S. Doplicher, R. Longo, J. E. Roberts, L. Zsido
Let H be a compact quantum group with faithful Haar measure and bounded counit. If H acts on a C*-algebra A, we show that A is nuclear if and only if its fixed-point subalgebra is nuclear. As a consequence H is a nuclear C*-algebra.
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We give a short proof of the nuclearity property of a class of Cuntz-Pimsner algebras associated with a Hilbert A-bimodule M, where A is a separable and nuclear C*-algebra. We assume that the left A-action on the bimodule M is given in terms of compact module operators and that M is direct summand of the standard Hilbert module over A.
We study partial actions of exact discrete groups on C*-algebras. We show that the partial crossed product of a commutative C*-algebra by an exact discrete group is nuclear whenever the full and reduced partial crossed products coincide. This generalises a result by Matsumura in the context of global actions. In general, we prove that a partial action of an exact discrete group on a C*-algebra $A$ has Exel's approximation property if and only if the full and reduced partial crossed products associated to the diagonal partial action on $A\otimes_{\max} A^\mathrm{op}$ coincide. We apply our resu…
We study C*-algebras associated with subsemigroups of groups. For a large class of such semigroups including positive cones in quasi-lattice ordered groups and left Ore semigroups, we describe the corresponding semigroup C*-algebras as C*-algebras of inverse semigroups, groupoid C*-algebras and full corners in associated group crossed products. These descriptions allow us to characterize nuclearity of semigroup C*-algebras in terms of faithfulness of left regular representations and amenability of group actions. Moreover, we also determine when boundary quotients of semigroup C*-algebras are U…
Essential groupoid amenability and nuclearity of groupoid C*-algebras
本文构造了étale群胚的本质C*-代数，引入其极大版本及本质顺从性条件（弱于拓扑顺从性），并将可消半群代数作用产生的Bruce-Li代数描述为奇异本质C*-代数。
We generalize Renault's notion of measurewise amenability to actions of second countable, Hausdorff, étale groupoids on separable $C^*$-algebras and show that measurewise amenability characterizes nuclearity of the crossed product whenever the $C^*$-algebra acted on is nuclear. In the more general context of Fell bundles over second countable, Hausdorff, étale groupoids, we introduce a version of Exel's approximation property. We prove that the approximation property implies nuclearity of the cross-sectional algebra whenever the unit bundle is nuclear. For Fell bundles associated to groupoid a…
Georges Skandalis exhibited in his work on $K$-nuclearity the first class of $C^*$-algebras $A$ for which the canonical map $K_*(A\otimes_{max}A)\to K_*(A\otimes_{min}A)$ is not an isomorphism. We show that it is the injectivity that fails (even rationally) in his examples.
Equivariant (co)module nuclearity of $C^*$-crossed products
本文定义了模核性的等变与等协变版本，即 $\mathcal E$-$Γ$-核性，并证明了约化交叉积的该性质与群顺从性及完全有界逼近性质之间的等价与传递关系。
On Toeplitz algebras of product systems
研究群可嵌入幺半群上积系统的Toeplitz C*-代数核性与系数代数核性的关系，在技术假设下证明相关等价性，结果适用于交换幺半群等特定情形。
$p$-nuclearity of $L^p$-operator crossed products
本文研究$L^p$-算子交叉积的$p$-核性问题，证明在$p$-完全等距作用下全$L^p$-算子交叉积的$p$-核性等价于原代数的$p$-核性，并应用该结论证明$L^p$-Cuntz代数等的$p$-核性，解决了Phillips提出的相关问题。
We study the notions of nuclearity and exactness for module maps on $C^{*}$-algebras which are $C^*$-module over another $C^*$-algebra with compatible actions and examine finite approximation properties of such $C^*$-modules. We prove module versions of the results of Kirchberg and Choi-Effros. As a concrete example we extend the finite dimensional approximation properties of reduced $C^*$-algebras and von Neumann algebras on countable discrete groups to these operator algebras on countable inverse semigroups with the module structure coming from the action of the $C^*$-algebras on the subsemi…
