The Connes embedding property for quantum group von Neumann algebras

Introduction

The Connes embedding problem asks whether any finite von Neumann algebra with separable predual embeds into an ultrapower of the hyperfinite II1-factor in a trace preserving way. This question was raised by Connes in . See for nice introductions on this topic. This central question in the theory of operator algebras is still open, and has ramifications in many other areas of mathematics, such as e.g. non-commutative probability theory, quantum information, and non-commutative algebraic geometry. In probabilistic terms, this question amounts to knowing whether any finite family of elements of a bounded tracial non-commutative probability space admits an asymptotic matrix model. In the framework of Voiculescu’s free entropy theory, this amounts to asking about the existence of matricial microstates, see .

The aim of this paper is to provide a new class of examples of Connes embeddable von Neumann algebras, namely von Neumann algebras arising from non-coamenable compact quantum groups of Kac type. Within the operator algebraic framework, arguably the most studied examples of compact quantum groups of Kac type include the free orthogonal quantum groups $O_{N}^{+}$ and the free unitary quantum groups $U_{N}^{+}$. Over the last two decades, this class of quantum groups has been extensively studied, and remarkable connections have emerged between these quantum groups and free probability theory. These connections occur at the level of quantum symmetries and asymptotic freeness results , and also at the operator algebra level . In particular, the von Neumann algebras $L^{\infty}(O_{N}^{+})$ and $L^{\infty}(U_{N}^{+})$ share many of the same structural properties with the free group factors: they are full type II1-factors; they are strongly solid, and in particular they are prime and have no Cartan subalgebra; they have the Haagerup property and are weakly amenable with Cowling-Haagerup constant $1$ (CMAP). But unlike the case of the free group factors, the II1-factors $L^{\infty}(O_{N}^{+})$ and $L^{\infty}(U_{N}^{+})$, $N\geq 3$ were not known to be Connes embeddable (i.e., to admit matricial microstates).

The paper is organized as follows. Section 2 contains preliminaries about compact and free quantum groups. Section 3 recalls facts about the Connes embedding property and relates this property for quantum group von Neumann algebras to the structure of quantum subgroups. In Section 4 the Connes embedding property for $L^{\infty}(O_{N}^{+})$ and $L^{\infty}(U_{N}^{+})$, $N\geq 4$, is derived through the study of specific quantum subgroups. Finally, in Section 5 we consider some applications of our results to free entropy dimension and to the problem of classifying the quantum subgroups of $O_{N}^{+}$ which contain the classical orthogonal group $O_{N}$ as a quantum subgroup.

Preliminaries

In this section we recall some basic facts on compact quantum groups. We follow and and refer to these papers for the facts stated below.

where $[S]$ denotes the norm-closed linear span of a subset $S\subset A\otimes A$. Here and in the rest of the paper, the symbol $\otimes$ will denote the minimal tensor product of C∗-algebras, $\overline{\otimes}$ will denote the spatial tensor product of von Neumann algebras, and $\odot$ will denote the algebraic tensor product of complex associative algebras. The homomorphism $\Delta$ is called a coproduct. The C∗-algebra $A$ together with the coproduct $\Delta$ is often called a Woronowicz C∗-algebra.

If $\dim H=n<\infty$, then (after fixing an orthonormal basis of $H$) we can identify $u$ with an invertible matrix $u=[u_{ij}]\in M_{n}(A)$ and (2) means exactly that

2. Free orthogonal and free unitary quantum groups.

We now introduce the free orthogonal and free unitary quantum groups, which form the central objects of study in this paper. These quantum groups were first introduced in the operator algebraic framework by Wang . Purely algebraic versions of these objects were also introduced by Dubois-Violette and Launer in .

Let $N\geq 2$. The free orthogonal quantum group $O_{N}^{+}$ is the compact quantum group given (in universal form) by the pair $(C^{u}(O_{N}^{+}),\Delta_{u})$, where

where $\bar{u}=[u_{ij}^{*}]$. The coproduct $\Delta_{u}$ is defined so that $u$ becomes a unitary representation of $O_{N}^{+}$. That is, $\Delta_{u}(u_{ij})=\sum_{k=1}^{N}u_{ik}\otimes u_{kj}$ for each $1\leq i,j\leq N$. Note that the abelianization of $C_{u}(O_{N}^{+})$ is naturally isomorphic to the C∗-algebra of continuous functions on the compact Lie group $O_{N}$. In particular, $O_{N}$ is a quantum subgroup of $O_{N}^{+}$.

