Classifying maps into uniform tracial sequence algebras
Jorge Castillejos, Samuel Evington, Aaron Tikuisis, Stuart White
本文利用迹数据对从核C*-代数到核Z-稳定C*-代数的一致迹序列代数的*-同态进行分类。
我们通过迹数据分类从核 $C^*$-代数到核 $\mathcal Z$-稳定 $C^*$-代数的一致迹序列代数的 $^*$-同态。
We classify $^*$-homomorphisms from nuclear $C^*$-algebras into uniform tracial sequence algebras of nuclear $\mathcal Z$-stable $C^*$-algebras via tracial data.
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Uniform property $Γ$ and finite dimensional tracial boundaries
证明了C$^*$-代数$A$在极值迹态集为有限覆盖维数的非空紧空间、且每个极值迹态对应的GNS表示生成的冯·诺依曼代数有性质Γ时，$A$具有一致性质Γ。
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Stable rank one, tracial local homogeneity and uniform property $Γ$
研究证明稳定秩一且在迹意义下局部有限核维度的分离、简单、单位、非元素、稳定有限的C*-代数具有统一性质Γ，推翻Villadsen代数和自由最小动作的交叉积满足统一性质Γ，从而验证Toms-Winter猜想。
We show that a simple separable unital nuclear nonelementary $C^*$-algebra whose tracial state space has a compact extreme boundary with finite covering dimension admits uniformly tracially large order zero maps from matrix algebras into its central sequence algebra. As a consequence, strict comparison implies $\Z$-stability for these algebras.
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Tracially amenable actions and purely infinite crossed products
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Equivariant property Gamma and the tracial local-to-global principle for C*-dynamics
本文研究可数离散群作用于带迹 C*-代数的等变一致性质 Gamma，证明在顺从群和紧迹状态空间条件下该性质蕴含迹局部到整体原理，并进一步证明对简单核 $\mathcal{Z}$-稳定 C*-代数，等变一致性质 Gamma 等价于等变 $\mathcal{Z}$-稳定性。
I combine recent results in the structure theory of nuclear C*-algebras and in topological dynamics to classify certain types of crossed products in terms of their Elliott invariants. In particular, transformation group C*-algebras associated to free minimal Z^d-actions on the Cantor set with compact space of ergodic measures are classified by their ordered K-theory. In fact, the respective statement holds for finite dimensional compact metrizable spaces, provided that projections of the crossed products separate tracial states. Moreover, C*-algebras associated to certain minimal homeomorphism…
Let $A$ be a unital separable simple amenable $C^*$-algebra with finite tracial rank which satisfies the Universal Coefficient Theorem (UCT). Suppose $\af$ and $\bt$ are two automorphisms with the Rokhlin property that {induce the same action on the $K$-theoretical data of $A$.} We show that $\af$ and $\bt$ are strongly cocycle conjugate and uniformly approximately conjugate, that is, there exists a sequence of unitaries $\{u_n\}\subset A$ and a sequence of strongly asymptotically inner automorphisms $σ_n$ such that $$ \af={\rm Ad}\, u_n\circ σ_n\circ \bt\circ σ_n^{-1}\andeqn \lim_{n\to\infty}…
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本文证明：若C*-代数A具有均匀性质Γ，则具有弱迹Rokhlin性质的有限群作用及具有带比较的迹Rokhlin性质的第二可数紧群作用所得到的交叉积与不动点代数均保持该性质。
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Amenability for groups can be extended to metric spaces, algebras over commutative fields and $C^*$-algebras by adapting the notion of Følner nets. In the present article we investigate the close ties among these extensions and show that these three pictures unify in the context of the uniform Roe algebra $C_u^*(X)$ over a metric space $(X,d)$ with bounded geometry. In particular, we show that the following conditions are equivalent: (1) $(X,d)$ is amenable; (2) the translation algebra generating $C_u^*(X)$ is algebraically amenable (3) $C_u^*(X)$ has a tracial state; (4) $C_u^*(X)$ is not pro…
Subquadratic growth and uniform property \(Γ\)
研究具有次二次增长的单位可分ASH代数，证明其无非零有限维表示时具一致性质\(Γ\)，得出二次维数增长尺度是\(Γ\)潜在失效阈值，还将相关工作扩展到最优次二次尺度。
Let A be a unital separable simple infinite-dimensional nuclear C*-algebra with at least one tracial state. We prove that if the trace space of A has compact finite-dimensional extreme boundary then there exist unital embeddings of matrix algebras into a certain central sequence algebra of A which is determined by the uniform topology on the trace space. As an application, it is shown that if furthermore A has strict comparison then A absorbs the Jiang-Su algebra tensorially.
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In this paper, we investigate the ideal structure of uniform Roe algebras for general metric spaces beyond the scope of Yu's property A. Inspired by the ideal of ghost operators coming from expander graphs and in contrast to the notion of geometric ideal, we introduce a notion of ghostly ideal in a uniform Roe algebra, whose elements are locally invisible in certain directions at infinity. We show that the geometric ideal and the ghostly ideal are respectively the smallest and the largest element in the lattice of ideals with a common invariant open subset of the unit space of the coarse group…
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Let $A$ be a separable, unital, simple, $\mathcal{Z}$-stable, nuclear $C^*$-algebra, and let $α\colon G\to \mathrm{Aut}(A)$ be an action of a discrete, countable, amenable group. Suppose that the orbits of the action of $G$ on $T(A)$ are finite and that their cardinality is bounded. We show that $α$ is strongly outer if and only if $α\otimes\mathrm{id}_{\mathcal{Z}}$ has the weak tracial Rokhlin property. If $G$ is moreover residually finite, these conditions are also equivalent to $α\otimes\mathrm{id}_{\mathcal{Z}}$ having finite Rokhlin dimension (in fact, at most 2). If $\partial_eT(A)$ is …
Crossed product C*-algebras associated with non-minimal actions on the circle
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We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable, unital) semigroups and corresponding semigroup rings. We consider also Følner's type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no Følner sequence. In the context of inverse semigroups $S$ we give a characterization of invariant measures on $S$ (in the sense of Day) in terms of two notions: $domain$ $measurability$ and $localization$. Given a unital representation of $S$ in terms of partial bijections on some set $X$ we define a n…
Upgraded free independence phenomena for random unitaries
本研究利用一致渐近自由性与体积分析，证明随机酉元的自由独立性可扩展到矩阵超积中更大的Pinsker代数，并通过嵌入论证推广到Connes可嵌入迹von Neumann代数自由积的超积，为相关经典结果提供了新证明与推广。
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