# A CONTINUUM OF C * -NORMS ON B(H) ⊗ B(H) AND RELATED TENSOR PRODUCTS

Narutaka Ozawa, Gilles Pisier

<figure><img src="image_1.png" /></figure>

► To cite this version:

Narutaka Ozawa, Gilles Pisier. A CONTINUUM OF C * -NORMS ON B(H) ⊗ B(H) AND RELATED TENSOR PRODUCTS. Glasgow Mathematical Journal, 2016, 58, pp.433-443. hal-05460838

HAL Id: hal-05460838

https://hal.science/hal-05460838v1

Submitted on 15 Jan 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# A CONTINUUM OF C*-NORMS ON $\mathbb{B}(H) \otimes \mathbb{B}(H)$ AND RELATED TENSOR PRODUCTS

NARUTAKA OZAWA AND GILLES PISIER

**ABSTRACT.** We prove that the set of C*-norms on the algebraic tensor product $\mathbb{B}(\ell_2) \otimes \mathbb{B}(\ell_2)$ is infinite with cardinality at least $2^{\aleph_0}$. Moreover there is a family with cardinality $2^{\aleph_0}$ of injective tensor product functors for C*-algebras in Kirchberg's sense. Let $B = \bigoplus_m M_{N(m)}$ for some sequence $N(1) < N(2) < \dots$. We also show that the set of C*-norms on $B \otimes \mathbb{B}(\ell_2)$ has cardinality equal to $2^{2^{\aleph_0}}$. The same result holds on $B \otimes M$ for any von Neumann algebra $M$ except for the nuclear ones i.e. those that are a finite direct sum of tensor products of a commutative algebra with a matrix algebra.

A norm $\alpha$ on an involutive algebra $A$ is called a C*-norm if it satisfies

$$ \forall x \in A \quad \alpha(x^*x) = \alpha(x)^2 $$

in addition to $\alpha(x^*) = \alpha(x)$ and $\alpha(xy) \le \alpha(x)\alpha(y)$ for all $x, y \in A$. After completion, $(A, \alpha)$ yields a C*-algebra. While it is well known that C*-algebras have a unique C*-norm, it is not so for involutive algebras before completion. In particular, it is well known that the algebraic tensor product $A \otimes B$ of two C*-algebras may admit distinct C*-norms. In general, there is a minimal C*-norm on $A \otimes B$ and a maximal one denoted respectively by $\|\|_{\min}$ and $\|\|_{\max}$. When two C*-norms on $A \otimes B$ are equivalent, they must coincide since the completion has a unique C*-norm. The C*-algebras $A$ such that $\|\|_{\min} = \|\|_{\max}$ on $A \otimes B$ (or equivalently $A \otimes B$ has a unique C*-norm) for any other C*-algebra $B$ are called nuclear. Since they were introduced in the 1950's, they have been extensively studied in the literature, notably in the works of Takesaki, Lance, Effros and Lance, Choi and Effros, Connes, Kirchberg, and many more. We refer to [15] or to [3] for an account of these developments.

In his 1976 paper [17], Simon Wassermann proved that $\mathbb{B}(H)$ is not nuclear when $H = \ell_2$ (or any infinite dimensional Hilbert space $H$). Here $\mathbb{B}(H)$ denotes the C*-algebra formed of all the bounded linear operators on $H$. This left open the question whether $\|\|_{\min} = \|\|_{\max}$ on $\mathbb{B}(H) \otimes \mathbb{B}(H)$. The latter was answered negatively in [7]. Curiously however the proofs in [7] only establish the existence of two inequivalent C*-norms on $\mathbb{B}(H) \otimes \mathbb{B}(H)$, namely the minimal and maximal ones, leaving open the likely existence of many more, which is the main result of this note.

<footnote>The author was partially supported by JSPS (23540233).</footnote>
It follows from [7] that the min and max norms are not equivalent on $M \otimes N$ for any pair $M, N$ of von Neumann algebras except if either $M$ or $N$ is nuclear, in which case, of course, the min and max norms are equal. In [17] Wassermann showed that a von Neumann algebra $M$ is nuclear iff it is “finite type I of bounded degree”. This means that $M$ is (isomorphic to ) a finite direct sum of tensor products of a commutative algebra with a matrix algebra.

In the first part of this note, we prove that there is at least a continuum of different (and hence inequivalent) $C^*$-norms on the algebraic tensor product $\mathbb{B}(\ell_2) \otimes \mathbb{B}(\ell_2)$. As a corollary, we obtain a continuum of injective tensor product functors for $C^*$-algebras in the sense of Kirchberg [9].

Let $B \subset \mathbb{B}(H)$ be any “block diagonal” subalgebra, with blocks formed of matrix algebras of sizes $N(m) \times N(m)$ with $N(m)$ unbounded. The assertion that there are at least two distinct $C^*$-norms on $\mathbb{B}(H) \otimes \mathbb{B}(H)$ (or on $M \otimes N$ with $M, N$ not nuclear) reduces to the same assertion on $B \otimes B$, and this is used in [7]. It turns out to be immediate to deduce from [7] (see Lemma 4) that the cardinality of the set of $C^*$-norms on $B \otimes B$ is $\ge c$. Unfortunately, however, we do not see how to pass from $B \otimes B$ to $\mathbb{B}(H) \otimes \mathbb{B}(H)$ in case of more than two $C^*$-norms. In any case we will show in §2 that the cardinality of the set of $C^*$-norms on $B \otimes B$ (or $B \otimes M$ with $M$ non-nuclear) is $2^c$ with $c$ denoting the continuum.

