Free probability studies probability-like calculations in algebras where products need not commute. Its central independence notion is *free independence*: a rule controlling alternating centered products rather than a rule built from commuting random variables. This course starts from ordered words, expectation functionals, moments, and joint laws, while using standard tools from linear algebra, elementary Hilbert-space language, finite posets, and bounded operator models when they clarify the foundations.

The course builds systematically from foundations to computational machinery. Chapters 1-3 establish noncommutative probability spaces, ordered moments, joint laws, and the first examples showing why free independence differs from classical independence. Chapters 4-6 introduce *noncrossing partitions* as the combinatorial skeleton of freeness and develop *free cumulants*, the coordinates in which free independence becomes a vanishing condition. This trio of concepts, freeness, noncrossing partitions, and cumulants, forms the computational core of the notes.

The final chapters move from foundations to models and computations. Chapters 7-8 study *semicircular variables*, the free analogue of Gaussian random variables, and prove the *[free central limit theorem](/theorems/7145)*. Chapters 9-10 construct free families in explicit algebraic and Hilbert-space models, then turn the moment-cumulant formulas into a practical toolkit for computing examples. Analytic and matrix-theoretic motivations remain in the background here; the page itself builds the algebraic language needed before those later directions.

# Introduction

This opening chapter explains what the course means by free probability and why its first foundations are algebraic rather than analytic. Classical probability studies random variables through expectations of products, but the products commute; free probability keeps the expectation language while allowing products to depend on order. The first goal is to replace a probability space by a noncommutative probability space, then to identify the correct analogue of independence.

The course deliberately postpones analytic transform methods and random matrix limits. Those topics motivate the subject, but the first layer of the theory is built from states, moments, words, partitions, and cumulants. By Chapter 8, the free [central limit theorem](/theorems/521) will emerge from these foundations in the same structural role played by the classical [central limit theorem](/theorems/1848).

## Why Noncommutative Probability Starts With Moments

What data should determine the law of a random object when multiplication is not commutative? In classical probability, a real [random variable](/page/Random%20Variable) $X$ is often studied through the numbers $\mathbb E[X^n]$, and several variables are studied through mixed moments such as $\mathbb E[X_1X_2X_1]$. If the variables commute, many words collapse to the same monomial; if they do not commute, the order of the letters becomes part of the data.

[definition: Noncommutative Polynomial] Let $x_1, \dots, x_n$ be formal noncommuting variables. The algebra $\mathbb C\langle x_1, \dots, x_n\rangle$ is the complex [vector space](/page/Vector%20Space) with basis all words in the letters $x_1, \dots, x_n$, equipped with multiplication by concatenation and extended linearly. [/definition]

This definition records the first structural change from commutative probability: $x_1x_2$ and $x_2x_1$ are different monomials. The next issue is whether this order-dependence can be seen in a concrete finite model, because the course needs examples where noncommutative words are actual products rather than formal symbols.

[example: Two Ordered Products] Let $A=e_{12}$ and $B=e_{21}$ in $M_2(\mathbb C)$, with $\operatorname{tr}_2(C)=\frac{1}{2}\operatorname{Tr}(C)$. For standard matrix units, $e_{ij}e_{kl}=\delta_{jk}e_{il}$, since the only possible nonzero entry of the product lies in position $(i,l)$ and occurs exactly when $j=k$. Therefore \begin{align*} AB=e_{12}e_{21}=\delta_{2,2}e_{11}=e_{11}. \end{align*} Similarly, \begin{align*} BA=e_{21}e_{12}=\delta_{1,1}e_{22}=e_{22}. \end{align*} The matrices are different: $e_{11}$ has diagonal entries $1,0$, while $e_{22}$ has diagonal entries $0,1$. Their normalized traces nevertheless agree, because \begin{align*} \operatorname{tr}_2(AB)=\operatorname{tr}_2(e_{11})=\frac{1}{2}\operatorname{Tr}(e_{11})=\frac{1}{2}. \end{align*} and \begin{align*} \operatorname{tr}_2(BA)=\operatorname{tr}_2(e_{22})=\frac{1}{2}\operatorname{Tr}(e_{22})=\frac{1}{2}. \end{align*} Thus the words $AB$ and $BA$ represent different products even though this particular trace cannot distinguish them; the order of multiplication is already real before any later theory of mixed moments is introduced. [/example]

The example also shows why traces are natural in this course. A trace does not erase all order information, but it allows cyclic rearrangements, so $\operatorname{tr}_2(ABC)=\operatorname{tr}_2(BCA)$. The next problem is to formulate the abstract expectation functional that will evaluate such words in every model at once.

## States as Expectations

What replaces the expectation operator when the random variables live in an algebra? The answer is a linear functional satisfying the same formal normalisation and positivity conditions as integration against a probability measure. Positivity is essential because it encodes the inequality $\mathbb E[|X|^2]\ge 0$ in a form that still makes sense for abstract algebras with involution.

[definition: Star Algebra] A unital complex algebra $\mathcal A$ is a $*$-algebra if it is equipped with a map $*:\mathcal A\to\mathcal A$, written $a\mapsto a^*$, such that, for all $a,b\in\mathcal A$ and $\lambda\in\mathbb C$, \begin{align*} (a+b)^* = a^*+b^*, \quad (\lambda a)^*=\overline{\lambda}a^*, \quad (ab)^*=b^*a^*, \quad (a^*)^*=a. \end{align*} [/definition]

The involution plays the role of complex conjugation or adjoint. Once it is present, positivity has a canonical algebraic form: elements of the shape $a^*a$ are the analogues of nonnegative random variables of the form $|X|^2$. This prepares the next definition, whose purpose is to make expectation an intrinsic part of the algebraic structure.

[definition: State] Let $\mathcal A$ be a unital $*$-algebra. A state on $\mathcal A$ is a linear functional $\varphi:\mathcal A\to\mathbb C$ such that \begin{align*} \varphi(1_{\mathcal A}) &= 1, & \varphi(a^*a)&\ge 0 \quad \text{for all } a\in\mathcal A. \end{align*} [/definition]

A state is the noncommutative expectation, but an expectation alone is not yet a probability model. The algebra and the expectation must be bundled into a single ambient object before the course can speak about variables, moments, laws, and independence. The next definition creates that ambient object.

[definition: Noncommutative Probability Space] A noncommutative probability space is a pair $(\mathcal A,\varphi)$, where $\mathcal A$ is a unital complex algebra and $\varphi:\mathcal A\to\mathbb C$ is a unital linear functional. [/definition]

When $\mathcal A$ is a $*$-algebra and $\varphi$ is a state, the space has positivity. Many algebraic constructions need only a unital linear functional, while laws of self-adjoint variables and Hilbert-space realizations use the stronger positive setting. The next examples show that this abstraction contains ordinary probability and matrix models, so it is not merely formal notation.