# The lifting property for C * -algebras: from local to global ?

Gilles Pisier

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Gilles Pisier. The lifting property for C * -algebras: from local to global?. Münster Journal of Mathematics, In press. hal-05412125

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# The lifting property for $C^*$-algebras : from local to global ?

by

Gilles Pisier

Texas A&M University

College Station, TX 77843, U. S. A.

August 21, 2023

## Abstract

This note is motivated by Kirchberg's conjecture that the local lifting property (LLP) implies the lifting property (LP) for separable $C^*$-algebras. The author recently constructed by a "local" method an example of $C^*$-algebra with the LLP and the weak expectation property (WEP) which might be a counterexample. We give here several conditions all equivalent to the validity of the implication $LLP \Rightarrow LP$ for WEP $C^*$-algebras, or equivalently for quotients of WEP $C^*$-algebras (QWEP), that hopefully clarify the nature of the problem. But unfortunately we cannot decide whether they hold true. These conditions highlight the notion of "controllable" finite dimensional (f.d.) operator space, in connection with certain $C^*$-tensor products. The latter led us to a closely related variant of local convexity.

MSC (2010): 46L06, 46L07, 46L09

Key words: $C^*$-algebras, von Neumann algebras, lifting property, tensor products

Let $A, C$ be $C^*$-algebras. Let $I \subset C$ be a closed self-adjoint two-sided ideal so that the quotient $C/I$ is a $C^*$-algebra. Assuming $A$ unital, let $S \subset A$ be an operator system, meaning a self-adjoint subspace containing the unit. Consider a map $u : S \to C/I$. We say that $u$ is liftable if there is a unital completely positive (u.c.p. in short) map $\hat{u} : S \to C$ lifting $u$. We say that $u$ is locally liftable if for any finite dimensional (f.d. in short) operator system $E \subset A$ the restriction $u|_E$ is liftable.

In [13] Kirchberg defined the lifting property (LP in short) and the local lifting property (LLP in short) for a unital $C^*$-algebra $A$ as follows:
The algebra $A$ has the LP (resp. LLP) if any u.c.p. map $u : A \to C/I$ into an arbitrary quotient is liftable (resp. locally liftable).
When $A$ is not unital, $A$ has the LP (resp. LLP) if its unitization does.

In some sense $C^*(\mathbb{F}_\infty)$ is the fundamental example of LP for $C^*$-algebras. It plays an analogous role to that of $\ell_1$-spaces in Banach space theory.

In the remarkable last section of his outstanding paper [13], Kirchberg formulated seven equivalent conjectures (labelled as B1, ..., B7) all equivalent to the validity of the Connes embedding problem (labelled as B1) which asks whether any finite von Neumann algebra on $\ell_2$ embeds in an ultraproduct of matrix algebras. One of these conjectures (namely B3) states that there is a unique $C^*$-norm on $C^*(\mathbb{F}_\infty) \otimes C^*(\mathbb{F}_\infty)$ where $\mathbb{F}_\infty$ denotes the free group with infinitely countably many generators. In the paper [11] (see also [27]) a negative solution is proposed, based on the equivalence of B3 with an equivalent conjecture of Tsirelson developed in quantum information theory. See [20] for a detailed account of the equivalence of these various conjectures.
In [13] Kirchberg proved that his (equivalent) B-conjectures imply his conjecture D which states that for separable $C^*$-algebras the LLP implies the LP (in other words they are equivalent). The latter conjecture is the motivation for the present paper. Although we do not reach a conclusion, we hope that our results may advance the situation. The difficulty is that, apart from the nuclear $C^*$-algebras, very few examples of LLP $C^*$-algebras are known except for $C^*(\mathbb{F}_\infty)$ and algebras derived from it. However, we recently constructed in [21] a new type of non-nuclear LLP $C^*$-algebras, for which it does not seem clear at all that they have the LP; the failure of all our attempts to either prove or disprove that is what led to the present paper. The special feature of the latter examples is that they have Lance's weak expectation property (WEP in short), which by another of Kirchberg's results in [13] is equivalent to the statement that, if A denotes the $C^*$-algebra, there is a unique $C^*$-norm on $A \otimes C^*(\mathbb{F}_\infty)$. Kirchberg's conjecture B3 states that $C^*(\mathbb{F}_\infty)$ has the WEP or equivalently that any separable $C^*$-algebra is a quotient of a WEP $C^*$-algebra (these are called QWEP). He also proved in [13] that A has the LLP if and only if there is a unique $C^*$-norm on $A \otimes B(\ell_2)$. Thus the question whether the algebras we constructed in [21] have the LP seemed like a very natural way to tackle Kirchberg's conjecture D. Two roads opened to us:

* The most ambitious one would be to prove that one of these algebras (which have LLP and WEP but are not nuclear) fails the LP. This would invalidate conjecture D and hence give an alternate and hopefully much simpler solution to the Connes embedding problem.

* A much less ambitious project would be to prove that conjecture D actually implies (conversely) the (equivalent) conjectures B. In that case one would deduce a negative answer for D from a negative solution to the Connes embedding problem.

This leaves aside the possibility that conjecture D might hold at least for all WEP (or all QWEP) $C^*$-algebras in which case our examples in [21] would have the LP. However, if it turns out that conjecture D is not valid, exhibiting a WEP $C^*$-algebra with the LP remains a quite interesting goal, considerably strengthening [21].

Unfortunately, we have been unable to decide any of these three alternatives. Nevertheless, we obtained several equivalent forms of the validity of LLP $\Rightarrow$ LP for WEP (and separable) $C^*$-algebras, which seem very close to the goal of at least the less ambitious project.

For instance, we show that if a single one of our examples in [21] has the LP then all WEP and LLP (separable) $C^*$-algebras also have LP. Moreover, denoting for simplicity

$$ \mathcal{C} = C^*(\mathbb{F}_\infty) $$

we prove the equivalence of the following (conjectural) assertions:

(i) For WEP (and separable) $C^*$-algebras, the LLP implies the LP.

(ii) There is a completely isometric embedding $f: \mathcal{C} \to \mathcal{C}$ such that the minimal and maximal $C^*$-norms coincide on the algebraic tensor product $f(\mathcal{C}) \otimes \mathcal{C}$.

(iii) Any (or some) completely isometric embedding $j: \mathcal{C} \to B(\ell_2)$ is such that the map

$$ \mathrm{Id}_{\ell_\infty(\mathcal{C})} \otimes j: \ell_\infty(\mathcal{C}) \otimes \mathcal{C} \to \ell_\infty(\mathcal{C}) \otimes B(\ell_2) $$

is contractive from the minimal tensor product to the maximal one.

We feel (ii) is very close to Kirchberg's B3 conjecture, which asserts that the min and max norms are equivalent on $\mathcal{C} \otimes \mathcal{C}$. Indeed, the latter is just (ii) for a surjective $f$.

On the discouraging side, even if there is a $\ast$-homomorphism $f$ as in (ii) we do not know whether B3 follows.

Our work also leads to some surprising classes of f.d. operator subspaces of $\mathcal{C}$. Let us say that a f.d. subspace $E \subset \mathcal{C}$ is rigid if any completely contractive map $u: E \to \mathcal{C}$ is the restriction to $E$ of a $\ast$-homomorphism. If $E \subset \mathcal{C}$ the dual operator space $E^*$ also embeds completely isometrically
in $\mathcal{C}$ (and conversely). The spaces that are completely isometric to a space with rigid dual are called *controllable*. We show that (i) is equivalent to the assertion that all f.d. subspaces $E \subset \mathcal{C}$ are completely isometric to rigid ones (see Lemma 2.14 and Theorem 3.1). Equivalently all subspaces $E \subset \mathcal{C}$ are controllable. The controllable spaces form an interesting class of f.d. operator subspaces of $\mathcal{C}$ that probably deserves more investigation. It can be shown that if the dual $E^*$ is 1-exact then $E$ is completely isometric to a rigid space but at the time of this writing we could not decide whether the latter is true when $E$ is the $n$-dimensional commutative $C^*$-algebra $\ell_\infty^n$. We conjecture it is not, already for $n=3$ i.e. for $\ell_\infty^3$. Equivalently, we conjecture that its operator space dual $\ell_1^3$ is not controllable. Actually, the conjecture that $M_k(\ell_1^3)$ is controllable for all $k \ge 1$ is equivalent to the validity of the assertion (i) by the linearization trick of [18] (see Remark 2.6).

We will always restrict our considerations of the LP to the separable case. In the non-separable case very little seems to be known on the LP. It is apparently open whether $C^*(\mathbb{F})$ has the LP when $\mathbb{F}$ is an uncountable free group. In the non-separable case, it is natural to consider (as e.g. in [24, §7]) the class of $C^*$-algebras $A$ such that all separable $C^*$-subalgebras $B \subset A$ are contained in a larger separable one with LP.

We end the paper with a generalization of the local convexity (LR in short) property on which our previous paper [24] is based.

## Notation and background

As often, we abbreviate completely bounded by c.b., completely positive by c.p. and completely contractive by c.c. We will also abbreviate operator space (or operator spaces) by o.s. and finite dimensional by f.d. We abbreviate completely isometric and completely isometrically by c.i. An o.s. is a closed subspace of a $C^*$-algebra or of $B(H)$. The duality of o.s. is a consequence of Ruan's characterization (see e.g. [9]) of the sequences of norms on $(M_n(E))_{n \ge 1}$ (where $M_n(E)$ denotes the space of $n \times n$-matrices with entries in a vector space $E$) that come from an embedding of $E$ into $B(H)$ for some $H$. Given an o.s. $E \subset B(H)$ with (Banach space sense) dual $E^*$ there is an $\mathcal{H}$ and an isometric embedding $j: E^* \to B(\mathcal{H})$ that induces isometric isomorphisms $M_n(j(E^*)) \simeq CB(E, M_n)$ for all $n$. The embedding $j$ allows one to consider $E^*$ as an o.s. This is what is called the dual o.s. structure on $E^*$. We will often refer to it as the o.s. dual of $E$. An important property due independently to Effros-Ruan and Blecher-Paulsen (see e.g. [9, p. 42] or [19, p. 41]) is that $E \simeq (E^*)^*$ c.i. for any f.d. $E$.

We denote by $Id_E$ the identity map on $E$.

We reserve the notation $E \otimes F$ for the *algebraic* tensor product of two o.s.

Recall that if $E, F$ are completely isomorphic operator spaces, we set

$$d_{cb}(E, F) = \inf\{\|u\|_{cb} \|u^{-1}\|_{cb}\}$$

where the inf runs over all the complete isomorphisms $u: E \to F$.

**Definition 0.1.** We will say that an o.s. $X$ locally embeds in an o.s. $Y$ if for any f.d. $E \subset X$ and any $\varepsilon > 0$ there is $F \subset Y$ such that $d_{cb}(E, F) < 1 + \varepsilon$.

We refer to [9, 19] for more background on operator spaces.