# 1. $C^*$-NORMS ON $\mathbb{B}(\ell_2) \otimes \mathbb{B}(\ell_2)$

We say a $C^*$-norm $\| \cdot \|_{\alpha}$ on $\mathbb{B}(\ell_2) \otimes \mathbb{B}(\ell_2)$ is *admissible* if it is invariant under the flip and tensorizes unital completely positive maps (i.e., for any unital completely positive maps $\phi: \mathbb{B}(\ell_2) \to \mathbb{B}(\ell_2)$ the corresponding map $\phi \otimes \text{id}$ extends to a completely positive map on the $C^*$-algebra $\mathbb{B}(\ell_2) \otimes_{\alpha} \mathbb{B}(\ell_2)$). For example, the min and max norms are admissible. Let $\| \cdot \|_{\alpha}$ be an admissible $C^*$-norm. We note that for every completely bounded map $\psi$ on $\mathbb{B}(\ell_2)$ one has

$$ \|\psi \otimes \text{id} : \mathbb{B}(\ell_2) \otimes_{\alpha} \mathbb{B}(\ell_2) \to \mathbb{B}(\ell_2) \otimes_{\alpha} \mathbb{B}(\ell_2)\|_{\text{cb}} = \|\psi\|_{\text{cb}} $$

(and likewise for the $\text{id} \otimes \psi$), since $\psi$ can be written as $\|\psi\|_{\text{cb}} S_1^* \phi(S_1 \cdot S_2^*) S_2$ for some unital completely positive map $\phi$ on $\mathbb{B}(\ell_2)$ and isometries $S_1, S_2$ on $\ell_2$ (see Theorem 1.6 in [10]). We recall the operator space duality which states that $F \otimes_{\min} E^* \subset \text{CB}(E, F)$ isometrically (see Theorem B.13 in [3] or [10, p. 40]). Namely, for any operator spaces $E, F$ and any tensor $z = \sum_k f_k \otimes e_k^* \in F \otimes E^*$, the corresponding map $\phi_z: E \to F$ given by $\phi_z(x) = \sum_k e_k^*(x) f_k$ satisfies $\|z\|_{\min} = \|\phi_z\|_{\text{cb}}$. For a finite dimensional operator space $E$, we denote by $j_E$ the “identity” element in $E \otimes E^*$. We note that $\|j_E\|_{\min} = 1$ and that $\|j_E\|_{\alpha}$ is independent of embeddings $E \hookrightarrow \mathbb{B}(\ell_2)$ and $E^* \hookrightarrow \mathbb{B}(\ell_2)$.

For each $d \in \mathbb{N}$, let $\mathcal{O}S_d$ denote the metric space of all $d$-dimensional operator spaces, equipped with the cb Banach–Mazur distance. We recall that by [7] the metric space $\mathcal{O}S_d$ is non-separable whenever $d \ge 3$. If $A$ is a separable $C^*$-algebra, then the set $\mathcal{O}S_d(A)$ of
all $d$-dimensional operator subspaces of $A$ is a separable subset of $\mathcal{OS}_d$. For an admissible $C^*$-norm $\| \cdot \|_{\alpha}$ on $\mathbb{B}(\ell_2) \otimes \mathbb{B}(\ell_2)$, we denote by $M_d^{\alpha}$ the subset of $\mathcal{OS}_d$ that consists of all $E \in \mathcal{OS}_d$ for which $\|j_E\|_{\alpha} = 1$. Note that $M_d^{\alpha}$ is a closed subset of $\mathcal{OS}_d$. For example, one has $M_d^{\max} = \mathcal{OS}_d(C^*F_{\infty})$ (see [7]).

**Theorem 1.** For every $d \in \mathbb{N}$ and every separable subset $\mathcal{L} \subset \mathcal{OS}_d$, there is an admissible $C^*$-norm $\alpha$ on $\mathbb{B}(\ell_2) \otimes \mathbb{B}(\ell_2)$ such that $M_d^{\alpha}$ is separable and contains $\mathcal{L}$. Consequently, there is a family, with the cardinality of the continuum, formed of admissible $C^*$-norms on $\mathbb{B}(\ell_2) \otimes \mathbb{B}(\ell_2)$.

We recall that the density character of a metric space $X$ is the smallest cardinality of a dense subset. Let $c$ denote the cardinality of the continuum.

**Lemma 2.** Let $\mathcal{H}$ be the Hilbert space with density character $c$ and consider $\ell_2 \subset \mathcal{H}$. Accordingly, let $\mathbb{B}(\ell_2) \subset \mathbb{B}(\mathcal{H})$ (non-unital embedding) and $\theta: \mathbb{B}(\mathcal{H}) \to \mathbb{B}(\ell_2)$ be the compression. Then for every unital completely positive map $\phi: \mathbb{B}(\ell_2) \to \mathbb{B}(\ell_2)$, there are a $*$-homomorphism $\pi: \mathbb{B}(\mathcal{H}) \to \mathbb{B}(\mathcal{H})$ and an isometry $V \in \mathbb{B}(\ell_2, \mathcal{H})$ such that $\phi(\theta(a)) = V^*\pi(a)V$ for every $a \in \mathbb{B}(\mathcal{H})$.

*Proof.* By Stinespring's Dilation Theorem (see [10, p. 24] or [3, p. 10]), there are a $*$-representation $\pi$ of $\mathbb{B}(\mathcal{H})$ on a Hilbert space $\mathcal{K}$ and an isometry $V \in \mathbb{B}(\ell_2, \mathcal{K})$ such that $\phi(\theta(a)) = V^*\pi(a)V$ for every $a \in \mathbb{B}(\mathcal{H})$. We may assume that $\pi(\mathbb{B}(\mathcal{H}))V\ell_2$ is dense in $\mathcal{K}$. Since $\phi(\theta(P_{\ell_2})) = 1$, one has $\pi(\mathbb{B}(\mathcal{H}))V\ell_2 = \pi(\mathbb{B}(\ell_2, \mathcal{H}))V\ell_2$. We claim that the density character of $\mathbb{B}(\ell_2, \mathcal{H})$ is $c$. Indeed, if we write $\mathcal{H} = \ell_2(I)$ with $|I| = c$, then $\mathbb{B}(\ell_2, \mathcal{H}) = \bigcup_{J \in [I]^N} \mathbb{B}(\ell_2, \ell_2(J))$, where $[I]^N$ is the family of countable subsets of $I$. Since $|[I]^N| = c$ and $\mathbb{B}(\ell_2)$ has density character $c$, our claim follows. It follows that $\mathcal{K}$ has density character $c$ and hence we may identify $\mathcal{K}$ with $\mathcal{H}$. $\square$

