The strong Haagerup inequality for $q$-circular systems

Haagerup [Haa79] proved the following inequality in the reduced $\mathrm{C}^{*}$-algebra $\mathrm{C}^{*}_{\mathrm{red}}(\mathbb{F}_{d})$ of the free group $\mathbb{F}_{d}$ on $d$ generators:

where $|g|$ is the length of $g\in\mathbb{F}_{d}$ as reduced words. The first author and Speicher [KS07] improved the Haagerup inequality for freely independent and identically distributed $\mathscr{R}$-diagonal elements $x=(x_{1},\ldots,x_{d})$:

where $w$ moves around the set of words with letters $1,\ldots,d$ and $|w|$ is the word length of $w$ and $x_{w}=x_{w_{1}}\cdots x_{w_{n}}$ for $w=w_{1}\cdots w_{n}$. The $\mathscr{R}$-diagonal elements are introduced by Nica and Speicher [NS97] which form a large family of non-self-adjoint random variables including Voiculescu’s circular elements and Haar unitaries. these elements are defined by using free cumulants and the proof of the strong Haagerup inequality is based on the moment-cumulant formula in free probability. Note that the strong Haagerup inequality stated above does not involve the adjoint of operators $x_{1}^{*},\ldots,x_{d}^{*}$ (it is called the holomorphic setting in [KS07]) while the Haagerup inequality involves the adjoint of generators. This difference reduces moments of $\sum_{|w|=n}\alpha_{w}x_{w}$ and produces the factor $\sqrt{n+1}$ in the strong Haagerup inequality.

There are some generalizations of Kemp-Speicher’s strong Haagerup inequality. In the paper [dlS09], de la Salle proved that freely independent $\mathscr{R}$-diagonal elements also satisfy the analog of this inequality for operator coefficients polynomials like $\sum_{|w|=n}\alpha_{w}\otimes x_{w}$. On the other hand, Brannan [Bra12, Theorem 1.4] generalized the strong Haagerup inequality for non-commutative random variables with some invariant properties that do not require the variables to be freely independent. In particular, he proved the strong Haagerup inequality for the free unitary quantum group $U_{N}^{+}$ (see also [You22] for non-Kac type orthogonal quantum groups).

In this paper, we prove that this type of estimate also holds for $q$-circular systems $c_{1}^{(q)},\ldots,c_{d}^{(q)}$, which are not freely independent and do not satisfy assumptions in Brannan’s paper [Bra12, Theorem 1.4.] (more precisely, the $q$-circular system is not invariant under free complexification). Namely, our main result is the following.

The concept of the $q$-circular systems is introduced by Mingo and Nica [MN01] as $q$-defomations of the circular systems ($q=0$) based on the work on the $q$-canonical commutation relations [FB70], [BS91]. These $q$-relations interpolate Bosonic $(q=1)$ and Fermionic $(q=-1)$ relations, and it gives a $q$-analog of quantum field theory. From this perspective, the algebra of polynomials $\sum_{w}\alpha_{w}c_{w}^{(q)}$ has been studied as a $q$-analog of the Segal-Bergman space (see [Kem05] and [CH18]). Regarding our main result, if we replace $\sqrt{n+1}$ with $n+1$, the inequality immediately follows from Bożejko’s Haagerup inequality [Boż99] for $q$-Gaussian system. In addition, the second author proved that the operator norm of $\sum_{|w|=n}\alpha_{w}c_{w}^{(q)}$ is continuous for $q$ [Miy23]. However, before our result, whether we can obtain $\sqrt{n+1}$ for $q$-circular system was unclear.

There is a heuristic observation of this $q$-deformation that the case $-1<q<1$ shares the same properties with the case $q=0$ which can be described by free probability. This is supported by the isomorphism of $q$-Gaussian von Neumann algebras [GS14] and $q$-Cuntz-Toeplitz algebras [DN93],[Kuz23], hypercontractivity and ultracontractivity of $q$-Ornstein-Uhlenbeck semigroup [Bia97], [Kem05],[Boż99],…etc. Our main result of the strong Haagerup inequality also follows this context.

