Stationary states on a $C^{\ast }$-algebra for an inner action

Abstract

We study stationary states for actions of countable discrete groups on unital separable $C^{\ast }$-algebras. We prove that the stationary state space associated with an inner action of a subgroup of the unitary group that generates the algebra is a Choquet simplex. We also give a locality criterion covering Bernoulli shifts. The simplex structure yields a canonical decomposition of stationary states into tracial and purely nontracial parts. For inner actions, we characterize extreme stationary states by factoriality of their GNS von Neumann algebras. We further show that a stationary state is tracial if and only if its GNS von Neumann algebra is finite, and that it is purely nontracial if and only if this algebra is of type~$\mathrm{III}$. As applications, for $2\le d\le \infty$ the stationary state simplex of $C^{\ast }({\mathbb{F}}_d)$ has a Poulsen face, while for a nontrivial property~$(T)$ group the stationary state simplex of $C^{\ast }(\Gamma )$ is not Poulsen. Finally, we give a sufficient spectral-gap condition for $S_{\mu }(A)$ to be a Bauer simplex and construct a family $(A_d,{\Gamma }_d,{\mu }_d)$ satisfying this condition.