The free unitary quantum group $U_{N}^{+}$ is defined in the same fashion as $O_{N}^{+}$, except that we no longer assume that the generators of $C^{u}(U_{N}^{+})$ are self-adjoint. More precisely, we define

Similarly, $U_{N}$ is a quantum subgroup of $U_{N}^{+}$.

where the middle symbol is $1$ if all strings of $p$ join pairs of equal indices, and is if not. We denote $TL_{N}(k,l)\subseteq\mathcal{B}(H^{\otimes k},H^{\otimes l})$ the subspace spanned by the maps $T_{p}$, $p\in NC_{2}(k,l)$. This subspace is related to $O_{N}^{+}$-intertwiners as follows.

Let $u$ be the fundamental representation of $O_{N}^{+}$. Then for all $N\geq 2$,

Moreover the family of linear maps $(T_{p})_{p\in NC_{2}(k,l)}$ is linearly independent as soon as $N\geq 2$.

If the first index is zero we omit it and we denote $NC_{2}(k)=NC_{2}(0,k)$, $TL(k)=TL(0,k)$. When there is no risk of confusion we will denote $\operatorname{Fix}_{k}=\operatorname{Fix}(u^{\otimes k})=\operatorname{Hom}_{O_{N}^{+}}(1,u^{\otimes k})\subseteq H^{\otimes k}$ the subspace of fixed vectors for the representation $u^{\otimes k}$ of $O_{N}^{+}$, and according to Theorem 2.1 we have for $N\geq 2$:

We also recall that Theorem 2.1 has a classical counterpart dating back to Brauer . We denote $P_{2}(k,l)$ the set of all pair partitions of $k$ upper points and $l$ lower points, and we observe that $T_{p}$ can still be defined for any $p\in P_{2}(k,l)$. Then we have

The Connes embedding property

Before specializing to quantum groups, let us first recall a few basic things about the Connes embedding property in the context of unital $\ast$-algebras.

The representation $\pi_{\tau}$ is usually called the GNS representation of $A$ with respect to the tracial state $\tau$. Taking double commutants, we obtain from $A$ a von Neumann algebra $\pi_{\tau}(A)^{\prime\prime}\subseteq\mathcal{B}(L^{2}(A,\tau))$, and the original state $\tau$ extends by continuity to a faithful normal tracial state on $\pi_{\tau}(A)^{\prime\prime}$ still denoted by $\tau$. Throughout this paper, we will always assume that our tracial $\ast$-algebras $(A,\tau)$ are such that $\pi_{\tau}$ exists and that the von Neumann algebra $\pi_{\tau}(A)^{\prime\prime}$ has a separable predual.

Since our point of view and motivation is that of matricial microstates, let us also recall the following definition.

The following von Neumann algebraic result connecting the existence of matricial microstates to the Connes embedding property is well known, see for example [14, Prop. 3.3]:

Let $M$ be a von Neumann algebra with separable predual equipped with a faithful normal tracial state $\tau$. Then the following are equivalent:

$\tau\in\operatorname{CEP}(M)$ (i.e., $M$ has the Connes embedding property).

Every finite subset $X\subset M_{sa}$ has matricial microstates relative to $\tau$.

If $Y\subset M_{sa}$ is a generating set for $M$, then every finite subset $X\subset Y$ has matricial microstates.

In particular, if $Y\subset M_{sa}$ is a finite generating set of $M$ then the above conditions are equivalent to $Y$ having matricial microstates.

The following lemma gives some important stability properties of $\operatorname{CEP}(A)$ that will be essential in the sequel.

Let $(A,\tau)$ and $(A_{i},\tau_{i})_{i=1,2}$ be unital $\ast$-algebras equipped with tracial states $\tau$ and $(\tau_{i})_{i=1,2}$ respectively. The following assertions are true.

If $B\subseteq A$ is a unital $\ast$-subalgebra and $\tau\in\operatorname{CEP}(A)$, then $\tau|_{B}\in\operatorname{CEP}(B)$.

If $\pi:A_{1}\to A_{2}$ is a unital $\ast$-homomorphism such that $\tau_{2}\circ\pi=\tau_{1}$ and $\tau_{1}\in\operatorname{CEP}(A_{1})$, then $\tau_{2}|_{\pi(A_{1})}\in\operatorname{CEP}(\pi(A_{1}))$.