Let $(D_i)_{i \in I}$ be a family of $C^*$-algebras. We will denote by $\ell_\infty(\{D_i \mid i \in I\})$ or simply by $\ell_\infty(\{D_i\})$ the $C^*$-algebra formed of the families $d = (d_i)_{i \in I}$ in $\prod_{i \in I} D_i$ such that $\sup_{i \in I} \|d_i\| < \infty$, equipped with the norm $d \mapsto \sup_{i \in I} \|d_i\|$.

The following classical fact will be invoked repeatedly.

**Lemma 0.2.** Let $\mathbb{F}$ be any uncountable free group and let $X \subset C^*(\mathbb{F})$ be a separable subspace. There is a $C^*$-subalgebra $\mathcal{C}' \subset C^*(\mathbb{F})$ such that $X \subset \mathcal{C}'$, the algebra $\mathcal{C}'$ is (unitally) isomorphic to
$C^*(\mathbb{F}_\infty)$ and there is a unital c.p. and contractive projection $P : C^*(\mathbb{F}) \to \mathscr{C}'$. Consequently, for any other $C^*$-algebra $A$ the norm induced on $A \otimes \mathscr{C}'$ (algebraic tensor product) by $A \otimes_{\max} C^*(\mathbb{F})$ coincides with the norm in $A \otimes_{\max} \mathscr{C}'$.

*Proof.* Since $X$ is separable it lies in the closure $\mathscr{C}'$ of a subgroup of $\mathbb{F}$ generated by a countably infinite set of free generators. The latter is a copy of $\mathbb{F}_\infty$. As is well known (see e.g. [20]) there is a conditional expectation operator $P : C^*(\mathbb{F}) \to \mathscr{C}'$. $\square$

*Remark* 0.3 (On Arveson's principle). We will often invoke Arveson's principle. This deals with certain classes $\mathcal{F}(X, C)$ (that we call admissible) formed of maps from a separable o.s. $X$ to a $C^*$-algebra $C$. Let $q: C \to Q$ be a surjective $*$-homomorphism between $C^*$-algebras. A map $u: A \to Q$ is called $\mathcal{F}$-liftable if there is $\hat{u} \in \mathcal{F}(X, C)$ such that $u = q\hat{u}$. Arveson's principle (see [1, p. 351]) says that pointwise limits of $\mathcal{F}$-liftable maps are $\mathcal{F}$-liftable. The typical example of admissible $\mathcal{F}$ is $\mathcal{F}(X, C) = B_{CB(X,C)}$. Assuming $X$ is a $C^*$-algebra, then $\mathcal{F}(X, C) = \{u \text{ c.c. and c.p.}\}$ is another basic example. By definition a class is admissible if $\mathcal{F}(X, C)$ is bounded in norm, pointwise closed and satisfies the following form of non-commutative convexity: for any pair $f, g$ in $\mathcal{F}$ and any $\sigma \in C_+$ with $\|\sigma\| \le 1$ the mapping

$$x \mapsto \sigma^{1/2} f(x) \sigma^{1/2} + (1 - \sigma)^{1/2} g(x) (1 - \sigma)^{1/2}$$

belongs to $\mathcal{F}$.

*Remark* 0.4 (Examples). Choi and Effros [6] proved that all separable nuclear $C^*$-algebras have the LP. Later on, Kirchberg [14] proved that $C^*(\mathbb{F})$ has the LP when $\mathbb{F}$ is any countable free group. While the basic idea to lift homomorphisms from $\mathbb{F}$ to the unitary group of a quotient $A/I$ is somewhat intuitive, the proof for general u.c.p. maps is non trivial. It uses Kasparov's version of the Stinespring dilation for such maps. See [5] for an account of that first proof. A different second proof can be derived from [24], using the criterion appearing in Theorem 1.1, the description of the unit ball of $E \otimes_{\max} C$ when $E \subset \mathscr{C}$ is the span of a finite set of free unitary generators (see (7) and (9) in [18]), and the linearization trick of [18]. That second proof can be equivalently rewritten by combining the proof of [18, Th. 17] with the criterion involving the nor-tensor product in [24, Th. 0.13]. These two proofs seem to be the only known ones that $C^*(\mathbb{F})$ has the LP.

*Remark* 0.5 (Stability under free product). In [18] we proved that the free product of an arbitrary family of unital $C^*$-algebras $(A_i)_{i \in I}$ with LLP also has the LLP. Here and below by “free product” we mean the full (or maximal) unital free product.

Let $A$ be a separable unital $C^*$-algebra and let $q: C^*(\mathbb{F}) \to A$ be a surjective unital $*$-homomorphism. Then $A$ has the LP if and only if there is a u.c.p. map $u: A \to C^*(\mathbb{F})$ lifting $q$. Indeed, if this holds the LP will be inherited by $A$ from $C^*(\mathbb{F})$. Taking the LP of $C^*(\mathbb{F})$ for granted, Boca proved in [4] that the LP is stable by full countable free products of unital $C^*$-algebra $(A_i)_{i \in I}$. He uses a simple application of his remarkable result on free products of u.c.p. maps in [2] (see also [3, 7]). This can be applied to an arbitrary family of unital $C^*$-algebras $(A_i)_{i \in I}$ and arbitrary free groups with surjective unital $*$-homomorphisms $q_i: C^*(\mathbb{F}_i) \to A_i$ admitting u.c.p. liftings $u_i: A_i \to C^*(\mathbb{F}_i)$. Boca's theorem from [2] yields a u.c.p. lifting for the quotient map $\ast_{i \in I} q_i: C^*(\ast_{i \in I} \mathbb{F}_i) \to \ast_{i \in I} A_i$. In the separable case this shows that the LP is stable by countable free products, but one first needs to know that $C^*(\mathbb{F}_\infty)$ has the LP (for which two proofs are described in Remark 0.4).
A direct proof (from scratch) can be derived from the more recent criterion in Theorem 1.1 and the linearization trick of [18]. This is similar to the second proof in Remark 0.4.

Given this, the best way to unify the Choi-Effros and Kirchberg theorems is perhaps to state that any countable free product of separable unital nuclear $C^*$-algebras has the LP.
Remark 0.6 (Counterexamples). It is natural to say that a discrete group $G$ has the LP if $C^*(G)$ has it. In the present remark all groups are assumed countable. By the preceding discussion we know that free groups and amenable groups have the LP. Thus it is not at all obvious to find groups failing the LP or the LLP. As it turns out all the known counterexamples to the LP or the LLP use Kazhdan’s property (T). Ozawa [17] proved that a continuum of (T) groups failing the LP exists but without producing an explicit example. Later on Thom [25] produced an explicit example failing the LLP. More recently, Ioana, Spaas and Wiersma showed that $SL(n, \mathbb{Z})$ fails the LLP for all $n \ge 3$. They also showed that any infinitely presented group with property (T) must fail the LP. By [17] there is a continuum of such groups. It remains an open question whether all discrete groups with property (T) fail the LP or even the LLP.

# 1 Preliminaries

We start by recalling the main result from [24] on the LP.

**Theorem 1.1.** Let $\mathcal{C} = C^*(\mathbb{F}_\infty)$. A separable $C^*$-algebra $A$ has the LP if and only if for any set $I$, any f.d. $E \subset A$ and any family $t = (t_i)_{i \in I}$ in $\mathcal{C} \otimes E$ we have

$$ (1.1) \qquad \|t\|_{\ell_\infty(I, \mathcal{C}) \otimes_{\max} A} \le \sup_{i \in I} \|t_i\|_{\mathcal{C} \otimes_{\max} A}. $$

For this to hold it suffices to check the case when $I = \mathbb{N}$.

Moreover, (1.1) holds whenever $A = C^*(\mathbb{F})$ for any (not necessarily countable) free group.

**Remark 1.2.** Assume that the Kirchberg conjecture (and the Connes embedding problem) holds. We will show that LLP $\Rightarrow$ LP for separable $C^*$-algebras (which is the same as conjecture (B3) $\Rightarrow$ (D) in Kirchberg’s [13]). The conjecture is that $\mathcal{C}$ has the WEP. By a general result $\ell_\infty(\mathcal{C})$ also has it (see [20, p. 193]). By Kirchberg’s fundamental theorem (see [20, p. 195]) we have $B \otimes_{\min} C = B \otimes_{\max} C$ whenever $C$ has LLP and $B$ has WEP. Therefore $\ell_\infty(\mathcal{C}) \otimes_{\min} C = \ell_\infty(\mathcal{C}) \otimes_{\max} C$. Using this last identity, Theorem 1.1 shows that $C$ has the LP if it is separable and has the LLP.

**Remark 1.3.** Note that $\ell_\infty(\mathcal{C})$ fails the LLP. Indeed, by [12] the algebra $\ell_\infty(\{M_n | n \ge 1\})$ (which clearly embeds c.i. in $\ell_\infty(\mathcal{C})$) does not locally embed in $\mathcal{C}$. More generally any $C^*$-algebra $C$ that contains c.i. copies of $M_n$ for all $n$ is such that $\ell_\infty(C)$ fails LLP.

In companion to $\mathcal{C} = C^*(\mathbb{F}_\infty)$, let $\mathcal{B} = B(\ell_2)$. We denote by

$$ j : \mathcal{C} \to \mathcal{B} $$

an isometric $*$-homomorphism.

Let $A, B, C$ be $C^*$-algebras. Let $E \subset A$. We say that a mapping $u : E \to B$ is $C$-nuclear if

$$ \|id_C \otimes u : C \otimes_{\min} E \to C \otimes_{\max} B\| = 1, $$

or equivalently if $\|u \otimes id_C : E \otimes_{\min} C \to B \otimes_{\max} C\| = 1$.

With this terminology, a $C^*$-algebra $A$ is WEP (resp. LLP) if and only if $Id_A$ is $\mathcal{C}$-nuclear (resp. $\mathcal{B}$-nuclear). See [20] for an exposition of these topics.

