Algebraic probability theory

Algebraic probability theory is the axiomatization of random variables as commutative algebras equipped with integration functionals, plus noncommutative extensions as found in quantum mechanics and random matrix theory. In the commutative case, the algebraic foundation is essentially equivalent to the standard measure-theoretic foundation, via Gelfand duality plus the Riesz-Markov representation theorem. That said, in addition to having wider scope for generalization, the algebraic foundation is arguably more faithful to statistical practice in prioritizing random variables over sample spaces.

Online resources

Series of blog posts by George Lowther:
1. Algebraic probability (1 ,2 ,3 )
2. *-Algebras and state on *-algebras (1 ,2 )
3. Noncommutative probability spaces (1 ,2 )
Blog posts by Terry Tao:
- Motivation: foundations in Tao’s review of probablity theory
- Free probability
- Algebraic probability spaces

Literature

Segal, 1965, AMS Bulletin: Algebraic integration theory (doi, pdf)
Segal & Kunze, 1978: Integrals and operators, 2nd ed., Chapter VIII: Algebraic integration theory
Whittle, 2000: Probability via expectation, 4th ed.
- Algebraic in spirit, without explicitly invoking abstract algebra

The literature on quantum theory includes much relevant material, such as:

Landsman, 2017: Foundations of quantum theory: From classical concepts to operator algebras (doi)