If $\tau_{1}\in\operatorname{CEP}(A_{1})$ and $\tau_{2}\in\operatorname{CEP}(A_{2})$, then $\tau_{1}\otimes\tau_{2}\in\operatorname{CEP}(A_{1}\odot A_{2})$ and $\tau_{1}*\tau_{2}\in\operatorname{CEP}(A_{1}*A_{2})$ where $*$ denotes the reduced free product of tracial unital $\ast$-algebras .

(1) and (2) follow from the fact that the Connes embedding property is stable under (trace-preserving) inclusions of von Neumann algebras. (3) follows from the fact that the Connes embedding property is stable under tensor products and free products of von Neumann algebras with respect to tracial states . (4) is a direct ultra product construction. Alternately, this readily follows from the definition of matricial microstates. ∎

2. Hyperlinear discrete quantum groups

As one might expect, duals of coamenable compact quantum groups of Kac type are always hyperlinear.

3. Quantum subgroups and a stability result for hyperlinearity

In this section we present a new stability result for hyperlinear discrete quantum groups (Theorem 3.6). The main conceptual tool here is a quantization of the notion of a compact group being topologically generated by a pair of closed subgroups. The results of this section will be applied to specific examples in the next section.

The classical orthogonal group $O_{N}\leq O_{N}^{+}$, given by the Woronowicz C∗-morphism $\pi_{O_{N}}:C^{u}(O_{N}^{+})\to C(O_{N})$ whose kernel is generated by commutators.

The classical permutation group $\mathfrak{S}_{N}\leq O_{N}^{+}$, given by the Woronowicz C∗-morphism $\pi_{\mathfrak{S}_{N}}:C^{u}(O_{N}^{+})\to C(\mathfrak{S}_{N})$ whose kernel is generated by the commutators together with the elements $(u_{ij}-u_{ij}^{2})_{1\leq i,j\leq N}$.

The free product quantum subgroups $O_{a}^{+}\mathbin{\hat{\ast}}O_{b}^{+}\leq O_{N}^{+}$ for $a+b=N$, given by the Woronowicz C∗-morphism $\pi_{a,b}:C^{u}(O_{N}^{+})\to C^{u}(O_{a}^{+}\mathbin{\hat{\ast}}O_{b}^{+})$ which sends the $a\times a$ upper left (resp. $b\times b$ lower right) corner of the fundamental representation of $O_{N}^{+}$ to the fundamental representation of $O^{+}_{a}$ (resp. $O^{+}_{b}$), and all other entries to .

The quantum stabilizer subgroups $O_{N-1}^{+,\xi}\leq O_{N}^{+}$ for $\xi\in S_{1}$, given by the Woronowicz C∗-morphisms $\pi_{\xi}:C^{u}(O_{N}^{+})\to C^{u}(O_{N-1}^{+,\xi})$ obtained by completing $\xi$ into an orthonormal basis and sending the corresponding generator $u_{11}$ to $1$. Note that $O_{N-1}^{+,\xi}\simeq O_{N-1}^{+}$ for all $\xi$.

The main theorems of this section are as follows.

Let $N\geq 4$, then the following assertions are true.

$O_{N}^{+}=\langle O_{N},O_{N-1}^{+,\xi}\rangle$ for each $\xi\in S_{1}$.

For any pair of linearly independent vectors $\xi_{1},\xi_{2}\in S_{1}$, $O_{N}^{+}=\langle O_{N-1}^{+,\xi_{1}},O_{N-1}^{+,\xi_{2}}\rangle$.

Let $n\geq 2$ be a non-negative integer. Then $O_{2n}^{+}=\langle\mathfrak{S}_{2n},O_{n}^{+}\mathbin{\hat{\ast}}O_{n}^{+}\rangle$.

Before proving Theorems 4.1 and 4.2 we state their applications to hyperlinearity.

Let $N=2$ or $N\geq 4$. Then $\widehat{O_{N}^{+}}$ is hyperlinear.

The hyperlinearity of $\widehat{O_{2}^{+}}$ follows from Lemma 3.4. For the case $N=4$, note that $O_{4}^{+}=\langle\mathfrak{S}_{4},O_{2}^{+}\hat{*}O_{2}^{+}\rangle$ by Theorem 4.2. Since $L^{\infty}(\mathfrak{S}_{4})$ and $L^{\infty}(O_{2}^{+}\hat{*}O_{2}^{+})=(L^{\infty}(O_{2}^{+}),h_{O_{2}^{+}})*(L^{\infty}(O_{2}^{+}),h_{O_{2}^{+}})$ are both Connes embeddable, we conclude that $\widehat{O_{4}^{+}}$ is hyperlinear by Theorem 3.6. Finally, the cases $N\geq 5$ follow by induction using Theorem 4.1 and Theorem 3.6. ∎

Using a structure result of Banica [3, Théorème 1], we can easily deduce the hyperlinearity of $\widehat{U_{N}^{+}}$ from the corresponding result for $\widehat{O_{N}^{+}}$.

