guide:F5c4fd7e1b

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With the above ingredients in hand, let us go back now to invariance questions with respect to the main quantum permutation and rotation groups, namely:

[[math]] \xymatrix@R=15mm@C=15mm{ S_N^+\ar[r]&O_N^+\\ S_N\ar[r]\ar[u]&O_N\ar[u] } [[/math]]

More generally, we would like in fact to have, ideally, de Finetti type theorems for all the easy quantum groups that we know, from the previous chapters. This is of course something quite technical, and time consuming, but we would like at least to understand what happens for the main quantum reflection and rotation groups, namely:

[[math]] \xymatrix@R=18pt@C=18pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

In order to discuss these questions, or at least some of them, let us start with a basic approximation result for the finite sequences, in the real case, from , as follows:

Theorem

Suppose that [math](x_1,\ldots,x_N)[/math] is [math]G_N[/math]-invariant, and that [math]G_N=O_N^+,S_N^+[/math], or that [math]G_N=O_N,S_N[/math] and [math](x_1,\ldots,x_N)[/math] commute. Let [math](y_1,\ldots,y_N)[/math] be a sequence of [math]B_N[/math]-valued random variables with [math]B_N[/math]-valued joint distribution determined as follows:

• [math]G=O^+[/math]: Free semicircular, centered with same variance as [math]x_1[/math].
• [math]G=S^+[/math]: Freely independent, [math]y_i[/math] has same distribution as [math]x_1[/math].
• [math]G=O[/math]: Independent Gaussian, centered with same variance as [math]x_1[/math].
• [math]G=S[/math]: Independent, [math]y_i[/math] has same distribution as [math]x_1[/math].

Then if [math]1\leq j_1,\ldots,j_k \leq N[/math] and [math]b_0,\ldots,b_k\in B_N[/math], we have the following estimate,

[[math]] \left|\left|E_N(b_0x_{j_1}\ldots x_{j_k}b_k)-E(b_0y_{j_1}\ldots y_{j_k}b_k)\right|\right|\leq\frac{C_k(G)}{N}||x_1||^k||b_0||\ldots||b_k|| [[/math]]

with [math]C_k(G)[/math] being a constant depending only on [math]k[/math] and [math]G[/math].

Show Proof

First we note that it suffices to prove the result for [math]N[/math] large enough. We will assume that [math]N[/math] is sufficiently large, as for the Gram matrix [math]G_{kN}[/math] to be invertible.

Let [math]1\leq j_1,\ldots,j_k\leq N[/math] and [math]b_0,\ldots,b_k\in B_N[/math]. We have then:

[[math]] \begin{eqnarray*} E_N(b_0x_{j_1}\ldots x_{j_k}b_k) &=&\sum_{i_1\ldots i_k}b_0x_{i_1}\ldots x_{i_k}b_k\int v_{i_1j_1}\ldots v_{i_kj_k}\\ &=&\sum_{i_1\ldots i_k}b_0x_{i_1}\ldots x_{i_k}b_k\sum_{\pi\leq\ker i}\sum_{\sigma\leq\ker j}W_{kN}(\pi,\sigma)\\ &=&\sum_{\sigma\leq\ker j}\sum_\pi W_{kN}(\pi,\sigma)\sum_{\pi\leq\ker i} b_0x_{i_1}\ldots x_{i_k}b_k \end{eqnarray*} [[/math]]

On the other hand, it follows from our assumptions on [math](y_1,\ldots,y_N)[/math], and from the various moment-cumulant formulae given before, that we have:

[[math]] E(b_0y_{j_1}\ldots y_{j_k}b_k)=\sum_{\sigma\leq\ker j}\xi_{E_N}^{(\sigma)}(b_0x_1b_1,\ldots,x_1b_k) [[/math]]

Here, and in what follows, [math]\xi[/math] denote the relevant free or classical cumulants.

The right hand side can be expanded, via the Möbius inversion formula, in terms of expectation functionals of the following type, with [math]\pi[/math] being a partition in [math]NC,P[/math] according to the cases (1,2) or (3,4) in the statement, and with [math]\pi\leq\sigma[/math] for some [math]\sigma\in D(k)[/math]:

[[math]] E_N^{(\pi)}(b_0x_1b_1,\ldots,x_1b_k) [[/math]]

Now if [math]\pi\notin D(k)[/math], we claim that this expectation functional is zero.

