In classical probability theory, we start with a *sample space* , a collection of *events*, which is a sigma-algebra on , and a probability measure on . The triple is a probability space, and the collection of bounded complex-valued random variables on the probability space forms a commutative algebra under pointwise addition and products. The measure defines an expectation, or integral with respect to , which is a linear map

In this post I provide definitions of probability spaces from the algebraic viewpoint. Statements of some of their first properties will be given in order to justify and clarify the definitions, although any proofs will be left until later posts. In the algebraic setting, we begin with a *-algebra , which takes the place of the collection of bounded random variables from the classical theory. It is not necessary for the algebra to be represented as a space of functions from an underlying sample space. Since the individual points constituting the sample space are not required in the theory, this is a *pointless* approach. By allowing multiplication of `random variables’ in to be noncommutative, we incorporate probability spaces which have no counterpart in the classical setting, such as are used in quantum theory. The second and final ingredient is a *state* on the algebra, taking the place of the classical expectation operator. This is a linear map satisfying the positivity constraint and, when is unitial, the normalisation condition . *Algebraic*, or *noncommutative* probability spaces are completely described by a pair consisting of a *-algebra and a state . Noncommutative examples include the *-algebra of bounded linear operators on a Hilbert space with *pure* state for a fixed unit vector . (more…)