\CB(E,E)$ such that $\n{T_\eta}_{\cb} \le C$ and $\n{T_\eta x - x} \to 0$ for every $x \in E$.
Note that by <cit.>, we also have $\n{(T_\eta)_n x - x} \to 0$ for every $x \in M_n(E)$.
In the particular case $C=1$, we say that $E$ has the completely metric approximation property (CMAP).

§.§ Relevant differences with the Banach space setting

It is well-known that the theory of Banach spaces cannot be painlessly transferred to the operator space framework, and the theory of tensor norms is no exception. Let us point out three important differences in this context.

§.§.§ Local reflexivity

A normed operator space $E$ is said to be  locally reflexive [17]  if for each finite-dimensional operator space $F$, any complete contraction $\varphi: F \to E''$ may be approximated in the point-weak$^*$ topology by a net of complete contractions $\varphi_\eta : F \to E$.
The class of all  locally reflexive normed operator spaces is denoted by OLOC.

Since local reflexivity is relevant to prove many properties related to Banach space tensor norms, it is often an additional hypothesis in the operator space versions of those results.
Moreover, in some cases this is known to be necessary: <cit.> is an example of an identity involving tensor norms whose validity turns out to characterize local reflexivity.
Nevertheless, there are some particular situations where we can omit this assumption (for instance, for $\lambda$-o.s. tensor norms, see Lemma <ref> and Lemma <ref>).

We point out, for future reference, that given a complete operator space $E$, the completely projective space $Z_E$ introduced in Section <ref> is in fact locally reflexive. Indeed,   its dual  $\ell_\infty(\{M_{n_i} : i \in I \})$ is clearly a $C^*$-algebra and thus a  von Neumann algebra, and preduals of von Neumann algebras are locally reflexive  (see e.g., <cit.>).

§.§.§ Exactness

Every finite-dimensional Banach space embeds $(1+\varepsilon)$-isomorphically into a space of the form $\ell_\infty^n$, but the analogous property in the operator space setting does not hold in general:
an operator space $E$ is said to be $C$-exact if for any $\varepsilon>0$ every finite-dimensional subspace of $E$ is $(C+\varepsilon)$-completely isomorphic to a subspace of an $M_n$ space.
When an operator space is $1$-exact we simply say that it is exact.

Remark <ref>  gives an unsurprising example where this causes a significant difference between the classical tensor norm theory and the one for operator spaces.

Another useful  property (for Banach tensor products arguments) which is not transferable to operator spaces due to exactness issues is the one proved in <cit.> about finite dimensional pieces of a quotient:

The following statement is false:

For every complete quotient between Banach
operator spaces $q : E \twoheadrightarrow F$, every $F_0 \in \OFIN(F)$, and every $\varepsilon>0$, there exists $E_0 \in \OFIN(E)$ such that $q(E_0)=F_0$ and for every $y \in M_n(F_0)$ there exists $x \in M_n(E_0)$ such that $q_n(x)=y$ and $\n{x}_{M_n(E_0)} \le (1+\varepsilon) \n{y}_{M_n(F_0)}$.

First let us remark that if $E_0 \in \OFIN(E)$ satisfies the conditions in the statement, then it is clear that $F_0$ is $(1+\varepsilon)$-completely isomorphic to $E_0/\ker(\restr{q}{E_0})$ via the canonical map $E_0/\ker(\restr{q}{E_0}) \to F_0$ induced by $\restr{q}{E_0}$.

Let $F_0$ be a finite-dimensional operator space.
By <cit.>, there exists a complete quotient $q : S_1 \twoheadrightarrow F_0$.
Given $\varepsilon>0$, suppose that there exists $E_0 \in \OFIN(S_1)$
satisfying the conditions in the statement.

Since $E_0 \in \OFIN(S_1)$, we can find $G_0 \in \OFIN(S_1)$ such that $E_0 \subseteq G_0$ and $G_0$ is $(1+\varepsilon)$-completely isomorphic to $S_1^N$ for some $N\in\N$ using perturbation arguments as in <cit.>.
First, take a basis of $E_0$ and approximate it by finitely supported matrices.
This yields a subspace $E_1 \subset S_1$ which is completely isomorphic to $E_0$ and naturally sits inside a subspace $G_1 \subset S_1$ that is completely isometric to $S_1^N$ for some $N$. Perturbing $G_1$ to “put $E_1$ back onto $E_0$” yields the desired $G_0$.

Because $G_0 \supseteq E_0$, it is clear that $G_0$ also satisfies the conditions in the statement.
This gives that $F$ is $(1+\varepsilon)^2$-completely isomorphic to a quotient of $S_1^N$, and therefore $F'$ is $(1+\varepsilon)^2$-completely isomorphic to a subspace of $M_N$.

But this is, in general, false: from <cit.>, for any $n \ge 2$ any complete isomorphism from $\Max(\ell_1^n)$ onto a subspace of an $M_N$ space has constant at least $\frac{n}{2\sqrt{n-1}}$.

As one would expect, this idea of approximating quotients using finite-dimensional pieces is useful when investigating projectivity of Banach space tensor norms.
For example, it is used in <cit.> to show that a tensor norm is projective when its dual tensor norm is injective.
Not only does this argument not work in the operator space setting, but the corresponding result is in fact not true: see Remark <ref> below, where we also point out that a related duality result for completely injective and completely projective hulls claimed without proof in [4] does not hold.

§.§.§ Approximation properties of completely  injective spaces

Each Banach space $E$  isometrically embeds into $\ell_\infty(B_{E'})$, which is an injective space in the Banach space setting. The fact that $\ell_\infty(B_{E'})$ has the approximation property plays a relevant role in many tensor product results in the Banach space framework, especially those related with the Approximation Lemma and completely injective tensor norms.
In contrast, for an operator space $E$ the standard complete isometry into a completely injective space is the one into $\mathcal B(H)$ for some Hilbert space $H$. The lack of the approximation property in $\mathcal B(H)$
is another important dissimilarity between both frameworks.

§ FINITE AND COFINITE HULLS

Given an o.s. tensor norm $\alpha$ on $\OFIN$, we can use the following two procedures to extend it to the class of all operator spaces.

Given operator spaces $E$ and $F$, and $u \in M_n(E \otimes F)$, let

\begin{align*}
\overrightarrow{\alpha}_n(u ; E, F) &= \inf\left\{\alpha_n(u; E_0,F_0) \colon E_0 \in \OFIN(E), F_0 \in\OFIN(F), u \in M_n(E_0 \otimes F_0)\right\}
\end{align*}

\begin{align*}
\overleftarrow{\alpha}_n(u ; E, F) = \sup\left\{\alpha_n\big(  (q_K^E \otimes q_L^F)_n(u); E/K,F/L\big) \colon K \in \OCOFIN(E), L \in\OCOFIN(F) \right\}
\end{align*}

An o.s. tensor norm $\alpha$ on $\ONORM$ is called
finitely-generated if $\alpha = \overrightarrow{\alpha}$,
and cofinitely-generated if $\alpha = \overleftarrow{\alpha}$.

Clearly, if $\alpha \le c \beta$ it follows that $\overleftarrow{\alpha} \le c \overleftarrow{\beta}$
$\overrightarrow{\alpha} \le c \overrightarrow{\beta}$.
Since $\min$ and $h$ are completely injective, $\min = \overrightarrow{\min}$ and $h = \overrightarrow{h}$.
Observe also that the description of the projective norm as an infimum gives $\proj = \overrightarrow{\proj}$.

For the same reason, any $\lambda$-o.s. tensor norm is finitely generated.

Let  $\alpha$ be an o.s. tensor norm on $\OFIN$. Then the finite hull $\overrightarrow{\alpha}$ of $\alpha$
and the cofinite hull $\overleftarrow{\alpha}$ of $\alpha$ are o.s. tensor norms on $\ONORM$ with

\min \le \overleftarrow{\alpha} \le \overrightarrow{\alpha} \le \proj, \qquad  \overleftarrow{\alpha}\vert_{\OFIN} = \overrightarrow{\alpha}\vert_{\OFIN} = \alpha.

If $\alpha$ is defined on $\ONORM$ (and not just on $\OFIN$) then

\overleftarrow{\alpha} \le \alpha \le \overrightarrow{\alpha}.

The complete metric mapping property of $\alpha$ yields $\overleftarrow{\alpha} \le \alpha \le \overrightarrow{\alpha}$;
the remarks before the statement of the Proposition then show $\min \le \overleftarrow{\alpha} \le \overrightarrow{\alpha} \le \proj$.
Since it is obvious that $\alpha$ agrees with $\overleftarrow{\alpha}$ and $\overrightarrow{\alpha}$ on finite-dimensional spaces, the only thing left to prove is that both hulls are in fact o.s. tensor norms.
But we have already proved that they are between $\min$ and $\proj$, so we just have to check the complete metric mapping property.

To that effect, let $S \in \CB(E_1,E_2)$, $T \in \CB(F_1,F_2)$ and $u \in M_n(E_1 \otimes F_1)$.
Then, using the complete metric mapping property of $\alpha$,

\begin{multline*}
\overrightarrow{\alpha}_n\big( (S\otimes T)_n u ; E_2, F_2\big) \\
\begin{aligned}[t]
\inf\Big\{\alpha_n\big((S\otimes T)_n u; E^2_0,F^2_0\big) \colon  E^2_0 \in \OFIN(E_2), F^2_0 \in\OFIN(F_2), (S\otimes T)_nu \in M_n(E^2_0 \otimes F^2_0)\Big\}
\end{aligned}
\\
\le
\begin{aligned}[t]
\inf\Big\{\alpha_n\big((S\otimes T)_nu&; SE^1_0,TF^1_0\big) \colon \\ &E^1_0 \in \OFIN(E_1), F^1_0 \in\OFIN(F_1), u \in M_n(E^1_0 \otimes F^1_0)\Big\}
\end{aligned}
\\
\le \n{S}_{\cb}\n{T}_{\cb}
\begin{aligned}[t]
\inf\Big\{\alpha_n\big(u; &E^1_0,F^1_0\big) \colon \\&E^1_0 \in \OFIN(E_1), F^1_0 \in\OFIN(F_1), u \in M_n(E^1_0 \otimes F^1_0)\Big\}
\end{aligned}
\\
= \n{S}_{\cb}\n{T}_{\cb} \overrightarrow{\alpha}_n\big(  u ; E_1, F_1\big),
\end{multline*}

which shows the complete metric mapping property for $\overrightarrow{\alpha}$.

