Noncommutative probability space

A $C^*$ -algebraic probability space is a pair $(A,\varphi)$ , where $A$ is a unital $C^*$ -algebra and $\varphi:A\to\mathbb C$ is a state:

\varphi(1)=1,\qquad\varphi(a^*a)\geq0\quad(a\in A).

It is called a noncommutative probability space when no commutativity assumption is imposed on $A$ . A von Neumann probability space uses a von Neumann algebra and a normal state; faithfulness or traciality is an additional hypothesis, stated when needed.

Classical model

For a probability space, $A=L^\infty(X,\mu)$ and $\varphi(f)=\int f\,d\mu$ . Projections are indicator functions modulo null sets and have state equal to the event's probability. In a matrix algebra, a density matrix $\rho$ supplies $\varphi(a)=\operatorname{Tr}(\rho a)$ .

Measurable and topological information

The classical $L^\infty$ algebra remembers events modulo null sets. The algebra $C(Y)$ of continuous functions on a compact Hausdorff space instead encodes topology through Gelfand duality. These models express different kinds of structure.