Let ${\mathcal{A}}$ be a ${\star}$-algebra over ${\mathbb{C}}$ with zero ${0}$, unit ${1}$, and involution ${a\mapsto a^*}$ such that ${(ab)^*=b^*a^*}$ for every ${a,b\in\mathcal{A}}$. Let ${\tau:\mathcal{A}\rightarrow\mathbb{C}}$ be a linear form such that ${\tau(aa^*)\geq0}$ for every ${a\in\mathcal{A}}$, and ${\tau(1)=1}$. We say then that ${(\mathcal{A},\tau)}$ is an algebraic probability space. We do not assume that ${\tau(ab)=\tau(ba)}$ for every ${a,b\in\mathcal{A}}$ or that ${\tau(aa^*)=0}$ iif ${a=0}$, even if it is the case in the following couple of examples. The simplest example, commutative, is given by

$\mathcal{A}=\bigcap_{1\leq p<\infty}\mathrm{L}^p(\Omega,\mathcal{F},\mathbb{P},\mathbb{C}) \quad\text{and}\quad \tau=\mathbb{E} \quad\text{and}\quad a^*=\bar{a}$

where ${(\Omega,\mathcal{F},\mathbb{P})}$ is a classical probability space. A non commutative example is given by

$\mathcal{A}=\mathcal{M}_n(\mathbb{C}) \quad\text{and}\quad \tau=\frac{1}{n}\mathrm{Trace} \quad\text{and}\quad a^*=\bar{a}^\top.$

One can mix the two by considering integrable random matrices equipped with ${\tau=\frac{1}{n}\mathbb{E}\mathrm{Trace}(\cdot)}$. Here we focus on the purely algebraic notion of ${\star}$-algebras, and we should not confuse this notion with the algebraic-analytic notions of ${C^*}$-algebras or von Neumann ${W^*}$-algebras.

Algebraic random variables. An element ${a\in\mathcal{A}}$ is called an algebraic random variable, and its ${\star}$-distribution is the collection of ${\star}$-moments

$\tau(a^{\varepsilon_1}\cdots a^{\varepsilon_n})$

for every ${n\geq1}$ and every ${\varepsilon_1,\ldots,\varepsilon_n}$ in ${\{1,\star\}}$. When ${a=a^*}$ (we say that ${a}$ is real) then the ${\star}$-distribution of ${a}$ is characterized by the sequence of moments ${\tau(a^n)}$, ${n\in\mathbb{N}}$. In this case, and thanks to the Hamburger moment theorem, this sequence of moments of ${a}$ is the sequence of moments of some probability measure ${\mu}$ on ${\mathbb{R}}$. This probability distribution ${\mu}$ is not unique in general, the Carleman condition says that uniqueness holds if

$\sum_n(\tau(a^{2n}))^{-1/(2n)}=\infty.$

Now we define four algebraic notions of independence, which corresponds actually to simplification rules for the computation of mixed moments. The first notion matches the classical notion of commutative probability theory, while the second notion is the one of free probability theory.

Commutative independence. A family ${(\mathcal{A}_i)_{i\in I}}$ of sub-${\star}$-algebras of ${\mathcal{A}}$ is commutative independent when for every ${i_1,\ldots,i_n\in I}$, and every ${a_1\in\mathcal{A}_{i_1},\ldots,a_n\in\mathcal{A}_{i_n}}$,

$\tau(a_1\cdots a_n) = \left\{ \begin{array}{ll} \tau(a_1)\tau(a_2\cdots a_n) & \mbox{if } i_1\not\in\{i_2,\ldots,i_n\} \\ \tau(a_2\cdots a_{r-1}(a_1a_r)a_{r+1}\cdots a_n) & \mbox{if } r=\min\{j>1:i_1=i_j\} \end{array} \right.$

This allows first to group the ${a_i}$ belonging to the same sub-${\star}$-algebra and then to break ${\tau}$. The ${\star}$-distribution of ${a}$ is in this case uniquely determined by the law of ${a}$ as a classical random variable (the converse is not true in general, since the moment problem may not have a unique solution).