Now, consider $K^2_0 \in \OCOFIN(E_2)$ and $L^2_0 \in\OCOFIN(F_2)$.
Observe that $S^{-1}K^2_0 \in \OCOFIN(E_1)$ and $T^{-1}L^2_0 \in\OCOFIN(F_1)$.
Moreover, by the basic properties of quotients there exist maps
$S_{K^2_0} : E_1/S^{-1}K^2_0 \to E_2/K^2_0$ and $T_{L^2_0} : F_1/T^{-1}L^2_0 \to F_2/L^2_0$
making the following diagrams commutative

\xymatrix{
E_1 \ar[d]_{ q^{E_1}_{S^{-1}K^2_0} } \ar[r]^{S}  & E_2 \ar[d]^{ q^{E_2}_{K^2_0} }\\
E_1/S^{-1}K^2_0 \ar[r]_{S_{K^2_0}}  & E_2/K^2_0
\qquad \qquad
\xymatrix{
F_1 \ar[d]_{ q^{F_1}_{T^{-1}L^2_0} } \ar[r]^{T}  & F_2 \ar[d]^{ q^{F_2}_{L^2_0} }\\
F_1/T^{-1}L^2_0 \ar[r]_{T_{L^2_0}}  & F_2/L^2_0

and moreover we have
that $\n{S_{K^2_0}}_{\cb} = \n{ q^{E_2}_{K^2_0} S }_{\cb} \le \n{S}_{\cb}$ and $\n{T_{L^2_0}}_{\cb} = \n{ q^{F_2}_{L^2_0} T }_{\cb} \le \n{T}_{\cb}$.
Therefore, once again using the complete metric mapping property of $\alpha$,

\begin{multline*}
\overleftarrow{\alpha}_n\big( (S\otimes T)_n u ; E_2, F_2\big) \\
\begin{aligned}[t]
\sup\Big\{\alpha_n\big(  \big(q^{E_2}_{K^2_0} \otimes q^{F_2}_{L^2_0}\big)_n (S\otimes T)_n (u) &; E_2/K^2_0,F_2/L^2_0\big) \colon \\
&K^2_0 \in \OCOFIN(E_2), L^2_0 \in\OCOFIN(F_2) \Big\}
\end{aligned}
\\
\begin{aligned}[t]
\sup\Big\{\alpha_n\big( (S_{K^2_0}\otimes T_{L^2_0})_n \big(q^{E_1}_{S^{-1}K^2_0} \otimes &q^{F_1}_{T^{-1}L^2_0} \big)_n  (u); E_2/K^2_0,F_2/L^2_0\big) \\
&\colon K^2_0 \in \OCOFIN(E_2), L^2_0 \in\OCOFIN(F_2) \Big\}
\end{aligned}
\\
\le \n{S}_{\cb}\n{T}_{\cb}
\begin{aligned}[t]
\sup\Big\{\alpha_n\big( \big(q^{E_1}_{K^1_0} \otimes q^{F_1}_{L^1_0} \big)_n  &(u); E_1/K^1_0,F_1/L^1_0\big) \colon\\
&K^1_0 \in \OCOFIN(E_1), L^1_0 \in\OCOFIN(F_1) \Big\}
\end{aligned}
\\
= \n{S}_{\cb}\n{T}_{\cb}  \overleftarrow{\alpha}_n\big( u ; E_1, F_1\big),
\end{multline*}

which shows the complete metric mapping property for $\overleftarrow{\alpha}$.

As a consequence of the previous proposition, since we already observed that $\min = \overrightarrow{\min}$ it follows that $\min = \overrightarrow{\min}= \overleftarrow{\min}$.

We will see also in Remark <ref> that the Haagerup o.s. tensor norm satisfies $h = \overrightarrow{h}= \overleftarrow{h}$.

Next we show that, just as in the classical theory, the projective norm is not cofinitely-generated.

* Let $W$ be a Banach space and $F$ a normed operator space. Then ${\Max(W) \otimes_{\overleftarrow{\proj}} F}$ and $W \otimes_{\overleftarrow{\pi}} F$ are isometrically isomorphic as Banach spaces.

* $\overleftarrow{\proj} \neq \proj$.

Let $u \in W \otimes F$. From the definitions,

\begin{multline*}
\overleftarrow{\proj}(u ; \Max(W), F) = \sup\big\{\proj\big(  (q_K^W \otimes q_L^F)(u); \Max(W)/K,F/L\big) \colon \\ K \in \OCOFIN(\Max(W)), L \in\OCOFIN(F) \big\}
\end{multline*}

\begin{align*}
\overleftarrow{\pi}(u ; W, F) =
\sup\left\{\pi\big(  (q_K^W \otimes q_L^F)(u); W/K,F/L\big) \colon K \in \COFIN(W), L \in \COFIN(F) \right\}.
\end{align*}

From <cit.> we have that $\Max(W/K) = \Max(W)/K$.
Therefore, it follows from <cit.> that

\proj\big(  (q_K^W \otimes q_L^F)(u); \Max(W)/K,F/L\big) = \pi\big(  (q_K^W \otimes q_L^F)(u); W/K,F/L\big)

and thus $\overleftarrow{\proj}(u ; \Max(W), F) = \overleftarrow{\pi}(u ; W, F)$.

Since $\overleftarrow{\pi} \neq \pi$, there exist Banach spaces $W$ and $V$ such that $W \otimes_\pi V$ and $W \otimes_{\overleftarrow{\pi}} V$ are different; it then follows from part $(a)$ and <cit.> that
${\Max(W) \otimes_{\overleftarrow{\proj}} \Max(V)}$ and
${\Max(W) \otimes_{\proj} \Max(V)}$ are different (even just as Banach spaces).

The right-finite hull is presented in Definition <ref>. The left-finite hull can be defined analogously.

For $1\le p \le \infty$, the o.s. tensor norms $d_p $ and $g_p $ are finitely generated.

By symmetry, it suffices to prove it for $d_p $.
Observe that the result will follow immediately if we can prove that for any $u \in M_n(E \otimes F)$ we have

\begin{equation}\label{eqn-dp-and-gp-finitely-generated}
(d_p )_n(u; E ,F) = \inf\{ \n{w}_{M_n( (S_{p'}^k \otimes_{\min}E) \otimes_{\proj} S_p^k[F]  )} \;:\; k\in \N, (q^{p,k})_nw = u \}
\end{equation}

where $q^{p,k} : (S_{p'}^k \otimes_{\min}E) \otimes_{\proj} S_p^k[F] \to E \otimes F$ is the tensor contraction.
Indeed, let $\varepsilon>0$ be given and choose $k\in\N$, $w \in M_n( (S_{p'}^k \otimes_{\min}E) \otimes_{\proj} S_p^k[F]  )$ such that $(q^{p,k})_nw = u$ and
$\n{w}_{M_n( (S_{p'}^k \otimes_{\min}E) \otimes_{\proj} S_p^k[F]  )}\le (1+\varepsilon)(d_p )_n(u; E ,F)$.
Since $\proj$ is finitely generated, we can $1+\varepsilon$ approximate the norm of $w$ by passing to finite-dimensional subspaces of $S_{p'}^k \otimes_{\min}E$ and $S_p^k[F]$. Without loss of generality, by enlarging the subspaces if necessary, we can assume that the aforementioned finite-dimensional subspaces are of the form $S_{p'}^k \otimes_{\min}E_0$ and $S_p^k[F_0]$ for some $E_0 \in\OFIN(E), F_0\in\OFIN(F)$, and also that $u \in M_n(E_0\otimes F_0)$.
It now follows that

(d_p )_n(u; E_0 ,F_0 ) \le (1+\varepsilon)^2 (d_p )_n(u; E ,F ),

showing that $d_p $ is finitely generated.

Let us now prove (<ref>).
Recall that the tensor contraction
${q}^p : \big( S_{p'}\widehat{\otimes}_{\min} E \big) \widehat{\otimes}_{\proj} S_p[F] \to E \widehat{\otimes}_{d_p } F$ is a complete 1-quotient.
By dualizing, there is a completely isometric embedding
$(E \otimes_{d_p } F)' \hookrightarrow \CB(S_{p'}\widehat{\otimes}_{\min} E , S_{p'}[F'])$.
Since the union of the spaces $S_{p'}^k \otimes_{\min} E$ is dense in $S_{p'}\widehat{\otimes}_{\min} E$, norms in $\CB(S_{p'}\widehat{\otimes}_{\min} E , S_{p'}[F'])$ (and more generally, norms of matrices over $\CB(S_{p'}\widehat{\otimes}_{\min} E , S_{p'}[F'])$) can be calculated by taking the supremum over the restrictions to $\CB(S_{p'}^k\otimes_{\min} E , S_{p'}^k[F'])$, as $k$ ranges over $\N$.
Thus, the norm of $u$ in $M_n(E \otimes_{d_p } F)$ can be calculated as the supremum of the norms of all pairings of $u$ against a $\varphi$ which is a matrix of norm at most one in $\CB(S_{p'}\widehat{\otimes}_{\min} E , S_{p'}[F'])$, and by the above we can consider instead  $\varphi$ to be a matrix of norm at most one in $\CB(S_{p'}^k \otimes_{\min} E , S_{p'}^k[F'])$, but this is exactly the same as the right-hand side of  (<ref>).

The symmetrized Haagerup o.s. tensor norms $h \cap h^t$ and $h + h^t$ are finitely generated.

Since $h$ and $h^t$ are completely injective, it is clear that so is $h\cap h^t$ and therefore the latter is finitely generated.

Consider operator spaces $E$ and $F$, $u \in M_n(E \otimes F)$, and $\varepsilon>0$.
Let $v,w \in M_n(E \otimes F)$
such that $u = v+w$ and

$$ \|(v,w)\|_{M_n\big( (E \otimes_h F)\oplus_1(E \otimes_{h^t} F) \big)} < (h+h^t)_n(u; E,F) + \varepsilon.$$

Let $E_0 \in \OFIN(E)$ and $F_0 \in\OFIN(F)$ such that $v, w \in M_n(E_0 \otimes F_0)$.
Since $h$ and $h^t$ are both completely injective,
and so are $\ell_1$-sums (this is clear from the description of $\ell_1$-sums in Section <ref> and the injectivity of $\mathcal{B}(H)$),
we have that

\|(v,w)\|_{M_n\big( (E_0 \otimes_h F_0)\oplus_1(E_0 \otimes_{h^t} F_0) \big)} = \|(v,w)\|_{M_n\big( (E \otimes_h F)\oplus_1(E \otimes_{h^t} F) \big)}.

$(h+h^t)_n(u; E_0, F_0) \le \|(v,w)\|_{M_n\big( (E_0 \otimes_h F_0)\oplus_1(E_0 \otimes_{h^t} F_0) \big)}$, combining the inequalities gives
$(h+h^t)_n(u; E_0, F_0) < (h+h^t)_n(u; E,F) + \varepsilon$ which implies that $h+h^t$ is finitely generated.

§ THE FIVE BASIC LEMMAS

In the theory of tensor products  of normed spaces, “The Five Basic Lemmas” (see Section 13 in Defant and Floret’s book [11]) are rather simple results which turn out to be “basic for the
understanding and use of tensor norms”. Namely, they are the Approximation Lemma, the
Extension Lemma, the Embedding Lemma, the Density Lemma and the Local Technique
Lemma.  We present here the
analogous results for the operator space setting and also exhibit some applications as example of
their potential. Our presentation follows the
lines of [11]. Although the proofs are similar to the Banach space case, the operator space nature of
our tensor products introduces some difficulties, as we can see, for example, in our
version of the Extension Lemma <ref>, whose proof involves additional hypothesis of local reflexivity.

§.§ The approximation lemma

Let $\alpha$ and $\beta$ be o.s. tensor norms (on $\ONORM$), $E$ and $F$ normed operator spaces, $c \ge 1$ and

\alpha \le c \beta \on E \otimes N

for cofinally many $N \in \OFIN(F)$.
If $F$ has the completely bounded approximation property with constant $C \ge 1$, then

\alpha \le  C c\beta \text{ on } E \otimes F.

Fix $n\in\N$ and $z\in M_n(E \otimes F)$, and take $\varepsilon >0$.
Since $F$ has the $C$-completely bounded approximation property,
there is a net $(T_{\eta})_\eta$ of finite rank operators in $\CB(F,F)$ with $\cb$-norm bounded by $C$ such that
$\n{(T_\eta)_n x - x} \to 0$ for every $x \in M_n(E)$.
Therefore $(id_E \otimes T_\eta)_n(z)$ converges to $z$ in the projective norm, and thus also in the  $\alpha$ norm.
Hence we have $\alpha_n(z-(id_E \otimes T_\eta)_n(z) ; E,F )<\varepsilon$ for some $\eta$ large enough.
If we take a subspace $N$ containing $T_\eta(E)$ satisfying the hypothesis of the lemma, by the complete metric mapping property of the o.s. tensor norm $\beta$ we have

\begin{align*}
\alpha_n(z;E,F) & \leq  \alpha_n(z-(id_E \otimes T_\eta)_n(z) ; E,F ) + \alpha_n((id_E \otimes T_\eta)_n(z) ; E,F ) \\
& \leq \varepsilon + \alpha_n((id_E \otimes T_\eta)_n(z) ; E,N ) \\
& \leq \varepsilon + c \beta_n((id_E \otimes T_\eta)_n(z) ; E,N ) \\
& \leq \varepsilon + c \n{T_\eta: E \to N }_{\cb} \beta_n(z; E,F) \\
& \leq \varepsilon + C c  \beta_n(z; E,F).
\end{align*}

Since this holds for every $\varepsilon > 0$,  we have $\alpha_n(z;E,F) \leq  C c  \beta_n(z; E,F).$

By Proposition <ref> and the previous lemma we obtain the following.