**Lemma 1.4.** Assume that $B$ satisfies (1.1) (e.g. $B = \mathcal{C}$ or $B = C^*(\mathbb{F})$ for any free group $\mathbb{F}$). Then any $\mathcal{C}$-nuclear mapping $u : E \to B$ is $\ell_\infty(\mathcal{C})$-nuclear.
Proof. Consider

$$ Id \otimes u : \ell_{\infty}(\mathscr{C}) \otimes E \to \ell_{\infty}(\mathscr{C}) \otimes B. $$

Note that we have an isometric inclusion

$$ \ell_{\infty}(\mathscr{C}) \otimes_{\min} E \subset \ell_{\infty}(\mathscr{C} \otimes_{\min} E) $$

while by (1.1) we have an isometric inclusion

$$ \ell_{\infty}(\mathscr{C}) \otimes_{\max} B \subset \ell_{\infty}(\mathscr{C} \otimes_{\max} B). $$

If $u$ is $\mathscr{C}$-nuclear, obviously $Id_{\mathscr{C}} \otimes u$ defines a contractive map

$$ \ell_{\infty}(\mathscr{C} \otimes_{\min} E) \to \ell_{\infty}(\mathscr{C} \otimes_{\max} B), $$

from which the lemma follows by restriction to $\ell_{\infty}(\mathscr{C}) \otimes E$. $\square$

Let $E \subset A$ and $u : E \to B$. We will denote the norm that controls $\ell_{\infty}(\mathscr{C})$-nuclearity by $\|u\|_{mM}$, i.e. we set

$$ (1.2) \qquad \|u\|_{mM} = \|Id_{\ell_{\infty}(\mathscr{C})} \otimes u : \ell_{\infty}(\mathscr{C}) \otimes_{\min} E \to \ell_{\infty}(\mathscr{C}) \otimes_{\max} B\|. $$

Thus Lemma 1.4 tells us that if $B$ satisfies (1.1)

$$ (1.3) \qquad \|u\|_{mM} = \|Id_{\mathscr{C}} \otimes u : \mathscr{C} \otimes_{\min} E \to \mathscr{C} \otimes_{\max} B\|. $$

If moreover $A$ has the WEP (with $B$ still satisfying (1.1)) then for any $u : A \to B$ we have

$$ (1.4) \qquad \|u\|_{mM} = \|Id_{\mathscr{C}} \otimes u : \mathscr{C} \otimes_{\max} A \to \mathscr{C} \otimes_{\max} B\|. $$

**Proposition 1.5.** A separable WEP $C^*$-algebra $A$ has the LP if and only if $Id_A$ is $\ell_{\infty}(\mathscr{C})$-nuclear, or equivalently if and only if

$$ \|Id_A\|_{mM} = 1. $$

Proof. Assume $\|Id_A\|_{mM} = 1$. A fortiori the pair $(A, \mathscr{C})$ is nuclear so $A$ is WEP. In addition we have isometrically

$$ \ell_{\infty}(\mathscr{C}) \otimes_{\max} A = \ell_{\infty}(\mathscr{C}) \otimes_{\min} A \subset \ell_{\infty}(\mathscr{C} \otimes_{\min} A) = \ell_{\infty}(\mathscr{C} \otimes_{\max} A), $$

which means $A$ has the LP by Theorem 1.1.

Conversely, if $A$ has both LP and WEP, then $(A, \mathscr{C})$ is a nuclear pair and

$$ \ell_{\infty}(\mathscr{C}) \otimes_{\max} A \subset \ell_{\infty}(\mathscr{C} \otimes_{\max} A) = \ell_{\infty}(\mathscr{C} \otimes_{\min} A) $$

and we also have isometrically

$$ \ell_{\infty}(\mathscr{C}) \otimes_{\min} A \subset \ell_{\infty}(\mathscr{C} \otimes_{\min} A), $$

so we conclude that $\ell_{\infty}(\mathscr{C}) \otimes_{\max} A = \ell_{\infty}(\mathscr{C}) \otimes_{\min} A$. $\square$

**Lemma 1.6.** Let $E \subset A$ be a f.d. subspace of a $C^*$-algebra. Let $q : C \to B$ be a surjective $*$-homomorphism. Let $\alpha$ be either $\min$ or $\max$. Assume $C \otimes_{\alpha} A / \ker(q) \otimes_{\alpha} A \simeq B \otimes_{\alpha} A$ (which always holds for $\alpha = \max$). Then $q \otimes Id_E$ is a metric surjection from $(C \otimes E, \| \cdot \|_{\alpha})$ onto $(B \otimes E, \| \cdot \|_{\alpha})$. More precisely, this map takes the closed unit ball onto the closed unit ball.
**Proof.** The argument for this is a classical one based on the existence of nice approximate units in ideals such as $ker(q)$, essentially going back to Arveson. For the first assertion full details can be found in [20, §4.7]. For the case $\alpha = \max$, see also [20, §7.2] on the “projectivity” of the max-tensor product. The second assertion can be proved by the same principle as in Lemma A.32 in [20] or [19, Lemma 2.4.7]. □

**Remark 1.7.** If a $C^*$-algebra $C$ has LLP and is QWEP then it is WEP. Indeed, let $q: W \to C$ be the quotient map from a WEP $C^*$-algebra $W$. Since $Id_C$ locally lifts up into $W$, the map $Id_C$ is $\mathscr{C}$-nuclear, and hence $C$ has the WEP.

Let $E \subset A$ be an operator space sitting in a $C^*$-algebra $A$. Let $D$ be another $C^*$-algebra. Recall we denote (absively) by $D \otimes_{\max} E$ the closure of $D \otimes E$ in $D \otimes_{\max} A$, and we denote by $\| \|_{\max}$ the norm induced on $D \otimes_{\max} E$ by $D \otimes_{\max} A$. We define similarly $E \otimes_{\max} D$. We should emphasize that $D \otimes_{\max} E$ (or $E \otimes_{\max} D$) depends on $A$ and on the embedding $E \subset A$, but there will be no risk of confusion.

For any linear map $u: E \to C$ We denote

$$ \|u\|_{mb} = \{\|Id_{\mathscr{C}} \otimes u : \mathscr{C} \otimes_{\max} E \to \mathscr{C} \otimes_{\max} C\|\} $$

Since any separable $C^*$-algebra is a quotient of $\mathscr{C}$ it is easy to see that

$$ \|u\|_{mb} = \sup\{\|u_D : D \otimes_{\max} E \to D \otimes_{\max} C\|\} $$

where the sup runs over all possible $C^*$-algebras $D$'s.

In general

$$ (1.5) \qquad \|u\|_{cb} \le \|u\|_{mb}. $$

However, if $C$ is WEP, we clearly have (since $\mathscr{C} \otimes_{\min} C = \mathscr{C} \otimes_{\max} C$)

$$ (1.6) \qquad \|u\|_{mb} = \|u\|_{cb} $$

for any $u: E \to C$.

We set

$$ MB(E, C) = \{u : E \to C \mid \|u\|_{mb} < \infty\}, $$

and we equip it with the mb-norm.

We recall (see [20, Th. 7.4 p. 137]) that

$$ \|u\|_{mb} = \inf\{\|\tilde{u} : A \to C^{**}\|_{dec}\} $$

where the inf runs over all $\tilde{u} : A \to C^{**}$ extending $i_C u : E \to C^{**}$. Here and throughout this note

$$ i_C : C \to C^{**} $$

denotes the canonical inclusion.

In particular if $C$ is a von Neumann algebra and $E = A$, we have $\|u\|_{mb} = \|u\|_{dec}$.

For any map $u: A \to C$ (defined on the whole of $A$) we have $\|i_C u\|_{mb} = \|i_C u\|_{dec} = \|u\|_{mb} \le \|u\|_{dec}$. Moreover, if $u$ is c.p. we have $\|u\|_{dec} = \|u\|$, and in the unital case this is $= \|u(1)\|$ (see e.g. [20]).

**Proposition 1.8.** Assume $A$ separable with LP. Let $C/\mathcal{I}$ be a quotient $C^*$-algebra, let $q: C \to C/\mathcal{I}$ denote the quotient map. Then any $u \in MB(A, C/\mathcal{I})$ admits a lifting $\hat{u} : A \to C$ with $\|\hat{u}\|_{MB(A,C)} = \|u\|_{MB(A,C/\mathcal{I})}$.
Proof. See [24, Th. 4.3]. □

It is worthwhile to recall here that a WEP $C^*$-algebra locally embeds in $\mathcal{C}$ (in the sense of Definition 0.1) if and only if it has the LLP (see [21, Prop. 3.7]).

We will use the following extension property for maps from a subspace of an LLP to a WEP due (in some variant) to Kirchberg. See [20, Th. 21.4 and Remark 21.5 p. 360] for a detailed proof.

**Proposition 1.9.** Let $X$ be a separable o.s. that locally embeds in $\mathcal{C}$ and let $A$ be a WEP $C^*$-algebra. Then, for any $\varepsilon > 0$, any $u : E \to A$ defined on a f.d. subspace $E \subset X$ admits an extension $\tilde{u} : X \to A$ with $\|\tilde{u}\|_{cb} \le (1 + \varepsilon)\|u\|_{cb}$.

We will say that a linear map $u : D \to C$ between $C^*$-algebras $D, C$ locally decomasably factors (resp. locally mb-factors) through another one $A$ if there is a net of maps $u_i : D \to A$, $v_i : A \to C$ with $\|u_i\|_{dec} \|v_i\|_{dec} \le 1$ (resp. $\|u_i\|_{mb} \|v_i\|_{mb} \le 1$) such that $v_i u_i : D \to C$ tends pointwise to $u$.

**Proposition 1.10.** Let $A$ be WEP, LLP and such that $\mathcal{C}$ locally embeds in $A$. Then for any WEP and LLP $C^*$-algebra $C$, the identity map $\mathrm{Id}_C$ locally mb-factors through $A$.

*Proof.* Let $C$ be LLP. Then $C$ locally embeds in $\mathcal{C}$ and hence in $A$. Let $E \subset C$ be a f.d. subspace. Let $\varepsilon > 0$ and let $F \subset A$ such that $d_{cb}(E, F) < 1 + \varepsilon$. Let $u : E \to F$ such that $\|u\|_{cb} < 1$ and $\|u^{-1}\|_{cb} < 1 + \varepsilon$. By Proposition 1.9 and the fact that $A$ and $C$ are WEP (which guarantees $\|U\|_{mb} = \|U\|_{cb}$ for maps with range either $A$ or $C$) there are maps $U : C \to A$ and $V : A \to C$ respectively extending $u$ and $u^{-1}$ such that $\|U\|_{mb} < 1$ and $\|V\|_{mb} < 1 + \varepsilon$. Then $VU|_E = \mathrm{Id}_E$. Taking for index set the set of pairs $(E, \varepsilon)$ and setting $u_i = U$ and $v_i = (1 + \varepsilon)^{-1}V$ for $i = (E, \varepsilon)$, we obtain the announced result. □

*Remark 1.11.* Let $D$ be another $C^*$-algebra. If $\mathrm{Id}_C$ locally decomasably factors (resp. locally mb-factors) through $A$, then

$$A \otimes_{\min} D = A \otimes_{\max} D \Rightarrow C \otimes_{\min} D = C \otimes_{\max} D.$$

This assertion is easy to check using the fact that decomposable maps are “tensorizable” for the maximal tensor product (see [20, p. 138]); or more precisely, that any map $u$ between $C^*$-algebras satisfies

$$\|u\|_{cb} \le \|u\|_{mb} \le \|u\|_{dec}.$$

Moreover, assuming $A$ and $C$ separable, if $A$ has the LP then so does $C$. This can be checked using Theorem 1.1.

By Remark 1.11 we deduce from the preceding proposition:

**Corollary 1.12.** Let $A_1, A_2$ be $C^*$-algebras with WEP and LLP. Assume that $\mathcal{C}$ locally embeds in each of them. Then for any $C^*$-algebra $D$

$$A_1 \otimes_{\min} D = A_1 \otimes_{\max} D \Leftrightarrow A_2 \otimes_{\min} D = A_2 \otimes_{\max} D.$$

Moreover, assuming both separable, $A_1$ has the LP if and only if so does $A_2$.