Free independence. A family ${(\mathcal{A}_i)_{i\in I}}$ of sub-${\star}$-algebras of ${\mathcal{A}}$ is free independent when for every ${i_1,\ldots,i_n\in I}$ with ${i_1\neq\cdots\neq i_n}$ (any two consecutive indices are different), and every ${a_1\in\mathcal{A}_{i_1},\ldots,a_n\in\mathcal{A}_{i_n}}$,

$\tau((a_1-\tau(a_1))\cdots(a_n-\tau(a_n))) =0.$

Note that if ${a,b}$ are free independent (i.e. their generated algebra are free independent), and if ${\tau(a)=\tau(b)=0}$, then ${\tau(abab)=0}$, while for the commutative independence, ${\tau(abab)=\tau(a^2b^2)=\tau(a^2)\tau(b^2)}$ which is not zero in general.

Boolean independence. A family ${(\mathcal{A}_i)_{i\in I}}$ of sub-sets of ${\mathcal{A}}$ closed for the algebraic operations and the involution ${\star}$, but which may not contain the unit ${1}$, is Boolean independent when for every ${i_1,\ldots,i_n\in I}$ with ${i_1\neq\cdots\neq i_n}$ (any two consecutive indices are different), and every ${a_1\in\mathcal{A}_{i_1},\ldots,a_n\in\mathcal{A}_{i_n}}$,

$\tau(a_1\cdots a_n)=\tau(a_1)\tau(a_2\cdots a_n).$

For instance, if ${a}$ and ${b}$ are Boolean free (i.e. their generated sub-sets are Boolean free) then ${\tau(abab)=\tau(a)^2\tau(b)^2}$ and ${\tau(aba^2b)=\tau(a)\tau(b)^2\tau(a^2)}$.

Monotone independence. Recall that if ${i_1\neq \cdots \neq i_n}$ is a sequence of integers where any two consecutive are different, then ${i_k}$ is a peak when either ${i_1>i_2}$ (if ${k=1}$), ${i_{n-1}<i_n}$ (if ${k=n}$), ${i_{k-1}<i_k>i_{k+1}}$ (if ${1<k<n}$). A family ${(\mathcal{A}_i)_{i\in\mathbb{N}}}$ of sub-sets of ${\mathcal{A}}$ closed for the algebraic operations and the involution ${\star}$, but which may not contain the unit ${1}$, is monotone independent when for every ${i_1,\ldots,i_n\in\mathbb{N}}$ with ${i_1\neq\cdots\neq i_n}$ (any two consecutive indices are different), and every ${a_1\in\mathcal{A}_{i_1},\ldots,a_n\in\mathcal{A}_{i_n}}$,

$\tau(a_1\cdots a_n)=\tau(a_k)\tau(a_1\cdots \check{a}_k \cdots a_n)$

when ${k}$ is a peak of the sequence ${i_1,\ldots,i_n}$, and where ${\check{a}_k}$ is the removal of ${a_k}$.

Note. The Boolean and monotone independence are trivial when ${1\in\mathcal{A}_i}$ i.e. when ${\mathcal{A}_i}$ is a sub-${\star}$-algebras. The notion of free independence was introduced by Voiculescu and is at the heart of free probability theory. The notion of Boolean independence was developed by Bozjeko and his followers. The notion of monotone independence is due to Lu and Muraki. Various other notions of independence, not considered here, are studied in the literature.

Convolutions. If ${a}$ and ${b}$ are independent for one of the four notions of independence, then the ${\star}$-distribution of ${a+b}$ depends only on the ${\star}$-distribution of ${a}$ and of ${b}$, and is called the convolution of these distributions. We recover the classical notion of convolution for the commutative independence, and the Voiculescu notion of free convolution for the free independence.