Let $\alpha$ be an o.s. tensor norm (on $\OFIN$), $E$ and $F$ normed operator spaces with the completely bounded approximation property with constants $C_E$ and $C_F$, respectively. Then

\overleftarrow{\alpha} \le  \overrightarrow{\alpha} \le C_E C_F \overleftarrow{\alpha} \on E \otimes F.

In particular, if $E$ and $F$ both have the completely metric approximation property then $\overleftarrow{\alpha} =  \overrightarrow{\alpha} $ on $E\otimes F$.

§.§ The extension lemma

Every $\varphi \in (E\otimes_{\proj} F)' = \CB(E, F')$ has a canonical extension $\varphi^\wedge \in (E \otimes_{\proj} F'')' = \CB(E, F'')$ which satisfies for every $x\in E$ and $y''\in F''$ the relation

\pair{\varphi^\wedge}{x \otimes y''} = \pair{L_\varphi x}{y''}_{F',F''},

where $L_\varphi \in \CB(E,F')$ is the linear map associated with $\varphi$.
The extension lemma deals with this situation for more general o.s. tensor norms.
Unlike in the Banach space case, local reflexivity does not come for free and we are forced to require it. The item $(a)$ of the following statement previously appeared  in [13]. We include the proof here for the reader's benefit. Also,  the proof of item $(b)$ is inspired by <cit.>) which presents the same argument just for the norm $\proj$.

(Right extension lemma)
Let $\alpha$ be a  o.s. tensor norm (on $\ONORM$), $E$ and $F$  be normed operator spaces and let $\varphi \in (E\otimes_{\proj} F)'$. Then

\varphi \in (E\otimes_{\alpha} F)' \text{ if and only if } \varphi^\wedge \in (E \otimes_\alpha F'')'

and the map $\varphi \mapsto \varphi^\wedge$ is a complete isometry from $(E\otimes_{\alpha} F)'$ to $(E \otimes_{\alpha} F'')'$ in the following cases:

* $\alpha$ is finitely-generated and $F$ is locally reflexive.

* $\alpha$ is a $\lambda$-o.s. tensor norm.

The complete metric mapping property implies that $id_E\otimes \kappa_F:E \otimes_\alpha F \hookrightarrow E \otimes_\alpha F''$ is a complete contraction,
and hence so is the restriction map $(id_E\otimes \kappa_F)':(E \otimes_{\alpha} F'')' \to (E\otimes_{\alpha} F)'$.
This shows that for any $(\varphi_{ij}) \in M_m\big( (E\otimes_{\proj} F)' \big)$,

\n{ (\varphi_{ij}) }_{ M_m( (E\otimes_{\alpha} F)' ) } \le \n{ (\varphi^\wedge_{ij}) }_{ M_m( (E\otimes_{\alpha} F'')' ) }.

$(a)$	Now consider $n\in\N$, $z_0 \in M_n(E \otimes F'')$, $M \in \OFIN(E)$ and $N \in \OFIN(F'')$ such that $z_0 \in M_n(M \otimes N)$.
Since $F$ is locally reflexive, by <cit.>
given $\varepsilon > 0$ there exists $R \in \CB(N,F)$ with $\n{R}_{\cb} \le 1 + \varepsilon$ such that for every $y'' \in N$, $x \in M$ and $1 \le i, j \le m$

\pair{y''}{L_{\varphi_{ij}} x}_{F'',F'} = \pair{Ry''}{L_{\varphi_{ij}} x}_{F,F'}.

This means that

\pair{\varphi_{ij}^\wedge}{x \otimes y''} = \pair{\varphi_{ij}}{(id_E \otimes R)(x \otimes y'')},

\mpair{(\varphi_{ij}^\wedge)}{z_0} = \mpair{(\varphi_{ij})}{(id_E \otimes R)z_0}

and hence

\begin{align*}
\n{ \mpair{(\varphi_{ij}^\wedge)}{z_0} }_{M_{mn}} &\le \n{ (\varphi_{ij}) }_{ M_m( (E\otimes_{\alpha} F)' ) } \alpha_n((id_E \otimes R)z_0; E,  F) \\
\n{ (\varphi_{ij}) }_{ M_m( (E\otimes_{\alpha} F)' ) }
\n{id_E \otimes R}_{\cb} \alpha_n(z_0 ; E, N) \\
\n{ (\varphi_{ij}) }_{ M_m( (E\otimes_{\alpha} F)' ) }
\n{R}_{\cb} \alpha_n(z_0 ; E, N) \\
\n{ (\varphi_{ij}) }_{ M_m( (E\otimes_{\alpha} F)' ) }
(1+\varepsilon) \alpha_n(z_0 ; E, N)
\end{align*}

which implies the result since $\alpha$ is finitely-generated.

$(b)$ Recall that by Theorem <ref> we can identify $(E\otimes_{\lambda} F)' = \CB_\lambda(E \times F)$, so
all we need to show is that the map $\varphi \mapsto \varphi^\wedge$ gives a completely isometric embedding $\CB_\lambda(E \times F) \to \CB_\lambda(E \times F'')$. Notice that at the beginning of the proof it has already been shown that this is a complete contraction. That we get a complete isometry is then clear from the definition of $\n{\cdot}_{\cb,\lambda}$ and the fact that the unit ball of $M_k(F)$ is weak$^*$-dense in that of $M_k(F'')=M_k(F)''$ by Goldstine's theorem.

In a similar way a left extension for each $\varphi \in (E\otimes_{\proj} F)'$ is defined: let  $^\wedge\varphi \in (E'' \otimes_\alpha F)'$  given by, for $x'' \in E''$ and $y\in F$,

\pair{^\wedge\varphi}{x'' \otimes y} = \pair{x''} { R_\varphi y}_{E'',E^{'}},
where $R_\varphi\in\CB(F,E') $ is the mapping $R_\varphi (y)(x)=\langle\varphi,x\otimes y\rangle$.

(Left extension lemma)
Let $\alpha$ be a  o.s. tensor norm (on $\ONORM$), $E$ and $F$  be normed operator spaces and let $\varphi \in (E\otimes_{\proj} F)'$. Then

\varphi \in (E\otimes_{\alpha} F)' \text{ if and only if } \ ^\wedge\varphi \in (E'' \otimes_\alpha F)'.

and the map $\varphi \mapsto \ ^\wedge\varphi$ is a complete isometry from $(E\otimes_{\alpha} F)'$ to $(E'' \otimes_{\alpha} F)'$ in the following cases:

* $\alpha$ is finitely-generated and $E$ is locally reflexive.

* $\alpha$ is a $\lambda$-o.s. tensor norm.

Now, both procedures can be applied to a given $\varphi \in (E\otimes_{\alpha} F)'$ to obtain two (possibly different) extensions to $(E''\otimes_{\alpha} F'')'$.

Let $\alpha$ be a  o.s. tensor norm (on $\ONORM$), $E$ and $F$  be normed operator spaces and let $\varphi \in (E\otimes_{\proj} F)'$. Then the following maps from $ (E\otimes_{\alpha} F)'$ to $ (E''\otimes_{\alpha} F'')'$ are complete isometries:

\varphi\mapsto \ ^\wedge(\varphi^\wedge)\quad \textrm{ and } \quad \varphi\mapsto (^\wedge\varphi)^\wedge
in the following cases:

* $\alpha$ is finitely-generated and $E$ and $F$ are locally reflexive.

* $\alpha$ is a $\lambda$-o.s. tensor norm.

As we have already mentioned, the fact that we do not need local reflexivity in the extension lemmas if the o.s. tensor norm involved is $\proj$ was proved in <cit.>.
In the same article this fact is also shown  for  the Haagerup o.s. tensor norm $h$ using that it satisfies the following property: for any operator spaces $E$ and $F$ and $\varphi \in (E \otimes_h F)'$, there is a unique extension $\widetilde\varphi \in (E'' \otimes_h F'')'$ which is separately $w^*$-continuous, see <cit.>.

A word can be said about the equality/non-equality of the two extensions. Repeating the arguments of <cit.> we obtain the analogous result in our setting. That is, given $\varphi \in (E\otimes_{\proj} F)'$, we have $ ^\wedge(\varphi^\wedge)=(^\wedge\varphi)^\wedge$ if and only if $L_\varphi:E\to F'$ is weakly compact. This is also equivalent to the existence of an extension of $\varphi$ to $ (E''\otimes_{\proj} F'')'$ that is separately $w^*$-continuous. Note that, by the comment above, if $\varphi \in (E \otimes_h F)'$ then $L_\varphi$ is weakly compact.

Note that we have the separating duality pair $\big\langle E \otimes F'', (E\otimes_{\alpha}F)' \big\rangle$ given by

$$ \langle z, \varphi \rangle = \varphi^{\wedge}(z).$$

Using the bipolar theorem it is easy to see that, in the Banach space realm, the unit ball
$B_{E\otimes_{\alpha}F}$ is $\sigma(E\otimes F'',(E\otimes_{\alpha}F)')$-dense in the unit ball $B_{E \otimes_\alpha F''}$, whenever $\alpha$ is a finitely generated tensor norm.
In other words, given $z \in B_{E \otimes\alpha F''}$ there is a net $(u_{\eta})_{\eta} \subset B_{E\otimes_{\alpha}F}$ such that for every $\varphi \in (E\otimes_{\alpha}F)'$,

$$\varphi(u_{\eta}) \to \varphi^{\wedge}(z).$$

In our setting, we can consider for each $n \in \mathbb{N}$, the following duality pair:

$$ \big\langle M_n(E \otimes F''),S_1^n[(E\otimes_{\alpha}F)'] \big\rangle$$
defined as

$$ \langle (z_{ij}),(\varphi_{ij}) \rangle = \sum_{i,j} \varphi_{ij}^{\wedge}(z_{ij}).$$

It is easy to see that this dual pairing system is separating.
Moreover, we have the following density result.

Let $\alpha$ be a  o.s. tensor norm (on $\ONORM$), $E$ and $F$  be normed operator spaces.

For every $n\in \mathbb{N}$, the unit ball $B_{M_n(E\otimes_{\alpha} F)}$ is $\sigma(M_n(E \otimes F''),S_1^n[(E\otimes_{\alpha}F)'])$-dense in the unit ball $B_{M_n(E\otimes_{\alpha} F'')}$
in the following cases:

* $\alpha$ is finitely-generated and $F$ is locally reflexive.

* $\alpha$ is a $\lambda$-o.s. tensor norm.

Since $\left(M_n(E\otimes_{\alpha} F)\right)'$ is $S_1^n[(E\otimes_{\alpha} F)']$ isometrically, it is clear that the polar  $\left(B_{M_n(E\otimes_{\alpha} F)}\right)^{\circ}$ (in this pairing system) is $B_{S_1^n[(E\otimes_{\alpha} F)']}.$
By the bipolar theorem it is enough to prove that $B_{M_n(E\otimes_{\alpha} F'')} \subset \left(B_{S_1^n[(E\otimes_{\alpha} F)']}\right)^{\circ}.$
Indeed, given $(z_{ij}) \in B_{M_n(E\otimes_{\alpha} F'')}$ and $(\varphi_{ij})\in B_{S_1^n[(E\otimes_{\alpha} F)']}$; by the Extension Lemma <ref> along with <cit.> we know that $(\varphi_{ij}^{\wedge})\in B_{S_1^n[(E\otimes_{\alpha} F'')']}$. Using again
$\left(M_n(E\otimes_{\alpha} F'')\right)'=S_1^n[(E\otimes_{\alpha} F'')']$ isometrically, we derive that that

$$ \left|\langle (z_{ij}),(\varphi_{ij}) \rangle\right| = \left|\sum_{i,j} \varphi_{ij}^{\wedge}(z_{ij})\right| \leq 1.$$
This concludes the proof.

