Non-commutative probability and non-commutative processes

A probability space is a pair ($\mathcal{A},\varphi$) where $\mathcal{A}$ is an algebra and $\varphi$ a state on the algebra. In classical probability $\mathcal{A}$ is the algebra of linear combinations of indicator functions on the sample space and in quantum probability $\mathcal{A}$ is the Heisenberg or Clifford algebra. However, other algebras are of interest in non-commutative probability. Here one discusses some other non-commutative probability spaces, in particular those associated to non-commutative space-time.