While previous works on the strong Haagerup inequality are based on combinatorial arguments of joint moments, we use several norm inequalities for creations and annihilation operators proved by Bożejko [Boż99]. He used these inequalities to prove the Haagerup inequality for the $q$-Gaussian system giving us some insights to prove the strong Haagerup inequality. As an application, our result gives the bounds of joint moments of the $q$-circular system without computing combinatorial moments directly. Since $\|c^{(q)n}\|_{2m}\leq\|c^{(q)n}\|$ for any $m\in\mathbb{N}$, by the formula of moments of a $q$-circular random variable $c^{(q)}$ [MN01, Definition 1.2], our main result implies

where $P_{\ast-\cdot}(\ast^{n},\ \cdot^{n},\ldots,\ast^{n},\ \cdot^{n})$ is the set of pair partitions on the $2mn$ vertices alternately aligned with $n$ stars $\ast^{n}$ and $n$ dots $\cdot^{n}$ which connect $\ast$ with $\cdot$ and $\mathrm{cr}(\pi)$ denotes the number of crossings of $\pi$. When $q=0$, the left-hand side of the inequality is equal to the Fuss-Catalan number $\frac{1}{m}\binom{m(n+1)}{m-1}$ (see [KS07, Corollary 3.2]). By the same proof in [KS07, Theorem 5.4.], we also have strong ultracontractivity of the $q$-Ornstein-Uhlenbeck semigroup $e^{-tN^{(q)}}$ affiliated with the $q$-circular system.

There are some possibilities of other generalizations of the strong Haagerup inequality since our approach is different from previous works. We expect that one can show the strong Haagerup inequality for the mixed $q_{ij}$ and twisted relations since they satisfy the Haagerup inequality (see [Kró05]). We also expect that one can show the strong Haagerup inequality for $q$-circular system with operator coefficients like de la Salle’s results in [dlS09]. We leave them for future work.

Throughout the paper, let $d\in\mathbb{N}$ and $-1<q<1$ be a parameter and we set a Hilbert space $H=\mathbb{C}^{d}\oplus\mathbb{C}^{d}$. Let $\{e_{i}\}_{i=1}^{d}$ and $\{e_{\overline{i}}\}_{i=1}^{d}$ be orthonormal basis of $\mathbb{C}^{d}$ in the first and second summands. We consider the algebraic Fock space of $H$ defined by

where we take the algebraic direct sum and we set $H^{\otimes 0}=\mathbb{C}\Omega$ with a unit vector $\Omega$. In [BS91], Bożejko and Speicher introduced the $q$-inner product of $\xi_{1}\otimes\cdots\otimes\xi_{m}\in H^{\otimes m}$ and $\eta_{1}\otimes\cdots\otimes\eta_{n}\in H^{\otimes n}$ is defined by

where $S_{m}$ is the symmetric group of degree $m$ and $P^{(m)}$ is a strictly positive operator defined by (note that $\mathrm{inv}(\pi)=\mathrm{inv}(\pi^{-1})$),

This $q$-inner product extends to the inner product on $\mathcal{F}_{\mathrm{alg}}(H)$ and by completing $\mathcal{F}_{\mathrm{alg}}(H)$ we obtain the Hilbert space $\mathcal{F}_{q}(H)$, so-called the $q$-Fock space.

For a vector $\xi\in H$, we define creation operator $a(\xi)$ and annihilation operator $a^{*}(\xi)$ by

Note that $a(\xi)$ and $a^{*}(\eta)$ ($\xi,\eta\in H$) satisfy the $q$-commutation relation:

The $q$-circular system $(c_{1},\ldots,c_{d})$ is a tuple of operators on the $q$-Fock space $\mathcal{F}_{q}(H)$ defined by

The von Neumann algebra $\mathrm{W}^{*}(c_{1}^{(q)},\ldots,c_{d}^{(q)})$ generated by a $q$-circular system admits a faithful tracial state $\tau(T)=\langle T\Omega,\Omega\rangle_{q}$, which naturally fits in the framework of non-commutative probability spaces. We defined $L^{2}$-norm $\|T\|_{2}$ of $T\in\mathrm{W}^{*}(c_{1}^{(q)},\ldots,c_{d}^{(q)})$ by

where $\|\cdot\|_{\mathcal{F}_{q}(H)}$ denotes the norm on the Hilbert space $\mathcal{F}_{q}(H)$.