§.§ The embedding lemma

Operator space tensor norms generally do not respect subspaces, but the embedding into the bidual is respected under certain conditions.

Let $\alpha$ be an o.s. tensor norm (on $\ONORM$).
The mapping

id_E \otimes \kappa_F : E \otimes_\alpha F \to E \otimes_\alpha F''

is a complete isometry in the following cases:

* $\alpha$ is finitely-generated and $F$ is locally reflexive.

* $\alpha$ is cofinitely-generated.

* $\alpha$ is a $\lambda$-o.s. tensor norm.

Let $n\in\N$ and $z \in M_n(E \otimes_\alpha F)$.
By the complete metric mapping property

id_E \otimes \kappa_F : E \otimes_\alpha F \to E \otimes_\alpha F''

is always a complete contraction, so

\alpha_n( z; E, F'') \le \alpha_n( z; E, F).

$(a)$ By the extension lemma,

\begin{align*}
\alpha_n(z ; E, F) &= \sup \big\{ \n{\mpair{z}{ \Phi }}_{M_{nm}} \; : \; \Phi \in M_m((E \otimes_\alpha F )'),  \n{\Phi}_{M_m( (E \otimes_\alpha F )')} \le 1 \big\} \\
&= \sup \big\{ \n{\mpair{z}{ \Phi^\wedge }}_{M_{nm}} \; : \; \Phi \in M_m((E \otimes_\alpha F )'),  \n{\Phi}_{M_m( (E \otimes_\alpha F )')} \le 1 \big\} \\
&\le \sup \big\{ \n{\mpair{z}{ \Psi }}_{M_{nm}} \; : \; \Psi \in M_m((E \otimes_\alpha F'' )'),  \n{\Psi}_{M_m( (E \otimes_\alpha F'' )')} \le 1 \big\} \\
&= \alpha_n(z ; E ,F''),
\end{align*}

which gives the reverse inequality.

$(b)$ Let $L \in \OCOFIN(F)$, then $L^{00}$ (formed in $F''$) is in $\OCOFIN(F'')$ and the map

\kappa_{F/L} : F/L \to (F/L)'' = F''/L^{00}

is completely isometric and surjective; moreover, $q^{F''}_{L^{00}} \circ \kappa_F = \kappa_{F/L} \circ q^F_L$.

\begin{align*}
\alpha_n\big((id_E \otimes q_L^F)_n(z); E,F/L\big)
&= \alpha_n\big((id_E \otimes (\kappa_{F/L} \circ q^F_L) )_n(z); E,(F/L)''\big) \\
&= \alpha_n\big((id_E \otimes q^{F''}_{L^{00}}) \circ (id_E \otimes \kappa_F) )_n(z); E, F''/L^{00} \big) \\
&\le \overleftarrow{\alpha}_n( (id_E \otimes \kappa_F) )_n(z); E, F'' ) \\
&=  \overleftarrow{\alpha}_n( z; E, F'' ) = \alpha_n(z ; E, F'')
\end{align*}

The same argument with $E/K$ instead of $E$ allows us to conclude $\alpha_n(z ; E, F) \le \alpha_n(z ; E, F'')$.

$(c)$ The same proof as in the case $(a)$ works here because a $\lambda$-o.s. tensor norm is finitely-generated and the hypothesis of local reflexivity of $F$ is not needed for the extension lemma with this norm.

We point out that part (c) above is a  generalization of <cit.>, where the spaces are assumed to be $C^*$-algebras and the embedding is only shown to be an isometry instead of a complete isometry.

Note that the completion of an operator space $F$ has the same bidual as $F$. Therefore, as a consequence of the previous result we have:

Let $\alpha$ be an o.s. tensor norm (on $\ONORM$) and let $\widetilde F$ the completion of a normed operator space $F$.
The mapping

id_E \otimes \iota_F : E \otimes_\alpha F \to E \otimes_\alpha \widetilde F,

where $\iota_F : F \to \widetilde F$ is the canonical inclusion,
is a complete isometry with dense range in the following cases:

* $\alpha$ is finitely-generated and $F$ is locally reflexive.

* $\alpha$ is cofinitely-generated.

* $\alpha$ is a $\lambda$-o.s. tensor norm.

§.§ The density lemma

We now state the so-called density lemma, which asserts that a given completely bounded mapping defined on the tensor product of certain normed operator spaces can be extended to a completely bounded mapping defined on the tensor product of the completions of the spaces. As usual, we need a local reflexivity hypothesis for the finitely-generated case. The proof easily follows from Corollary <ref>.

Let $\alpha$ be an o.s. tensor norm  (on $\ONORM$), $E, F$ and $G$ normed operator spaces, $E_0$ and $F_0$ dense subspaces of $E$ and $F$ respectively. Suppose $T \in CB(E \otimes_{\proj}F; G)$ such that $T|_{E_0 \otimes_{\alpha}F_0} \in CB(E_0 \otimes_{\alpha}F_0; G)$.  Then $T \in CB(E \otimes_{\alpha}F; G)$ and

\Vert T \Vert_{ CB(E_0 \otimes_{\alpha}F_0; G)}=\Vert T \Vert_{ CB(E \otimes_{\alpha}F; G)},

in the following cases:

* $\alpha$ is finitely-generated and $F$ is locally reflexive.

* $\alpha$ is cofinitely-generated.

* $\alpha$ is a $\lambda$-o.s. tensor norm.

§.§ The $\mathscr{OS}_{p}$-local technique lemma

In the Banach space setting the $\mathscr L_p$-local technique lemma allow us to transfer results from tensor products of $\ell_p^n$ spaces to general $\mathscr L_p$-spaces (which, loosely speaking, are space whose finite dimensional subspaces “uniformly look”  like $\ell_p^n$'s).
In the operator space setting, one of the possible extension of the $\mathscr L_p$-spaces are the $\mathscr{OS}_{p}$-spaces  (see Section <ref> for the definition).
For this extension, we have the following local technique lemma.

Let $\alpha$ and $\beta$ be o.s. tensor norms, $c \ge 0$ and $E$ a normed operator space such that for all $k \in \N$,

\alpha \le c \beta \on E \otimes S_p^k.

Then $\alpha \le C c \overrightarrow{\beta} \on E \otimes F$
for every $\mathscr{OS}_{p,C}$-space $F$.

For $N \in \OFIN(F)$, by <cit.> we can take a factorization

\xymatrix{
N \ar[rr]^{id_N} \ar[dr]_{R} & &F\\
& S_p^k \ar[ru]_{S} &

with $\n{R}_{\cb} \n{S}_{\cb} \le C(1+\varepsilon)$. Let $M := S(S_p^k)$.
Then for every $z \in M_n(E \otimes N)$,

\begin{multline*}
\alpha_n(z ; E, M) = \alpha_n( (id_E \otimes (S \circ R))_n z; E , M )
\le \n{S}_{\cb} \alpha_n( (id_E \otimes R)_nz ; E, S_p^k ) \\
\le \n{S}_{\cb} c \beta_n( (id_E \otimes R)_n z ; E , S_p^k )
\le \n{S}_{\cb} \n{R}_{\cb} c \beta_n(z ; E , N).
\end{multline*}

The statement follows.

Note that more generally the proof above applies whenever $F$ has the $\gamma_p$-AP <cit.>, thanks to <cit.>.
Also note that the same argument yields versions of the local technique lemma for other operator space versions of the $\mathscr{L}_p$ spaces, e.g. the $\mathscr{OL}_p$ spaces from [25].

§ DUAL O.S. TENSOR NORMS

In a nutshell, the dual o.s. tensor norm  $\alpha'$ for the o.s. tensor norm  $\alpha$ is the one that makes

E \otimes_{\alpha'} F = (E' \otimes_\alpha F')'

into a complete isometry for any $E,F \in \OFIN$.
More explicitly, when $E,F \in\OFIN$ for $z \in M_n( E \otimes F)$ we define

\alpha'_n(z ; E ,F) := \sup \big\{  \n{ \mpair{u}{z} }_{M_{mn}}  \mid \alpha_m(u ; E' , F') \le 1 \big\}.

Let us check that, for example, the minimal and projective o.s. tensor norms  are in duality with each other.
Consider finite-dimensional operator spaces $E$ and $F$. The operator space structure on $E \otimes_{\min} F$ is, by definition, the one induced by the embedding $E \otimes_{\min} F \to\CB(E',F)$. Now, $(E' \otimes_{\proj} F')'$ is completely isometric to $\CB(E', F'') = \CB(E',F)$, from where we conclude that the canonical identification

E \otimes_{\min} F = (E' \otimes_{\proj} F')'

is a complete isometry. Taking duals and exchanging the roles of $E,F$ and $E',F'$ we also have a complete isometry

E \otimes_{\proj} F = (E' \otimes_{\min} F')'.

If $\alpha$ is an o.s. tensor norm on $\OFIN$, then so is $\alpha'$.

Let $E, F$ be finite-dimensional operator spaces.
It is clear that $\alpha'$ is an operator space structure on $E \otimes F$.
From $\min \le \alpha \le \proj$, taking duals we conclude $\min = \proj' \le \alpha' \le \min'=\proj$ (for finite-dimensional spaces).
The same sort of duality argument shows that $\alpha'$ is uniform.

The finite hull $\overrightarrow{\alpha'}$ of $\alpha'$ will be called the dual o.s. tensor norm $\alpha'$ (on $\ONORM$) of the o.s. tensor norm  $\alpha$ (on $\OFIN$ or $\ONORM$).

The following properties are easy to check:

* If $\alpha \le c \beta$, then $\beta' \le c \alpha'$.

* $\alpha = \alpha''$ on $\OFIN$, and $\overrightarrow{\alpha}= \alpha''$.

* $\alpha = \alpha''$ on $\ONORM$ if and only if $\alpha$ is finitely-generated.

Since $\min$ and $\proj$ are finitely-generated, $\min'=\proj$ and $\proj'=\min$. Also, $h'=h$ (see, for instance, <cit.>).
Additionally, the symmetrized Haagerup o.s. tensor norms are in duality with each other.

$(h \cap h^t)' = h + h^t$ and $(h+h^t)' = h \cap h^t$.

By Proposition <ref> both $h \cap h^t$ and $h+h^t$ are finitely-generated, so it suffices to check that they are in duality for finite-dimensional spaces. Let $E, F \in \OFIN$. By taking the adjoint of the canonical complete isometry $E \otimes_{h \cap h^t} F \hookrightarrow (E \otimes_h F) \oplus_\infty (E \otimes_{h^t} F)$ we get a canonical complete quotient
$(E \otimes_h F)' \oplus_1 (E \otimes_{h^t} F)' \twoheadrightarrow (E \otimes_{h \cap h^t} F)'$.
But the self-duality of $h$ and $h^t$ implies $(E \otimes_h F)' = E' \otimes_h F'$ and $(E \otimes_{h^t} F)' = E' \otimes_{h^t} F'$, so we get a complete quotient
$(E' \otimes_h F') \oplus_1 (E' \otimes_{h^t} F') \twoheadrightarrow (E \otimes_{h \cap h^t} F)'$. This is exactly the same quotient giving the o.s. structure to $E' \otimes_{h+h^t} F'$, and thus $E' \otimes_{h+h^t} F' = (E \otimes_{h \cap h^t} F)'$ as claimed.