We construct explicit approximating nets for crossed products of C*-algebras by actions of discrete quantum groups. This implies that C*-algebraic approximation properties such as nuclearity, exactness or completely bounded approximation property are preserved by taking crossed products by actions of amenable discrete quantum groups. We also show that the noncommutative topological entropy of a transformation commuting with the quantum group action does not change when we pass to the canonical extension to the crossed product. Both these results are extended to twisted crossed products via a s…
Amenable actions of locally compact groups on von Neumann algebras are investigated by exploiting the natural module structure of the crossed product over the Fourier algebra of the acting group. The resulting characterisation of injectivity for crossed products generalises a result of Anantharaman-Delaroche on discrete groups. Amenable actions of locally compact groups on $C^*$-algebras are investigated in the same way, and amenability of the action is related to nuclearity of the corresponding crossed product. A survey is given to show that this notion of amenable action for $C^*$-algebras s…
We start from Rieffel data (A,f,X) where A is a C*-algebra, X is an action of an abelian group H on A and f is a 2-cocycle on the dual group. Using Landstad theory of crossed product we get a deformed C*-algebra A(f). In the case of H being the n-th Cartesian product of the real numbers we obtain a very simple proof of invariance of K-groups under the deformation. In the general case we also get a very simple proof that nuclearity is preserved under the deformation. We show how our approach leads to quantum groups and investigate their duality. The general theory is illustrated by an example o…
We give a new sufficient condition on a spectral triple to ensure that the quantum group of orientation and volume preserving isometries defined in \cite{qorient} has a $C^*$-action on the underlying $C^*$ algebra.
Crossed products by compact group actions with the weak tracial Rokhlin property
本文研究了具有弱迹罗克林性质的紧群作用，证明了相关稳定性性质可从前代数转移到交叉积，并给出了非平凡例子。
Consider a product system over the positive cone of a quasi-lattice ordered group. We construct a Fell bundle over an associated groupoid so that the cross-sectional algebra of the bundle is isomorphic to the Nica-Toeplitz algebra of the product system. Under the additional hypothesis that the left actions in the product system are implemented by injective homomorphisms, we show that the cross-sectional algebra of the restriction of the bundle to a natural boundary subgroupoid coincides with the Cuntz-Nica-Pimsner algebra of the product system. We apply these results to improve on existing suf…
We extend a result about the gauge action on noncommutative solitons by showing that a family of functions can be gauged away to a Gaussian using the quantification condition given in "On a gauge action on sigma model solitons" IDAQP(2018).
We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of C*-algebras, including: $D$-absorbing C*-algebras, where $D$ is a strongly self-absorbing C*-algebra, stable C*-algebras, C*-algebras with finite …
We introduce and study the continuous Rokhlin property for actions of compact groups on C*-algebras. An important technical result is a characterization of the continuous Rokhlin property in terms of asymptotic retracts. As a consequence, we derive strong KK-theoretical obstructions to the continuous Rokhlin property. Using these, we show that the UCT is preserved under formation of crossed products and passage to fixed point algebras by such actions, even in the absence of nuclearity. Our analysis of the KK-theory of the crossed product allows us to prove a $\mathbb{T}$-equivariant version of…
Fourier--Stieltjes category for twisted groupoid actions
该研究将傅里叶—斯蒂尔杰斯代数理论推广到étale广群对任意C*-丛的扭曲作用范畴，开发了相关工具并建立了KSGNS型膨胀结果，还提供了若干应用。