Indeed this is only possible if [math]D= NC_2,P_2[/math], and if [math]\pi[/math] has a block with an odd number of legs. But it is easy to see that in these cases [math]x_1[/math] has an even distribution with respect to [math]E_N[/math], and therefore we have, as claimed, the following formula:

[[math]] E_N^{(\pi)}(b_0x_1b_1,\ldots,x_1b_k)=0 [[/math]]

Now this observation allows to to rewrite the above equation as follows:

[[math]] E(b_0y_{j_1}\ldots y_{j_k}b_k)=\sum_{\sigma\leq\ker j}\sum_{\pi\leq \sigma} \mu_{D(k)}(\pi,\sigma)E_N^{(\pi)}(b_0x_1b_1,\ldots,x_1b_k) [[/math]]

We therefore obtain the following formula:

[[math]] E(b_0y_{j_1}\ldots y_{j_k}b_k)=\sum_{\sigma\leq\ker j}\sum_{\pi\leq\sigma} \mu_{D(k)}(\pi,\sigma)N^{-|\pi|}\sum_{\pi\leq\ker i}b_0x_{i_1}\ldots x_{i_k}b_k [[/math]]

Comparing the above two equations, we find that:

[[math]] \begin{eqnarray*} &&E_N(b_0x_{j_1}\ldots x_{j_k}b_k)-E(b_0y_{j_1}\ldots y_{j_k}b_k)\\ &=&\sum_{\sigma\leq\ker j}\sum_\pi\left(W_{kN}(\pi,\sigma)-\mu_{D(k)}(\pi,\sigma)N^{-|\pi|}\right)\sum_{\pi\leq\ker i}b_0x_{i_1}\ldots x_{i_k}b_k \end{eqnarray*} [[/math]]

Now since [math]x_1,\ldots,x_N[/math] are identically distributed with respect to the faithful state [math]\varphi[/math], it follows that these variables have the same norm. Thus, for any [math]\pi \in D(k)[/math]:

[[math]] \left|\left|\sum_{\pi\leq\ker i}b_0x_{i_1}\ldots x_{i_k}b_k\right|\right|\leq N^{|\pi|}||x_1||^k||b_0||\ldots||b_k|| [[/math]]

Combining this with the former equation, we obtain the following estimate:

[[math]] \begin{eqnarray*} &&\left|\left|E_N(b_0x_{j_1}\ldots x_{j_k}b_k)-E(b_0y_{j_1}\ldots y_{j_k}b_k)\right|\right|\\ &\leq&\sum_{\sigma\leq\ker j}\sum_\pi\left|W_{kN}(\pi,\sigma)N^{|\pi|}-\mu_{D(k)}(\pi,\sigma)\right|||x_1||^k||b_0||\ldots||b_k|| \end{eqnarray*} [[/math]]

Let us set now, according to the above:

[[math]] C_k(G)=\sup_{N\in\mathbb N}\left(N\times\sum_{\sigma,\pi \in D(k)}\left|W_{kN}(\pi,\sigma)N^{|\pi|}-\mu_{D(k)}(\pi,\sigma)\right|\right) [[/math]]

But this number is finite by our main estimate, which completes the proof.

■

We will use in what follows the inclusions [math]G_N\subset G_M[/math] for [math]N \lt M[/math], which correspond to the Hopf algebra morphisms [math]\omega_{N,M}:C(G_M)\to C(G_N)[/math] given by:

[[math]] \omega_{N,M}(u_{ij})= \begin{cases} u_{ij}&{\rm if}\ 1\leq i,j\leq N\\ \delta_{ij}&{\rm if}\ \max(i,j) \gt N\ \end{cases} [[/math]]

Still following , we begin by extending the notion of [math]G_N[/math]-invariance to the infinite sequences of variables, in the following way:

Definition

Let [math](x_i)_{i\in\mathbb N}[/math] be a sequence in a noncommutative probability space [math](A,\varphi)[/math]. We say that [math](x_i)_{i\in\mathbb N}[/math] is [math]G[/math]-invariant if

[[math]] (x_1,\ldots,x_N) [[/math]]

is [math]G_N[/math]-invariant for each [math]N\in\mathbb N[/math].

In other words, the condition is that the joint distribution of [math](x_1,\ldots,x_N)[/math] should be invariant under the following coaction map, for each [math]N\in\mathbb N[/math]:

[[math]] \alpha_N:\mathbb C \lt t_1,\ldots,t_N \gt \to\mathbb C \lt t_1,\ldots,t_N \gt \otimes\,C(G_N) [[/math]]

It is convenient to extend these coactions to a coaction on the algebra of noncommutative polynomials on an infinite number of variables, in the following way:

[[math]] \beta_N:\mathbb C \lt t_i|i\in\mathbb N \gt \to\mathbb C \lt t_i|i\in\mathbb N \gt \otimes\,C(G_N) [[/math]]

Indeed, we can define [math]\beta_N[/math] to be the unique unital morphism satisfying:

[[math]] \beta_N(t_j)= \begin{cases} \sum_{i=1}^Nt_i\otimes v_{ij}&{\rm if}\ 1\leq j\leq N\\ t_j\otimes 1&{\rm if}\ j \gt N \end{cases} [[/math]]

