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

Łuczak, A.

doi:10.1090/S0002-9939-1986-0826491-2

Transformations induced in the state space of a $C^ \ast$-algebra and related ergodic theorems
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Proc. Amer. Math. Soc. 96 (1986), 617-625 Request permission

Abstract:

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

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