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Let      A A       be a norm-separable      C ∗  {C^*}      -algebra with unit  1 ,      σ  \sigma      -weakly dense in a      W ∗  {W^*}      -algebra      M M      , and let      α  \alpha       be a positive linear mapping of      M M       into itself leaving  1  invariant. We show that      α  \alpha       induces a transformation      α  ~  \tilde \alpha       defined "almost everywhere" on the state space      σ  \sigma       of      A A       with values in      σ  \sigma      . If      α  \alpha       is a      ∗  ^*      -automorphism of      M M      , then there exists  \[    lim N →  ∞  1 N ∑  n = 0 N −  1 α  ~  n ψ  \lim \limits _{N \to \infty } \frac {1}{N}\sum \limits _{n = 0}^{N - 1} {{{\tilde \alpha }_n}\psi }     \]  for "almost all" states      ψ  \psi       of      A A      , where      α  ~  n {\tilde \alpha _n}       are transformations on      σ  \sigma       induced by      α  n {\alpha ^n}      .

Transformations induced in the state space of a 𝐶*-algebra and related ergodic theorems