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My focus in my free probability notes is on scalar random variables, in which one needs both the algebra structure and the expectation structure in order to define the concepts mentioned above. Neither structure is necessary to define the other, but they enjoy some compatibility conditions (e.g. ${\bf E} X^2 \geq 0$ for any real-valued random variable $X$). In my notes, I also restricted largely to the case of bounded random variables $X \in L^\infty$ for simplicity (or at least with random variables $X \in L^{\infty-}$ in which all moments were finite), but one can certainly study unbounded noncommutative random variables as well, though the theory becomes significantly more delicate (much as the spectral theorem becomes significantly more subtle when working with unbounded operators rather than bounded operators).

When teaching classical probability theory, one usually focuses first on the scalar case, and then perhaps moves on to the general case in more advanced portions of the course. Similarly, noncommutative probability (of which free probability is a subfield) usually focuses first on the case of scalar noncommutative variables, which was the also the focus of my post. For instance, random $n \times n$ matrices, using the normalised expected trace $X \mapsto \frac{1}{n} {\bf E} \mathrm{tr} X$ as the trace structure, would be examples of scalar noncommutative random values (note that the normalised expected trace of a random matrix is a scalar, not a matrix). It is true that random $n \times n$ matrices, when equipped with the classical expectation ${\bf E}$ instead of the normalised expected trace $\frac{1}{n} {\bf E} \mathrm{tr}$, can also be viewed as classical non-scalar random variables, but this is a rather different structure (note now that the expectation is a matrix rather than a scalar) and should not be confused with the scalar noncommutative probability structure one can place here.

My focus in my free probability notes is on scalar random variables (commutative or noncommutative), in which one needs both the algebra structure and the expectation structure in order to define the concepts mentioned above. Neither structure is necessary to define the other, but they enjoy some compatibility conditions (e.g. ${\bf E} X^2 \geq 0$ for any real random variable $X$, in both the commutative and noncommutative settings). In my notes, I also restricted largely to the case of bounded random variables $X \in L^\infty$ for simplicity (or at least with random variables $X \in L^{\infty-}$ in which all moments were finite), but one can certainly study unbounded noncommutative random variables as well, though the theory becomes significantly more delicate (much as the spectral theorem becomes significantly more subtle when working with unbounded operators rather than bounded operators).

When teaching classical probability theory, one usually focuses first on the scalar case, and then perhaps moves on to the general case in more advanced portions of the course. Similarly, noncommutative probability (of which free probability is a subfield) usually focuses first on the case of scalar noncommutative variables, which was the also the focus of my post. For instance, random $n \times n$ matrices, using the normalised expected trace $X \mapsto \frac{1}{n} {\bf E} \mathrm{tr} X$ as the trace structure, would be examples of scalar noncommutative random variables (note that the normalised expected trace of a random matrix is a scalar, not a matrix). It is true that random $n \times n$ matrices, when equipped with the classical expectation ${\bf E}$ instead of the normalised expected trace $\frac{1}{n} {\bf E} \mathrm{tr}$, can also be viewed as classical non-scalar random variables, but this is a rather different structure (note now that the expectation is a matrix rather than a scalar) and should not be confused with the scalar noncommutative probability structure one can place here.

It is certainly possible to consider non-scalar noncommutative random variables, such as a matrix in which the entries are themselves elements of some noncommutative tracial von Neumann algebra (e.g. a matrix of random matrices); see e.g. Section 5 of these slides of Speicher. Similarly, there is certainly literature on free point processes (see e.g. this paper), noncommutative white noise (see e.g. this paper), etc., but these are rather advanced topics and beyond the scope of the scalar noncommutative probability theory discussed in my notes. I would not recommend trying to think about these objects until one is completely comfortable conceptually both with non-scalar classical random variables and with scalar noncommutative random variables, as one is likely to become rather confused otherwise when dealing with them. (This is analogous to how one should not attempt to study quantum field theory until one is completely comfortable conceptually both with classical field theory and with the quantum theory of particles.)

It is certainly possible to consider non-scalar noncommutative random variables, such as a matrix in which the entries are themselves elements of some noncommutative tracial von Neumann algebra (e.g. a matrix of random matrices); see e.g. Section 5 of these slides of Speicher. Similarly, there is certainly literature on free point processes (see e.g. this paper), noncommutative white noise (see e.g. this paper), etc., but these are rather advanced topics and beyond the scope of the scalar noncommutative probability theory discussed in my notes. I would not recommend trying to think about these objects until one is completely comfortable conceptually both with non-scalar classical random variables and with scalar noncommutative random variables, as one is likely to become rather confused otherwise when dealing with them. (This is analogous to how one should not attempt to study quantum field theory until one is completely comfortable conceptually both with classical field theory and with the quantum theory of particles. Much as one should not conflate the superficially similar notions of a classical field and a quantum wave function, one should also not conflate the superficially similar notions of a non-scalar classical random variable and a scalar noncommutative random variable.)

