arXiv subjects

Benjamin Anderson-Sackaney

Publications and source records attributed to Benjamin Anderson-Sackaney.

7 recordsLinked to original sources

arXiv2024

$C^*$-simplicity and boundary actions of discrete quantum groups

We introduce and investigate several quantum group dynamical notions for the purpose of studying $C^*$-simplicity of discrete quantum groups via the theory of boundary actions. In particular we define a quantum analogue of Powers' Averaging Property (PAP) and a quantum analogue of strongly faithful actions. We show that our quantum PAP implies $C^*$-simplicity and the uniqueness of $\sigma$-KMS states, and that the existence of a strongly $C^*$-faithful quantum boundary action also implies $C^*$-simplicity and, in the unimodular case, the quantum PAP. We illustrate these results in the case of the unitary free quantum groups $\mathbb{F} U_F$ by showing that they satisfy the quantum PAP and that they act strongly $C^*$-faithfully on their quantum Gromov boundary. Moreover we prove that this particular action of $\mathbb{F} U_F$ is a quantum boundary action.

math.OA↗

arXiv2024

Entropies and Poisson boundaries of random walks on groups with rapid decay

Let $G$ be a countable group and $\mu$ a probability measure on $G$. We build a new framework to compute asymptotic quantities associated with the $\mu$-random walk on $G$, using methods from harmonic analysis on groups and Banach space theory, most notably complex interpolation. It is shown that under mild conditions, the Lyapunov exponent of the $\mu$-random walk with respect to a weight $\omega$ on $G$ can be computed in terms of the asymptotic behavior of the spectral radius of $\mu$ in an ascending class of weighted group algebras, and we prove that for natural choices of $\omega$ and $\mu$, the Lyapunov exponent vanishes. Also, we show that the Avez entropy of the $\mu$-random walk can be realized as the Lyapunov exponent of $\mu$ with respect to a suitable weight. We apply our results to stationary dynamical systems consisting of an action of a group with the property of rapid decay on a probability space. We prove that whenever the associated Koopman representation is weakly contained in the left-regular representation of the group, then the Avez entropy coincides with the Furstenberg entropy of the stationary space. This gives a characterization of (Zimmer) amenability for actions of rapid decay groups on stationary spaces. Next, by considering the spectral radius in the algebras of $p$-pseudofunctions on $G$, we introduce a new asymptotic quantity, which we call convolution entropy. We show that for groups with the property of rapid decay, the convolution entropy coincides with the Avez entropy.

math.DS↗

arXiv2023

Topological Boundaries of Representations and Coideals

For a locally compact quantum group $\mathbb{G}$, a (left) coideal is a (left) $\mathbb{G}$-invariant von Neumann subalgebra of $L^\infty(\mathbb{G})$. We introduce and analyze various generalizations of amenability and coamenability to coideals of discrete and compact quantum groups. We focus on a particular class of coideals found in the category of compact quantum groups, which are associated with a compact quasi-subgroup. This class includes all coideals of the quotient type. We also introduce the notion of a Furstenberg-Hamana boundary for representations of discrete quantum groups and use it to study amenability and coamenability properties of coideals. We then prove that a coideal of a compact quantum group that is associated with a compact quasi-subgroup is coamenable if and only if its codual coideal is $\mathbb{G}$-injective. If $\mathbb{G}$ is a unimodular or an exact discrete quantum group, we can replace $\mathbb{G}$-injectivity in the latter statement with the weaker condition of relative amenability. This result leads to a complete characterization of the unique trace property. Specifically, a unimodular discrete quantum group $\mathbb{G}$ has the unique trace property if and only if the action of $\mathbb{G}$ on its noncommutative Furstenberg boundary is faithful. We also demonstrated that if a unimodular discrete quantum group $\mathbb{G}$ is $C^*$-simple then it has the unique trace property. These findings are the quantum analogs of the groundbreaking results of Breuillard, Kalantar, Kennedy, and Ozawa and they provide answers to questions posted by Kalantar, Kasprzak, Skalski, and Vergnioux.

math.OA↗

arXiv2023

Fusion modules and amenability of coideals of compact and discrete quantum groups