*Proof of Theorem.* Let $\mathcal{L}^* = \{E^*: E \in \mathcal{L}\}$ and take a separable unital $C^*$-algebra $A_0$ such that $\mathcal{OS}_d(A_0)$ contains a dense subset of $\mathcal{L} \cup \mathcal{L}^*$. Let $B = C^*F_{\infty} * *_{\mathbb{N}} A_0$ be the unital full free product of the full free group algebra $C^*F_{\infty}$ and countably many copies of $A_0$. Let $\{\sigma_i\}$ be the set of all unital $*$-homomorphisms from $B$ into $\mathbb{B}(\mathcal{H})$ and $\sigma = *_i \sigma_i$ be the $*$-homomorphism from $\tilde{B} = *_i B$ to $\mathbb{B}(\mathcal{H})$, which is surjective. Note that $\mathcal{OS}_d(\tilde{B}) = \mathcal{OS}_d(B)$ and hence it is separable. Denote $J = \ker \sigma$. We induce the $C^*$-norm $\beta$ on $\mathbb{B}(\mathcal{H}) \otimes \mathbb{B}(\ell_2)$ from $\tilde{B} \otimes_{\min} \mathbb{B}(\ell_2)$ through $\sigma \otimes \mathrm{id}$, i.e., for every $z \in \mathbb{B}(\mathcal{H}) \otimes \mathbb{B}(\ell_2)$ one defines

$$ \|z\|_{\beta} = \inf\{\|\tilde{z}\|_{\tilde{B} \otimes_{\min} \mathbb{B}(\ell_2)} : \tilde{z} \in \tilde{B} \otimes_{\min} \mathbb{B}(\ell_2), (\sigma \otimes \mathrm{id})(\tilde{z}) = z\}. $$

We note that for $z \in E \otimes F \subset \mathbb{B}(\mathcal{H}) \otimes \mathbb{B}(\ell_2)$, one has an isometric embedding $E \otimes_{\beta} F \hookrightarrow (B \otimes_{\min} F)/(J \otimes_{\min} F)$. Moreover, there is a lift $\tilde{z} \in B \otimes F$ such that $\|\tilde{z}\|_{\min} = \|z\|_{\beta}$. See Section 2.4 in [10].

Consider $\ell_2 \hookrightarrow \mathcal{H}$ and restrict $\beta$ to $\mathbb{B}(\ell_2) \otimes \mathbb{B}(\ell_2)$, which is still denoted by $\beta$. We claim that for every unital completely positive $\phi$ on $\mathbb{B}(\ell_2)$, the corresponding maps $\phi \otimes \mathrm{id}$ and
id $\otimes \phi$ are completely positive on $\mathbb{B}(\ell_2) \otimes_{\beta} \mathbb{B}(\ell_2)$. The latter is trivial. For the former, we use the above lemma. The $*$-homomorphism $\pi$ on $\mathbb{B}(\mathcal{H})$ induces a map on $\{\sigma_i\}$ and thus a $*$-homomorphism $\tilde{\pi}$ from $\tilde{\mathbb{B}}$ into $\tilde{\mathbb{B}}$ such that $\sigma \circ \tilde{\pi} = \pi \circ \sigma$. It follows that $\pi \otimes \text{id}$ is a continuous $*$-homomorphism on $\mathbb{B}(\mathcal{H}) \otimes_{\beta} \mathbb{B}(\ell_2)$ and hence that $\phi \otimes \text{id}$ is completely positive.

We note that $\beta = \min$ on $E \otimes \mathbb{B}(\ell_2)$ for any $E \in \mathcal{OS}_d(B)$. Let $E \subset \mathbb{B}(\ell_2) \subset \mathbb{B}(\mathcal{H})$ and consider the element $j_E \in E \otimes E^* \subset \mathbb{B}(\mathcal{H}) \otimes \mathbb{B}(\ell_2)$. If $\|j_E\|_{\beta} = 1$, then $\text{id}_E: E \to \mathbb{B}(\ell_2)$ has a completely contractive lift into $\tilde{\mathbb{B}}$. Indeed, an isometric lifting $\tilde{j}_E \in \tilde{\mathbb{B}} \otimes_{\min} E^*$ corresponds to a complete contraction $\theta: E \to \tilde{\mathbb{B}}$ for which $\sigma \circ \theta = \text{id}_E: E \hookrightarrow \mathbb{B}(\mathcal{H})$. It follows that $M_d^{\beta} \subset \mathcal{OS}_d(B)$. Finally, take the flip $\beta^{\text{op}}$ of $\beta$ and let $\alpha = \max\{\beta, \beta^{\text{op}}\}$. $\square$

We recall that a tensor product functor is a bifunctor $(A, B) \mapsto A \otimes_{\alpha} B$ which assigns in a functorial way a $C^*$-completion of each algebraic tensor product $A \otimes B$ of $C^*$-algebras $A$ and $B$. It is said to be *injective* if $A_0 \hookrightarrow A_1$ and $B_0 \hookrightarrow B_1$ gives rise to a faithful embedding $A_0 \otimes_{\alpha} B_0 \hookrightarrow A_1 \otimes_{\alpha} B_1$. See [9]. For example, the spatial tensor product functor $\min$ is injective, while the maximal one $\max$ is not.

**Corollary 3.** *There is a family with cardinality $2^{\aleph_0}$ of different injective tensor product functors.*

*Proof.* Let $\alpha$ be an admissible $C^*$-norm $\| \cdot \|_{\alpha}$ on $\mathbb{B}(\ell_2) \otimes \mathbb{B}(\ell_2)$. We will extend it to a tensor product functor. For every finite dimensional operator spaces $E$ and $F$, the norm $\| \cdot \|_{\alpha}$ is unambiguously defined (by the injectivity of $\mathbb{B}(\ell_2)$) via embeddings $E \hookrightarrow \mathbb{B}(\ell_2)$ and $F \hookrightarrow \mathbb{B}(\ell_2)$. For every $C^*$-algebras $A$ and $B$ and $z \in A \otimes B$, we find finite dimensional operator subspaces $E$ and $F$ such that $z \in E \otimes F$ and define $\|z\|_{\alpha}$ to be the $\alpha$-norm of $z$ in $E \otimes F$. $\square$

## 2. $C^*$-NORMS ON $\mathbb{B} \otimes \mathbb{B}(\ell_2)$ OR $\mathbb{B} \otimes M$

Let $(N(m))$ be any sequence of positive integers tending to $\infty$ and let

$$B = \bigoplus_m M_{N(m)}.$$

Actually, the existence of a continuum of distinct $C^*$-norms on $B \otimes B$ can be proved very simply, as a consequence of [7].

