15a. Invariance questions

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An interesting question, which often appears in theoretical probability, as well in connection with certain questions coming from physics, is the study of the sequences of random variables [math]x_1,x_2,x_3,\ldots\in L^\infty(X)[/math] which are exchangeable, in the sense that their joint distribution is invariant under the infinite permutations [math]\sigma\in S_\infty[/math]:

[[math]] \mu_{x_1,x_2,x_3,\ldots}=\mu_{x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)},\ldots} [[/math]]

This question is solved by the classical De Finetti theorem, which basically says that the variables [math]x_1,x_2,x_3,\ldots[/math] must be i.i.d., in some asymptotic sense. We will see a precise statement of this theorem, along with a complete proof, in a minute.

The De Finetti theorem has many generalizations. One can replace for instance the action of the group [math]S_\infty=\cup_NS_N[/math] by the action of the bigger group [math]O_\infty=\cup_NO_N[/math], and the sequences [math]x_1,x_2,x_3,\ldots\in L^\infty(X)[/math] which are invariant in this stronger sense, which are called “rotatable”, can be characterized as well, via a De Finetti type theorem.

All this is interesting for us, in connection with what we have been doing so far, in this book. On one hand the groups [math]S_N,O_N[/math] are easy, and we would like to understand how the above-mentioned De Finetti theorems, involving [math]S_N,O_N[/math], as well as their various technical generalizations, follow from the easiness property of [math]S_N,O_N[/math]. On the other hand, we would like to understand as well what happens for [math]S_N^+,O_N^+[/math].

Long story short, we would like to discuss here probabilistic invariance questions with respect to the basic 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]]

As a second objective, in tune with what we have been doing so far in this book, we would like as well to understand what happens to the invariance questions with respect to the basic quantum reflection and rotation groups, from our beloved cube, 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]]

We will discuss here most of these questions, following the classical theory of the De Finetti theorem, then the foundational paper of Köstler and Speicher , in the free case, and then the more advanced paper , dealing with both the classical and free De Finetti theorems, and their other easy quantum group generalizations.

Let us start by fixing some notations. In order to deal with our first question above, we will use here the formalism of the orthogonal quantum groups, which best covers the main quantum groups that we are interested in. We first have the following definition:

Definition

Given a closed subgroup [math]G\subset O_N^+[/math], we denote by

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

[[math]] t_i\to\sum_jt_j\otimes v_{ji} [[/math]]

the standard coaction of [math]C(G)[/math] on the free complex algebra on [math]N[/math] variables.

Observe that the map [math]\alpha[/math] constructed above is indeed a coaction, in the sense that it satisfies the following standard coassociativity and counitality conditions:

[[math]] (id\otimes\Delta)\alpha=(\alpha\otimes id)\alpha [[/math]]

[[math]] (id\otimes\varepsilon)\alpha=id [[/math]]

With the above notion of coaction in hand, we can now talk about invariant sequences of classical or noncommutative random variables, in the following way:

Definition

Let [math](B,tr)[/math] be a [math]C^*[/math]-algebra with a trace, and [math]x_1,\ldots,x_N\in B[/math]. We say that [math]x=(x_1,\ldots,x_N)[/math] is invariant under [math]G\subset O_N^+[/math] if the distribution functional

[[math]] \mu_x:\mathbb C \lt t_1,\ldots,t_N \gt \to\mathbb C [[/math]]

[[math]] P\to tr(P(x_1,\ldots,x_N)) [[/math]]

is invariant under the coaction [math]\alpha[/math], in the sense that we have

[[math]] (\mu_x\otimes id)\alpha(P)=\mu_x(P) [[/math]]

for any noncommuting polynomial [math]P\in\mathbb C \lt t_1,\ldots,t_N \gt [/math].

In the classical case, where [math]G\subset O_N[/math] is a usual group, we recover in this way the usual invariance notion from classical probability. In the general case, where [math]G\subset O_N^+[/math] is arbitrary, what we have is a natural generalization of this. For further comments on all this, including examples, and motivations too, we refer to , , , , .