**Proposition 1.13.** If there is an $A$ with WEP and LP such that $\mathcal{C}$ locally embeds in $A$, then for any WEP $C^*$-algebra $C$ the implication LLP $\Rightarrow$ LP holds.

*Proof.* This follows immediately from Proposition 1.10 and Remarks 1.11 and 1.7. □
**Lemma 1.14.** Let C, D be C*-algebras. Let I be any index set.
If D is separable then for any s ∈ ℓ∞(I; D) ⊗ C we have

$$
(1.7) \qquad \|s\|_{\max} = \sup \|(p_J \otimes Id_C)s\|_{\ell_\infty(J;D) \otimes \max C}
$$

where the sup runs over all countable subsets $J \subset I$ and $p_J : \ell_\infty(I;D) \to \ell_\infty(J;D)$ denotes the canonical map of restriction to $J$.

Proof. Fix ε > 0. Let x ∈ ℓ∞(I; D) viewed as a bounded function x : I → D. Let {Bn | n ∈ N} be a covering of D by disjoint sets of diameter < ε. Note that the disjoint sets Ix = x⁻¹(Bn) (n ∈ N) cover I. We choose and fix in ∈ Iin. Let J = {in}. Note that J depends on x. Let fJ : ℓ∞(J; D) → ℓ∞(I; D) be the embedding defined by

$$
f_J(y) = \sum_n 1_{I_n^x} y(i_n).
$$

Clearly ||f_J p_J(x) - x|| ≤ ε. We now apply the same idea with x replaced by a finite set {x | x ∈ S}.
We replace {I_n^x | n ∈ N} by the still countable partition

$$
\{\cap_{x \in S} I_{n(x)}^{x} \mid n \in \mathbb{N}^{S}\}.
$$

With the obvious analogue of $f_J$ we now have

$$
\forall x \in S \quad \|f_J p_J(x) - x\| \le \varepsilon.
$$

Let $s \in \ell_\infty(I; D) \otimes C$. By what precedes we can find for any $\varepsilon > 0$ a countable $J$ such that

$$
\|(f_J p_J \otimes \mathrm{Id}_C)(s) - s\|_{\max} \le \varepsilon.
$$

Therefore

$$
\|s\|_{\max} \le \|(f_J p_J \otimes \mathrm{Id}_C)(s)\|_{\max} + \varepsilon \le \|(p_J \otimes \mathrm{Id}_C)(s)\|_{\max} + \varepsilon.
$$

This proves (1.7).

<figure><img src="image_3.png" /></figure>

**Proposition 1.15.** Let X be an o.s. and C, D C*-algebras. If a linear map $u : X \to C$ is $\ell_\infty(D)$-nuclear with D separable then it is $\ell_\infty(I; D)$-nuclear for any index set I.

Proof. This is immediate by (1.7).

<figure><img src="image_4.png" /></figure>

Remark 1.16. By Proposition 1.15 we may replace $\ell_\infty(\mathcal{C})$ by $\ell_\infty(I; \mathcal{C})$ for any infinite set $I$ in (1.2).

## 2 Controllable operator spaces

To abbreviate, we introduce a new definition.

**Definition 2.1.** Let $\alpha$ be a $C^*$-norm on $\mathcal{C} \otimes A$ (e.g. $\alpha = \min$ or $\alpha = \max$). A f.d. subspace $E \subset A$ will be called $\alpha$-controllable if there is $t_E \in B_{\mathcal{C} \otimes \alpha E}$ such that for any $t \in B_{\mathcal{C} \otimes \alpha E}$ there is a unital $*$-homomorphism $\pi : \mathcal{C} \to \mathcal{C}$ such that $t = [\pi \otimes \mathrm{Id}_E](t_E)$.

The next result was stated in [24, Th. 10.5]. We include the proof for convenience.

**Proposition 2.2.** A (resp. separable) $C^*$-algebra $A$ satisfies (1.1) (resp. has the LP) if and only if any f.d. subspace $E \subset A$ is max-controllable.
**Proof.** Assume A satisfies (1.1). Let $I = B_{\mathscr{C}\otimes_{\max} E}$. Let $X = (X_T)_{T \in I} \in \ell_\infty(I, \mathscr{C} \otimes_{\max} E)$ be defined by $X_T = T$. Then clearly $\|X\| = 1$. By (1.1) we have $\|X\|_{\ell_\infty(I, \mathscr{C})\otimes_{\max} E} = 1$. Let $C = C^*(\mathbb{F})$ with $\mathbb{F}$ a sufficiently big free group so that we have a quotient $*-\text{homomorphism } q : C \to \ell_\infty(I; \mathscr{C})$. By Lemma 1.6 there is $T_E \in B_{C\otimes_{\max} E}$ lifting $X$. Since a finite set of elements of $C = C^*(\mathbb{F})$ is "supported" by a countable free subgroup, there is a copy of $\mathscr{C}$, say $\mathscr{C} \simeq C' \subset C$ such that $T_E \in C' \otimes E$ and $C' \otimes_{\max} \mathscr{C} \subset C \otimes_{\max} \mathscr{C}$ isometrically (see Lemma 0.2). Thus $T_E$ can be identified with an element $T_E \in B_{C'\otimes_{\max} E}$. Let $\pi_i : \ell_\infty(I; \mathscr{C}) \to \mathscr{C}$ denote the $i$-th coordinate. Then it is easy to check that for any $T \in I$ we have $T = [\pi \otimes Id_E](T_E)$. With $\pi = \pi_T q|_{C'}$. Since $C'$ is a copy of $\mathscr{C}$, this proves the only if part.

To check the if part, assume $E$ max-controllable and controled by $t_E \in B_{\mathscr{C}\otimes_{\max} E}$; if $t$ and $I$ are as in Theorem 1.1, with $t_i \in B_{\mathscr{C}\otimes_{\max} E}$ then for any $i \in I$ there is a unital $*-\text{homomorphism } \pi_i : \mathscr{C} \to \mathscr{C}$ such that $t_i = [\pi_i \otimes Id_E](t_E)$ and hence $t = [\pi \otimes Id_E](t_E)$ where $\pi = (\pi_i)_{i \in I} : \mathscr{C} \to \ell_\infty(I, \mathscr{C})$ is also a unital $*-\text{homomorphism}$. Then $\|t\|_{\ell_\infty(I, \mathscr{C})\otimes_{\max} E} \le \|t_E\|_{\max} \le 1$. By homogeneity (1.1) follows. $\square$

We will need to consider the analogue of the property in Proposition 2.2 for the minimal tensor product.

**Definition 2.3.** A f.d. o.s. $E$ will be called controllable if it is min-controllable. In that case we will say that $t_E$ controls $B_{\mathscr{C}\otimes_{\min} E}$ or simply that it controls $E$.

By Proposition 2.2, if A satisfies (1.1) (e.g. is separable with LP) and has the WEP then any f.d. $E \subset A$ is controllable.

**Proposition 2.4.** Let $E$ be a f.d.o.s. The following are equivalent:

(i) The space $E$ is controllable.

(ii) For any set $I$, any free group $\mathbb{F}$ and any surjective unital $*-\text{homomorphism } q : C^*(\mathbb{F}) \to \ell_\infty(I, \mathscr{C})$ the mapping $q \otimes Id_E$ is a metric surjection from $C^*(\mathbb{F}) \otimes_{\min} E$ to $\ell_\infty(I, \mathscr{C}) \otimes_{\min} E$.

(iii) Same as (ii) for $I = \mathbb{N}$ and some $q$.

**Proof.** (i) $\Rightarrow$ (ii) : For any $i \in I$ and $t_i \in B_{\mathscr{C}\otimes_{\min} E}$ there is a unital $*-\text{homomorphism } \pi_i : \mathscr{C} \to \mathscr{C}$ such that $t_i = (\pi_i \otimes Id_E)(t_E)$. Let $\pi = (\pi_i) : \mathscr{C} \to \ell_\infty(I, \mathscr{C})$. By the LP of $\mathscr{C}$ there is a unital c.p. map $\hat{\pi} : \mathscr{C} \to C^*(\mathbb{F})$ lifting $q$. Let $\hat{t} = (\hat{\pi} \otimes Id_E)(t_E)$. Then $(q \otimes Id_E)(\hat{t}) = (t_i)$. This shows that (ii) holds.

(ii) $\Rightarrow$ (i) : Let $I = B_{\mathscr{C}\otimes_{\min} E}$, and let $t \in \ell_\infty(I, \mathscr{C} \otimes E) \simeq \ell_\infty(I, \mathscr{C}) \otimes E$ be defined by $t_i = i$. Let $t_E \in B_{C^*(\mathbb{F})\otimes_{\min} E}$ be a lifting of $t$. Let $q_i : C^*(\mathbb{F}) \to \mathscr{C}$ be the $i$-th coordinate of $q$. We have $(q_i \otimes Id_E)(t_E) = t_i$. Using Lemma 0.2 we may assume $\mathbb{F} = \mathbb{F}_\infty$ and $t_E \in \mathscr{C}$ and we obtain (i).

(ii) $\Rightarrow$ (iii) is trivial.

(iii) $\Rightarrow$ (ii) : Assume (iii). By the same reasoning as for (ii) $\Rightarrow$ (i) we obtain $t_E \in B_{\mathscr{C}\otimes_{\min} E}$ such that $\{(q_n \otimes Id_E)(t_E) \mid n \in \mathbb{N}\}$ is dense in $B_{\mathscr{C}\otimes_{\min} E}$. Let $I$ be an arbitrary set. Observe that by the separability of $\mathscr{C}$ any finite set of elements of $\ell_\infty(I, \mathscr{C})$ can be approximated by ones that are constant on a common countable partition of $I$. With this observation, the argument for (i) $\Rightarrow$ (ii) now shows that the image under $q \otimes Id_E$ of $B_{\mathscr{C}\otimes_{\min} E}$ is dense in the unit ball of $\ell_\infty(I, \mathscr{C})$. By the open mapping theorem, this means that (ii) holds. $\square$

By well known properties of ideals in $C^*$-algebras (see e.g. [20, p. 103]) it can be shown that if $E$ satisfies (ii) (or (iii)) in Proposition 2.4 then any subspace of $E$ also does. This implies the next statement which can also be derived from Theorem 2.9 below.