Let $N=2$ or $N\geq 4$. Then $\widehat{U_{N}^{+}}$ is hyperlinear.

We expect that Theorem 4.1 holds when $N=3$ (and therefore that $\widehat{O^{+}_{3}},\widehat{U^{+}_{3}}$ are hyperlinear). However, the following proof method seems to break down in this case. See also Remark 6.

The remainder of this section is devoted to proving the above quantum subgroup generation results for $O_{N}^{+}$.

We begin by developing some tools for the proof of Theorem 4.1.

Recall that we denote $\operatorname{Fix}_{k}=\operatorname{Hom}_{O_{N}^{+}}(1,u^{\otimes k})$ where $u$ is the fundamental representation of $O_{N}^{+}$, and let us denote similarly

According to Proposition 3.5, Theorem 4.1 is equivalent to the equalities $\operatorname{Fix}_{k}^{\xi}\cap\operatorname{Fix}_{k}^{O_{N}}$ $=$ $\operatorname{Fix}_{k}^{\xi_{1}}\cap\operatorname{Fix}_{k}^{\xi_{2}}$ $=$ $\operatorname{Fix}_{k}$ for all $k$. Hence we start by describing the subspaces $\operatorname{Fix}_{k}^{\xi}$.

Let $NC_{2,1}(k)$ be the set of non-crossing partitions of $\{1,\ldots,k\}$ consisting of blocks with cardinality at most $2$. In what follows, a block of $p\in NC_{2,1}(k)$ with cardinality equal to $1$ will be called a singleton, and a block with cardinality equal to $2$ will be called a pair. We also denote by $NC_{2,1}^{s}(k)\subset NC_{2,1}(k)$ the subset of non-crossing partitions containing exactly $s$ singletons, so that $NC_{2}(k)=NC_{2,1}^{0}(k)$. For $p\in NC_{2,1}(k)$ and $i$ a $k$-tuple we put $\delta^{p}_{i}=1$ if $i_{l}=i_{m}$ for all pairs $\{l,m\}\in p$, and $\delta^{p}_{i}=0$ else. Then we associate to $p\in NC_{2,1}^{s}(k)$ a linear map $T_{p}:H^{\otimes s}\to H^{\otimes k}$ as follows:

where we put a term $\xi_{i}$ at position $l$ if $\{l\}$ is the $i^{\text{th}}$ singleton in $p$, and a term $e_{i_{l}}$ else. In other words, $T_{p}$ is the usual map associated to the pair partition (possibly with crossings) $p^{\prime}\in P_{2}(s,k)$ obtained from $p$ by attaching a vertical segment to each singleton. We will also denote $T_{p}=T_{p}^{1}$ and consider the variant $T_{p}^{2}$ (resp. $T_{p}^{3}$) where the indices $i_{j}$ range from $2$ to $N$ (resp. $3$ to $N$). Finally we denote $S:H^{\otimes l}\to H^{\otimes l}$ the “symmetrizing” operators

Denote $v=(\iota\otimes\pi_{O_{N}})(u)$ the fundamental representation of $O_{N}$ and fix $\xi\in S_{1}$.

The vectors $T_{p}(\xi^{\otimes s})$ for $p\in NC_{2,1}(k)$ are linearly independent if $N\geq 3$.

We have $T_{p}\in\operatorname{Hom}_{O_{N}}(v^{\otimes s},v^{\otimes k})$ for all $p\in NC_{2,1}^{s}(k)$.