We have the following equivalent formulation of the above invariance condition:

Proposition

Let [math](B,tr)[/math] be a [math]C^*[/math]-algebra with a trace, and [math]x_1,\ldots,x_N\in B[/math]. Then [math]x=(x_1,\ldots,x_N)[/math] is invariant under [math]G\subset O_N^+[/math] precisely when

[[math]] tr(x_{i_1}\ldots x_{i_k})=\sum_{j_1\ldots j_k}tr(x_{j_1}\ldots x_{j_k})v_{j_1i_1}\ldots v_{j_ki_k} [[/math]]

as an equality in [math]C(G)[/math], for any [math]k\in\mathbb N[/math], and any [math]i_1,\ldots,i_k\in\{1,\ldots,N\}[/math].

Show Proof

By linearity, in order for a sequence [math]x=(x_1,\ldots,x_N)[/math] to be [math]G[/math]-invariant in the sense of Definition 15.2, the formula there must be satisfied for any noncommuting monomial [math]P\in\mathbb C \lt t_1,\ldots,t_N \gt [/math]. But an arbitrary such monomial can be written as follows, for a certain [math]k\in\mathbb N[/math], and certain indices [math]i_1,\ldots,i_k\in\{1,\ldots,N\}[/math]:

[[math]] P=t_{i_1}\ldots t_{i_k} [[/math]]

Now with this formula for [math]P[/math] in hand, we have the following computation:

[[math]] \begin{eqnarray*} (\mu_x\otimes id)\alpha(P) &=&(\mu_x\otimes id)\sum_{j_1,\ldots,j_k}t_{j_1}\ldots t_{j_k}\otimes v_{j_1i_1}\ldots v_{j_ki_k}\\ &=&\sum_{j_1,\ldots,j_k}\mu_x(t_{j_1}\ldots t_{j_k})v_{j_1i_1}\ldots v_{j_ki_k}\\ &=&\sum_{j_1\ldots j_k}tr(x_{j_1}\ldots x_{j_k})v_{j_1i_1}\ldots v_{j_ki_k} \end{eqnarray*} [[/math]]

On the other hand, by definition of the distribution [math]\mu_x[/math], we have:

[[math]] \mu_x(P) =\mu_x(t_{i_1}\ldots t_{i_k}) =tr(x_{i_1}\ldots x_{i_k}) [[/math]]

Thus, we are led to the conclusion in the statement.

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As already mentioned after Definition 15.2, in the classical case, where [math]G\subset O_N[/math] is a usual compact group, our notion of [math]G[/math]-invariance coincides with the usual [math]G[/math]-invariance notion from classical probability. We have in fact the following result:

Proposition

In the classical group case, [math]G\subset O_N[/math], a sequence [math](x_1,\ldots,x_N)[/math] is [math]G[/math]-invariant in the above sense if and only if

[[math]] tr(x_{i_1}\ldots x_{i_k})=\sum_{j_1\ldots j_k}g_{j_1i_1}\ldots g_{j_ki_k}tr(x_{j_1}\ldots x_{j_k}) [[/math]]

for any [math]k\in\mathbb N[/math], any [math]i_1,\ldots,i_k\in\{1,\ldots,N\}[/math], and any [math]g=(g_{ij})\in G[/math], and this coincides with the usual notion of [math]G[/math]-invariance for a sequence of classical random variables.

Show Proof

According to Proposition 15.3, the invariance property happens precisely when we have the following equality, for any [math]k\in\mathbb N[/math], and any [math]i_1,\ldots,i_k\in\{1,\ldots,N\}[/math]:

[[math]] tr(x_{i_1}\ldots x_{i_k})=\sum_{j_1\ldots j_k}tr(x_{j_1}\ldots x_{j_k})v_{j_1i_1}\ldots v_{j_ki_k} [[/math]]

Now by evaluating both sides of this equation at a given [math]g\in G[/math], we obtain:

[[math]] tr(x_{i_1}\ldots x_{i_k})=\sum_{j_1\ldots j_k}g_{j_1i_1}\ldots g_{j_ki_k}tr(x_{j_1}\ldots x_{j_k}) [[/math]]

Thus, we are led to the conclusion in the statement.