**Lemma 4.** Let $M$ be any $C^*$-algebra such that $B \otimes_{\min} M \neq B \otimes_{\max} M$. Then there is a continuum of distinct $C^*$-norms on $B \otimes M$.

*Proof.* For any infinite subset $s \subset \mathbb{N}$ we can define a $C^*$-norm $\gamma_s$ on $B \otimes M$ by setting

$$\gamma_s(x) = \max\{\|x\|_{\min}, \|(q_s \otimes \text{Id})(x)\|_{B_s \otimes_{\max} B}\},$$

where $B_s = \bigoplus_{m \in s} M_{N(m)}$ and where $q_s: B \to B_s$ denotes the canonical projection (which is a $*$-homomorphism). Let $\tilde{B}_s = B_s \oplus \{0\} \subset B$ be the corresponding ideal in $B$. We
claim that if $s' \subset \mathbb{N}$ is another infinite subset such that $s \cap s' = \emptyset$, or merely such that $t = s \setminus s'$ is infinite, then $\gamma_s \neq \gamma_{s'}$. Indeed, otherwise we would find that the minimal and maximal norms coincide on $\hat{B}_t \otimes M$, and hence (since $B$ embeds in $\hat{B}_t$ and is the range of a unital completely positive projection) on $B \otimes M$, contradicting our assumption. $\square$

By [7] this gives a continuum ($\gamma_s$) of distinct $C^*$-norms on $B \otimes B$ or on $B \otimes M$ whenever $M$ is not nuclear. Apparently, producing a family of cardinality $2^{2^{\aleph_0}}$ requires a bit more.

**Theorem 5.** There is a family of cardinality $2^{2^{\aleph_0}}$ of mutually distinct (and hence inequivalent) $C^*$-norms on $M \otimes B$ for any von Neumann algebra $M$ that is not nuclear.

**Remark 6.** Assuming $M \subset B(\ell_2)$ non-nuclear, we note that the cardinality of $B(\ell_2)$ and hence of $M \otimes B(\ell_2)$ is $c = 2^{\aleph_0}$, so the set of all real valued functions of $M \otimes B(\ell_2)$ into $\mathbb{R}$ has the same cardinal $2^{2^{\aleph_0}}$ as the set of $C^*$-tensor norms.

**Remark 7.** In the sequel, the complex conjugate $\bar{a}$ of a matrix $a$ in $M_N$ is meant in the usual way, i.e. $(\bar{a})_{ij} = \bar{a}_{ji}$. In general, we will need to consider the conjugate $\bar{A}$ of a $C^*$-algebra $A$. This is the same object but with the complex multiplication changed to $(\lambda, a) \to \bar{\lambda}a$, so that $\bar{A}$ is anti-isomorphic to $A$. For any $a \in A$, we denote by $\bar{a}$ the same element viewed as an element of $\bar{A}$. Note that $\bar{A}$ can also be identified with the opposite $C^*$-algebra $A^{op}$ which is defined as the same object but with the product changed to $(a, b) \to ba$. It is easy to check that the mapping $a \to a^*$ (or more properly $\bar{a} \to a^*$) is a (linear) $*$-isomorphism from $\bar{A}$ to $A^{op}$.

The distinction between $A$ and $\bar{A}$ is necessary in general, but not when $A = B(H)$ since in that case, using $H \simeq \bar{H}$, we have $\bar{B}(H) \simeq B(H) \simeq B(H)$, and in particular $\bar{M}_N \simeq M_N$. As a consequence, it is easy to see that for any matrix $a$ in $M_N(A)$ we have

$$ \|[\bar{a}_{ij}]\|_{M_N(\bar{A})} = \|[a_{ij}]\|_{M_N(A)}. $$

Note however that $H \simeq \bar{H}$ depends on the choice of a basis so the isomorphism $\bar{B}(H) \simeq B(H)$ is not canonical. Nevertheless, this shows that the problem whether the min and max norms are the same is identical for $B(H) \otimes B(H)$ (resp. $B \otimes B$) and for $B(H) \otimes \bar{B}(H)$ (resp. $B \otimes \bar{B}$). Let $A, C$ be any pair of $C^*$-algebra. Consider $a_1, \dots, a_n$ in $A$ and $b_1, \dots, b_n$ in $C$. Using the preceding remark we have

$$ \left\| \sum a_k \otimes \bar{b}_k \right\|_{A \otimes_{\alpha} \bar{C}} = \left\| \sum a_k \otimes b_k^* \right\|_{A \otimes_{\alpha} C^{op}} $$

for either $\alpha = \min$ or $\alpha = \max$. Moreover, we have

$$ \left\| \sum a_k \otimes \bar{b}_k \right\|_{A \otimes_{\max} \bar{C}} = \left\| \sum a_k \otimes b_k^* \right\|_{A \otimes_{\max} C^{op}} = \sup \left\{ \left\| \sum \pi(a_k) \sigma(b_k)^* \right\| \right\} $$

where the supremum runs over all commuting range pairs $\pi: A \to B(H)$, $\sigma: C \to B(H)$ with $\pi$ a representation and $\sigma$ an anti-representation on the same (arbitrary) Hilbert space $H$.
As in [7, 11], our main ingredient will be the fact that the numbers $C(n)$ defined below are smaller than $n$. More precisely, it was proved in [6] that $C(n) = 2\sqrt{n-1}$ for any $n$. However, it suffices to know for our present purpose that $C(n) < n$ for infinitely many $n$'s or even merely for some $n$. This can be proved in several ways for which we refer the reader to [7] or [10]. See also [12] for a more recent-somewhat more refined-approach.

