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Tensor algebras of product systems and their C⁎-envelopes
Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2020-04-01 , DOI: 10.1016/j.jfa.2019.108416
Adam Dor-On , Elias Katsoulis

Let $(G, P)$ be an abelian, lattice ordered group and let $X$ be a compactly aligned product system over $P$. We show that the C*-envelope of the Nica tensor algebra $\mathcal{N}\mathcal{T}^+_X$ coincides with both Sehnem's covariance algebra $\mathcal{A} \times_X P$ and the co-universal C*-algebra $\mathcal{N}\mathcal{O}^r_X$ for injective, gauge compatible, Nica-covariant representations of Carlsen, Larsen, Sims and Vittadello. We give several applications of this result on both the selfadjoint and non-selfadjoint operator algebra theory. First we guarantee the existence of $\mathcal{N}\mathcal{O}^r_X$, thus settling a problem of Carlsen, Larsen, Sims and Vittadello which was open even for abelian, lattice ordered groups. As a second application, we resolve a problem posed by Skalski and Zacharias on dilating isometric representations of product systems to unitary representations. As a third application we characterize the C*-envelope of the tensor algebra of a finitely aligned higher-rank graph which also holds for topological higher-rank graphs. As a final application we prove reduced Hao-Ng isomorphisms for generalized gauge actions of discrete groups on C*-algebras of product systems. This generalizes recent results that were obtained by various authors in the case where $(G, P) =(\mathbb{Z},\mathbb{N})$.

中文翻译:

乘积系统的张量代数及其 C⁎-包络

令 $(G, P)$ 是一个阿贝尔格有序群,令 $X$ 是一个在 $P$ 上紧密对齐的产品系统。我们证明了 Nica 张量代数 $\mathcal{N}\mathcal{T}^+_X$ 的 C* 包络与 Sehnem 的协方差代数 $\mathcal{A} \times_X P$ 和共同普遍 C *-代数 $\mathcal{N}\mathcal{O}^r_X$ 用于 Carlsen、Larsen、Sims 和 Vittadello 的单射、规范兼容、尼卡协变表示。我们给出了这个结果在自伴随和非自伴随算子代数理论中的几种应用。首先我们保证 $\mathcal{N}\mathcal{O}^r_X$ 的存在,从而解决了 Carlsen、Larsen、Sims 和 Vittadello 的问题,该问题甚至对阿贝尔格有序群也是开放的。作为第二个应用程序,我们解决了 Skalski 和 Zacharias 提出的将产品系统的等距表示扩展为单一表示的问题。作为第三个应用,我们描述了有限对齐的高阶图的张量代数的 C* 包络,该图也适用于拓扑高阶图。作为最终应用,我们证明了离散群在乘积系统的 C*-代数上的广义规范作用的简化 Hao-Ng 同构。这概括了不同作者在 $(G, P) =(\mathbb{Z},\mathbb{N})$ 的情况下获得的最新结果。作为最终应用,我们证明了离散群在乘积系统的 C*-代数上的广义规范作用的简化 Hao-Ng 同构。这概括了不同作者在 $(G, P) =(\mathbb{Z},\mathbb{N})$ 的情况下获得的最新结果。作为最终应用,我们证明了离散群在乘积系统的 C*-代数上的广义规范作用的简化 Hao-Ng 同构。这概括了不同作者在 $(G, P) =(\mathbb{Z},\mathbb{N})$ 的情况下获得的最新结果。
更新日期:2020-04-01
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