Given an o.s. tensor norm  $\alpha$, its adjoint or contragradient o.s. tensor norm is defined as $\alpha^* = (\alpha^t)' = (\alpha')^t$.

As in the classical case, the following theorem tells us in what sense the completely isometric embedding

E \otimes_{\alpha'} F \to (E' \otimes_\alpha F')',

valid for finite-dimensional spaces, extends to infinite-dimensional ones.

Let $\alpha$ be an o.s. tensor norm  (on $\OFIN$) and let $E$, $F$ be normed operator spaces. The following natural mappings are complete isometries:

\begin{align}
\label{eqn-duality-3}E \otimes_{\overleftarrow{\alpha}} F  &\hookrightarrow (E' \otimes_{\alpha'} F')'
\\
\label{eqn-duality-2}E' \otimes_{\overleftarrow{\alpha}} F  &\hookrightarrow (E \otimes_{\alpha'} F')' \qquad \textrm{ whenever } E \textrm{ is locally reflexive}\\
\label{eqn-duality-1}E' \otimes_{\overleftarrow{\alpha}} F'  &\hookrightarrow (E \otimes_{\alpha'} F)' \qquad \textrm{ whenever } E \textrm{ and } F \textrm{ are locally reflexive}
\end{align}

To prove (<ref>), observe first that

\OFIN(E') = \{ K^0 \mid K \in \OCOFIN(E) \}

and that for $(K,L) \in \OCOFIN(E) \times \OCOFIN(F)$, $z \in M_n(E \otimes F)$ and $u \in M_m(K^0 \otimes L^0 ) \subset M_m(E' \otimes F' )$,
we have

\mpair{z}{u} = \mpair{ (q_K^E \otimes q_L^F)(z) }{u}

Now, by the valid duality relation for finite-dimensional spaces,

\begin{align*}
\overleftarrow{\alpha}_n(z ; E, F) &= \sup_{K,L}\left\{\alpha_n\big(  (q_K^E \otimes q_L^F)(z); E/K,F/L\big)  \right\}\\
&= \sup_{K,L} \sup \left\{ \n{  \mpair{(q_K^E \otimes q_L^F)(z)}{ u} }_{M_{mn}} \mid \alpha'_m(u ; K^0, L^0) <1 \right\} \\
&= \sup \left\{ \n{  \mpair{z}{ u} }_{M_{mn}} \mid \overrightarrow{\alpha'}_m(u ; E', F') <1 \right\} \\
\end{align*}

and this is precisely (<ref>).

Consider now the commutative diagram

\xymatrix{
E' \otimes_{\overleftarrow{\alpha}} F\ar[r]^{\phi_1} \ar[dr]  & (E'' \otimes_{\alpha'} F')'\\
& (E \otimes_{\alpha'} F')' \ar[u]_{\phi_2}  \\

Note that $\phi_1$ is a complete isometry by (<ref>), whereas $\phi_2$ is a complete isometry by
the Extension Lemma <ref>; (<ref>) follows.
In the same way, in the diagram

\xymatrix{
E' \otimes_{\overleftarrow{\alpha}} F' \ar[r]^{\psi_1} \ar[dr]  & (E \otimes_{\alpha'} F'')'\\
& (E \otimes_{\alpha'} F)' \ar[u]_{\psi_2}  \\

$\psi_1$ is a complete isometry by (<ref>) and $\psi_2$ is a complete isometry the Extension Lemma <ref>; (<ref>) follows.

The Extension Lemma <ref> is valid without the hypothesis of local reflexivity for $\lambda$-o.s. tensor norms.
Hence, in equations (<ref>) and (<ref>) of the Duality Theorem, if $\alpha'$ is a $\lambda$-tensor norm the local reflexivity of the involved spaces is not needed.

In particular, for $\alpha'=\proj$, recalling that $\overleftarrow{\min}=\min$  we obtain for any normed operator spaces $E$ and $F$  the following known completely isometric embeddings

E' \otimes_{\min} F  \hookrightarrow (E \otimes_{\proj} F')'\quad \textrm{ and }\quad E' \otimes_{\min} F'  \hookrightarrow (E \otimes_{\proj} F)'.

One nice consequence of the duality theorem is the following:

If $E$ and $F$ are normed operator spaces with the completely metric approximation property,

then the natural mappings

E \otimes_\alpha F \to E\otimes_{\overleftarrow{\alpha}} F \qquad \text{and} \qquad E \otimes_{\alpha} F \to (E' \otimes_{\alpha'} F')'

are complete isometries.

Proposition <ref> and Corollary <ref> imply the complete isometry between  $E \otimes_\alpha F$ and $E\otimes_{\overleftarrow{\alpha}} F$, now  (<ref>) implies the rest.

Observe that if we replace the CMAP property by the CBAP property the above identifications become complete isomorphisms (with norms controlled by the CBAP constant).

We now show that the Haagerup o.s. tensor norm  is both finitely and cofinitely-generated.

Let $h$ be the Haagerup o.s. tensor norm. Then,

The fact that $h=\overrightarrow{h}$ easily follows since the Haagerup o.s. tensor norm respects complete isometries <cit.>.

To see that $h=\overleftarrow{h}$ notice that by the Duality Theorem <ref> the canonical embedding below
is a complete isometry

\begin{equation}\label{inclusion h1}
E \otimes_{\overleftarrow{h}} F  \hookrightarrow (E' \otimes_{h'} F')'=(E' \otimes_{h} F')',
\end{equation}

where we have used in the last equality the self-dual property of the o.s. tensor norm  $h$  <cit.>.

On the other hand,  by <cit.> and the Embedding Lemma <ref> we have that

\begin{equation}\label{inclusion h2}
E \otimes_{h} F  \hookrightarrow  E'' \otimes_{h} F'' \hookrightarrow (E' \otimes_{h} F')',
\end{equation}

is a complete isometry.

Since both inclusions in equations (<ref>), (<ref>) are complete isometries, $h$ and $\overleftarrow{h}$ must coincide as o.s. tensor norms.

In general, o.s. tensor norms are not both finitely and cofinitely-generated, as we have just seen for the Haagerup o.s. tensor norm and the minimal o.s. tensor norm.
For many purposes it is often enough that the equality $\overrightarrow{\alpha}=\overleftarrow{\alpha}$
holds just for finite dimensional spaces or locally reflexive spaces. This motivates the following definitions that we will use in the sequel.

An o.s. tensor norm  $\alpha$ is called right-accessible if

\overrightarrow{\alpha}_n(\cdot ; M, F) = \overleftarrow{\alpha}_n(\cdot ; M, F)  \qquad \text{for all }  M \in \OFIN, F \in \ONORM, n\in \N

and left-accessible if

\overrightarrow{\alpha}_n(\cdot ; E, M) = \overleftarrow{\alpha}_n(\cdot ; E, M)  \qquad \text{for all }  M \in \OFIN, E \in \ONORM, n \in \N.

An o.s. tensor norm that is both left and right-accessible is called accessible, and totally accessible means that $\overrightarrow{\alpha} = \overleftarrow{\alpha}$.

In many cases, local reflexivity  naturally appears when dealing with accessibility. Based on this we introduce the following:

An o.s. tensor norm $\alpha$ is called locally right-accessible if

\overrightarrow{\alpha}_n(\cdot ; M, F) = \overleftarrow{\alpha}_n(\cdot ; M, F)  \qquad \text{for all }  M \in \OFIN, F \in \OLOC, n\in \N

and locally left-accessible if

\overrightarrow{\alpha}_n(\cdot ; E, M) = \overleftarrow{\alpha}_n(\cdot ; E, M)  \qquad \text{for all }  M \in \OFIN,E \in \OLOC, n \in \N.

An o.s. tensor norm that is both locally left and locally right-accessible is called locally accessible, and locally totally accessible means that $\overrightarrow{\alpha} = \overleftarrow{\alpha}$ in $\OLOC \otimes \OLOC$.

We have already seen that the minimal and the Haagerup o.s. tensor norms are both finitely and cofinitely-generated, or in other words they are totally accessible.
Recall that, as we saw in Proposition <ref>, the projective o.s. tensor norm  is not cofinitely-generated, hence it is not totally accessible. Nevertheless, as in the Banach space framework, $\proj$ is indeed accessible.
To see this, let  $E\in \ONORM$ and $M \in \OFIN$. We always have that the natural mapping

\begin{equation} \label{inclusion natural proj}
E \otimes_{\proj}M \to (E' \otimes_{\min}M')'
\end{equation}

is a complete isometry (see <cit.> or Remark <ref>). On the other hand, as a consequence of the Duality Theorem <ref> we also have that the mapping

E \otimes_{\overleftarrow{\proj}}M \to (E' \otimes_{\min}M')'

is a complete isometry. Thus, both o.s. tensor norms, $\overrightarrow{\proj}=\proj$ and $\overleftarrow{\proj}$, must coincide on $E \otimes M$. This shows the left-accessibility of $\proj$. Analogously, we have that $\proj$ is also right-accessible and therefore accessible.

More generally, a similar argument will show that any $\lambda$-o.s. tensor norm is accessible.
Let us start with a formula for calculating the dual of a $\lambda$-o.s. tensor norm, which is essentially <cit.>.

Let $E$ and $F$ be operator spaces, and $u \in M_n(E \otimes F)$. For any $\lambda$-o.s. tensor norm,

\lambda'_n(u; E,F) = \sup\Big\{ \n{ \big( \otimes_{\bl_k}(\phi,\psi)\big)_n(u) }_{M_{n\tau(k)}} \;:\; k\in\N,  \n{\phi}_{M_k(E')}, \n{\psi}_{M_k(F')} \le 1 \Big\}

Assume first that $E$ and $F$ are finite-dimensional.
By the definition of $\lambda'$ and Theorem <ref>,

E \otimes_{\lambda'}F = (E' \otimes_\lambda F')' = \CB_\lambda(E'\times F').

Therefore, $\lambda'_n(u; E,F)$ is the norm $\n{\Phi_u}_{\cb,\lambda}$ of the bilinear map $\Phi_u : E' \times F' \to M_n$ associated to $u$.
By definition, said norm is

\n{\Phi_u}_{\cb,\lambda} = \sup_{k\in\N} \big\{\n{(\Phi_u)_{\bl_k}(\phi,\psi)}_{M_{n\tau(k)}} \;:\; \n{\phi}_{M_k(E')}, \n{\psi}_{M_k(F')} \le 1 \},

so it suffices to check

\big( \otimes_{\bl_k}(\phi,\psi)\big)_n(u) = (\Phi_u)_{\bl_k}(\phi,\psi)

which follows easily from the definitions.

Let us now consider the case of arbitrary $E$ and $F$.
Let $E_0 \in \OFIN(E)$ and $F_0 \in \OFIN(F)$ such that
$u \in M_n(E_0 \otimes F_0)$.
By the previous argument,

\lambda'_n(u; E_0,F_0) = \sup\Big\{ \n{ \big( \otimes_{\bl_k}(\phi_0,\psi_0)\big)_n(u) }_{M_{n\tau(k)}} \;:\; k\in\N,  \n{\phi_0}_{M_k(E_0')}, \n{\psi_0}_{M_k(F_0')} \le 1 \Big\}.

By the Arveson extension theorem, $\phi_0 \in M_k(E_0') = \CB(E_0,M_k)$ and $\psi_0 \in M_k(F_0') = \CB(F_0,M_k)$
admit respective norm-preserving extensions $\phi \in \CB(E,M_k)$ and $\psi \in \CB(F,M_k)$.
Since clearly

\big( \otimes_{\bl_k}(\phi,\psi)\big)_n(u) = \big( \otimes_{\bl_k}(\phi_0,\psi_0)\big)_n(u),

it follows that

\lambda'_n(u; E_0,F_0) = \sup\Big\{ \n{ \big( \otimes_{\bl_k}(\phi,\psi)\big)_n(u) }_{M_{n\tau(k)}} \;:\; k\in\N,  \n{\phi}_{M_k(E')}, \n{\psi}_{M_k(F')} \le 1 \Big\},

and therefore $\lambda'_n(u; E,F)$ is also equal to the same quantity.