For any integer $n \ge 1$, the constant $C(n)$ is defined as follows: $C(n)$ is the smallest constant $C$ such that for each $m \ge 1$, there is $N_m \ge 1$ and an $n$-tuple $[u_1(m), \dots, u_n(m)]$ of unitary $N_m \times N_m$ matrices such that

$$ (1) \qquad \sup_{m \neq m'} \left\| \sum_{i=1}^{n} u_k(m) \otimes \overline{u_k(m')} \right\|_{\min} \le C. $$

Throughout this note we fix $n > 2$ and a constant $C < n$ and we assume given a sequence of $n$-tuples $[u_1(m), \dots, u_n(m)]$ of unitary $N_m \times N_m$ matrices satisfying (1). By compactness (see e.g. [11]) we may assume (after passing to a subsequence) that the $n$-tuples $[u_1(m), \dots, u_n(m)]$ converge in distribution (i.e. in moments) to an $n$-tuple $[u_1, \dots, u_n]$ of unitaries in a von Neumann algebra $M$ equipped with a faithful normal trace $\tau$. In fact, if $\omega$ is any ultrafilter refining the selected subsequence, we can take for $M, \tau$ the associated ultraproduct $M_\omega$ of the family $\{M_{N(m)}\} (m \to \infty)$ equipped with normalized traces.

For any subset $s \subset \mathbb{N}$ and any $1 \le k \le n$ we denote by $u_k(s) = \oplus_m u_k(s)(m)$ the element of $B$ defined by $u_k(s)(m) = u_k(m)$ if $m \in s$ and $u_k(m) = 0$ otherwise. Thus with a slight abuse of notation we have

$$ u_k(s) = \oplus_{m \in s} u_k(m) \oplus_{m \notin s} 0 \in B. $$

Let $\tau_N$ denote the normalized trace on $M_N$. To any free ultrafilter $\omega$ on $\mathbb{N}$ is associated a tracial state on $B$ defined for any $x = (x_m) \in B$ by

$$ \varphi_\omega(x) = \lim_{\omega} \tau_{N(m)}(x_m). $$

The GNS construction applied to that state produces a representation

$$ \pi_\omega : B \to B(H_\omega). $$

It is classical that $M_\omega = \pi_\omega(B)$ is a $II_1$-factor and that $\varphi_\omega$ allows to define a trace $\tau_\omega$ on $M_\omega$ such that

$$ \forall b \in B \qquad \tau_\omega(\pi_\omega(b)) = \varphi_\omega(b). $$

For any pair $\omega, \omega'$ of free ultrafilters on $\mathbb{N}$ we denote by $\{\alpha_{\omega, \omega'}\}$ the norm defined on $B \otimes \bar{B}$ by

$$ \forall t = \sum_{k=1}^{n} a_k \otimes b_k \in B \otimes \bar{B} \qquad \alpha_{\omega, \omega'}(t) = \max\{\|t\|_{B \otimes_{\min} B}, \|[\pi_\omega \otimes \overline{\pi_{\omega'}}](t)\|_{M_\omega \otimes_{\max} M_{\omega'}}\}. $$
**Remark 8.** Let $M$ be a $C^*$-algebra equipped with a tracial state $\tau$. Then the GNS construction associated to $(M, \tau)$ produces a Hilbert space $H$, a cyclic unit vector $\xi$ in $H$ and commuting left and right actions of $M$ on $H$ (we denote the latter simply by $a \cdot h \cdot b$ for $h \in H$, $a, b \in M$) so that $x \cdot \xi = \xi \cdot x$ and $\tau(xy) = \langle x \cdot \xi \cdot y, \xi \rangle$ ($x, y \in M$). If we denote $L(a)h = a \cdot h$ and $R(a)h = h \cdot a$ then $L$ (resp. $R$) is a representation of $M$ resp. $(M^{op})$ on $B(H)$ and the ranges of $L$ and $R$ commute.

We have then for any $n$-tuple $(u_1, \dots, u_n)$ of unitaries in $M$

$$ (2) \qquad \left\| \sum_{1}^{n} u_k \otimes \bar{u}_k \right\|_{M \otimes_{\max} \overline{M}} = \left\| \sum_{1}^{n} u_k \otimes u_k^* \right\|_{M \otimes_{\max} M^{op}} = n. $$

Indeed, this is

$$ \geq \left\| \sum_{1}^{n} L(u_k) R(u_k^*) \right\| \geq \left\| \sum_{1}^{n} u_k \cdot \xi \cdot u_k^* \right\| $$

but $u_k \cdot \xi \cdot u_k^* = \xi$ hence (2) is $\geq n$ and $\leq n$ is trivial by the triangle inequality.

In particular:

**Lemma 9.** Let $\omega \neq \omega'$. Consider disjoint subsets $s \subset \mathbb{N}$ and $s' \subset \mathbb{N}$ with $s \in \omega$ and $s' \in \omega'$, and let

$$ t(s, s') = \sum_{k=1}^{n} u_k(s) \otimes \overline{u_k(s')} \in B \otimes \bar{B}. $$

Then

$$ \|t(s, s')\|_{B \otimes_{\min} \bar{B}} \leq C \quad \text{and} \quad \|[\pi_{\omega} \otimes \overline{\pi_{\omega'}}](t(s, s'))\|_{M_{\omega} \otimes_{\max} \overline{M_{\omega'}}} = n. $$

*Proof.* We have obviously

$$ \|t\|_{\min} = \sup_{(m,m') \in s \times s'} \left\| \sum u_k(m) \otimes \overline{u_k(m')} \right\| $$

hence $\|t\|_{\min} \leq C$. We now turn to the max tensor product.