**Corollary 2.5.** Any subspace of a f.d. controllable $E$ is controllable.
Remark 2.6. Let $q: C^*(\mathbb{F}) \to \ell_\infty(I, \mathscr{C})$ be as in Proposition 2.4. Let $\mathcal{I} = \ker(q)$ so that $\ell_\infty(I, \mathscr{C}) = C^*(\mathbb{F})/\mathcal{I}$. With the latter identification, we have a natural map

$$ \Phi_E : \ell_\infty(I, \mathscr{C}) \otimes_{\min} E \to [C^*(\mathbb{F}) \otimes_{\min} E]/[\mathcal{I} \otimes_{\min} E], $$

and $E$ is controllable if and only if $\|\Phi_E\| = 1$. Assume that $E \subset B(H)$ is a unital subspace spanned by unitary operators. Let $A$ be the $C^*$-algebra generated by $E$. Note that $[C^*(\mathbb{F}) \otimes_{\min} E]/[\mathcal{I} \otimes_{\min} E]$ naturally embeds in $[C^*(\mathbb{F}) \otimes_{\min} A]/[\mathcal{I} \otimes_{\min} A]$ so that $\Phi_A$ can be identified with the unital $*$-homomorphism extending $\Phi_E$. Then the linearization trick of [18] shows that $\|\Phi_E\|_{cb} = 1$ if and only if $\|\Phi_A\| = 1$, or equivalently if and only if every f.d. subspace of $A$ is controllable. It is easy to check that $\|Id_{M_k} \otimes \Phi_E\| = 1$ if and only if $M_k(E)$ is controllable. Thus $M_k(E)$ is controllable for all $k \ge 1$ if and only if every f.d. subspace of $A$ is controllable.

Remark 2.7. Let $A$ be a $C^*$-algebra. The assertion that (ii) holds for any f.d. subspace $E \subset A$ is equivalent to:

(ii)' The sequence

$$ \{0\} \to \mathcal{I} \otimes_{\min} A \to C^*(\mathbb{F}) \otimes_{\min} A \to \ell_\infty(I, \mathscr{C}) \otimes_{\min} A $$

is exact where $\mathcal{I} = \ker(q)$. Therefore, any f.d. subspace of an exact $C^*$-algebra is controllable.

**Lemma 2.8.** Let $u: E \to B$ be a linear map from a f.d.o.s. to a $C^*$-algebra.

For any $C^*$-algebra $C$, we denote $\|u\|_C = \|Id_C \otimes u: C \otimes_{\min} E \to C \otimes_{\max} B\|$.

If $E$ is controllable with $t_E$ controlling $B_{\mathscr{C} \otimes_{\min} E}$ we have $\|u\|_{\mathscr{C}} = \|[Id_{\mathscr{C}} \otimes u](t_E)\|_{\max}$, and $\|u\|_{\ell_\infty(\mathscr{C})} = \|u\|_{\mathscr{C}}$ or equivalently by (1.2)

$$ (2.1) \qquad \|u\|_{mM} = \|u\|_{\mathscr{C}} = \|[Id_{\mathscr{C}} \otimes u](t_E)\|_{\max}. $$

If $E$ is controllable and $B$ WEP then $\|u\|_{mM} = \|u\|_{cb}$.

*Proof.* Assume that $t_E \in B_{\mathscr{C} \otimes_{\min} E}$ controls $E$. For any $t \in B_{\mathscr{C} \otimes_{\min} E}$ there is a unital $*$-homomorphism $\pi_t: \mathscr{C} \to \mathscr{C}$ such that $t = [\pi_t \otimes Id_E](t_E)$. We have $[Id_{\mathscr{C}} \otimes u](t) = [\pi_t \otimes u](t_E) = [\pi_t \otimes Id_B][Id_{\mathscr{C}} \otimes u](t_E)$ and hence $\|[Id_{\mathscr{C}} \otimes u](t)\|_{\max} \le \|[Id_{\mathscr{C}} \otimes u](t_E)\|_{\max}$, which proves the first equality.

Let $I$ be any set, let $t$ be in the unit ball of $\ell_\infty(I, \mathscr{C}) \otimes_{\min} E$ or equivalently of $\ell_\infty(I, \mathscr{C} \otimes_{\min} E)$. We have $t = (t_i)_{i \in I}$ with $t_i \in B_{\mathscr{C} \otimes_{\min} E}$. For any $i \in I$ there is a unital $*$-homomorphism $\pi_i: \mathscr{C} \to \mathscr{C}$ such that $t_i = [\pi_i \otimes Id](t_E)$ and hence $t = [\pi \otimes Id_E](t_E)$ where $\pi = (\pi_i)_{i \in I}: \mathscr{C} \to \ell_\infty(I, \mathscr{C})$ is a unital $*$-homomorphism. Now $(Id_{\ell_\infty(I, \mathscr{C})} \otimes u)(t) = [\pi \otimes u](t_E) = [\pi \otimes Id_B][Id_{\mathscr{C}} \otimes u](t_E)$ and hence $\|(Id_{\ell_\infty(I, \mathscr{C})} \otimes u)(t)\|_{\ell_\infty(I, \mathscr{C}) \otimes_{\max} B} \le \|[Id_{\mathscr{C}} \otimes u](t_E)\|_{\mathscr{C} \otimes_{\max} B} = \|u\|_{\mathscr{C}}$. This shows that $\|u\|_{mM} \le \|u\|_{\mathscr{C}}$. The converse is obvious. If $B$ has the WEP then $\mathscr{C} \otimes_{\min} B = \mathscr{C} \otimes_{\max} B$ and hence $\|u\|_{\mathscr{C}} = \|u\|_{cb}$. $\square$

**Theorem 2.9.** Let $E \subset B(H)$ be a f.d.o.s. The following assertions are equivalent:

(i) $E$ is controllable.

(ii) The inclusion $i_E: E \to B(H)$ satisfies $\|i_E\|_{mM} = 1$.

(iii) There is a completely isometric map $u$ from $E$ to a $C^*$-algebra $B$ such that $\|u\|_{mM} \le 1$.

(iv) There is a completely isometric map $\hat{u}$ from $E$ to $\mathscr{C}$ such that $\|\hat{u}\|_{mM} \le 1$.

(v) There is a c.i. embedding $f: E \to \mathscr{C}$ such that the min and max norms coincide on $\mathscr{C} \otimes f(E)$.
Moreover, if E is controllable and B WEP (e.g. $B = B(H)$) then any completely isometric $u: E \to B$ satisfies $\|u\|_{mM} = 1$.

**Proof.** (i) ⇒ (ii): Since $B(H)$ has the WEP this follows from Lemma 2.8.

(ii) ⇒ (iii) is trivial.

(iii) ⇒ (iv): Assume (iii). Using an embedding $B \subset B(\mathcal{H})$ we may assume $B = B(\mathcal{H})$. Let $u: E \to B$ be completely isometric. Let $\mathbb{F}$ be a big enough free group so that there is a quotient map $q: C^*(\mathbb{F}) \to B$. Let $v: u(E) \to C^*(\mathbb{F})$ be a linear lifting. Let $T \in B_{\mathcal{C}\otimes_{\min}E}$ controlling $B_{\mathcal{C}\otimes_{\min}E}$. Then $\mathcal{C}\otimes_{\min}B = \mathcal{C}\otimes_{\max}B$ (which holds since $B = B(\mathcal{H})$ has the WEP) implies $\|(Id_{\mathcal{C}} \otimes u)(T)\|_{\max} \le 1$. Let $F \subset \mathcal{C}$ be f.d. such that $T \in F \otimes E$. By Lemma 1.6 there is $\hat{T} \in B_{F \otimes_{\max}C^*(\mathbb{F})}$ lifting $(Id \otimes u)(T)$. Let $0 \le \sigma_i \le 1$ be an approximate unit in $I = \ker(q)$. Let $w_i(x) := (1-\sigma_i)x$, $x \in C^*(\mathbb{F})$. Then $(Id_{\mathcal{C}} \otimes vu)(T) - \hat{T} \in F \otimes I$ and hence $\lim_i \|(Id_{\mathcal{C}} \otimes w_i)[(Id_{\mathcal{C}} \otimes vu)(T) - \hat{T}]\|_{\max} = 0$. Let $u_i = w_ivu: E \to C^*(\mathbb{F})$. We have $\lim_i \|(Id_{\mathcal{C}} \otimes u_i)(T)\|_{\max} \le 1$. Since $T$ controls $E$ it follows recalling (1.3) and (2.1)

$$ \lim_i \|u_i\|_{mM} \le 1, $$

and since $u_i$ lifts $u$, Arveson's principle gives us a lifting $\hat{u}: E \to C^*(\mathbb{F})$ with $\|\hat{u}\|_{mM} = 1$. Since $u$ is c.i. so is $\hat{u}$. Lastly, being separable the range of $\hat{u}$ is included in a copy $\mathcal{C}'$ of $\mathcal{C}$ sitting in $C^*(\mathbb{F})$ and by Lemma 0.2 we obtain (iv).

(iv) ⇒ (v): Assume (iv). A fortiori we have $\|\hat{u}\|_{\mathcal{C}} \le 1$. This means that $f = \hat{u}$ satisfies (v).

(v) ⇒ (i): Note $f(E) \subset \mathcal{C}$ is max-controllable by Proposition 2.2. If (v) holds it must be min-controllable, i.e. controllable, and $f(E)$ is c.i. to $E$, so $E$ is controllable.

When $B$ is WEP, the last assertion follows from the last one in Lemma 2.8. □

**Remark 2.10.** An alternate proof of (iii) ⇒ (iv) can be based on the local convexity in Theorem 4.1.

**Corollary 2.11.** Any controllable f.d.o.s. embeds completely isometrically in $\mathcal{C}$.

**Remark 2.12.** By [12] there are 3-dimensional o.s. that do not embed c.i. in $\mathcal{C}$ and hence are not controllable.

**Remark 2.13.** Let $E$ be a f.d.o.s. By [12] (see also [19, p. 351-352] or [20, p. 351]) $E$ is c.i. to a subspace of $\mathcal{C}$ if and only if the same is true for $E^*$. Incidentally, Corollary 2.11 can be deduced from Proposition 2.4 after observing that any f.d.o.s. $F$ embeds c.i. in $\ell_\infty(\mathcal{C})$ and applying this to $F = E^*$. The next statement shows that $E$ is controllable if and only if its dual o.s. $E^*$ is c.i. to a rigid subspace of $\mathcal{C}$.

**Lemma 2.14.** Let $E \subset \mathcal{C}$ be a f.d. subspace. Then $E$ is controllable if and only if there is a completely isometric (c.i. in short) embedding $f: E^* \to \mathcal{C}$ on the dual o.s. such that any c.c. $v: f(E^*) \to \mathcal{C}$ is the restriction of a unital $*-\text{homomorphism } \pi: \mathcal{C} \to \mathcal{C}$.