As an immediate consequence, we have the following result (which in <cit.> is taken as the definition for the dual of $\lambda$).

Let $E$ and $F$ be operator spaces. For any $\lambda$-o.s. tensor norm, the natural map

E \otimes_{\lambda'} F \hookrightarrow \CB_\lambda(E' \times F')

is a complete isometry.

Let $E$ and $F$ be operator spaces. For any $\lambda$-o.s. tensor norm, the natural map

E' \otimes_{\lambda'} F' \hookrightarrow \CB_\lambda(E \times F)

is a complete isometry.

From the previous result, the natural map

\begin{equation}\label{eqn-embedding-lambda-prime}
E' \otimes_{\lambda'} F' \hookrightarrow \CB_\lambda(E'' \times F'')
\end{equation}

is a complete isometry.
As we have already observed (see the proof of the Extension Lemma <ref>), we also have a canonical completely isometric embedding

\begin{equation}\label{eqn-embedding-lambda-prime-2}
\CB_\lambda(E \times F) \hookrightarrow \CB_\lambda(E'' \times F'').
\end{equation}

It is clear that the image of the map in (<ref>) is contained in that of the map in (<ref>), which gives the desired result.

Let $E$ and $F$ be operator spaces, with $E$ finite-dimensional. For any $\lambda$-o.s. tensor norm, the natural map

E \otimes_{\lambda} F \hookrightarrow (E' \otimes_{\lambda'} F')'

is a complete isometry.

From the previous result, the natural map

E' \otimes_{\lambda'} F' \hookrightarrow \CB_\lambda(E'' \times F'')

is a complete isometry, and since $E$ is finite-dimensional it follows that it is surjective and therefore
$E' \otimes_{\lambda'} F' = \CB_\lambda(E'' \times F'')$.
Taking duals and using Theorem <ref>  we have

(E \otimes_\lambda F'')''=\CB_\lambda(E'' \times F'')'=(E' \otimes_{\lambda'} F')',

which, together with  the Embedding Lemma <ref>, the desired result follows.

Any $\lambda$-o.s. tensor norm is accessible.

Let $E$ and $F$ be operator spaces, with $E$ finite-dimensional. From the previous result the natural map

E \otimes_{\lambda} F \hookrightarrow (E' \otimes_{\lambda'} F')'

is a complete isometry, and from the Duality Theorem <ref> so is

E \otimes_{\overleftarrow{\lambda}} F  \hookrightarrow (E' \otimes_{\lambda'} F')',

which means that  $E \otimes_{\lambda} F = E \otimes_{\overleftarrow{\lambda}} F$ and therefore $\lambda$ is right-accessible.
The proof for left-accessibility is analogous.

Left-accessibility of an o.s. tensor norm  $\alpha$ provides a complete isometry analogous to the one that appears (for the  o.s. tensor norm $\proj$) in Equation (<ref>). This fact combined with a CMAP hypothesis imply the following result.

* Let $E  \in \ONORM$, $F$ a normed operator space with the CMAP and $\alpha$ a left-accessible o.s. tensor norm. Then,

E \otimes_{\overrightarrow{\alpha}} F \to E\otimes_{\overleftarrow{\alpha}} F \qquad \text{and} \qquad E \otimes_{\overrightarrow{\alpha}} F \to (E' \otimes_{\alpha'} F')'
are complete isometries.

* Let $E  \in \OLOC$, $F$ a normed operator space with the CMAP and $\alpha$ a locally left-accessible o.s. tensor norm.  Then,

E \otimes_{\overrightarrow{\alpha}} F \to E\otimes_{\overleftarrow{\alpha}} F  \qquad \text{and}  \qquad E \otimes_{\overrightarrow{\alpha}} F \to (E' \otimes_{\alpha'} F')'
are complete isometries.

* The analogous results hold with right-accessibility/local right-accesibility.

We will only prove $(a)$ since the proofs of the  other statements are similar. Note that, by the left-accesibility of the o.s. tensor norm we have

\begin{equation}
\overrightarrow{\alpha} = \overleftarrow{\alpha} \qquad  \text{on} \qquad  E \otimes M
\end{equation}

for every $M \in \OFIN(F).$ Now by the Approximation Lemma <ref>, since $F$ has the CMAP then $\overrightarrow{\alpha}$ and $\overleftarrow{\alpha}$ coincide on $E \otimes F$, and therefore

E \otimes_{\overrightarrow{\alpha}} F \to E\otimes_{\overleftarrow{\alpha}} F

is a complete isometry.
The fact that $E \otimes_{\overrightarrow{\alpha}} F \to (E' \otimes_{\alpha'} F')'$ is also  a complete isometry follows from the previous identification and the Duality Theorem <ref>.

Note that by Proposition <ref> in the statement of the previous corollary we can change $\overrightarrow{\alpha}$ by $\alpha$ and, of course, everything remains the same.
Let us now consider the issue of the accessibility of dual o.s. tensor norms.

Let $\alpha$ be an o.s. tensor norm on $\ONORM$.

* $\alpha$ is right-accessible (resp.  left-accessible,  accessible) then $\alpha'$ is locally right-accessible (resp. locally left-accessible, locally accessible).

* If $\alpha$ is accessible then the transposed o.s. tensor norm $\alpha^t$  is accessible and the adjoint o.s. tensor norm $\alpha^*$ is locally accessible.

Assume that $\alpha$ is  right-accessible. Then for all $M \in \OFIN$ and $F \in \OLOC$ we have complete isometries

M' \otimes_{\alpha''} F' = M' \otimes_{\overrightarrow{\alpha}} F' = M' \otimes_{\overleftarrow{\alpha}} F' = (M \otimes_{\alpha'} F)'

where the first equality follows from Proposition <ref>,
the second one from right-accessibility, and the third one from the Duality Theorem <ref>.
Therefore we also have complete isometries

M \otimes_{\alpha'} F \hookrightarrow (M \otimes_{\alpha'} F)'' = (M' \otimes_{\alpha''} F')',

which means that $\overleftarrow{\alpha'} = \alpha' = \overrightarrow{\alpha'}$ (again by the Duality Theorem) in $\OFIN \otimes \OLOC$.
This shows that $\alpha'$ is locally right-accessible.
Part (b) follows trivially.

In the Banach space setting the dual of a right(left)-accessible tensor norm is again right(left)-accessible. For operator spaces we obtain in the previous proposition a weaker statement since we have proved that the dual norm is  locally right(left)-accessible. However, we do not know whether this notion is truly weaker since we do not have any example of an o.s. tensor norm which is locally right(left)-accessible but not right(left)-accessible.

§ THE COMPLETELY BOUNDED APPROXIMATION PROPERTY

Recall that given $C \ge 1$, a normed operator space is said to have the $C$-completely bounded approximation property ($C$-CBAP for short)  if there exists a net of finite-rank mappings $\phi_i : E \to E$ such that $\n{\phi_i}_{\cb} \le C$ for all $i$ and for every $x \in E$, $\n{\phi_i(x) -  x} \to 0$.
One of the goals in this section is to study the CBAP via various conditions involving tensor products, in the spirit of the characterizations for the operator space approximation property appearing in <cit.>. We also deal with a weaker version, called W*CBAP, which carries over many of the interesting equivalences that appear in the classical theory of tensor products <cit.>.

For a normed operator space $E$, we use the notation $K_\infty(E) = \mathcal{K} \otimes_{\min} E$ to emphasize that we think of this space as consisting of infinite $E$-valued matrices; see <cit.> for the specific details of how to interpret $K_\infty(E)$ as a completion of the union of the spaces $M_n(E)$ of $E$-valued matrices.
Given normed operator spaces $E$ and $F$,
recall that a net $(\varphi^\alpha)$ in $\CB(E,F)$ converges to $\varphi$ in the stable point-norm topology if  for every $x \in K_\infty(E)$ we have that
\varphi^\alpha_\infty(x) \to \varphi_\infty(x)
in the norm topology of $K_\infty(F)$

Our first characterization of CBAP (Proposition <ref> below) corresponds to <cit.>, but instead of the stable point-norm topology we simply use the point-norm topology (which we denote by $\tau$).
The reason is clarified in the following remark.

Observe that for a bounded net in $\CB(E,F)$, convergence in the stable point-norm topology coincides with convergence in the point-norm topology. Indeed, suppose $\varphi^\alpha, \varphi \in \CB(E,F)$ and $x \in K_\infty(E)$. If $P^m : K_\infty(E) \to K_\infty(E)$ denotes the truncation operator,

\begin{multline*}
\n{ \varphi^\alpha_\infty(x) - \varphi_\infty(x) }_{K_\infty(F)}
\le
\n{\varphi^\alpha}_{\cb}\n{ x - P^m(x) }_{K_\infty(E)} \\+
\n{ \varphi^\alpha_m( P^m(x) ) - \varphi_m( P^m(x) ) }_{M_m(F)}
+ \n{\varphi}_{\cb} \n{ P^m(x) - x }_{K_\infty(E)},
\end{multline*}

which shows that when $\varphi^\alpha$ is bounded, convergence in the point-norm topology implies convergence in the stable point-norm topology (the other implication is trivial and holds in general).

Suppose that $E$ is a normed operator space and $C \ge 1$. The following are equivalent:

* $E$ has the $C$-CBAP.

* $C\cdot B_{E' \otimes_{\min} E}$ is  $\tau$-dense in $B_{\CB(E,E)}$.

* For any normed operator space $F$, $C\cdot B_{E' \otimes_{\min} F}$ is  $\tau$-dense in $B_{\CB(E,F)}$.

* For any normed operator space $F$, $C\cdot B_{F' \otimes_{\min} E}$ is  $\tau$-dense in $B_{\CB(F,E)}$.

$(a) \Rightarrow (b)$:
Suppose that $E$ has $C$-CBAP, so that there exists a net of finite-rank mappings $\phi_i : E \to E$ such that $\n{\phi_i}_{\cb} \le C$ for all $i$ and for every $x \in E$, $\n{x - \phi_i x} \to 0$. This precisely means that $\phi_i \xrightarrow{\tau} id_E$, and
note that each $\phi_i$ can be identified with an element of $C\cdot B_{E' \otimes_{\min} E}$.
For any other $\varphi \in B_{\CB(E,E)}$, the net $(\phi_i \circ \varphi)_i$ does the job.
Reversing the argument gives $(b) \Rightarrow (a)$.

Now, if $(\phi_i)$ is a net in $C\cdot B_{E' \otimes_{\min} E}$ converging to $id_E$ in the $\tau$ topology, then for any $\psi \in B_{\CB(E,F)}$ the net $\psi \circ \phi_i \in C\cdot B_{E' \otimes_{\min} F}$ converges to $\psi$ in the $\tau$ topology.
Similarly, for $\psi \in B_{\CB(F,E)}$ the net $\phi_i \circ \psi \in C\cdot B_{F' \otimes_{\min} E}$ converges to $\psi$ in the $\tau$ topology. This proves $(b) \Rightarrow (c)$ and $(b) \Rightarrow (d)$.
The implications $(c) \Rightarrow (b)$ and $(d) \Rightarrow (b)$ are clear by specializing to $F=E$.

It should be noted that although completeness plays an important role in  <cit.>, we do not assume it.
This was to be expected, as the same is true of the Banach space case:
note that in [11] the section on BAP deals with normed spaces, whereas the one on AP deals with Banach spaces.
In this regard we point out the following easy lemma, which shows that when it comes to the CBAP it makes no difference to work with a space or with its completion.

As usual, $\widetilde{E}$ denotes the completion of $E$.
Moreover, given $\varphi \in \CB(E,F)$ we denote by $\widetilde{\varphi}$ its unique extension to a map in $\CB(\widetilde{E},\widetilde{F})$.