Let $u_k = \pi_{\omega}(u_k(s))$ and $v_k = \pi_{\omega'}(u_k(s'))$ so that we have

$$ \|[\pi_{\omega} \otimes \overline{\pi_{\omega'}}](t(s, s'))\|_{M_{\omega} \otimes_{\max} \overline{M_{\omega'}}} = \left\| \sum u_k \otimes \bar{v}_k \right\|_{M_{\omega} \otimes_{\max} \overline{M_{\omega'}}}. $$

Now, since we assume that $[u_1(m), \dots, u_n(m)]$ converges in distribution, $(u_1, \dots, u_n)$ and $(v_1, \dots, v_n)$ must have the same distribution relative respectively to $\tau_{\omega}$ and $\tau_{\omega'}$. But this implies that there is a $*$-isomorphism $\pi$ from the von Neumann algebra $M(v) \subset M_{\omega'}$ generated by $(v_1, \dots, v_n)$ to the one $M(u) \subset M_{\omega}$ generated by $(u_1, \dots, u_n)$, defined simply by $\pi(v_k) = u_k$. Moreover, since we are dealing here with finite traces, there is a conditional expectation $P$ from $M_{\omega'}$ onto $M(v)$. Therefore the composition $Q = \pi P$ is a unital
completely positive map from $M_{\omega'}$ to $M(u)$ such that $Q(v_k) = u_k$. Since such maps preserve the max tensor products (see e.g. [3] or [10]) we have

$$ \left\| \sum u_k \otimes \bar{v}_k \right\|_{\max} \geq \left\| \sum u_k \otimes \overline{Q(v_k)} \right\|_{M(u) \otimes_{\max} \overline{M(u)}} = \left\| \sum u_k \otimes \bar{u}_k \right\|_{M(u) \otimes_{\max} \overline{M(u)}}. $$

But then by the preceding Remark we conclude that $\|t(s, s')\|_{\max} = n$. $\square$

**Remark 10.** With the preceding notation let $f_{\omega, \omega'}$ be the state defined on $B \otimes_{\max} \bar{B}$ by

$$ f_{\omega, \omega'}(a \otimes \bar{b}) = \tau_{\omega}(\pi_{\omega}(a)(Q(\pi_{\omega'}(b)))^*). $$

The preceding argument shows that

$$ f_{\omega, \omega'}(t(s, s')) = n. $$

**Theorem 11.** There is a continuum (actually a family of cardinality $2^{2^{\aleph_0}}$) of mutually distinct (and hence inequivalent) $C^*$-norms on $B \otimes B$. More precisely, the family $\{\alpha_{\omega, \omega'}\}$ indexed by pairs of free ultrafilters on $\mathbb{N}$ with $\omega \neq \omega'$ is such a family on $B \otimes \bar{B}$.

*Proof.* Let $(\omega_1, \omega'_1)$ and $(\omega_2, \omega'_2)$ be two distinct pairs of free ultrafilters on $\mathbb{N}$, with $\omega_1 \neq \omega'_1$ and $\omega_2 \neq \omega'_2$. Then either $\omega_1 \neq \omega_2$ or $\omega'_1 \neq \omega'_2$. By symmetry, we may assume $\omega_1 \neq \omega_2$. Let $s_1 \subset \mathbb{N}$ and $s_2 \subset \mathbb{N}$ be disjoint subsets such that $s_1 \in \omega_1$ and $s_2 \in \omega_2$. Let $s'_1 \subset \mathbb{N}$ be any subset disjoint from $s_1$ such that $s'_1 \in \omega'_1$. Then by Lemma 9 we have

$$ \alpha_{\omega_1, \omega'_1}(t(s_1, s'_1)) = n $$

but

$$ \alpha_{\omega_2, \omega'_2}(t(s_1, s'_1)) \leq C $$

because since $s_1 \in \omega_2$ we have $[\pi_{\omega_2} \otimes \pi_{\omega'_2}](t(s_1, s'_1)) = 0$. Since $C < n$, we conclude that $\alpha_{\omega_1, \omega'_1}$ and $\alpha_{\omega_2, \omega'_2}$ are different, and hence (automatically for $C^*$-norms) inequivalent. $\square$

**Remark 12.** It is well known (see e.g. [5, p. 146]) that the cardinality of the set of free ultrafilters on $\mathbb{N}$ is $2^{2^{\aleph_0}}$.

*Proof of Theorem 5.* If $M$ is not nuclear, by [17, Cor.1.9] there is an embedding $B \subset M$. Moreover, since $B$ is injective, there is a conditional expectation from $M$ to $B$.

Fix a free ultrafilter $\omega_0$. Recall we have defined a state $f_{\omega, \omega_0}$ on $B \otimes_{\max} \bar{B}$ in Remark 10. By the injectivity of $B$, there is a state $g_{\omega, \omega_0}$ extending $f_{\omega, \omega_0}$ on $B \otimes_{\max} \bar{M}$, obtained by composing $f_{\omega, \omega_0}$ with a conditional expectation (of the form $Id \otimes \overline{E^B}$) onto $B \otimes_{\max} \bar{B}$. Let $I_s \subset B$ be the two sided ideal formed of all $x = (x_k) \in B$ such that $k \mapsto a_k$ is supported on $s$. Note that if $s \notin \omega'$ we have

$$ (3) \qquad \forall x \in I_s \quad \forall y \in M \quad g_{\omega', \omega_0}(x \otimes \bar{y}) = 0, $$

and a fortiori

$$ (4) \qquad \forall b \in B \quad \forall x \in I_s \quad \forall y \in M \quad g_{\omega', \omega_0}(b^*x^*xb \otimes \bar{y}) = 0. $$
Let $\sigma_{\omega, \omega_0}$ be the GNS representation on $B \otimes_{\max} \bar{M}$ associated to $g_{\omega, \omega_0}$. We then define for $T \in B \otimes \bar{M}$

$$ \beta_{\omega, \omega_0}(T) = \max\{\|T\|_{\min}, \|\sigma_{\omega, \omega_0}(T)\|\}. $$

We claim that these are distinct norms on $B \otimes \bar{M}$. By the preceding Remark, this leads to the desired conclusion.

Let $\omega' \neq \omega$ and let $s$ be again such that $s \notin \omega'$, and let $s_0$ be disjoint from $s$ and such that $s_0 \in \omega_0$. Clearly, on one hand, by Remark 10, we have $\beta_{\omega, \omega_0}(t(s, s_0)) = n$. But on the other hand, by (4), $\sigma_{\omega', \omega_0}$ vanishes on $u_k(s) \otimes \bar{M}$. Therefore $\sigma_{\omega', \omega_0}$ vanishes on $t(s, s_0)$, so that $\beta_{\omega', \omega_0}(t(s, s_0)) \le C$. This proves our claim. $\square$

### 3. ADDITIONAL REMARKS

**Remark 13.** It is easy to see that any unital simple $C^*$-algebra is what algebraists call “central simple”. A unital algebra over a field is called central simple (or centrally simple) if it is simple and its centre is reduced to the field of scalars. It is classical (see e.g. [4, p. 151]) that the tensor product of two such algebras is again central simple, and a fortiori simple. The kernel of a $C^*$-seminorm on (the algebraic tensor product) $A \otimes B$ of two $C^*$-algebras is clearly an ideal. Therefore, if $A, B$ are both simple and unital, any $C^*$-seminorm on (the algebraic tensor product) $A \otimes B$ is a norm as soon as it induces a norm on each of its two factors.