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Summarizing, what we have so far is a general notion of probabilistic invariance, generalizing the classical notions of exchangeability and rotatability, than we can use for reformulating the classical De Finetti problematics, and its various generalizations.

In order to formulate De Finetti type theorems, that we can try to prove afterwards, we are still in need of a few pieces of general theory. Indeed, in the classical De Finetti theorem, the independence occurs after conditioning. Likewise, we can expect the free De Finetti theorem to be a statement about freeness with amalgamation.

Both these concepts may be expressed in terms of operator-valued probability theory, that we will recall now. There are many things to be said here, and in what follows we will mainly present the main definitions and theorems, with some brief explanations. Following Speicher's paper , we first have the following definition:

Definition

An operator-valued probability space consists of:

• A unital algebra [math]A[/math].
• A unital subalgebra [math]B\subset A[/math].
• An expectation [math]E:A\to B[/math], which must be unital, [math]E(1)=1[/math], and satisfying
  [[math]] E(b_1ab_2)=b_1E(a)b_2 [[/math]]
  for any [math]a\in A[/math], and any [math]b_1,b_2\in B[/math].

As a basic example, which motivates the whole theory, we have the case where [math]A=L^\infty(X)[/math] is a usual algebra of classical random variables, and [math]B=L^\infty(Y)[/math] is a subalgebra. Here the expectation [math]E:A\to B[/math] is the usual one from classical probability.

Given an operator-valued probability space as above, the joint distribution of a family of variables [math](x_i)_{i\in I}[/math] in the algebra [math]A[/math] is by definition the following functional:

[[math]] \mu_x:B \lt (t_i)_{i\in I} \gt \to B [[/math]]

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

We refer to Speicher's paper and related papers for more on all this, general results and examples, in relation with the operator-valued probability theory.

Next in line, we have the following key definition, also from :

Definition

Let [math](A,B,E)[/math] be as above, and [math](x_i)_{i\in I}[/math] be a family of variables.

• These variables are called independent if the following algebra is commutative
  [[math]] \lt B,(x_i)_{i\in I} \gt \subset A [[/math]]
  and for [math]i_1,\ldots,i_k\in I[/math] distinct and [math]P_1,\ldots,P_k\in B \lt t \gt [/math], we have:
  [[math]] E(P_1(x_{i_1})\ldots P_k(x_{i_k}))=E(P_1(x_{i_1}))\ldots E(P_k(x_{i_k})) [[/math]]
• These variables are called free if for any [math]i_1,\ldots,i_k\in I[/math] such that [math]i_l\neq i_{l+1}[/math], and any [math]P_1,\ldots,P_k\in B \lt t \gt [/math] such that [math]E(P_l(x_{i_l}))= 0[/math], we have:
  [[math]] E(P_1(x_{i_1})\ldots P_k(x_{i_k}))=0 [[/math]]

The above notions are straighforward extensions of the usual notions of independence and freeness, that we discussed in chapter 9, which correspond to the case [math]B=\mathbb C[/math].

As in the scalar case, [math]B=\mathbb C[/math], in order to deal with invariance questions, we will need the theory of classical and free cumulants, in the present setting. Let us start with:

Definition

Let [math](A,B,E)[/math] be an operator-valued probability space.

