Stationary $C^*$-dynamical systems (with an appendix by Uri Bader, Yair Hartman, and Mehrdad Kalantar)

Subalgebras, subgroups, and singularity

“…To prove coamenability in our setup, we needed to use instead, the singular hereditary property. The following proposition establishes the link between an invariant algebra and its relative commutant in the crossed product if we know that the second object is singular (also see [18, Lemma 2.2] and [19, Proposition 3.7]). Proposition Let $$(X,\nu )$$ be a nonsingular $$\Gamma$$‐space.…”

“…We now briefly recall the notion of stationary states in the context of unital 𝐶 * -algebras and refer the readers to [18] for more details. We now proceed to prove the following lemma.…”

“…We now briefly recall the notion of stationary states in the context of unital $$C^*$$‐algebras and refer the readers to [18] for more details. Definition Let $$\mathcal {A}$$ be a unital $$\Gamma$$‐$$C^*$$‐algebra.…”

“…The notion of “singularity” has been used to prove rigidity results for $$\Gamma$$‐operator algebras in various settings, where $$\Gamma$$ is a discrete countable group. It appears in the works of [4, 15, 18, 23], and so on, where the authors put singular states into use. More recently, [2, 3] used singularity in the context of $$\Gamma$$‐equivariant ucp maps $$\Phi :\mathcal {M}\rightarrow L^{\infty }(B,\nu )$$, where $$\mathcal {M}$$ is a $$\Gamma$$‐von Neumann algebra and $$(B,\nu )$$ is a nonsingular probability $$\Gamma$$‐space.…”

Ratio-limit boundaries of random walks on relatively hyperbolic groups

“…Recent applications of operator algebraic techniques to representation theory of infinite discrete groups [49,13] stem from studying non-commutative boundaries of non-self-adjoint operator algebras in the sense of Arveson [4,20]. These works eventually led to the resolution of open problems in group theory [51], exciting new techniques in stationary dynamics [44] and progress in ergodic theory of lattices in semisimple Lie groups [9,6].…”