**Remark 14.** Let $G$ be a discrete group such that its reduced $C^*$-algebra $A$ is simple. We can associate to any unitary representation $\pi : G \to B(H_\pi)$ a $C^*$-norm $\alpha_\pi$ on $A \otimes A$ as follows. Let $\lambda : A \to B(\ell_2(G))$ and $\rho : A \to B(\ell_2(G))$ be the left and right regular representations of $G$ linearly extended to $A$. This gives us a pair of commuting representations of $A$ on $\ell_2(G)$. By the Fell absorption principle (see e.g. [3, p. 44] or [10, p. 149]) the representation $\pi \otimes \lambda : G \to B(H_\pi \otimes \ell_2(G))$ is unitarily equivalent to $I \otimes \lambda$, and hence (since $A$ is assumed simple) it extends to a faithful representation on $A$. Similarly $I \otimes \rho : G \to B(H_\pi \otimes \ell_2(G))$ extends to a faithful representation on $A$. We define

$$ \forall a, b \in A \times A \quad \tilde{\pi}(a \otimes b) = (\pi \otimes \lambda)(a) \cdot (I \otimes \rho)(b), $$

and we denote by $\tilde{\pi}$ the canonical extension to $A \otimes A$. Then for any $x \in A \otimes A$ we set

$$ \alpha_{\pi}(x) = \|\tilde{\pi}(x)\|. $$

By the preceding remark, this is a $C^*$-norm on $A \otimes A$. However, if we restrict it to the diagonal subalgebra $D \subset A \otimes A$ spanned by $\{\lambda(t) \otimes \lambda(t) \mid t \in G\}$ we find for any $x = \sum x(t)\lambda(t) \otimes \lambda(t)$

$$ \|\tilde{\pi}(x)\| = \|\sum x(t)\pi(t) \otimes \sigma(t)\| $$

where $\sigma(t)\delta_s = \delta_{tst-1}$.
Now, if $G$ is any non-Abelian free group, $\sigma$ is weakly equivalent to $1 \oplus \lambda$ (see [2]), so we have for any such diagonal $x$ (using again $\pi \otimes \lambda \simeq I \otimes \lambda$)

$$ (5) \qquad \|\tilde{\pi}(x)\| = \max\{\|\sum x(t)\pi(t)\|, \|\sum x(t)\lambda(t)\|\}. $$

But it is known (see [14]) that there is a continuum of unitary representations on a non-Abelian free group $G$ that are “intermediate” between $\lambda$ and the universal unitary representation of $G$. More precisely, let $G = \mathbb{F}_k$ be the free group with $k > 1$ generators $g_1, \dots, g_k$. Let $S_k = \sum_{j=1}^k \delta_{g_j} + \delta_{g_j^{-1}}$. By [14, Th. 5], for any number $r \in ((2k-1)^{-1/2}, 1)$, $G$ admits a unitary representation $\pi_r$ such that

$$ \|\pi_r(S_k)\| = (2k-1)r + 1/r > 2\sqrt{2k-1}. $$

By (5) we have

$$ \|\tilde{\pi}_r(S_k)\| = (2k-1)r + 1/r, $$

and hence if we define $x_k = \sum \lambda(g_k) \otimes \lambda(g_k) \in A \otimes A$ we find

$$ \alpha_{\pi_r}(x_k) = (2k-1)r + 1/r $$

which shows that the family of $C^*$-norms $\{\alpha_{\pi_r} \mid (2k-1)^{-1/2} < r < 1\}$ are mutually distinct. Thus we obtain in this case a continuum of distinct $C^*$-norms on $A \otimes A$.

Let $M$ denote the von Neumann algebra generated by $A$ in $B(\ell_2(G))$. Since $G$ is i.c.c. $M$ is a finite factor and hence (see [15, p. 349]) is a simple $C^*$-algebra, thus again automatically central simple. The representation $\tilde{\pi}$ clearly extends to a $*$-homomorphism on $M \otimes M$ which is isometric when restricted either to $M \otimes 1$ or $1 \otimes M$. Thus we also obtain a continuum of distinct $C^*$-norms on $M \otimes M$, extending the preceding ones on $A \otimes A$.

**Remark 15.** Let $I$ be a closed ideal in a $C^*$-algebra $A$. It is well known that the maximal $C^*$-norm is “projective” in the following sense (see e.g. [3, p. 92] or [10, p. 237]): for any other $C^*$-algebra $B$, $I \otimes_{\max} B$ embeds naturally (isometrically) in $A \otimes_{\max} B$ we have a natural (isometric) identification

$$ (6) \qquad (A/I) \otimes_{\max} B = (A \otimes_{\max} B)/(I \otimes_{\max} B). $$

Let $Q(H) = B(H)/K(H)$ be the Calkin algebra. By Kirchberg's well known work [9] (see [10, p. 289] or [3, p. 105] for more details) a $C^*$-algebra $A$ is exact iff

$$ (7) \qquad Q(H) \otimes_{\min} A = (B(H) \otimes_{\min} A)/(K(H) \otimes_{\min} A). $$

Note that $K(H) \otimes_{\min} A = K(H) \otimes_{\max} A$ since $K(H)$ is nuclear. Thus, by (6), if $A$ is not exact, the minimal and maximal $C^*$-norms must differ on $Q(H) \otimes A$.
**Remark 16.** We can define a $C^*$-norm on $(A/I) \otimes B$ by setting, for any $x \in (A/I) \otimes B$,

$$ \alpha(x) = \|x\|_{(A \otimes_{\min} B)/(I \otimes_{\min} B)}. $$

More precisely, if $y \in A \otimes B$ is any element lifting $x$ i.e. such that $(q \otimes Id)(y) = x$ where $q: A \to A/I$ denotes the quotient map, we have

$$ \alpha(x) = \inf\{\|y + z\|_{\min} \mid z \in I \otimes_{\min} B\}. $$

Since $(I \otimes_{\min} B) \cap (A \otimes B) = I \otimes B$, this is indeed a norm on $(A/I) \otimes B$.