• A [math]B[/math]-functional is a [math]N[/math]-linear map [math]\rho:A^N\to B[/math] such that:
  [[math]] \rho(b_0a_1b_1,a_2b_2\ldots,a_Nb_N)=b_0\rho(a_1,b_1a_2,\ldots,b_{N-1}a_N)b_N [[/math]]
  Equivalently, [math]\rho[/math] is a linear map of the following type
  [[math]] A^{\otimes_BN}\to B [[/math]]
  where the tensor product is taken with respect to the natural [math]B-B[/math] bimodule structure on the algebra [math]A[/math].
• Suppose that [math]B[/math] is commutative. For [math]k\in\mathbb N[/math] let [math]\rho^{(k)}[/math] be a [math]B[/math]-functional. Given [math]\pi\in P(n)[/math], we define a [math]B[/math]-functional [math]\rho^{(\pi)}:A^N\to B[/math] by the formula
  [[math]] \rho^{(\pi)}(a_1,\ldots,a_N)=\prod_{V\in\pi}\rho(V)(a_1,\ldots,a_N) [[/math]]
  where if [math]V=(i_1 \lt \ldots \lt i_s)[/math] is a block of [math]\pi[/math] then:
  [[math]] \rho(V)(a_1,\ldots,a_N)=\rho_s(a_{i_1},\ldots,a_{i_s}) [[/math]]

As before with the notions of independence and freeness, these are classical extensions of the notions that we discussed in chapter 12 above. See .

When [math]B[/math] is not commutative, there is no natural order in which to compute the product appearing in the above formula for [math]\rho^{(\pi)}[/math]. However, the nesting property of the noncrossing partitions allows for a natural definition of [math]\rho^{(\pi)}[/math] for [math]\pi\in NC(N)[/math], which we now recall:

Definition

For [math]k\in\mathbb N[/math] let [math]\rho^{(k)}:A^k\to B[/math] be a [math]B[/math]-functional. Given [math]\pi \in NC(N)[/math], define a [math]B[/math]-functional [math]\rho^{(N)}:A^N\to B[/math] recursively as follows:

• If [math]\pi=1_N[/math] is the partition having one block, define [math]\rho^{(\pi)}=\rho^{(N)}[/math].
• Otherwise, let [math]V=\{l+1,\ldots,l+s\}[/math] be an interval of [math]\pi[/math] and define:
  [[math]] \rho^{(\pi)}(a_1,\ldots,a_N)=\rho^{(\pi-V)}(a_1,\ldots,a_l\rho^{(s)}(a_{l+1},\ldots,a_{l+s}),a_{l+s+1},\ldots,a_N) [[/math]]

As before, we refer to , and related work for more on all this.

Finally, we have the following definition:

Definition

Let [math](x_i)_{i\in I}[/math] be a family of random variables in [math]A[/math].

• The operator-valued classical cumulants [math]c_E^{(k)}:A^k\to B[/math] are the [math]B[/math]-functionals defined by the following classical moment-cumulant formula:
  [[math]] E(a_1\ldots a_N)=\sum_{\pi\in P(N)}c_E^{(\pi)}(a_1,\ldots,a_N) [[/math]]
• The operator-valued free cumulants [math]\kappa_E^{(k)}:A^k\to B[/math] are the [math]B[/math]-functionals defined by the following free moment-cumulant formula:
  [[math]] E(a_1,\ldots,a_N)=\sum_{\pi\in NC(N)}\kappa_E^{(\pi)}(a_1,\ldots,a_N) [[/math]]

As basic illustrations here, in the scalar case, where the subalgebra is [math]B=\mathbb C[/math], we recover in this way the classical and free cumulants, as discussed in chapter 12 above. In general, we refer to for more on the above notions.

We have the following result, which is well-known in the classical case, due to Rota, and which in the free case is due to Speicher :

Theorem

Let [math](x_i)_{i \in I}[/math] a family of random variables in [math]A[/math].

• If the algebra [math] \lt B,(x_i)_{i\in I} \gt [/math] is commutative, then [math](x_i)_{i \in I}[/math] are conditionally independent given [math]B[/math] if and only if when there are [math]1\leq k,l\leq N[/math] such that [math]i_k\neq i_l[/math]:
  [[math]] c_E^{(N)}(b_0x_{i_1}b_1,\ldots,x_{i_N}b_N)=0 [[/math]]
• The variables [math](x_i)_{i \in I}[/math] are free with amalgamation over [math]B[/math] if and only if when there are [math]1\leq k,l\leq N[/math] such that [math]i_k\neq i_l[/math]:
  [[math]] \kappa_E^{(N)}(b_0x_{i_1}b_1,\ldots,x_{i_N}b_N)=0 [[/math]]