It is clear that [math]\beta_N[/math] as constructed above is a coaction of [math]G_N[/math]. Also, we have the following relations, where [math]\iota_N:\mathbb C \lt t_1,\ldots,t_N \gt \to\mathbb C \lt t_i|i\in\mathbb N \gt [/math] is the natural inclusion:

[[math]] (id\otimes \omega_{N,M})\beta_M=\beta_N [[/math]]

[[math]] (\iota_N\otimes id)\alpha_N=\beta_N\iota_N [[/math]]

By using these compatibility relations, we obtain the following result:

Proposition

An infinite sequence of random variables [math](x_i)_{i\in\mathbb N}[/math] is [math]G[/math]-invariant if and only if the joint distribution functional

[[math]] \mu_x:\mathbb C \lt t_i|i\in\mathbb N \gt \to\mathbb C [[/math]]

[[math]] P\to tr(P(x)) [[/math]]

is invariant under the coaction [math]\beta_N[/math], for each [math]N\in\mathbb N[/math].

Show Proof

This is clear indeed from the above discussion.

■

In what follows [math](x_i)_{i\in\mathbb N}[/math] will be a sequence of self-adjoint random variables in a von Neumann algebra [math](M,tr)[/math]. We will assume that [math]M[/math] is generated by [math](x_i)_{i\in\mathbb N}[/math]. We denote by [math]L^2(M,tr)[/math] the corresponding GNS Hilbert space, with inner product as follows:

[[math]] \lt m_1,m_2 \gt =tr(m_1m_2^*) [[/math]]

Also, the strong topology on [math]M[/math], that we will use in what follows, will be taken by definition with respect to the faithful representation on the space [math]L^2(M,tr)[/math].

We let [math]P_N[/math] be the fixed point algebra of the action [math]\beta_N[/math], and we set:

[[math]] B_N=\left\{p(x)\Big|p\in P_N\right\}'' [[/math]]

We have then an inclusion [math]B_{N+1}\subset B_N[/math], for any [math]N\geq1[/math], and we can then define the [math]G[/math]-invariant subalgebra as the common intersection of these algebras:

[[math]] B=\bigcap_{N\geq1}B_N [[/math]]

With these conventions, we have the following result, from :

Proposition

If an infinite sequence of random variables [math](x_i)_{i\in\mathbb N}[/math] is [math]G[/math]-invariant, then for each [math]N\in\mathbb N[/math] there is a coaction

[[math]] \widetilde{\beta}_N:M\to M\otimes L^\infty(G_N) [[/math]]

determined by the following formula, for any [math]p\in\mathcal P_\infty[/math]:

[[math]] \widetilde{\beta}_N(p(x))=(ev_x\otimes\pi_N)\beta_N(p) [[/math]]

The fixed point algebra of [math]\widetilde{\beta}_N[/math] is then [math]B_N[/math].

Show Proof

This is indeed clear from definitions, and from the various compatibility formulae given above, between the coactions [math]\alpha_N[/math] and [math]\beta_N[/math].

■

We have as well the following result, which is clear as well:

Proposition

In the above context, that of an infinite sequence of random variables belonging to an arbitrary von Neumann algebra [math]M[/math] with a trace

[[math]] (x_i)_{i\in\mathbb N} [[/math]]

which is [math]G[/math]-invariant, for each [math]N\in\mathbb N[/math] there is a trace-preserving conditional expectation [math]E_N:M\to B_N[/math] given by integrating the action [math]\widetilde{\beta}_N[/math]:

[[math]] E_N(m)=\left(id\otimes\int_G\right)\widetilde{\beta}_N(m) [[/math]]

By taking the limit of these expectations as [math]N\to \infty[/math], we obtain a trace-preserving conditional expectation onto the [math]G[/math]-invariant subalgebra.

Show Proof

Once again, this is clear from definitions, and from the various compatibility formulae given above, between the coactions [math]\alpha_N[/math] and [math]\beta_N[/math].

■

We are now prepared to state and prove the main theorem, from , which comes as a complement to the reverse De Finetti theorem that we already established:

Theorem

Let [math](x_i)_{i\in\mathbb N}[/math] be a [math]G[/math]-invariant sequence of self-adjoint random variables in [math](M,tr)[/math], and assume that [math]M= \lt (x_i)_{i\in\mathbb N} \gt [/math]. Then there exists a subalgebra [math]B\subset M[/math] and a trace-preserving conditional expectation [math]E:M\to B[/math] such that:

• If [math]G=(S_N)[/math], then [math](x_i)_{i\in\mathbb N}[/math] are conditionally independent and identically distributed given [math]B[/math].
• If [math]G=(S_N^+)[/math], then [math](x_i)_{i\in\mathbb N}[/math] are freely independent and identically distributed with amalgamation over [math]B[/math].
• If [math]G=(O_N)[/math], then [math](x_i)_{i\in\mathbb N}[/math] are conditionally independent, and have Gaussian distributions with mean zero and common variance, given [math]B[/math].
• If [math]G=(O_N^+)[/math], then [math](x_i)_{i\in\mathbb N}[/math] form a [math]B[/math]-valued free semicircular family with mean zero and common variance.