**Proof.** Assume $E$ controllable. We know by Remark 2.13 that there is a c.i. map $h: E^* \to \mathcal{C}$. The latter corresponds to a tensor $t \in \mathcal{C} \otimes E$ with $\|t\|_{\min} = 1$. Since $E$ is controllable, there is $t_0 \in B_{\mathcal{C}\otimes_{\min}E}$ such that for any $s \in B_{\mathcal{C}\otimes_{\min}E}$ there is a unital $*-\text{homomorphism } \pi_s: \mathcal{C} \to \mathcal{C}$ such that $s = \pi_s \otimes Id_E(t_0)$. Let $f: E^* \to \mathcal{C}$ be associated to $t_0$. In particular we have $t = \pi_t \otimes Id_E(t_0)$, or equivalently $h = \pi_t f$. This shows that $f: E^* \to \mathcal{C}$ is also a c.i. embedding. Now for any c.c. $v: f(E^*) \to \mathcal{C}$, $vf: E^* \to \mathcal{C}$ is c.c. Let $t_{vf}: E \otimes \mathcal{C}$ be the associated tensor so that $\|t_{vf}\|_{\min} = 1$. Using $s = t_{vf}$ we find $\pi_s$ such that $s = \pi_s \otimes Id_E(t_0)$, or equivalently $vf = \pi_s f$ which means

$$ v|_{f(E^*)} = \pi_s|_{f(E^*)}. $$
Conversely, given $f$, we set $F = f(E^*)^*$, and let $t_0 \in \mathcal{C} \otimes F$ be associated to the inclusion $f(E^*) \to \mathcal{C}$. Let $t \in B_{\mathcal{C}\otimes_{\min F}}$, then $t$ is associated to a c.c. map $v : f(E^*) \to \mathcal{C}$, so it follows easily that $F$ is controllable. Since $F$ and $E$ are c.i. the same is true for $E$. $\square$

Let $E$ be a f.d. operator space. Recall

$$ex(E) = \inf\{\|u\|_{cb}\|u^{-1}\|_{cb}\}$$

where the inf runs over all $n \ge \dim(E)$ and all isomorphisms $u$ from $E$ to a subspace of $M_n$.

Let $C_u^*(X)$ denote the unital universal $C^*$-algebra of an operator space $X$ with the canonical inclusion $k_X : X \to C_u^*(X)$. This unital $C^*$-algebra is characterized by the property that any c.c. map $v : X \to D$ into an arbitrary unital $C^*$-algebra admits a unique extension to a unital $*$-homomorphism $\dot{v} : C_u^*(X) \to D$. The algebra $C_u^*(X)$ is generated by $X$, i.e. it is the smallest unital $C^*$-subalgebra containing $k_X(X)$. See e.g. [19, p.160] or [20, §2.7] for details.

A separable operator space $X$ is said to have the OLP if any c.c. $u : X \to C/I$ into an arbitrary quotient $C^*$-algebra admits a c.c. lifting. Ozawa proved in [16] that this holds if and only if $C_u^*(X)$ has the LP. The next statement is but a simple reformulation of the OLP. We state this here to emphasize the comparison with the subspaces that we call rigid.

**Proposition 2.15.** A separable operator space $X$ has the OLP if and only if there is a c.i. embedding $f : X \to \mathcal{C}$ such that any c.c. $v : f(X) \to D$ into an arbitrary unital $C^*$-algebra admits an extension $\hat{v} : \mathcal{C} \to D$ that is a unital $*$-homomorphism.

*Proof.* Assume $X$ has the OLP. By [16] this holds if and only if $C_u^*(X)$ has the LP. We set $k = k_X : X \to C_u^*(X)$ for simplicity. Let $q : \mathcal{C} \to C_u^*(X)$ be a surjective unital $*$-homomorphism. Then $k$ admits a c.c. lifting $f : X \to \mathcal{C}$, and the latter extends to a unital $*$-homomorphism $\sigma : C_u^*(X) \to \mathcal{C}$. Note that $\sigma$ is a lifting of the identity on $C_u^*(X)$, and hence $\sigma$ defines a $*$-isomorphism between $C_u^*(X)$ and $\sigma(C_u^*(X))$. Let $\mu = \sigma^{-1} : \sigma(C_u^*(X)) \to C_u^*(X)$. Let $P = \sigma q : \mathcal{C} \to \mathcal{C}$. Note that $P$ is a projection onto $\sigma(C_u^*(X))$ and $P$ is a unital $*$-homomorphism. Let $v : f(X) \to D$ be a c.c. map. Since $vf : X \to D$ is c.c. it admits an extension $\pi : C_u^*(X) \to D$ which is a unital $*$-homomorphism. The composition $\hat{v} := \pi \mu P : \mathcal{C} \to D$ is a unital $*$-homomorphism that extends $v : f(X) \to D$. Conversely, assume there is $f$ as in Proposition 2.15. We have a unital $*$-homomorphism $\hat{f} : C_u^*(X) \to \mathcal{C}$ extending $f$, and also a unital $*$-homomorphism $\pi : \mathcal{C} \to C_u^*(X)$ extending $f|_{f(X)}^{-1}$. Clearly the composition $\pi \hat{f}$ is the identity on $C_u^*(X)$, which shows the latter factors through $\mathcal{C}$ via unital $*$-homomorphisms. Since $\mathcal{C}$ has the LP, $C_u^*(X)$ also does. Equivalently, $X$ has the OLP. $\square$

**Corollary 2.16.** (i) Any f.d. $E$ with $ex(E) = 1$ (e.g. $E = \ell_\infty^n$ or $E = M_n$) is controllable. (ii) More generally, if $E$ locally embeds in a WEP and LP $C^*$-algebra (e.g. in a nuclear algebra), then $E$ is controllable.

*Proof.* By [16] a f.d. operator space $E^*$ satisfies the OLP if and only if $ex(E) = 1$. Thus (i) follows from Proposition 2.15 and Lemma 2.14. Let $u : E \to B$ be an inclusion with $B$ having WEP and LP. By Proposition 1.5 we have $\|u\|_{mM} = 1$ and hence (ii) follows from Theorem 2.9. $\square$

*Remark 2.17.* Let $n \ge 3$. We conjecture that $\ell_1^n$ is not controllable. Equivalently (see Lemma 2.14) we conjecture that $\ell_\infty^n$ is not c.i. to a rigid subspace of $\mathcal{C}$. Note that $\ell_\infty^n$ fails the OLP and hence fails the stronger rigidity property in Proposition 2.15. Our conjecture boils down to the failure of the latter for $D = \mathcal{C}$.

Nevertheless, the next statement shows that $\ell_\infty^n$ is somewhat “close” to being c.i. to a rigid o.s.
**Proposition 2.18.** There is a c.i. and c.p. map $f : \ell_{\infty}^n \to \mathcal{C}$ such that for any unital $C^*$-algebra $D$ any contractive and positive map $u : f(\ell_{\infty}^n) \to D$ extends to a unital $*$-homomorphism $\pi : \mathcal{C} \to D$.

*Proof.* Let $I = \{w : \ell_{\infty}^n \to D \mid \text{positive contraction into } D \text{ unital separable}\}$. Let $v : \ell_{\infty}^n \to \ell_{\infty}(\{D_w\})$ be defined by $v(x) = \oplus_{w \in I} w(x)$. Then $v$ is a positive isometry. Let $q : C^*(\mathbb{F}) \to \ell_{\infty}(\{D_w\})$ be a surjective unital $*$-homomorphism, for some large enough free group $\mathbb{F}$. By a result due to Vesterstrøm [26] there is a positive (still isometric) lifting $\tilde{v} : \ell_{\infty}^n \to C^*(\mathbb{F})$. Note that $\tilde{v}$ is c.i. and c.p. because on one hand for maps into $\ell_{\infty}^n$ the norm and the cb-norm are equal, and on the other hand this also holds for positive maps on $\ell_{\infty}^n$ and the latter are c.p. It follows that any positive contraction $w : \tilde{v}(\ell_{\infty}^n) \to D$ is the restriction of a unital $*$-homomorphism. Indeed, by the same argument as in Proposition 2.2 $w\tilde{v} : \ell_{\infty}^n \to D_w$ is of the form $\pi\tilde{v}$ for some unital $*$-homomorphism $\pi : C^*(\mathbb{F}) \to D_w$. Thus we may take $f = \tilde{v}$ except that we must replace $C^*(\mathbb{F})$ by $\mathcal{C}$ using Lemma 0.2. $\square$

*Remark 2.19.* We could replace positive by $n$-positive for any fixed $n \ge 1$.

*Remark 2.20.* If we drop positive and set $I = \{w : \ell_{\infty}^n \to D \mid \text{contraction into } D \text{ unital separable}\}$, we obtain an isometric $f : \ell_{\infty}^n \to \mathcal{C}$ such that for any unital separable $C^*$-algebra $D$ any contractive map $u : f(\ell_{\infty}^n) \to D$ extends to a unital $*$-homomorphism $\pi : \mathcal{C} \to D$. But this is nothing but the embedding of $X = \ell_{\infty}^n$ equipped with its maximal o.s. structure (which has the OLP) provided by Proposition 2.15.

# 3 Main point

In the next theorem we relate the problem whether LLP $\Rightarrow$ LP for WEP $C^*$-algebras to the notion of controllable subspace. Note that (i) $\Rightarrow$ (vii) has already been proved in Proposition 1.13.

**Theorem 3.1.** *The following assertions are equivalent:*

(i) There is a separable $C^*$-algebra $A$ with WEP and LP such that $\mathcal{C}$ locally embeds in $A$ (and hence $A$ is not nuclear).

(ii) For any f.d. subspace $E \subset \mathcal{C}$ and any $\varepsilon > 0$ there is a subspace $F \subset \mathcal{C}$ with $d_{cb}(E, F) < 1 + \varepsilon$ such that the min and max norms coincide on $F \otimes \mathcal{C}$. In other words the inclusion $F \to \mathcal{C}$ is $\mathcal{C}$-nuclear.

(iii) For any f.d. subspace $E \subset \mathcal{C}$, any complete contraction $v : E \to B$ with values in a WEP $C^*$-algebra $B$ is $\ell_{\infty}(\mathcal{C})$-nuclear.

(iv) Any complete contraction $u : \mathcal{C} \to \mathcal{B}$ is $\ell_{\infty}(\mathcal{C})$-nuclear.

(v) The mapping $j$ is $\ell_{\infty}(\mathcal{C})$-nuclear.

(vi) Any f.d. subspace $E \subset \mathcal{C}$ is controllable.

(vii) For separable WEP (or QWEP) $C^*$-algebras) the LLP implies the LP.

*Proof.* (i) $\Rightarrow$ (ii) : After unitization, we may assume $A$ unital. Let $q : \mathcal{C} \to A$ be a surjective $*$-homomorphism. By the LP of $A$ we have a u.c.p. lifting $r : A \to \mathcal{C}$. Let $Y = r(A)$. Then $Y \subset \mathcal{C}$ is completely isometric to $A$. Moreover, we have a u.c.p. projection $P : \mathcal{C} \to Y$ given by $P = rq$. Therefore the min and max norms are identical on $Y \otimes \mathcal{C}$. Indeed, this follows from the WEP of $A$ once we observe

$$Y \otimes_{\max} \mathcal{C} \approx A \otimes_{\max} \mathcal{C}.$$
To check the latter note that the isomorphism $q|_Y : Y \to A$ with inverse $r : A \to Y$ induces an isomorphism $q \otimes Id_C : Y \otimes_{\max} C \to A \otimes_{\max} C$ with inverse $r \otimes Id_C : A \otimes_{\max} C \to Y \otimes_{\max} C$.
Now for any $E \subset C$ there is $F \subset Y$ with $d_{cb}(E, F) < 1 + \varepsilon$, whence $u : E \to F$ with $\|u\|_{cb}\|u^{-1}\|_{cb} < 1 + \varepsilon$. Now the min and max norms are identical on $F \otimes C \subset Y \otimes C$. This yields (ii).

