Boundary quotient C*‐algebras of semigroups
Boundary quotient C*‐algebras of semigroups
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半群的边界商 C*→ 代数
DOI：
10.1112/jlms.12557
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Li, Xin
中科院分区：
文献类型：
--
作者：
Kakariadis, Evgenios T.;Katsoulis, Elias G.;Laca, Marcelo;Li, Xin
We study two classes of operator algebras associated with a unital subsemigroup P$P$ of a discrete group G$G$: one related to universal structures and one related to co‐universal structures. First we provide connections between universal C*‐algebras that arise variously from isometric representations of P$P$ that reflect the space J$\mathcal {J}$ of constructible right ideals, from associated Fell bundles, and from induced partial actions. This includes connections of appropriate quotients with the strong covariance relations in the sense of Sehnem. We then pass to the reduced representation Cλ∗(P)$\mathrm{C}^*_\lambda (P)$, and we consider the boundary quotient ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$ related to the minimal boundary space. We show that ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$ is co‐universal in two different classes: (a) with respect to the equivariant constructible isometric representations of P$P$; and (b) with respect to the equivariant C*‐covers of the reduced non‐selfadjoint semigroup algebra A(P)$\mathcal {A}(P)$. If P$P$ is an Ore semigroup, or if G$G$ acts topologically freely on the minimal boundary space, then ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$ coincides with the usual C*‐envelope Cenv∗(A(P))$\mathrm{C}^*_{\text{env}}(\mathcal {A}(P))$ in the sense of Arveson. This covers total orders, finite type and right‐angled Artin monoids, the Thompson monoid, multiplicative semigroups of non‐zero algebraic integers, and the ax+b$ax+b$‐semigroups over integral domains that are not a field. In particular, we show that P$P$ is an Ore semigroup if and only if there exists a canonical ∗$*$‐isomorphism from ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$, or from Cenv∗(A(P))$\mathrm{C}^*_{\text{env}}(\mathcal {A}(P))$, onto Cλ∗(G)$\mathrm{C}^*_\lambda (G)$. If any of the above holds, then A(P)$\mathcal {A}(P)$ is shown to be hyperrigid.
We study two classes of operator algebras associated with a unital subsemigroup P$P$ of a discrete group G$G$: one related to universal structures and one related to co‐universal structures. First we provide connections between universal C*‐algebras that arise variously from isometric representations of P$P$ that reflect the space J$\mathcal {J}$ of constructible right ideals, from associated Fell bundles, and from induced partial actions. This includes connections of appropriate quotients with the strong covariance relations in the sense of Sehnem. We then pass to the reduced representation Cλ∗(P)$\mathrm{C}^*_\lambda (P)$, and we consider the boundary quotient ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$ related to the minimal boundary space. We show that ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$ is co‐universal in two different classes: (a) with respect to the equivariant constructible isometric representations of P$P$; and (b) with respect to the equivariant C*‐covers of the reduced non‐selfadjoint semigroup algebra A(P)$\mathcal {A}(P)$. If P$P$ is an Ore semigroup, or if G$G$ acts topologically freely on the minimal boundary space, then ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$ coincides with the usual C*‐envelope Cenv∗(A(P))$\mathrm{C}^*_{\text{env}}(\mathcal {A}(P))$ in the sense of Arveson. This covers total orders, finite type and right‐angled Artin monoids, the Thompson monoid, multiplicative semigroups of non‐zero algebraic integers, and the ax+b$ax+b$‐semigroups over integral domains that are not a field. In particular, we show that P$P$ is an Ore semigroup if and only if there exists a canonical ∗$*$‐isomorphism from ∂Cλ∗(P)$\partial \mathrm{C}^*_\lambda (P)$, or from Cenv∗(A(P))$\mathrm{C}^*_{\text{env}}(\mathcal {A}(P))$, onto Cλ∗(G)$\mathrm{C}^*_\lambda (G)$. If any of the above holds, then A(P)$\mathcal {A}(P)$ is shown to be hyperrigid.
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