Singleton property. The four notion of independence satisfy to the singleton property: if ${a_1\in\mathcal{A}_{i_1}\ldots,a_n\in\mathcal{A}_{i_n}}$ where ${\mathcal{A}_1,\ldots,\mathcal{A}_n}$ are independent (for any of these four notions), and if ${a_i=a_i^*}$ for any ${1\leq i\leq n}$, and if ${\tau(a_1)=\cdots=\tau(a_n)=0}$, and if there exists ${1\leq k\leq n}$ such that ${\{1\leq j\leq n:i_j=i_k\}=\{k\}}$ (i.e. ${a_k}$ is the unique element of ${\mathcal{A}_{i_k}}$ in the sequence ${a_1,\ldots,a_n}$), then ${\tau(a_1\cdots a_n)=0}$.

Central limit theorems. Let ${a_1,a_2,\ldots\in\mathcal{A}}$. We have, for every ${n,m\geq1}$,

$\tau((a_1+\cdots+a_n)^m)=\sum_{i_1,\ldots,i_m=1}^n\tau(a_{i_1}\cdots a_{i_m}).$

The mixed moment ${\tau(a_{i_1}\cdots a_{i_m})}$ can be computed using the notions of independence. Let us make the following assumptions on the variables:

• the variables are real: ${a_i=a_i^*}$ for any ${i\geq1}$
• the variables are centered and normalized: ${\tau(a_i)=0}$ and ${\tau(a_i^2)=1}$ for all ${i\geq1}$
• the variables have bounded mixed moments: for all ${n\geq1}$,

$\sup_{i_1\geq1,\ldots,i_n\geq1}|\tau(a_{i_1}\cdots a_{i_n})|<\infty$

• the variables are independent (for one of the four notions of independence).

Then it can be shown that for every ${m\geq1}$,

$\lim_{n\rightarrow\infty}\tau\left(\left(\frac{a_1+\cdots+a_n}{\sqrt{n}}\right)^m\right) = \int\!x^m\,d\mu$

where the limiting distribution ${\mu}$ is${\ldots}$

• for commutative independence, the standard Gaussian distribution

$\frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\,dx$

Its moments are given by ${m_{2k+1}=0}$ and ${m_{2k}=\frac{(2k)!}{2(k!)}=(2k-1)!!}$

• for free independence, the Wigner semi-circle distribution on ${[-2,2]}$

$\sqrt{4-x^2}\frac{\mathbf{1}_{[-2,2]}(x)}{2\pi}\,dx$

Its moments are given by ${m_{2k+1}=0}$ and ${m_{2k}=\frac{1}{k+1}\binom{2k}{k}=\frac{(2k)!}{k!(k+1)!}}$ (Catalan numbers)

• for Boolean independence, the symmetric Bernoulli distribution on ${\{-1,1\}}$

$\frac{1}{2}(\delta_{-1}+\delta_1)$

Its moments are given by ${m_{2k+1}=0}$ and ${m_{2k}=1}$

• for monotone independence, the arc-sine distribution on ${[-\sqrt{2},\sqrt{2}]}$

$\frac{\mathbf{1}_{[-\sqrt{2},\sqrt{2}]}(x)}{\pi\sqrt{2-x^2}}\,dx$

Its moments are given by ${m_{2k+1}=0}$ and ${m_{2k}=2^{-k}\binom{2k}{k}=\frac{(2k)!}{k!^22^k}}$

Stability. The Gaussian distribution is stable by the commutative convolution, the Wigner semi-circle distribution is stable by the free convolution, the Bernoulli distribution is stable by the Bernoulli convolution, while the arc-sine distribution is stable by the monotone convolution.

Open problem. Note that in the four cases, the second moment is constantly equal to ${1}$ along the central limit theorem (conservation law). In the case of commutative independence, it has been conjectured by Shannon and proved few years ago that the Boltzmann-Shannon entropy is monotonic along the central limit theorem (additionally, its maximum under a second moment constraint is achieved by the standard Gaussian law). Similarly, in the case of free independence, it has been proved by Shlyakhtenko few years ago that the Voiculescu entropy is monotonic along the central limit theorem (additionally, its maximum under a second moment constraint is achieved by the Wigner semi-circle distribution). Both entropies are additive for tensor products of random variables. The existence of such entropies for the Boolean and monotonic independence constitutes a natural problem (still open at the time of writing – any ideas?).