(ii) $\Rightarrow$ (iii) : Let $u : E \to F$ be as before and $\varepsilon > 0$. By Lemma 1.4 the inclusion $i_F : F \to C$ is $\ell_\infty(C)$-nuclear. Then, by Proposition 1.9 and (1.6), since B has the WEP, $vu^{-1} : F \to B$ extends to a map $g : C \to B$ with $\|g\|_{mb} \le \|u^{-1}\|_{cb}(1+\varepsilon)$ such that $gi_F = vu^{-1}$. We have $v|_E = gi_F u$. It follows that if $c = \|u\|_{cb}\|u^{-1}\|_{cb}(1+\varepsilon)$ the map $(1/c)v|_E$ is $\ell_\infty(C)$-nuclear. Since this holds for any $c > 1$ we obtain (iii).

(iii) $\Rightarrow$ (iv) $\Rightarrow$ (v) : obvious.

Theorem 2.9 shows that (v) $\Rightarrow$ (vi).

(vi) $\Rightarrow$ (vii) : Assume (vi). Let $A$ be QWEP and LLP. By Remark 1.7 $A$ is automatically WEP and hence $C \otimes_{\min} E = C \otimes_{\max} E$ for any $E \subset A$. Then $A$ has the LP by the criterion in Proposition 2.2.

(vii) $\Rightarrow$ (i) : obvious by [21]. Indeed, it is proved there that there is an $A$ with WEP and LLP such that $C$ locally embeds in $A$. $\square$

*Remark 3.2.* By Lemma 2.14 the properties in Theorem 3.1 are also equivalent to

(vi)* Any f.d. subspace $E \subset C$ is c.i. to a rigid subspace of $C$.

**Proposition 3.3.** Let $A$ be any separable WEP, LLP $C^*$-algebra such that $C$ locally embeds in $A$. Let $X$ be an o.s. that locally embeds in $C$ and let $j_X : X \to B(H)$ be a c.i. embedding. There is a net of c.c. maps $v_i : X \to A$, $w_i : A \to B(H)$ such that the composition $w_i v_i : X \to B(H)$ tends pointwise to $j_X$.

*Proof.* Let $E \subset X$ be a f.d. subspace and let $\varepsilon > 0$. We have an embedding $v : E \to A$ with $\|v\|_{cb} \le 1$ and $\|v^{-1}|_{v(E)}\|_{cb} \le 1 + \varepsilon$. By Proposition 1.9 the map $v$ admits an extension $\tilde{v} : X \to A$ with $\|\tilde{v}\|_{cb} \le 1 + \varepsilon$. By the injectivity of $B(H)$ there is a map $w : A \to B(H)$ extending $j_{X|E}v^{-1}|_{v(E)} : v(E) \to B(H)$ with $\|w\|_{cb} \le 1 + \varepsilon$. Clearly $w\tilde{v}|_E = j_{X|E}$. Thus we may index the desired net by pairs $(E, \varepsilon)$ and for $i = (E, \varepsilon)$ setting $v_i = (1+\varepsilon)^{-1}\tilde{v}$ and $w_i = (1+\varepsilon)^{-1}w$ gives the announced net. $\square$

**Proposition 3.4.** Assume that the equivalent properties in Theorem 3.1 hold. Let $X$ be a separable o.s. that locally embeds in $C$. Then there is a c.i. embedding $f : X \to C$ such that the min and max norms agree on $f(X) \otimes C$.

*Proof.* Let $j_X : X \to B(H)$ be a c.i. embedding. Let $q : C^*(\mathbb{F}) \to B(H)$ be a surjective $*$-homomorphism. Our goal is to show that $j_X : X \to B(H)$ admits a lifting $f : X \to C^*(\mathbb{F})$ such that $\|f\|_{mM} = 1$. Since $j_X$ is completely isometric, so will be $f$. By Arveson's principle it suffices to show the following claim: there is a net of maps $f_i : X \to C^*(\mathbb{F})$ with $\lim_i \|f_i\|_{mM} \le 1$ such that $qf_i$ tends pointwise to $j_X$. Let $v_i, w_i$ be as in Proposition 3.3. Note $\|w_i\|_{mb} = \|w_i\|_{cb}$ by (1.6). By the LP of $A$ (see Proposition 1.8 and (1.5)) the map $w_i$ admits a lifting $\hat{w}_i : A \to C^*(\mathbb{F})$ with $\|\hat{w}_i\|_{mb} \le 1$. Moreover, $\|\hat{w}_i\|_{mM} = \|\hat{w}_i\|_{mb} \le 1$ by (1.4). Let $f_i = \hat{w}_i v_i : X \to C^*(\mathbb{F})$. Note $qf_i = w_i v_i$ and $\|f_i\|_{mM} \le \|\hat{w}_i\|_{mM} \|v_i\|_{cb} \le 1$. Thus the maps $(f_i)$ form a net such that $qf_i \to j_X$ pointwise and $\lim \|f_i\|_{mM} \le 1$. This proves our claim. By Arveson's principle, there is a c.c. map $f : X \to C^*(\mathbb{F})$ lifting $j_X$. Then $f$ is a c.i. embedding and as before (by Lemma 0.2) $\mathbb{F}$ can be replaced by $\mathbb{F}_\infty$. $\square$
**Corollary 3.5.** If the equivalent properties in Theorem 3.1 hold then any separable o.s. that locally embeds in $\mathcal{C}$ actually embeds completely isometrically in $\mathcal{C}$.

The next statement shows that the equivalent assertions in Theorem 3.1 are actually equivalent to a “global” version of (vi).

**Corollary 3.6.** For convenience let $\mathcal{C}_1$ be a copy of $\mathcal{C}$. The equivalent properties in Theorem 3.1 hold if and only if there is a c.i. embedding $f: \mathcal{C} \to \mathcal{C}_1$ such that the min and max norms (induced by the respective ones on $\mathcal{C}_1 \otimes \mathcal{C}$) coincide on $f(\mathcal{C}) \otimes \mathcal{C} \subset \mathcal{C}_1 \otimes \mathcal{C}$.

*Proof.* Assume that $f(\mathcal{C}) \otimes_{\min} \mathcal{C} = f(\mathcal{C}) \otimes_{\max} \mathcal{C}$. By (i) $\Leftrightarrow$ (v) in Theorem 2.9, any f.d. $E \subset \mathcal{C}$ is controllable, which is (vi) in Theorem 3.1. This proves the if part. Conversely, by Proposition 3.4, the assertion (vi) in Theorem 3.1 implies the existence of $f$ as in Corollary 3.6. $\square$

For convenience, let us say that a $C^*$-algebra $D$ is $j$-nuclear if $j$ is $D$-nuclear.

**Theorem 3.7.** Let $A$ be any separable WEP, LLP $C^*$-algebra such that $\mathcal{C}$ locally embeds in $A$. Let $(D_\alpha)_{\alpha \in I}$ be an arbitrary family of $j$-nuclear separable $C^*$-algebras. If $A$ has the LP then $(A, \ell_\infty(\{D_\alpha\}))$ is a nuclear pair and $j$ is $\ell_\infty(\{D_\alpha\})$-nuclear. Equivalently $\ell_\infty(\{D_\alpha\})$ is $j$-nuclear.

*Proof.* Fix $\alpha \in I$. Assume that $D_\alpha$ is separable and $j$-nuclear. By the construction in [21] there is a WEP and LLP separable $C^*$-algebra $A_\alpha$ such that $\mathcal{C}$ locally embeds in $A_\alpha$ and is such that the pair $(A_\alpha, D_\alpha)$ is nuclear. By Corollary 1.12 the pairs $(A, D_\alpha)$ are nuclear for any $\alpha \in I$. By Theorem 1.1 the pair $(A, \ell_\infty(\{D_\alpha\}))$ is nuclear. We now apply Proposition 3.3 to $X = \mathcal{C}$ with $j$ in place of $j_X$. Recalling that $\|w_i\|_{cb} = \|w_i\|_{mb}$ (see (1.6)) we observe that the map $w_i v_i: \mathcal{C} \to \mathcal{C}$ is $\ell_\infty(\{D_\alpha\})$ nuclear (since $\mathrm{Id}_A$ is so). Taking the limit it follows that $j$ itself is $\ell_\infty(\{D_\alpha\})$ nuclear. $\square$

*Remark 3.8.* Thus if there is an $A$ with WEP and LP such that $\mathcal{C}$ locally embeds in $A$, the class of $j$-nuclear $C^*$-algebras is stable by arbitrary $\ell_\infty$-sums.

# 4 A “local reflexivity” theorem

In Banach space theory, the local reflexivity (LR) principle from [15] plays an important role. For $C^*$-algebras, following seminal work by Archbold and Batty, Effros and Haagerup studied in [8] the cb-version of that principle. In [24] we proved an mb-version of the same principle. What follows is a more general version of the one in [24], suggested by our use of the norm $mM$ in the preceding section.

Let $E \subset A$ be a finite dimensional subspace of a $C^*$-algebra $A$.

**Theorem 4.1.** Let $E \subset A$ be any f.d. subspace of a $C^*$-algebra $A$. Then for any separable $C^*$-algebra $C$ we have a contractive inclusion

$$ (4.1) \qquad mM(E, C^{**}) \to mM(E, C)^{**}. $$

In other words any $u$ in the unit ball of $mM(E, C^{**})$ is the weak* limit of a net $(u_i)$ in the unit ball of $mM(E, C)$.

More generally, consider a class of $C^*$-algebras $\mathcal{D}$ that is stable by arbitrary $\ell_\infty$-sum, i.e. for any family $(D_i)_{i \in I}$ in $\mathcal{D}$ we have $\ell_\infty(\{D_i\}) \in \mathcal{D}$.

Then we define

$$ \|u\|_{mM^{\mathcal{D}}} = \sup_{D \in \mathcal{D}} \|id_D \otimes u : D \otimes_{\min} E \to D \otimes_{\max} C\|, $$
and we denote by $mM_D(E, C)$ the space of all $u : E \to C$ equipped with this norm.