Show Proof

We use the various partial results and formulae established above. Let [math]j_1,\ldots,j_k \in \mathbb N[/math] and [math]b_0,\ldots,b_k\in B[/math]. We have then the following computation:

[[math]] \begin{eqnarray*} E(b_0x_{j_1}\ldots x_{j_k}b_k) &=&\lim_{N\to\infty}E_N(b_0x_{j_1}\ldots x_{j_k}b_k)\\ &=&\lim_{N\to\infty}\sum_{\sigma\leq\ker j}\sum_\pi W_{kN}(\pi,\sigma) \sum_{\pi\leq\ker i}b_0x_{i_1}\ldots x_{i_k}b_k\\ &=&\lim_{N\to\infty}\sum_{\sigma\leq\ker j}\sum_{\pi\leq\sigma}\mu_{D(k)}(\pi,\sigma)N^{-|\pi|}\sum_{\pi\leq\ker i}b_0x_{i_1}\ldots x_{i_k}b_k \end{eqnarray*} [[/math]]

Let us recall now from the above that we have the following compatibility formula, where [math]\widetilde{\iota}_N:W^*(x_1,\ldots,x_N)\to M[/math] is the canonical inclusion, and [math]\widetilde{\alpha}_N[/math] is as before:

[[math]] (\widetilde{\iota}_N\otimes id)\widetilde{\alpha}_N=\widetilde{\beta}_N\widetilde{\iota}_N [[/math]]

By using this formula, and the above cumulant results, we have:

[[math]] E(b_0x_{j_1}\ldots x_{j_k}b_k)=\lim_{N\to\infty}\sum_{\sigma\leq\ker j} \sum_{\pi\leq\sigma}\mu_{D(k)}(\pi,\sigma)E_N^{(\pi)}(b_0x_1b_1,\ldots,x_1b_k) [[/math]]

We therefore obtain the following formula:

[[math]] E(b_0x_{j_1}\ldots x_{j_k}b_k)=\sum_{\sigma\leq\ker j}\sum_{\pi\leq\sigma} \mu_{D(k)}(\pi,\sigma)E^{(\pi)}(b_0x_1b_1,\ldots,x_1b_k) [[/math]]

We can replace the sum of expectation functionals by cumulants, as to obtain:

[[math]] E(b_0x_{j_1}\ldots x_{j_k}b_k)=\sum_{\sigma\leq\ker j}\xi_E^{(\sigma)}(b_0x_1b_1,\ldots,x_1b_k) [[/math]]

Here and in what follows [math]\xi[/math] denotes as usual the relevant free or classical cumulants, depending on the quantum group that we are dealing with, free or classical.

Now since the cumulants are determined by the moment-cumulant formulae, we conclude that we have the following formula:

[[math]] \xi_E^{(\sigma)}(b_0x_{j_1}b_1,\ldots,x_{j_k}b_k) =\begin{cases} \item[a]i_E^{(\sigma)}(b_0x_1b_1,\ldots,x_1b_k)&{\rm if}\ \sigma\in D(k)\ {\rm and}\ \sigma\leq\ker j\\ 0&{\rm otherwise} \end{cases} [[/math]]

With this formula in hand, the result then follows from the characterizations of these joint distributions in terms of cumulants, and we are done.

■

Summarizing, we are done with our first and main objective, namely establishing De Finetti theorems for the main quantum permutation and rotation groups, namely:

[[math]] \xymatrix@R=15mm@C=15mm{ S_N^+\ar[r]&O_N^+\\ S_N\ar[r]\ar[u]&O_N\ar[u] } [[/math]]

The story is of course not over here, and there are many related interesting questions left, which are more technical, in relation with the invariance questions with respect to these quantum groups. We refer here to , , , , and related papers.

Regarding now our second objective, which appears as a variation of this, fully in tune with the present book, we would like to understand as well what happens to the invariance questions with respect to the basic quantum reflection and rotation groups, namely:

[[math]] \xymatrix@R=18pt@C=18pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

Here the answer is more or less known as well from , but with the problem however that the paper is extremely general, and in relation with our cube question, more general than needed. In any case, for this and for further aspects of invariance questions, we refer as before to , , , , and related papers.

General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].