We know by Theorem 2.1 that the vectors $T_{q}^{2}(1)$, $q\in NC_{2}(k-s)$, are linearly independent for $N-1\geq 2$. Since we have decomposed $(\iota\otimes\pi_{\xi})(u)^{\otimes k}$ into orthogonal subrepresentations, this implies that the family of vectors $T_{r}^{2}(\xi^{\otimes s})$, $r\in NC_{2,1}(k)$, is linearly independent. Now we observe that the vectors $T_{p}(\xi^{\otimes s})=T_{p}^{1}(\xi^{\otimes s})$, $p\in NC_{2,1}(k)$, can be decomposed as linear combinations of the vectors $T_{r}^{2}(\xi^{\otimes s})$ by writing

at each pair of legs of $H^{\otimes k}$ determined by the pairs in $p$. Note that the partitions $r\neq p$ used to decompose in this way a vector $T_{p}(\xi^{\otimes s})$ have strictly more singletons than $p$, so that the decomposition matrix is block triangular (with respect to the value of $s$) with identity blocks on the diagonal. This implies that the family $T_{p}(\xi^{\otimes s})$, $p\in NC_{2,1}(k)$, is linearly independant, and spans the same subspace as the vectors $T_{r}^{2}(\xi^{\otimes s})$. This proves the first two assertions.

Finally, for any $p\in NC_{2,1}^{s}(k)$ we know that $T_{p}=T_{p^{\prime}}$ for a suitable partition $p^{\prime}\in P_{2}(s,k)$, see above, and that the maps $T_{p^{\prime}}$ are $O_{N}$-intertwiners, see the end of Section 2.3. ∎

We now reduce Theorem 4.1 to a linear independence problem:

We have $\operatorname{Fix}_{k}^{\xi_{1}}\cap\operatorname{Fix}_{k}^{\xi_{2}}=\operatorname{Fix}_{k}$ for some (or any) pair of linearly independent vectors $\xi_{1},\xi_{2}\in S_{1}$;

We have $\operatorname{Fix}_{k}^{\xi}\cap\operatorname{Fix}_{k}^{O_{N}}=\operatorname{Fix}_{k}$ for some (or any) $\xi\in S_{1}$;

The vectors $T_{p}(S(e_{1}\otimes\cdots\otimes e_{1}\otimes e_{2}))$, $p\in NC_{2,1}(k)\setminus NC_{2}(k)$, are linearly independent.

We first recall that two different stabilizer subgroups $O_{N-1}<O_{N}$ generate $O_{N}$. Indeed, for $1\leq i<j\leq N$, calling $R_{i,j,\theta}$ the rotation of angle $\theta$, between the canonical basis vectors $e_{i}$ and $e_{j}$ it is known that $R_{i,j,\theta}$ generate $SO_{N}$ if one takes all $1\leq i<j\leq N,\theta\in[0,2\pi)$.

Without loss of generality – at the possible cost of involving conjugation by rotations – we can assume that the first copy of $O_{N-1}$ fixes $e_{N}$ and the second copy fixes $e_{1}$. One can check that $R_{1,N,\theta}$ can be obtained as a conjugation of $R_{1,N-1,\theta}$ by $R_{N-1,N,\pi}$. This implies that any two copies of $SO_{N-1}<SO_{N}$ generate $SO_{N}$. The fact that two different copies of $O_{N-1}<O_{N}$ generate $O_{N}$ follows from the fact that we can find in $O_{N-1}$ an isometry that takes $SO_{N-1}$ to $O_{N-1}\backslash SO_{N-1}$ by left multiplication.

Now let $x\in\operatorname{Fix}_{k}^{\xi_{1}}\cap\operatorname{Fix}_{k}^{\xi_{2}}$ be given. Then $x$ is fixed by the two copies of $O_{N-1}$ inside the quantum subgroups $O_{N-1}^{+,\xi_{1}}$ and $O_{N-1}^{+,\xi_{2}}$, hence it is fixed by $O_{N}$ by the previous paragraph. On the other hand, for $g\in O_{N}$ we have $\operatorname{Fix}_{k}^{g\xi}=g\cdot\operatorname{Fix}_{k}^{\xi}:=v(g)^{\otimes k}\operatorname{Fix}_{k}^{\xi}$, where $v=u^{O_{N}}$ is the fundamental representation of $O_{N}$. As a result, if $x\in\operatorname{Fix}_{k}^{O_{N}}\cap\operatorname{Fix}_{k}^{\xi}$, then $x$ lies in $\operatorname{Fix}_{k}^{\zeta}$ for any $\zeta\in S_{1}$. This shows that $\operatorname{Fix}_{k}^{\xi_{1}}\cap\operatorname{Fix}_{k}^{\xi_{2}}$ and $\operatorname{Fix}_{k}^{O_{N}}\cap\operatorname{Fix}_{k}^{\xi}$ are equal and independent of the choice of $\xi$ and the linearly independent pair $\xi_{1},\xi_{2}$ in $S_{1}$.