When $\mathcal{D} = \{\ell_\infty(I, \mathcal{C}) | I \text{ arbitrary}\}$ and $C$ has LP then

$$ \|u\|_{mM^{\mathcal{D}}} = \|u\|_{mM}. $$

**Theorem 4.2.** Assume $\mathcal{D}$ stable by arbitrary $\ell_\infty$-sums. Let $E \subset A$ be any f.d. subspace of a $C^*$-algebra $A$. Then for any $C^*$-algebra $C$ we have a contractive inclusion

$$ (4.2) \qquad mM^{\mathcal{D}}(E, C^{**}) \to mM^{\mathcal{D}}(E, C)^{**}. $$

*Proof.* This will follow from the bipolar theorem. We first need to identify the dual of $mM^{\mathcal{D}}(E, C)$. As a vector space $mM^{\mathcal{D}}(E, C) \simeq C \otimes E^*$ and hence $mM^{\mathcal{D}}(E, C)^* \simeq C^* \otimes E$ (or say $(C^*)^{\dim(E)}$). We equip $C^* \otimes E$ with the norm $\alpha$ defined as follows. We call “admissible” any $C^*$-algebra $D$ in $\mathcal{D}$. Let $K \subset mM^{\mathcal{D}}(E, C)^*$ denote the set of those $f \in mM^{\mathcal{D}}(E, C)^*$ for which there is an admissible $D$, a functional $w$ in the unit ball of $(D \otimes_{\max} C)^*$ and $t \in B_{D \otimes_{\min} E}$, so that

$$ \forall u \in mM^{\mathcal{D}}(E, C) \quad f(u) = \langle w, [Id_D \otimes u](t) \rangle. $$

By the very definition of $\|u\|_{mM^{\mathcal{D}}(E, C)}$ we have

$$ (4.3) \qquad \|u\|_{mM^{\mathcal{D}}} = \sup\{|f(u)| \mid f \in K\}. $$

This implies that the unit ball of the dual of $mM^{\mathcal{D}}(E, C)$ is the bipolar of $K$. We will show that actually $K$ is the unit ball of $mM^{\mathcal{D}}(E, C)^*$.

First we show that $K$ is convex. Let $D_1, D_2$ be admissible $C^*$-algebras. Let $D = D_1 \oplus D_2$ with the usual $C^*$-norm. Note that $D$ is admissible. Using the easily checked identities (here the direct sum is in the $\ell_\infty$-sense)

$$ D \otimes_{\min} E = (D_1 \otimes_{\min} E) \oplus (D_2 \otimes_{\min} E), \text{ and } D \otimes_{\max} C = (D_1 \otimes_{\max} C) \oplus (D_2 \otimes_{\max} C), $$

and hence $(D \otimes_{\max} C)^* = (D_1 \otimes_{\max} C)^* \oplus_1 (D_2 \otimes_{\max} C)^*$ (direct sum in the $\ell_1$-sense), it is easy to check that $K$ is convex and hence that $K$ is the unit ball of some norm $\alpha$ on $mM^{\mathcal{D}}(E, C)^*$.

Our main point is the claim that $K$ is weak* closed. To prove this, let $(f_i)$ be a net in $K$ converging weak* to some $f \in mM^{\mathcal{D}}(E, C)^*$. Let $D_i$ be admissible $C^*$-algebras, $w_i \in B_{(D_i \otimes_{\max} C)^*}$ and $t_i \in B_{D_i \otimes_{\min} E}$ such that we have

$$ \forall u \in mM^{\mathcal{D}}(E, C) \quad f_i(u) = \langle w_i, [Id_{D_i} \otimes u](t_i) \rangle. $$

Let $D = \ell_\infty(\{D_i\})$ and let $t \in D \otimes E$ be associated to $(t_i)$. Clearly $\|t\|_{\min} \le 1$. Let $p_i : D \to D_i$ denote the canonical coordinate projection, and let $v_i \in (D \otimes_{\max} C)^*$ be the functional defined by $v_i(x) = w_i([p_i \otimes Id_C](x))$. Clearly $v_i \in B_{(D \otimes_{\max} C)^*}$ and $f_i(u) = \langle v_i, [Id_D \otimes u](t) \rangle$. Let $w$ be a weak* limit point of $(v_i)$ (or the limit point if we pass to an ultrafilter refining our net). By weak* compactness, $w \in B_{(D \otimes_{\max} C)^*}$. Then $f(u) = \lim f_i(u) = \langle w, [Id_D \otimes u](t) \rangle$. Thus we conclude $f \in K$, which proves our claim.

By (4.3) the unit ball of the dual of $mM^{\mathcal{D}}(E, C)$ is the bipolar of $K$, which is equal to its weak* closure. By what precedes, the latter coincides with $K$. Thus the gauge of $K$ is the announced dual norm $\alpha = \| \cdot \|_{mM^{\mathcal{D}}}^*$.

Let $u'' \in mM^{\mathcal{D}}(E, C^{**})$ with $\|u''\|_{mM^{\mathcal{D}}} \le 1$. By the bipolar theorem, to complete the proof it suffices to show that $u''$ belongs to the polar of $K$, or equivalently that $|f(u'')| \le 1$ for any $f \in K$. To show this consider $f \in K$ taking $u \in mM^{\mathcal{D}}(E, C)$ to $f(u) = \langle w, [Id_D \otimes u](t) \rangle$ with $D$ admissible, $w \in B_{(D \otimes_{\max} C)^*}$ and $t \in B_{D \otimes_{\min} E}$.
Observe that $[Id_D \otimes u''](t) \in D \otimes C^{**} \subset (D \otimes_{\max} C)^{**}$. Recall that $mM^D(E, C^{**}) \simeq mM^D(E, C)^{**} \simeq (C^{**})^{\dim(E)}$ as vector spaces. Thus we may view $f \in mM^D(E, C)^*$ as a weak* continuous functional on $mM^D(E, C^{**})$ to define $f(u'')$. We claim that

$$ (4.4) \qquad f(u'') = \langle w, [Id_D \otimes u''](t) \rangle, $$

where the duality is relative to the pair $((D \otimes_{\max} C)^*, (D \otimes_{\max} C)^{**})$. From this claim the conclusion is immediate. Indeed, we have $\|[Id_D \otimes u''](t)\|_{D \otimes_{\max} C^{**}} \le \|u''\|_{mM^D} \le 1$, and by the maximality of the max-norm on $D \otimes C^{**}$ we have a fortiori

$$ \|[Id_D \otimes u''](t)\|_{(D \otimes_{\max} C)^{**}} \le \|[Id_D \otimes u''](t)\|_{D \otimes_{\max} C^{**}} \le 1. $$

Therefore $|f(u'')| = |\langle w, [Id_D \otimes u''](t) \rangle| \le \|w\|_{(D \otimes_{\max} C)^*} \le 1$, which completes the proof modulo our claim (4.4).

To verify the claim, note that the identity (4.4) holds for any $u'' \in mM^D(E, C)$. Thus it suffices to prove that the right-hand side of (4.4) is a weak* continuous function of $u''$ (which is obvious for the left-hand side). To check this one way is to note that $t \in D \otimes E$ can be written as a finite sum $t = \sum d_k \otimes e_k$ ($d_k \in D, e_k \in E$) and if we denote by $\dot{w} : D \to C^*$ the linear map associated to $w$ we have

$$ \langle w, [Id_D \otimes u''](t) \rangle = \sum_k \langle w, [d_k \otimes u''(e_k)] \rangle = \sum_k \langle \dot{w}(d_k), u''(e_k) \rangle, $$

and since $\dot{w}(d_k) \in C^*$ the weak* continuity as a function of $u''$ is obvious, completing the proof. $\square$

**Remark 4.3.** Let $E \subset A$ be f.d. and let $\mathcal{D}$ be as before stable by arbitrary $\ell_\infty$-sums. For the sake of the comparison we wish to make below, we define for any $u : E \to C$,

$$ \gamma_1(u) = \sup_{D \in \mathcal{D}} \|Id_D \otimes u : D \otimes_{\min} E \to D \otimes_{\max} C\|. $$

For instance, if $\mathcal{D}$ is formed of WEP $C^*$-algebras and if $\{M_n | n \ge 1\}$ locally embeds in $\mathcal{D}$, then

$$ \gamma_1(u) = \|Id_{\mathcal{B}} \otimes u : \mathcal{B} \otimes_{\min} E \to \mathcal{B} \otimes_{\max} C\|. $$

In that case $\gamma_1$ is related to the LLP. We also wish to consider

$$ \gamma_2(u) = \sup_{D \in \mathcal{D}} \|Id_D \otimes u : D \otimes_{\max} E \to D \otimes_{\max} C\|. $$

The preceding proof (resp. the similar one in [24]) shows that, without any additional assumption on $A, D, C$, for $\gamma = \gamma_1$ (resp. for $\gamma = \gamma_2$) we have a contractive inclusion

$$ (4.5) \qquad \Gamma(E, C^{**}) \to \Gamma(E, C)^{**}, $$

where $\Gamma(E, C)$ denotes the space of all maps $u : E \to C$ equipped with the norm $\gamma$.

Assume $A$ has the LP or satisfies (1.1). Then when either $\mathcal{D} = \{\mathcal{C}\}$ or $\mathcal{D} = \{\ell_\infty(I, \mathcal{C}) | I \text{ arbitrary}\}$ (or when $\mathcal{D}$ is the class of all $C^*$-algebras) we have $\gamma_2(u) = \|u\|_{mb}$. In that case (4.5) is nothing but the LR principle from [24, Th. 5.2], which is equivalent to the LP of $A$.

If $\mathcal{D}$ is the collection formed of all $\ell_\infty(I)$'s (i.e. we replace $\mathcal{C}$ by $\mathbb{C}$!) then $\gamma_1(u) = \gamma_2(u) = \|u\|$ and we recover the classical LR principle for Banach spaces (see [15]).

If $C$ (resp. $A$) is nuclear and if $\{M_n\}$ locally embeds in $\mathcal{D}$, then $\gamma_1(u) = \|u\|_{cb}$ (resp. $\gamma_2(u) = \gamma_1(u)$ for all $\mathcal{D}$) which is consistent with the fact that nuclearity implies LR (resp. LP).
**Corollary 4.4.** Let $\mathcal{D}$ be as in Theorem 4.2. Let $C/\mathcal{I}$ be a quotient $C^*$-algebra with quotient map $q: C \to C/\mathcal{I}$. Let $E \subset A$ a f.d. subspace. Then any $u: E \to C/\mathcal{I}$ admits a lifting $\hat{u}: E \to C$ such that

$$ \|\hat{u}\|_{mM^{\mathcal{D}}} = \|u\|_{mM^{\mathcal{D}}}. $$

*Proof.* Replacing $C/\mathcal{I}$ by $(C/\mathcal{I})^{**}$ and using the morphism lifting $(C/\mathcal{I})^{**}$ up to $C^{**}$ we can lift $u$ to a map $v: E \to C^{**}$ with $\|v\|_{mM^{\mathcal{D}}} = \|u\|_{mM^{\mathcal{D}}}$. Then by Theorem 4.2 there is a net $v_i: E \to C$ with $\|v_i\|_{mM^{\mathcal{D}}} \le 1$ tending pointwise weak* to $v$. By the usual Mazur argument passing to suitable convex hulls we get a net such that $qv_i$ tends pointwise (in norm) to $u$. For the final point we use Arveson's principle, observing that the classes

$$ \mathcal{F}(E, C) = \{u: E \to C \mid \|u\|_{mM^{\mathcal{D}}} \le 1\} $$

are admissible.

□

**Acknowledgement.** I am grateful to Jean Roydor for stimulating conversations.

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