State-preserving operator-algebra dynamical system

A state-preserving dynamical system is a triple $(A,\varphi,\alpha)$ , where $(A,\varphi)$ is a unital $C^*$ -algebraic probability space and $\alpha:A\to A$ is a unital $*$ -endomorphism satisfying

\varphi\circ\alpha=\varphi.

The system is invertible when $\alpha$ is a $*$ -automorphism. In the von Neumann setting, require $\alpha$ and $\varphi$ to be normal. For a group action, each $\alpha_g$ is an automorphism and $\alpha_{gh}=\alpha_g\alpha_h$ .

Preservation of events and probabilities

A $*$ -homomorphism preserves multiplication, involution, and hence projections. State preservation keeps their probabilities unchanged. More general quantum channels need not be multiplicative; they belong to a broader setting than this definition.

Classical convention

For invertible $T$ , use $\alpha(f)=f\circ T^{-1}$ . For noninvertible $T$ , use $\beta(f)=f\circ T$ , an injective state-preserving endomorphism of $L^\infty$ . If $T$ is invertible, $\beta=\alpha^{-1}$ .