$E$ has $C$-CBAP if and only if $\widetilde{E}$ has $C$-CBAP.

$(\Rightarrow)$ Let $(\varphi^\alpha)$ be a net of finite-rank maps in $\CB(E,E)$ of $\cb$-norm at most $C$ converging to the identity of $E$ in the point-norm topology.
Let $\widetilde{x} \in \widetilde{E}$. By a triangle inequality argument, for $x\in E$

\n{\widetilde{\varphi}^\alpha (\widetilde{x}) - \widetilde{x}} \le
\n{\widetilde{\varphi}^\alpha} \n{\widetilde{x}-x} + \n{ \varphi^\alpha(x) - x } + \n{ x - \widetilde{x} },

which shows $(\widetilde{\varphi}^\alpha)$ is a net of finite-rank maps of $\cb$-norm at most $C$ converging to the identity of $\widetilde{E}$ in the point-norm topology.

$(\Leftarrow)$ Let $(\psi^\alpha)$ be a net of finite-rank maps in $\CB(\widetilde{E},\widetilde{E})$ of $\cb$-norm at most $C$ converging to the identity of $\widetilde{E}$ in the point-norm topology.
For each $\alpha$ let $E_\alpha$ be the range of $\psi^\alpha$, and for each $\delta>0$
use perturbation to find a linear map $R_\delta : E_\alpha \to E$ with
$\n{R_\delta-Id_{E_\alpha}}_{\cb}< \delta$.
Let $\varphi^{\alpha,\delta}$ be the restriction to $E$ of $\frac{1}{1+\delta}R_\delta \circ \psi^\alpha$.
Then for any $x \in E$,

\n{\varphi^{\alpha,\delta}(x)-x} \le \frac{1}{1+\delta}\n{R_\delta \psi^\alpha(x) - \psi^\alpha(x)} + \frac{\delta}{1+\delta}\n{\psi^\alpha(x)} +  \n{\psi^\alpha(x)-x},

so  $(\varphi^{\alpha,\delta})$ is a net of finite-rank maps in $\CB(E,E)$ of $\cb$-norm at most $C$ converging to the identity of $E$ in the point-norm topology.

For normed operator spaces $E$ and $F$,
recall that a net $(\varphi^\alpha)$ in $\CB(E,F)$ converges to $\varphi$ in the stable point-weak topology (denoted $\tau_w$) if  for every $x \in K_\infty(E)$ we have that
\varphi^\alpha_\infty(x) \to \varphi_\infty(x)
in the weak topology of $K_\infty(F)$

Let us remark that although <cit.> works with Banach operator spaces, the proof of <cit.> applies more generally to normed operator spaces.
Therefore, the proof of <cit.> together with Proposition <ref> yield the following result.

Suppose that $E$ is a normed operator space and $C \ge 1$. The following are equivalent:

* $E$ has the $C$-CBAP.

* $ C \cdot B_{E' \otimes_{\min} E}$ is  $\tau_w$-dense in $B_{\CB(E,E)}$.

* For any operator space $F$, $C \cdot  B_{E' \otimes_{\min} F}$ is  $\tau_w$-dense in $B_{\CB(E,F)}$.

* For any operator space $F$,  $C \cdot B_{F' \otimes_{\min} E}$ is  $\tau_w$-dense in $B_{\CB(F,E)}$.

Consider now the natural embedding

\CB(E,F) \subseteq \CB(E, F'') = (E \otimes_{\proj} F')'.

In  <cit.> it is proved that  when $E$ and $F$ are complete,
the stable point-weak topology on $\CB(E,F)$ is just the relative weak$^*$ topology determined by $E \otimes_{\proj} F'$, and thus each stable point-weakly continuous functional on $\CB(E,F)$ is determined by an element of $E \widehat{\otimes}_{\proj} F'$.
Let us once again remark that the proof of <cit.> applies more generally to normed operator spaces.

The bounded approximation property, in the context of normed spaces, has many important equivalences in the tensor product realm. Indeed, a Banach space $E$ has the $C$-approximation property if and only if $E\otimes_{\pi}F \to (E'\otimes_{\varepsilon}F')'$ is a $C$-isomorphism onto its image. Moreover, these properties are also equivalent to the following inequality:
$\pi \leq C \overleftarrow{\pi}$ on $E\otimes F$ (for all normed spaces $F$).
At first glance the CBAP seems to be a nice extension of the bounded approximation property to operator spaces, and the literature shows a number of examples where the analogy works very well.
However, the lack of local reflexivity does not directly allow the equivalence between CBAP and the natural operator space versions of the mentioned results related to tensor products. Nevertheless, we will see that a weak version of CBAP introduced in [19] will allow us to translate those approximation properties relative to tensor products of normed spaces into this context.

The following is the obvious adaptation of the notion of W*MAP defined in [19].

An operator space $E$ has the $C$-W*CBAP if there exists a net of finite-rank maps $\phi_i : E \to E''$ such that $\n{\phi_i}_{\cb} \le C$ for all $i$ and for every $x \in E$, $\phi_i (x) \to \kappa_E(x)$ in the weak* topology.

Note that clearly the $C$-CBAP implies the $C$-W*CBAP, as convergence in norm implies convergence in the weak* topology.
The same argument as in [19]  shows that if $E$ has the $C$-W*CBAP, then so does $M_n(E)$.

Similarly as in [19] it is easy to prove (taking adjoints) that $E$ has the $C$-CBAP if and only if there is a net of weak* continuous finite-rank maps $\psi_i:E' \to E'$ such that $\Vert \psi_i \Vert_{cb}\le C$ for all $i$ and $\psi_i(x')\to x'$ in the weak* topology for every $x'\in E'$. Instead, $E$ has the $C$-W*CBAP if and only if the same condition as above holds with the exception that the weak* continuity of the mappings $\psi_i$ is not assumed/required.

We now prove a variant of Corollary <ref>.

Suppose that $E$ is a normed operator space and $C \ge 1$. The following are equivalent:

* $E$ has the $C$-W*CBAP.

* $ C \cdot B_{E' \otimes_{\min} E''}$ is  point-weak*-dense in $B_{\CB(E,E'')}$.

* For any operator space $F$, $C \cdot  B_{E' \otimes_{\min} F'}$ is  point-weak*-dense in $B_{\CB(E,F')}$.

The implication $(a) \Rightarrow (c)$ is implicit in the proof of <cit.>. Suppose that $T : E \to F'$ is a complete contraction. Consider the commutative diagram

\xymatrix{
E'' \ar[r]^{T''} &F''' \ar[d]^{\kappa_F'}\\
E \ar[u]^{\kappa_E} \ar[r]_{T} &F'

Note that both $T''$ and $\kappa_F'$ are weak* continuous.
Therefore, if $\phi_i : E \to E''$ is the net of maps in $C \cdot B_{E' \otimes_{\min} E}$ converging to $\kappa_E$ in the point-weak* topology given by the definition of $C$-W*CBAP, it is clear that $\kappa_F'T''\phi_i : E \to F'$ converges to $T$ in the point-weak* topology.

$(c)\Rightarrow (b)$  is trivial.

Finally, $(b)$ implies that the canonical inclusion $\kappa_E : E \to E''$ can be approximated in the point-weak* topology by a net of finite-rank maps whose cb-norms are at most $C$, which is exactly the definition of $C$-W*CBAP, yielding $(a)$.

The following result is, in a sense, a quantitative version of <cit.>.

Let $E$ be an operator space and $C \ge 1$. The following are equivalent:

* For any operator space $F$, the natural map $E \otimes_{\proj} F \to (E' \otimes_{\min} F')'$ is a complete $C$-isomorphism onto its image.

* For any operator space $F$, the natural map $E \otimes_{\proj} F \to (E' \otimes_{\min} F')'$ is a $C$-isomorphism onto its image.

* $E$ has the $C$-W*CBAP.

* For any complete contraction $T: E \to E''$ there exists a net of finite-rank maps $\phi_i : E \to E''$ such that $\n{\phi_i}_{\cb} \le C$ for all $i$ and for every $x \in E$, $\phi_i (x) \to Tx$ in the weak* topology.

* For any operator space $F$ and any complete contraction $T: E \to F'$, there is a net of finite-rank maps $\psi_i : E \to F'$ such that $\n{\psi_i}_{\cb} \le C$ for all $i$ and for every $x \in E$, $\psi_i x \to Tx$ in the weak* topology.

* The natural map $E \otimes_{\proj} E' \to (E' \otimes_{\min} E'')'$ is a complete $C$-isomorphism onto its image.

* The natural map $E \otimes_{\proj} E' \to (E' \otimes_{\min} E'')'$ is a $C$-isomorphism onto its image.

$(a) \Rightarrow (b)$ is obvious. Let us assume $(b)$, and let
$F$ be an operator space. In order to show that the natural map $E \otimes_{\proj} F \to (E' \otimes_{\min} F')'$ is a complete $C$-isomorphism onto its image, it suffices to show that its adjoint $(E' \otimes_{\min} F')'' \to (E \otimes_{\proj} F)' = \CB(E,F')$ is a complete $C$-quotient. In turn, this will follow from proving that for any $n$ the amplification $M_n((E' \otimes_{\min} F')'') \to M_n(\CB(E,F'))$ is a $C$-quotient. Considering the standard identifications
$M_n((E' \otimes_{\min} F')'') = (M_n(E' \otimes_{\min} F'))'' = (E' \otimes_{\min} M_n(F'))''= (E' \otimes_{\min} S_1^n[F]')''$ and
$\CB(E,M_n(F')) = \CB(E,S_1^n[F]')$, we need to show that the natural map
$(E' \otimes_{\min} (S_1^n[F])')'' \to \CB(E,S_1^n[F]')$ is a $C$-quotient. This follows from $(b)$ with $S_1^n[F]$ in place of $F$ and taking adjoints. Thus, $(a) \Leftrightarrow (b)$.

Now, $(c) \Leftrightarrow (d) \Leftrightarrow (e)$ is just a restatement of Proposition <ref>.

Let us now prove that $(b) \Leftrightarrow (e)$.
First observe that the natural map $E \otimes_{\proj} F \to (E' \otimes_{\min} F')'$ always has cb-norm one.
Therefore, $(b)$ is equivalent to having

(E \otimes F) \cap B_{( E' \otimes_{\min} F' )'} \subset C \cdot B_{E \otimes_{\proj} F}

By the bipolar theorem, this in turn is equivalent to

B_{\CB(E,F')} \subset \overline{C \cdot B_{ E' \otimes_{\min} F' }}^{w^*}.

where the closure is in the weak* topology associated with the identification $\CB(E,F') = (E \otimes_{\proj} F)'$. And this is clearly equivalent to the statement of $(e)$.

Finally, it is evident that $(a)\Rightarrow (f)\Rightarrow (g)$ and arguing as in $(b) \Leftrightarrow (e)$ we easily obtain that $(g)\Leftrightarrow (d)$.

In view of our preceding results, it seems plausible to extend the usual concept of the bounded approximation property with respect to a tensor norm (see e.g. <cit.>) to our operator space setting. Namely, a natural definition to consider should be the following: an operator space $E$ has the $\alpha$-W*CBAP (with constant $C>0$) if the natural map

F\otimes_{\alpha}E \to (F'\otimes_{\alpha'}E')'$$
is a complete $C$-isomorphism onto its image for every operator space $F$.
Nevertheless,  we do not tackle this here and leave it for future research.

The following two results follow similarly as <cit.> and <cit.>. We include somewhat alternative proofs for completeness.

If an operator space $E$ has the $C$-W*CBAP and is locally reflexive, then $E$ has the $C$-CBAP.