My focus in my free probability notes is on scalar random variables, in which one needs both the algebra structure and the expectation structure in order to define the concepts mentioned above. Neither structure is necessary to define the other, but they enjoy some compatibility conditions (e.g. ${\bf E} X^2 \geq 0$ for any real-valued random variable $X$). In my notes, I also restricted to the bounded case for simplicity, but one can certainly study unbounded noncommutative random variables as well, though the theory becomes significantly more delicate (much as the spectral theorem becomes significantly more subtle when working with unbounded operators rather than bounded operators).

When teaching classical probability theory, one usually focuses first on the scalar case, and then perhaps moves on to the general case in more advanced portions of the course. Similarly, noncommutative probability (of which free probability is a subfield) usually focuses first on the case of scalar noncommutative variables, which was the also the focus of my post. For instance, random $n \times n$ matrices, using the normalised expected trace $X \mapsto \frac{1}{n} {\bf E} \mathrm{tr} X$ as the trace structure, would be examples of scalar noncommutative random values (note that the normalised expected trace of a random matrix is a scalar, not a matrix). It is true that random $n \times n$ matrices, when equipped with the classical expectation ${\bf E}$ instead of the normalised expected trace, can also be viewed as classical non-scalar random variables, but this is a rather different structure (note now that the expectation is a matrix rather than a scalar) and should not be confused with the scalar noncommutative probability structure one can place here.

It is certainly possible to consider non-scalar noncommutative random variables, such as a matrix in which the entries are themselves elements of some noncommutative tracial von Neumann algebra (e.g. a matrix of random matrices); see e.g. Section 5 of these slides of Speicher. Similarly, there is certainly literature on free point processes (see e.g. this paper), noncommutative white noise (see e.g. this paper), etc., but these are rather advanced topics and beyond the scope of the scalar noncommutative probability theory discussed in my notes. I would not recommend trying to think about these objects until one is completely comfortable conceptually both with non-scalar classical random variables and with scalar noncommutative random variables, as one is likely to become rather confused otherwise when dealing with them.

My focus in my free probability notes is on scalar random variables, in which one needs both the algebra structure and the expectation structure in order to define the concepts mentioned above. Neither structure is necessary to define the other, but they enjoy some compatibility conditions (e.g. ${\bf E} X^2 \geq 0$ for any real-valued random variable $X$). In my notes, I also restricted largely to the case of bounded random variables $X \in L^\infty$ for simplicity (or at least with random variables $X \in L^{\infty-}$ in which all moments were finite), but one can certainly study unbounded noncommutative random variables as well, though the theory becomes significantly more delicate (much as the spectral theorem becomes significantly more subtle when working with unbounded operators rather than bounded operators).

When teaching classical probability theory, one usually focuses first on the scalar case, and then perhaps moves on to the general case in more advanced portions of the course. Similarly, noncommutative probability (of which free probability is a subfield) usually focuses first on the case of scalar noncommutative variables, which was the also the focus of my post. For instance, random $n \times n$ matrices, using the normalised expected trace $X \mapsto \frac{1}{n} {\bf E} \mathrm{tr} X$ as the trace structure, would be examples of scalar noncommutative random values (note that the normalised expected trace of a random matrix is a scalar, not a matrix). It is true that random $n \times n$ matrices, when equipped with the classical expectation ${\bf E}$ instead of the normalised expected trace $\frac{1}{n} {\bf E} \mathrm{tr}$, can also be viewed as classical non-scalar random variables, but this is a rather different structure (note now that the expectation is a matrix rather than a scalar) and should not be confused with the scalar noncommutative probability structure one can place here.

It is certainly possible to consider non-scalar noncommutative random variables, such as a matrix in which the entries are themselves elements of some noncommutative tracial von Neumann algebra (e.g. a matrix of random matrices); see e.g. Section 5 of these slides of Speicher. Similarly, there is certainly literature on free point processes (see e.g. this paper), noncommutative white noise (see e.g. this paper), etc., but these are rather advanced topics and beyond the scope of the scalar noncommutative probability theory discussed in my notes. I would not recommend trying to think about these objects until one is completely comfortable conceptually both with non-scalar classical random variables and with scalar noncommutative random variables, as one is likely to become rather confused otherwise when dealing with them. (This is analogous to how one should not attempt to study quantum field theory until one is completely comfortable conceptually both with classical field theory and with the quantum theory of particles.)