GNS implementation of state-preserving dynamics

Core idea

For a state-preserving system $(A,\varphi,\alpha)$ , let $(H_\varphi,\pi_\varphi,\Omega_\varphi)$ be its GNS representation. The rule

V_\alpha\pi_\varphi(a)\Omega_\varphi =\pi_\varphi(\alpha(a))\Omega_\varphi

extends uniquely to an isometry of $H_\varphi$ . If $\alpha$ is an automorphism, this isometry is unitary and fixes $\Omega_\varphi$ .

Why it is well defined

The squared norm of the vector on the right is $\varphi(\alpha(a)^*\alpha(a))=\varphi(\alpha(a^*a))=\varphi(a^*a)$ . Thus null vectors remain null, and the isometry extends from the dense cyclic subspace. Invertibility supplies the inverse implementation.

Covariance and classical recovery

For an automorphism,

V_\alpha\pi_\varphi(a)V_\alpha^*=\pi_\varphi(\alpha(a)).

Group actions give unitary representations by the same construction. For integration on $L^\infty(X,\mu)$ , the GNS space is $L^2(X,\mu)$ ; inverse pullback recovers the classical Koopman representation.