Now we differentiate: $\mathcal{T}_{\lambda}$ is constant on $S_{1}$ iff $d_{\xi}\mathcal{T}_{\lambda}(\eta)=0$ for all $\xi\in S_{1}$, $\eta\bot\xi$. Moreover by $O_{N}$-covariance of $\mathcal{T}_{\lambda}$ we have $g\cdot d_{\xi}\mathcal{T}_{\lambda}(\eta)=d_{g\xi}\mathcal{T}_{\lambda}(g\eta)$, and since $O_{N}$ acts transitively on pairs of normed orthogonal vectors, $\mathcal{T}_{\lambda}$ is constant on $S_{1}$ iff $d_{e_{1}}\mathcal{T}_{\lambda}(e_{2})=0$. Then we compute $d_{\xi}(\xi^{\otimes s})(\eta)=S(\xi\otimes\cdots\otimes\xi\otimes\eta)/(s-1)!$, hence

This shows the equivalence of the last assertion in the statement with the condition (I) above. ∎

We will verify the linear independence condition given in Part (3) of Proposition 4.6. Consider the vectors $y_{p,i}=T_{p}^{3}(e_{i_{1}}\otimes\cdots\otimes e_{i_{s}})\in H^{\otimes k}$ with $i_{l}=1,2$ and $p\in NC_{2,1}(k)$. They form a linearly independent family, which we shall denote by $\mathcal{C}$. Indeed if $\langle y_{p,i}|y_{q,j}\rangle\neq 0$, then $p$, $q$ must have the same singletons and $i=j$. Moreover when this is the case, then $\langle y_{p,i}|y_{q,j}\rangle$ coincides with the scalar product $\langle T^{\prime}_{p^{\prime}}(1)|T^{\prime}_{q^{\prime}}(1)\rangle$ associated with the partitions $p^{\prime}$, $q^{\prime}\in NC_{2}(k-s)$ obtained from $p$ and $q$ by removing singletons, where $T^{\prime}_{p^{\prime}}$ is the map analogous to $T_{p^{\prime}}$, but in dimension $N-2$. Since $N-2\geq 2$, the vectors $T^{\prime}_{p^{\prime}}(1)$ are linearly independent, and we can deduce that the Gram matrix of the family $\mathcal{C}$ is invertible (cf. Theorem 2.1).

Now consider the vectors $x_{p}=T_{p}(S(e_{1}\otimes\cdots\otimes e_{1}\otimes e_{2}))/(s-1)!$ from Proposition 4.6. Note that each $x_{p}$ can be written as a (unique) linear combination of elements in $\mathcal{C}$. This follows from the definition of $S$ and by writing $\sum_{i=1}^{N}e_{i}\otimes e_{i}=e_{1}\otimes e_{1}+e_{2}\otimes e_{2}+\sum_{i=3}^{N}e_{i}\otimes e_{i}$ as in the proof of Lemma 4.5. More precisely, if $p\in NC_{1,2}^{s}(k)$ then $x_{p}$ decomposes into the sum of the $s$ vectors $y_{p,i}$ with $i$ taking the value $2$ only once, and a linear combination of vectors $y_{q,j}$ with $q$ having strictly more singletons that $p$. As a result, if we partially order the families $\mathcal{B}=(x_{p})$ and $\mathcal{C}=(y_{p,i})$ according to the number of singletons $s$ in $p$, the corresponding decomposition matrix for $\mathcal{B}$ in terms of the basis $\mathcal{C}$ will be block lower-triangular (with rectangular blocks), and each diagonal sub-block (one for each integer $s$) is itself block diagonal, with diagonal blocks which are non-zero columns (one for each partition $p\in NC_{2,1}^{s}(k)$). In particular, this decomposition matrix has maximal rank and therefore $\mathcal{B}$ is linearly independent. ∎

Although the proof above only applies for $N\geq 4$, it seems very likely that the linear independence condition introduced in Proposition 4.6, and hence Theorem 4.1, also hold at $N=3$. This would imply the hyperlinearity of $\hat{O}_{N}^{+}$ and $\hat{U}_{N}^{+}$ for all $N\geq 2$, without relying on Theorem 4.2. In fact, we have strong numerical evidence that the family of vectors $T_{p}(e_{1})$ associated to “one singleton” partitions $p\in NC_{2,1}^{1}(k)$ is linearly independant for all $N\geq 2$, and using similar techniques as above this would imply Theorem 4.1 for all $N\geq 3$.