If $E$ has the $C$-W*CBAP  there is a net of finite-rank maps $\phi_i : E \to E''$ such that $\n{\phi_i}_{\cb} \le C$ for all $i$ and for every $x \in E$, $\phi_i (x) \to \kappa_E(x)$ in the weak* topology. For each $i$, let us denote the range of the mapping $\phi_i$ by $F_i\in \OFIN(E'')$. Since $E$ is locally reflexive for each $i$, there is a net of complete contractions $(\psi_\eta^i)_\eta\subset\CB(F_i,E)$ approximating the inclusion $F_i\hookrightarrow E''$ in the point-weak* topology. By composing we obtain a net of finite rank mappings $(\psi_\eta^i\circ \phi_i)\subset\CB(E,E)$ which converges in the point-weak topology to the identity $id:E\to E$ satisfying $\n{\psi_\eta^i\circ\phi_i}_{\cb} \le C$. A classical convex combinations argument allow us to obtain a net with the same properties but which converges in the point-norm topology to the identity; thus concluding that $E$ has the C-CBAP.

If $E$ is an operator space such that $E'$ has the $C$-W*CBAP, then so does $E$.

If $E'$ has the $C$-W*CBAP, by Theorem <ref> $(a)$ (applied to $F=E$) we obtain that  the map $E' \otimes_{\proj} E \to (E'' \otimes_{\min} E')'$ is a complete $C$-isomorphism with its image. Since both $\proj$ and $\min$ are symmetric we can flip the spaces (by transposition) to obtain that $E \otimes_{\proj} E' \to (E' \otimes_{\min} E'')'$ is a complete $C$-isomorphism with its image. This is exactly what is stated in Theorem <ref> $(f)$. Hence, $E$ has the $C$-W*CBAP.

As a corollary we see that under local reflexivity the $C$-CBAP can be transferred from the dual $E'$ to $E$.

Let $E$ be an operator space which is locally reflexive. If $E'$ has the $C$-CBAP, then so does $E$.
(in fact, if $E'$ has the $C$-W*CBAP then $E$ has $C$-CBAP).

As a consequence of Theorem <ref> we obtain the following analogue of <cit.>:

Let $E$ be an operator space and $C \ge 1$. The following are equivalent:

* $E$ has the $C$-W*CBAP.

* For every operator space $F$ (or only $F=E'$), the  identity $E \otimes_{\overleftarrow{\proj}} F \to E \otimes_{\proj} F$ has $\cb$-norm at most $C$.

* For every operator space $F$, the  identity $E \otimes_{\overleftarrow{\proj}} F \to E \otimes_{\proj} F$ has norm at most $C$.

In particular, $E$ has the W*CMAP if and only if

\proj = \overleftarrow{\proj} \quad \text{ on } E \otimes F

for every operator space $F$ (or only $F=E'$).

By the Duality Theorem <ref> we have the complete isometry $E \otimes_{\overleftarrow{\proj}} F \hookrightarrow (E' \otimes_{\min} F')'$. Also, we know that $\overleftarrow{\proj} \leq \proj$. Therefore,
$E \otimes_{\overleftarrow{\proj}} F \to E \otimes_{\proj} F$ has $\cb$-norm at most $C$ (resp. norm at most $C$)

if and only if the mapping
$E \otimes_{\proj} F \to (E' \otimes_{\min} F')'$
is a complete $C$-isomorphism (resp. $C$-isomorphism) onto its image. Now,  Theorem <ref> gives the result.

Now we present an operator space version of <cit.> which enables us to prove the subsequent corollary.

Let $\alpha$ be an o.s. tensor norm, $E$ and $F$ be normed operator spaces, and $n\in\N$. Then $\alpha_n \le C \overleftarrow{\alpha}_n$ on $E \otimes F$ if and only if

B_{M_n\big((E \otimes_\alpha F)'\big)} \subset C\cdot \overline{ B_{M_n( E' \otimes_{\alpha'} F' )} }^{\tau_w} \subset M_n\big(\CB(E,F') \big) = \CB\big(E , M_n(F')\big).

Let $z \in M_n(E \otimes F)$. By the Duality Theorem <ref>,

\overleftarrow{\alpha}_n(z; E, F) =  \sup \big\{  \n{ \mpair{T}{z} }_{M_{nm}} \mid  T \in B_{M_m( E' \otimes_{\alpha'} F' )} \big\},

whereas clearly

\alpha_n(z; E, F) = \sup  \big\{  \n{ \mpair{T}{z} }_{M_{nm}} \mid  T \in B_{M_m(( E \otimes_{\alpha} F )')} \big\}

By the bipolar theorem, the inclusion

B_{M_n\big((E \otimes_\alpha F)'\big)} \subset C \cdot\overline{ B_{M_n( E' \otimes_{\alpha'} F' )} }^{\tau_w}

is then equivalent to $\alpha_n \le C \overleftarrow{\alpha}_n$ on $M_n(E \otimes F)$.

From Theorem <ref> and Lemma <ref> we derive the following corollary. Compare to Proposition <ref>.

Suppose that $E$ is a normed operator space and $C \ge 1$. The following are equivalent:

* $E$ has the $C$-W*CBAP.

* $ C \cdot B_{E' \otimes_{\min} E''}$ is  $\tau_w$-dense in $B_{\CB(E,E'')}$.

* For any operator space $F$, $C \cdot  B_{E' \otimes_{\min} F'}$ is $\tau_w$-dense in $B_{\CB(E,F')}$.

The equivalence between $(a)$ and $(c)$ follows from Theorem <ref> $(c)$ and Lemma <ref>.
Clearly, $(c)$ implies $(b)$.
Assuming $(b)$, any $T \in B_{\CB(E,E'')}$ is the $\tau_w$-limit of a net in $ C \cdot B_{E' \otimes_{\min} E''}$, which means $T$ is also the point-weak limit of the same net (by an argument analogous to that of Remark <ref>), and hence also the point-weak* limit. Now Proposition  <ref> implies $(a)$.

§ MAPPING IDEALS

We introduce now the definition of a mapping ideal, the operator space version of Banach operator ideals defined by Pietsch and also widely developed by Defant and Floret.
The notion in our framework comes from <cit.>, though we have to strengthen Effros and Ruan's definition in order to keep the essence of the traditional Banach space concept and allow the natural relationship  with tensor norms.
As in [11] the definition of ideal is restricted to the category of complete spaces. Let us  denote by $\OBAN$ the class of all Banach operator spaces.

A (normed) mapping ideal $(\mathfrak{A},\mathbf{A})$ is an assignment, for each pair of operator spaces $E, F \in \OBAN$, of a linear space $\mathfrak{A}(E,F) \subseteq \CB(E,F)$  together with an operator space structure $\mathbf{A}$ on $\mathfrak{A}(E,F)$ such that

* The identity map $\mathfrak{A}(E,F) \to \CB(E,F)$ is a complete contraction.

* For every $x'\in M_n(E')$ and $y\in M_m(F)$ the mapping $x'\otimes y$ belongs to $M_{nm}(\mathfrak{A}(E,F))$ and $\mathbf{A}_{nm}(x'\otimes y)=\|x'\|_{M_n(E')} \|y\|_{M_m(F)}$.

* The ideal property: whenever $T \in M_n(\mathfrak{A}(E,F))$, $r \in \CB(E_0,E)$ and $s \in \CB(F,F_0)$, it follows that $s_n \circ T \circ r$ belongs to $ M_n(\mathfrak{A}(E_0,F_0))$ with

\mathbf{A}_n( s_n \circ T \circ r ) \le \n{s}_{\cb} \mathbf{A}_n(T) \n{r}_{\cb}.

Item (b) of the previous definition is the new requirement, not considered in <cit.>.
In the Banach space realm, the definition of normed operator ideal includes that the ideal norm of the identity $id_{\mathbb  C}:  \mathbb C \to \mathbb C$  is one <cit.>. This requisite is in fact the key to relate normed ideals and finitely generated  tensor norms.
Unfortunately there is no  immediate  translation of this requirement into the operator space framework, but nevertheless there is a way to tackle this issue.
By <cit.> the aforementioned condition about the norm of the identity (together with the other properties of normed operator ideals) implies that —and in fact it is equivalent to— the ideal norm of any elementary tensor $x'\otimes y$ (for $x' \in E'$ and $y \in F$) is exactly $\Vert x' \Vert \Vert y \Vert$. This has inspired  item (b) on our definition. Note the similarity between this statement and equation (<ref>) in the definition of o.s. tensor norms.

Let us show now that (b) can be somehow relaxed.

First, note that since $\|x'\otimes y\|_{cb}=\|x'\|_{M_n(E')} \|y\|_{M_m(F)}$, condition (a) implies that $\mathbf{A}_{nm}(x'\otimes y)\ge\|x'\|_{M_n(E')} \|y\|_{M_m(F)}$. Hence, we can replace in (b) the symbol $=$ by $\le$.

Also, it is enough that this inequality is satisfied for every $E,F\in\OFIN$. Indeed, let us show that the inequality is valid  for every Banach operator space when it is so for every finite-dimensional operator space. Let $x'=(x'_{ij})\in M_n(E')$ and $y=(y_{ij})\in M_m(F)$ with $E,F\in\OBAN$. Consider $L=\bigcap_{i,j}\ker(x'_{ij})\in\OCOFIN(E)$ and $u'\in M_n((E/L)')$ such that $x'=(q_L^E)'_n(u')$. Since $(q_L^E)'$ is a complete isometry, we have $\|u'\|_{M_n((E/L)')}=\|x'\|_{M_n(E')}$. Choosing $N\in\OFIN(F)$ such that $y\in M_m(N)$ by means of condition (c) of the definition we obtain

\mathbf{A}_{nm}(x'\otimes y;E,F)\le \mathbf{A}_{nm}(u'\otimes y; E/L,N)\le
\|u'\|_{M_n((E/L)')}\|y\|_{M_m(N)}=\|x'\|_{M_n(E')}\|y\|_{M_m(F)}.

Therefore, we can exchange condition (b) of the previous definition by the following:

(b') For every $E,F\in\OFIN$, $x'\in M_n(E')$ and $y\in M_m(F)$ the mapping $x'\otimes y$ belongs to $M_{nm}(\mathfrak{A}(E,F))$ and $\mathbf{A}_{nm}(x'\otimes y)\le\|x'\|_{M_n(E')} \|y\|_{M_m(F)}$.

Of course, the class of completely bounded mappings $\CB$ is obviously a mapping ideal.

Let us recall the definitions of some other mapping ideals that have already appeared in the literature. We explain in each case (whenever it is not obvious) why condition (b') is satisfied. Let $E$ and $F$ be Banach  operator spaces.

* Finite rank mappings $\mathcal F$:
the space $\mathcal{F}(E,F)$ consists of finite rank mappings from $E$ to $F$. This is a mapping ideal with the $\cb$-norm.

* Completely approximable mappings $\mathcal{A}$:  the space $\mathcal{A}(E,F)$ is the closure of $\mathcal F(E,F)$ inside $\CB(E,F).$

* Completely nuclear mappings $\mathcal N$ <cit.>:  the space $\mathcal{N}(E,F)$ is the image of the mapping

\begin{equation}\label{def nucleares}
\Phi: E'\widehat{\otimes}_{\proj}F \to E'\widehat{\otimes}_{\min}F \subset \CB(E,F)
\end{equation}

with the quotient operator space structure given by the canonical identification

\mathcal{N}(E,F) =\frac{ E' \widehat{\otimes}_{\proj} F}{\ker  \Phi}.

The nuclear operator space norm is denoted by $\nu$. Equation (<ref>) for the $\proj$ norm shows that (b') is satisfied for $\mathcal{N}$.

* Completely integral mappings  $\mathcal I$ <cit.>:  the space $\mathcal I(E,F)$ is formed by those mappings such that

\begin{equation*}
\iota(T)=\sup \{(\nu(T|_M):M \in \OFIN(E) \}<\infty.
\end{equation*}