Hilbert, in his 1900 so-called "problems lecture", formulated as 6th problem the challenge to find an axiomatic basis for mechanics and probability. Kolmogorov's 1933 "Grundbegriffe" monograph was widely accepted as an adequate answer to this challenge regarding the axiomatisation of --- one has to say now --- "classical" probability. Coincidentally, 1900 is also the year when Planck formulated his thesis of energy quanta, which would give rise to quantum theory, and which requires a new probability theory. This became clear after Heisenberg's 1925 paper and his Göttingen colleagues' works afterwards, which together with Dirac introduced the algebraic point of view, culminating in von Neumann's 1932 monograph on the widely known Hilbert space representation of quantum mechanics; which actually appeared before Kolmogorov's monograph. The main difference between the classical and the "new" probability lay in the non-commutativity of random variables. Today there are also other areas where such quantum like behaviour (QLB) seems to occur.
The algebraic view offers a way on how to treat both classical and quantum like phenomena in a unified mathematical setting. And although probabilists today seem to be happy with Kolmogorov's approach based on measure theory, it may be interesting to look at the subject through a different pair of glasses. This algebraic view also offers a more direct way to address random variables with values in infinite dimensional spaces, something which with classical measure theory can only be done in a somewhat circumlocutory fashion. It also helps to separate purely algebraic questions from analytical ones, but of course thrives in the interplay of both.
Without wanting to present a strict axiomatic derivation, the start will be an early --- and in the light of modern theory also abstract algebraic --- view on random variables, as can be found implicitly in the work of early probabilists like the Bernoullis. Their properties are sketched as emanating from simple operational requirements regarding random variables, the mean or expectation, as well as sampling or observations. Concrete representations of this abstract setting connect it with algebras of linear mappings and the spectral theory of these, and one may recover Kolmogorov's classical characterisation as one particular representation.
Striking differences between classical or commutative probability and non-commutative probability appear already with simple linear algebra. As this is a subject which nowadays all engineering and science students learn at a very early stage, it may also be an interesting approach to teaching probability. And possible novel devices like quantum computers can be described in this setting.
This algebraic view has also a functional analytic extension, which can be used to construct generalised random variables and "ideal elements". It allows the specification of not only analogues of all the classical spaces of random variables, but to go beyond this and address questions of "smoothness" on the one hand, and the definition of idealised elements resp. "generalised" random variables on the other hand. This very much echoes the construction of distributions resp. generalised functions in the sense of Sobolev and Schwartz.