Show Proof

As a first observation, the condition in (1) is equivalent to the statement that if [math]\pi\in P(N)[/math], then the following happens, unless [math]\pi\leq\ker i[/math]:

[[math]] c_E^{(\pi)}(b_0x_{i_1}b_1,\ldots,x_{i_N}b_N)=0 [[/math]]

Similarly, the condition (2) above is equivalent to the statement that if [math]\pi\in NC(N)[/math], then the following happens, unless [math]\pi\leq\ker i[/math]:

[[math]] \kappa_E^{(\pi)}(b_0x_{i_1}b_1,\ldots,x_{i_N}b_N)=0 [[/math]]

Observe also that in the case [math]B=\mathbb C[/math] we obtain the usual notions of independence and freeness. In general now, the proof is via standard combinatorics, following the proof from the case [math]B=\mathbb C[/math], and as before, we refer to , for more on all this.

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Stronger characterizations of the joint distribution of [math](x_i)_{i\in I}[/math] can be given by specifying what types of partitions may contribute to the nonzero cumulants.

To be more precise, we have here the following result, also from :

Theorem

Let [math](x_i)_{i \in I}[/math] be a family of random variables in [math]A[/math].

• Suppose that [math] \lt B,(x_i)_{i\in I} \gt [/math] is commutative. The [math]B[/math]-valued joint distribution of [math](x_i)_{i\in I}[/math] is independent for [math]D=P[/math] and independent centered Gaussian for [math]D=P_2[/math] if and only if, for any [math]\pi\in P(N)[/math], unless [math]\pi \in D(N)[/math] and [math]\pi\leq\ker i[/math]:
  [[math]] c_E^{(\pi)}(b_0x_{i_1}b_1,\ldots,x_{i_N}b_N)=0 [[/math]]
• The [math]B[/math]-valued joint distribution of [math](x_i)_{i\in I}[/math] is freely independent for [math]D=NC[/math] and freely independent centered semicircular for [math]D=NC_2[/math] if and only if, for any [math]\pi\in NC(N)[/math], unless [math]\pi\in D(N)[/math] and [math]\pi\leq\ker i[/math]:
  [[math]] \kappa_E^{(\pi)}(b_0x_{i_1}b_1,\ldots,x_{i_N}b_N)=0 [[/math]]

Show Proof

These results are indeed well-known, coming from the definition of the classical and free cumulants, in the present setting, via some combinatorics. See .

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Finally, here is one more basic result that we will need:

Theorem

Let [math](x_i)_{i \in I}[/math] be a family of random variables. Define the [math]B[/math]-valued moment functionals [math]E^{(N)}[/math] by the following formula:

[[math]] E^{(N)}(a_1,\ldots,a_N)=E(a_1\ldots a_N) [[/math]]

• If [math]B[/math] is commutative, then for any [math]\sigma\in P(N)[/math] and [math]a_1,\ldots,a_N\in A[/math] we have:
  [[math]] c_E^{(\sigma)}(a_1,\ldots,a_N)=\sum_{\pi\in P(N),\pi\leq\sigma}\mu_{P(N)}(\pi,\sigma)E^{(\pi)}(a_1,\ldots,a_N) [[/math]]
• For any [math]\sigma\in NC(N)[/math] and [math]a_1,\ldots,a_N\in A[/math] we have:
  [[math]] \kappa_E^{(\sigma)}(a_1,\ldots,a_N)=\sum_{\pi\in NC(N),\pi\leq\sigma} \mu_{NC(N)}(\pi,\sigma)E^{(\pi)}(a_1,\ldots,a_N) [[/math]]

Show Proof

This follows indeed from the Mòbius inversion formula. See , .

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This was the general operator-valued free probability theory that we will need, in what follows. For the detailed proofs, examples and comments on all the above, and for more operator-valued free probability in general, we refer to , .

General references

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