内作用在$C^*$-代数上的平稳态
Stationary states on a $C^*$-algebra for an inner action
AI总结：
本文研究可数离散群在$C^*$-代数上内作用的平稳态，证明其状态空间为Choquet单纯形，给出迹态与纯非迹态的分解及判别，并应用于自由群和性质(T)群，最后给出Bauer单纯形的充分条件。
AI中文摘要：
我们研究可数离散群在单位可分$C^*$-代数上的作用的平稳态。我们证明了由酉群的子群（该子群生成该代数）的内作用所关联的平稳态空间是一个Choquet单纯形。我们还给出了一个覆盖Bernoulli平移的局部性判据。单纯形结构给出了平稳态到迹态和纯非迹态部分的典范分解。对于内作用，我们通过其GNS von Neumann代数的因子性来刻画极端平稳态。我们进一步证明，一个平稳态是迹态当且仅当其GNS von Neumann代数是有限的，并且它是纯非迹态当且仅当该代数是$\textrm{III}$型。作为应用，对于$2\leq d\leq\infty$，$C^*(\mathbb{F}_d)$的平稳态单纯形不是Bauer的，而对于非平凡性质$(T)$群，$C^*(\Gamma)$的平稳态单纯形不是Poulsen的。最后，我们给出了$S_{\mu}(A)$为Bauer单纯形的一个充分谱间隙条件，并构造了一个满足该条件的族$(A_d,\Gamma_d,\mu_d)$。
英文摘要：
We study stationary states for actions of countable discrete groups on unital separable $C^*$-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 $2\leq d\leq\infty$ the stationary state simplex of $C^*(\mathbb{F}_d)$ has a Poulsen face, while for a nontrivial property~$(T)$ group the stationary state simplex of $C^*(Γ)$ is not Poulsen. Finally, we give a sufficient spectral-gap condition for $S_μ(A)$ to be a Bauer simplex and construct a family $(A_d,Γ_d,μ_d)$ satisfying this condition.