One can actually even show more, namely that for any $d\geq 2$, $U_{d}$ is (algebraically) generated by the subgroups $\mathfrak{S}_{d}$ and $U_{2}$, where $U_{2}$ is viewed as sitting on the upper left corner of $d\times d$ matrices. This is trivial for $d=2$. For general $d$, we proceed by induction over $d$: given an element of $U_{d}$, it is possible to multiply it on the left by $d-1$ elements of type $\sigma U\sigma^{-1}$ where $\sigma\in\mathfrak{S}_{d}$ and $U\in U_{2}$, and ensure that the bottom element of the last column of the new element of $U_{d}$ is $1$. Indeed, the action by left multiplication leaves the columns invariant, and successive operations of the groups $U_{2},(13)U_{2}(13),\ldots,(1d)U_{2}(1d)$ can be performed to ensure that the element of respective indices $(1,d),(2,d),\ldots,(d-1,d)$ is sent to zero, and in turn, that the entry of index $(d,d)$ is sent to $1$. By orthogonality relations, the new matrix obtained has also zeros on all entries of the last row apart from the last one, therefore it sits in $U_{d-1}$ viewed as the upper left corner of $U_{d}$. The general result follows by induction.

Applying fact (7), we finally conclude that $x\in\operatorname{Fix}(w_{2k}^{U_{n}^{+}\mathbin{\hat{\ast}}U_{n}^{+}})\cap\operatorname{Fix}(w_{2k}^{U_{2n}})$. To finish the proof, we appeal to the following lemma, which is a special case of a very recent result of Chirvasitu . We include a detailed proof for the convenience of the reader. ∎

With the notation and conventions as above, we have the equalities

given by identifying an elementary tensor $e_{i_{1}}\otimes\overline{e_{i_{2}}}\otimes\ldots\otimes e_{i_{2k-1}}\otimes\overline{e_{i_{2k}}}\in(H\otimes\bar{H})^{\otimes k}$ with the rank-one operator

Recall that $\operatorname{Fix}(w_{2k}^{U_{2n}})$ is spanned by the vectors $T_{p}=\sum_{i}\delta_{i}^{p}(e_{i_{1}}\otimes\overline{e_{i_{2}}}\otimes\ldots\otimes e_{i_{2k-1}}\otimes\overline{e_{i_{2k}}})$ where $p\in P_{2}(2k)$ is a pair partition respecting the additional requirement that even points are connected to odd points.

When $\Phi$ is restricted to the subspace $\operatorname{Fix}(w_{2k}^{U_{2n}})$, we obtain an isomorphism

which maps the vector $T_{p}$ to a map $T_{q}$, where $q\in P_{2}(k,k)$ is obtained by connecting the even (respectively, odd) points of $p$ to the input (respectively, output) points of $q$. We denote by $q\in S(k,k)$ the pair partitions obtained in this way: these are exactly the ones where input points are connected to output points via the permutation specified by $q$. Note that one recovers the Schur-Weyl duality for unitary groups describing $\operatorname{Hom}((w^{U_{2n}})^{\otimes k},(w^{U_{2n}})^{\otimes k})$ as the linear span of the operators $(T_{q})_{q\in S(k,k)}$ which permute the tensor factors of $H^{\otimes k}$.

On the other hand, the image by $\Phi$ of the subspace $\operatorname{Fix}(w_{2k})$ is spanned by the maps $T_{q}$ where $q$ belongs to a subset $S^{\prime}(k,k)\subset S(k,k)\subset P_{2}(k,k)$: namely the one corresponding to $p\in NC_{2}(2k)\subset P_{2}(2k)$. The only thing we will need to know about $S^{\prime}(k,k)$ is that the family $T_{q}$, $q\in S^{\prime}(k,k)$, is linearly independent as soon as $n\geq 2$. This is indeed the case since $(T_{p})_{p\in NC_{2}(2k)}$ is linearly independent and $\Phi$ is an isomorphism.