We give a definition of an amenable fusion module over a fusion algebra. A notion of relative integrability for the `coduals' of coideals of compact quantum groups was recently introduced in the joint work of de Commer and Dzokou Talla. We use this property to construct an analogue of the quasi-regular representation. Then, we characterize a certain coamenability property of quasi-regular representations with amenability of their associated fusion modules. Afterwards, we obtain a duality result that generalizes Tomatsu's theorem for this coamenability property and an amenability property of their `codual' coideals (under an additional assumption). As an example, we apply this result to show the fusion modules associated to certain non-standard Podleś spheres are amenable.

math.QA↗

arXiv2023

On Amenable and Coamenable Coideals

We study relative amenability and amenability of a right coideal $\widetilde{N}_P\subseteq \ell^\infty(\mathbb{G})$ of a discrete quantum group in terms of its group-like projection $P$. We establish a notion of a $P$-left invariant state and use it to characterize relative amenability. We also develop a notion of coamenability of a compact quasi-subgroup $N_ω\subseteq L^\infty(\widehat{\mathbb{G}})$ that generalizes coamenability of a quotient as defined by Kalantar, Kasprzak, Skalski, and Vergnioux, where $\widehat{\mathbb{G}}$ is the compact dual of $\mathbb{G}$. In particular, we establish that the coamenable compact quasi-subgroups of $\widehat{\mathbb{G}}$ are in one-to-one correspondence with the idempotent states on the reduced $C^*$-algebra $C_r(\widehat{\mathbb{G}})$. We use this work to obtain results for the duality between relative amenability and amenability of coideals in $\ell^\infty(\mathbb{G})$ and coamenability of their codual coideals in $L^\infty(\widehat{\mathbb{G}})$, making progress towards a question of Kalantar et al{.}.

math.OA↗

arXiv2022

On Ideals of $L^1$-algebras of Compact Quantum Groups

We develop a notion of a non-commutative hull for a left ideal of the $L^1$-algebra of a compact quantum group $\mathbb{G}$. A notion of non-commutative spectral synthesis for compact quantum groups is proposed as well. It is shown that a certain Ditkin's property at infinity (which includes those $\mathbb{G}$ where the dual quantum group $\widehat{\mathbb{G}}$ has the approximation property) is equivalent to every hull having synthesis. We use this work to extend recent work of White that characterizes the weak$^*$ closed ideals of a measure algebra of a compact group to those of the measure algebra of a coamenable compact quantum group. In the sequel, we use this work to study bounded right approximate identities of certain left ideals of $L^1(\mathbb{G})$ in relation to coamenability of $\mathbb{G}$.

math.OA↗

arXiv2022

Tracial States and $\mathbb{G}$-Invariant States of Discrete Quantum Groups

We investigate the tracial states and $\mathbb{G}$-invariant states on the reduced $C^*$-algebra $C_r(\widehat{\mathbb{G}})$ of a discrete quantum group $\mathbb{G}$. Here, we denote its dual compact quantum group by $\widehat{\mathbb{G}}$. Our main result is that a state on $C_r(\widehat{\mathbb{G}})$ is tracial if and only if it is $\mathbb{G}$-invariant. This generalizes a known fact for unimodular discrete quantum groups and builds upon the work of Kalantar, Kasprzak, Skalski, and Vergnioux. As one consequence of this, we find that $C_r(\widehat{\mathbb{G}})$ is nuclear and admits a tracial state if and only if $\mathbb{G}$ is amenable. This resolves an open problem due to C.-K. Ng and Viselter, and Crann, in the discrete case. As another consequence, we prove that tracial states on $C_r(\widehat{\mathbb{G}})$ "concentrate" on $\widehat{\mathbb{G}}_F$, where $\mathbb{G}_F$ is the cokernel of the Furstenberg boundary. Furthermore, given certain assumptions, we characterize the existence of traces on $C_r(\widehat{\mathbb{G}})$ in terms of whether or not $\widehat{\mathbb{G}}_F$ is Kac type. We also characterize the uniqueness of (idempotent) traces in terms of whether not $\widehat{\mathbb{G}}_F$ is equal to the canonical Kac quotient of $\widehat{\mathbb{G}}$. These results rely on the following, of which we give proofs: So\l tan's canonical Kac quotient construction, whether it is applied to the universal or the reduced CQG $C^*$-algebra of $\widehat{\mathbb{G}}$ (when the latter admits a trace), yields the maximal Kac type closed quantum subgroup of $\widehat{\mathbb{G}}$.

math.OA↗