Now assume that $I$ is a nuclear ideal in $A$ so that the min and max norms coincide on $I \otimes B$. Then

$$ (8) \qquad (A/I) \otimes_{\min} B = (A/I) \otimes_{\max} B \Rightarrow A \otimes_{\min} B = A \otimes_{\max} B. $$

More precisely, it suffices to assume that $\alpha = \| \|_{\max}$, i.e. we have

$$ (9) \qquad (A \otimes_{\min} B)/(I \otimes_{\min} B) = (A/I) \otimes_{\max} B \Rightarrow A \otimes_{\min} B = A \otimes_{\max} B. $$

Indeed, this follows from (6) and $I \otimes_{\min} B = I \otimes_{\max} B$.

**Remark 17.** (Communicated by S. Wassermann) It follows from [7] that there are at least 3 distinct $C^*$-norms on $Q(H) \otimes B(H)$ or on $Q(H) \otimes Q(H)$. For $Q(H) \otimes B(H)$ this can be proved as follows. First the min and max norm must be distinct because otherwise, by Remark 15, $B(H)$ would be exact, which is well known to be false, essentially since [17]. But we can define a third $C^*$-norm $\alpha$ on $Q(H) \otimes B(H)$ by setting for any $x \in Q(H) \otimes B(H)$

$$ \alpha(x) = \|x\|_{(B(H) \otimes_{\min} B(H))/(K(H) \otimes_{\min} B(H))}. $$

Again, since $B(H)$ is not exact, by Kirchberg's criterion (7), this norm must be different from the min-norm on $Q(H) \otimes B(H)$. By (9) it is different from the max-norm, because otherwise, since $K(H)$ is nuclear, we would conclude that the min and max norms coincide on $B(H) \otimes B(H)$ contradicting the main result in [7].

The argument for $Q(H) \otimes Q(H)$ is similar. First note that $Q(H)$ is not exact. Indeed, by [1] there is a quotient of $Q(H)$ which is a $II_\infty$ factor, and hence is not exact, since exactness passes to subquotients (due to Kirchberg, see [3, p. 297]) and $B(\ell_2)$ is not exact. We define a $C^*$-norm $\alpha$, by setting for any $x \in Q(H) \otimes Q(H)$

$$ \alpha(x) = \|x\|_{(B(H) \otimes_{\min} Q(H))/(K(H) \otimes_{\min} Q(H))}. $$

Since $Q(H)$ is not exact we already know by Remark 15 that this differs from the min-norm. If it were equal to the max-norm we would deduce arguing as before by (9) that $B(H) \otimes_{\min} Q(H) = B(H) \otimes_{\max} Q(H)$, and by (8) this would imply again $B(H) \otimes_{\min} B(H) = B(H) \otimes_{\max} B(H)$ contradicting [7].

**Acknowledgments.** IHP
REFERENCES

[1] J. Anderson, Extreme points in sets of positive linear maps in $B(H)$, *J. Funct. Anal.* **31** (1979), 195-217.

[2] M. Bożejko, Some aspects of harmonic analysis on free groups, *Colloq. Math.* **41** (1979) 265-271.

[3] N.P. Brown and N. Ozawa, *C*-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, 88, American Mathematical Society, Providence, RI, 2008.

[4] P.M. Cohn, Basic algebra, Springer, 2003.

[5] W.W. Comfort, S. Negrepontis, The Theory of Ultrafilters, Springer, New York Heidelberg, 1974.

[6] U. Haagerup and S. Thorbjoernsen, Random matrices and K-theory for exact *C*-algebras, *Doc. Math.* **4** (1999), 341–450 (electronic).

[7] M. Junge and G. Pisier, Bilinear forms on exact operator spaces and $B(H) \otimes B(H)$, *Geom. Funct. Anal.* **5** (1995), no. 2, 329–363.

[8] E. Kirchberg, *On nonsemisplit extensions, tensor products and exactness of group C*-algebras*. Invent. Math. **112** (1993), 449–489.

[9] E. Kirchberg, Exact $C^*$-algebras, tensor products, and the classification of purely infinite algebras, *Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994)*, 943-954, Birkhäuser, Basel, 1995.

[10] G. Pisier, *Introduction to operator space theory*, Cambridge University Press, Cambridge, 2003.

[11] G. Pisier, Remarks on $B(H) \otimes B(H)$, *Proc. Indian Acad. Sci. (Math. Sci.)* Vol. 116, No. 4, November 2006, pp. 423428.

[12] G. Pisier, Quantum Expanders and Geometry of Operator Spaces, *J. Europ. Math. Soc.* To appear.

[13] T. Pytlik and R. Szwarc. An analytic family of uniformly bounded representations of free groups, *Acta Math.* **157** (1986), no. 3-4, 287–309.

[14] R. Szwarc. An analytic series of irreducible representations of the free group. *Annales de l'institut Fourier*, 38 (1988), 87-110.

[15] M. Takesaki, Theory of Operator algebras, vol. III. Springer-Verlag, Berlin, Heidelberg, New York, 2003.

[16] D. Voiculescu, K. Dykema and A. Nica, *Free random variables*, American Mathematical Society, Providence, RI, 1992.

[17] S. Wassermann, On tensor products of certain group $C^*$-algebras, *J. Funct. Anal.* **23** (1976), 239-254.

RIMS, KYOTO UNIVERSITY, 606-8502 JAPAN

E-mail address: narutaka@kurims.kyoto-u.ac.jp

TEXAS A&M UNIVERSITY, COLLEGE STATION, TX 77843, U.S.A.

AND

UNIVERSITÉ PARIS VI, IMJ, EQUIPE D'ANALYSE FONCTIONNELLE, PARIS 75252, FRANCE