Finally, for each $q\in S^{\prime}(k,k)$ and each $k$-tuple $i=(i_{1},\ldots,i_{k})\in\{1,2\}^{k}$, we define a linear map $T_{q,i}:=T_{q}P_{i}$, where $P_{i}:=\otimes_{r=1}^{k}P_{i_{r}}$ is the orthogonal projection whose range is $H_{i}:=\otimes_{r=1}^{k}H_{i_{r}}$. From the description of the intertwiner spaces of $U_{N}^{+}$ ($N\geq 2$) and their free products given in [6, Section 9] and [27, Proposition 2.15], respectively, it follows that the family $T_{q,i}$, $q\in S^{\prime}(k,k)$, $i\in\{1,2\}^{k}$, forms a basis of the intertwiner space

In view of the above isomorphisms, it remains to demonstrate that any linear map $T\in$ $\operatorname{Hom}$ $((w^{U_{n}^{+}\hat{*}U_{n}^{+}})^{\otimes k},(w^{U_{n}^{+}\hat{*}U_{n}^{+}})^{\otimes k})\cap\operatorname{Hom}((w^{U_{2n}})^{\otimes k},(w^{U_{2n}})^{\otimes k})$ lies in fact in $\operatorname{Hom}(w^{\otimes k},w^{\otimes k})$. By linear independence, $T$ can be uniquely expressed as the sum

Denote by $1^{k}=(1,1,\ldots,1)$ the constant $k$-tuple. Since we can decompose each $T_{q}$ as $T_{q}=\sum_{i\in\{1,2\}^{k}}T_{q,i}$, we may subtract $T_{1}:=\sum_{q\in S^{\prime}(k,k)}\lambda_{q,1^{k}}T_{q}\in\operatorname{Hom}(w^{\otimes k},w^{\otimes k})$ from $T$ and consequently assume for the remainder that $TP_{1^{k}}=0$.

Now consider an arbitrary $k$-tuple $i$. We claim that $T|_{H_{i}}=0$. To see this, consider the linear map $g:H\to H_{1}$, $e_{r}\mapsto e_{r}$, $e_{r+n}\mapsto e_{r}$ for all $r=1,\ldots,n$. Observe that the restriction $g^{\otimes k}:H_{i}\to H_{1^{k}}$ is an isomorphism and that $g^{\otimes k}T_{q,i}=T_{q,1^{k}}g^{\otimes k}P_{i}$. Moreover, since $T$ is an $U_{2n}$-intertwiner it is a linear combination of maps $T_{r}$, $r\in S(k,k)$, and as each of these maps verifies the relation $h^{\otimes k}T=Th^{\otimes k}$ for any $h\in\mathcal{B}(H)$. We have then

Since the family $T_{q,1^{k}}$, $q\in S^{\prime}(k,k)$ is linearly independent and $g^{\otimes k}P_{i}:H_{i}\to H_{1^{k}}$ is an isomorphism, we conclude that $\lambda_{q,i}=0$ for each $q$. Finally, since $H^{\otimes k}=\oplus_{i}H_{i}$ and $i$ was arbitrary, this implies $T=0$. I.e., $T=T_{1}\in\operatorname{Hom}(w^{\otimes k},w^{\otimes k})$. ∎

Applications

In this section we present an application of our hyperlinearity results to the computation of the free entropy dimension of the canonical generators of $L^{\infty}(O_{N}^{+})$. We refer the reader to the survey for details on the various notions of free entropy dimension and related concepts.

Let $\Gamma$ be a finitely generated discrete group with a finite symmetric system of generators $(g_{i})_{i=1}^{n}$, and put $x_{i}=\Re\lambda(g_{i}),y_{i}=\Im\lambda(g_{i})\in\mathcal{L}(\Gamma)$. In [17, Corollary 4.9], Connes and Shlyakhtenko showed that the (non-microstates) free entropy dimension $\delta^{*}(x_{i},y_{i})$ verifies the inequality

Finally, if $\mathcal{L}(\Gamma)$ is diffuse and has the Connes embedding property, it was shown in [25, Corollary 4.7] that

The microstates (and non-microstates) free entropy dimension of $L^{\infty}(O_{N}^{+})$ associated to the canonical generators $(u_{ij})_{1\leq i,j\leq N}$ is $1$ for all $N\geq 4$.

It was proved in that the right hand side of inequality (13) is exactly $1$. Together with the above discussion, the proof is complete. ∎

From Theorem 4.1 it follows by a simple induction on $k=\dim E\geq 3$ that $O_{N}^{+}$ is generated by $O_{N}$ and any subgroup $O_{k,E}^{+}$. In particular it is generated by $O_{N}^{*}$ and $O_{3,E}^{+}$. ∎

Acknowledgments

We would like to thank Teo Banica, Julien Bichon, Marius Junge and Reiji Tomatsu for enlightening conversations. M.B.’s research was partially supported by an NSERC postdoctoral fellowship. B.C.’s research was partially supported by NSERC, ERA, Kakenhi and ANR-14-CE25-0003 funding.

References