We also consider the set of words $[d,\overline{d}]^{*}$ which consist of letters in $1,\ldots,d$ and $\overline{1},\ldots,\overline{d}$ and the empty word $\Omega$. For $w\in[d,\overline{d}]^{*}$, $|w|$ denotes the word length of $w$ and we set $|\Omega|=0$. We define $\overline{w}=\overline{w_{1}}\cdot\overline{w_{2}}\cdots\overline{w_{n}}$ for $w=w_{1}w_{2}\cdots w_{n}\in[d]^{*}$ and $w^{*}=w_{n}w_{n-1}\cdots w_{1}$ for $w=w_{1}w_{2}\cdots w_{n}\in[d,\overline{d}]^{*}$. In our convention, for each $w=w_{1}w_{2}\cdots w_{n}\in[d,\overline{d}]^{*}$ and a family of operators $\{T_{i}\}_{i\in[d,\overline{d}]}$, we write $T_{w}=T_{w_{1}}T_{w_{2}}\cdots T_{w_{n}}$ and $T_{\Omega}=1$. We set a linear basis $\{e_{w}\}_{w\in[d,\overline{d}]}$ in $\mathcal{F}_{\mathrm{alg}}(H)$ by $e_{w}=e_{w_{1}}\otimes\cdots\otimes e_{w_{n}}$ for $w=w_{1}\cdots w_{n}$ and $e_{\Omega}=\Omega$.

We identify $\pi\in S_{m}$ with the permutation operator on $H^{\otimes m}$ and the permutation of letters in a word $w$ as follows; we defined an operator $\pi$ on $H^{\otimes m}$ by

and we define a permutation $\pi(w)$ of a word $w=w_{1}\cdots w_{m}$ by

which is a left action of $S_{m}$. Note that we have

Let $S_{k}\times S_{m-k}$ is the subgroup of $S_{m}$ such that $\pi\in S_{k}\times S_{m-k}$ maps $\{1,\ldots,k\}$ to $\{1,\ldots,k\}$ and $\{k+1,\ldots,m\}$ to $\{k+1,\ldots,m\}$, and $S_{m}$/$S_{k}\times S_{m-k}$ denote the left cosets of $S_{k}\times S_{m-k}$ in $S_{m}$. In this note, we always take the unique representative of $\sigma\in\mathchoice{\text{\raise 4.30554pt\hbox{$S_{m}$}\!\Big{/}\!\lower 4.30554pt\hbox{$S_{k}\times S_{m-k}$}}}{\text{\raise 1.0pt\hbox{$S_{m}$}\big{/}\lower 1.0pt\hbox{$S_{k}\times S_{m-k}$}}}{{S_{m}}\,/\,{S_{k}\times S_{m-k}}}{{S_{m}}\,/\,{S_{k}\times S_{m-k}}}$ so that $\mathrm{inv}(\sigma)$ is minimal. Such permutations can be described as a permutation $\sigma\in S_{m}$ such that $\sigma(1)<\sigma(2)<\cdots<\sigma(k)$ and $\sigma(k+1)<\sigma(k+2)\ldots<\sigma(m)$. If we take $\pi\in S_{m}$, then we have the unique factorization $\pi=\sigma(\tau_{1}\times\tau_{2})$ where $\sigma\in\mathchoice{\text{\raise 4.30554pt\hbox{$S_{m}$}\!\Big{/}\!\lower 4.30554pt\hbox{$S_{k}\times S_{m-k}$}}}{\text{\raise 1.0pt\hbox{$S_{m}$}\big{/}\lower 1.0pt\hbox{$S_{k}\times S_{m-k}$}}}{{S_{m}}\,/\,{S_{k}\times S_{m-k}}}{{S_{m}}\,/\,{S_{k}\times S_{m-k}}}$ is a permutation such that for each $1\leq i\leq k$, $\sigma(i)$ is the $i$-th smallest number in $\{\pi(j)\}_{j=1}^{k}$ and for each $k+1\leq i\leq m$, $\sigma(i)$ is the $(i-k)$-th smallest number in $\{\pi(j)\}_{j=k+1}^{m}$. Note that under this factorization, we have $\mathrm{inv}(\pi)=\mathrm{inv}(\sigma)+\mathrm{inv}(\tau_{1}\times\tau_{2})$.