In this work we introduce and study a new notion of amenability for actions of locally compact groups on $C^*$-algebras. Our definition extends the definition of amenability for actions of discrete groups due to Claire Anantharaman-Delaroche. We show that our definition has several characterizations and permanence properties analogous to those known in the discrete case. For example, for actions on commutative $C^*$-algebras, we show that our notion of amenability is equivalent to measurewise amenability. Combined with a recent result of Alex Bearden and Jason Crann, this also settles a long s…
For a group $G$ and $ω\in Z^{3}(G, \text{U}(1))$, an $ω$-anomalous action on a C*-algebra $B$ is a $\text{U}(1)$-linear monoidal functor between 2-groups $\text{2-Gr}(G, \text{U}(1), ω)\rightarrow \underline{\text{Aut}}(B)$, where the latter denotes the 2-group of $*$-automorphisms of $B$. The class $[ω]\in H^{3}(G, \text{U}(1))$ is called the anomaly of the action. We show for every $n\ge 2$ and every finite group $G$, every anomaly can be realized on the stabilization of a commutative C*-algebra $C(M)\otimes \mathcal{K}$ for some closed connected $n$-manifold $M$. We also show that although …
Classifiability of crossed products
本综述系统研究拓扑动力系统交叉积在 Elliott 纲领下的可分类性，总结核性、单性及 Z-稳定性的动力学刻画，并指出两个核心猜想与开放问题。
We prove that for a weakly mixing algebraic action $σ: G\curvearrowright(X,ν)$, the $n$-cohomology group $H^n(G\curvearrowright X; \mathbb{T})$, after quotienting out the natural subgroup $H^n(G,\mathbb{T})$, contains $H^n(G,\widehat{X})$ as a natural subgroup for $n=1$. If we further assume the diagonal actions $σ^2, σ^4$ are $\mathbb{T}$-cocycle superrigid and $H^2(G, \widehat{X})$ is torsion free as an abelian group, then the above also holds true for $n=2$. Applying it for principal algebraic actions when $n=1$, we show that $H^2(G,\mathbb{Z}G)$ is torsion free as an abelian group when $G$…
Recently C. Houdayer and Y. Isono have proved among other things that every biexact group $Γ$ has the property that for any non-singular strongly ergodic action $Γ\curvearrowright (X,μ)$ on a standard measure space the group measure space von Neumann algebra $Γ\ltimes L^\infty(X)$ is full. In this note, we prove the same property for a wider class of groups, notably including $\mathrm{SL}(3,{\mathbb Z})$. We also prove that for any connected simple Lie group $G$ with finite center, any lattice $Γ\le G$, and any closed non-amenable subgroup $H\le G$, the non-singular action $Γ\curvearrowright G…
Let $Γ$ be a discrete group satisfying the approximation property (AP). Let $X$, $Y$ be $Γ$-spaces and $π\colon Y \to X$ be a proper factor map which is injective on the non-free part. We prove the one-to-one correspondence between intermediate ${\rm C}^\ast$-algebras of $C_0(X) \rtimes_r Γ\subset C_0(Y) \rtimes Γ$ and intermediate $Γ$-${\rm C}^\ast$-algebras of $C_0(X) \subset C_0(Y)$. This is a generalization of Suzuki's theorem that proves the statement for free actions.
We show that if $p:\B\to G$ is a Fell bundle over a locally compact groupoid $G$ and that $A=Γ_{0}(G^{(0)};\B)$ is the \cs-algebra sitting over $G^{(0)}$, then there is a continuous $G$-action on $\Prim A$ that reduces to the usual action when $\B$ comes from a dynamical system. As an application, we show that if $I$ is a $G$-invariant ideal in $A$, then there is a short exact sequence of \cs-algebras \xymatrix{0\ar[r]&\cs(G,\BI)\ar[r] &\cs(G,\B)\ar[r]&\cs(G,\BqI)\ar[r]&0,} where $\cs(G,\B)$ is the Fell bundle \cs-algebra and $\BI$ and $\BqI$ are naturally defined Fell bundles corresponding to…
We introduce the notion of fibred action of a group bundle on a C(X)-algebra. By using such a notion, a characterization in terms of induced C*-bundles is given for C*-dynamical systems such that the relative commutant of the fixed-point algebra is minimal (i.e., it is generated by the centre of the given C*-algebra and the centre of the fixed-point algebra). A class of examples in the setting of the Cuntz algebra is given, and connections with superselection structures with nontrivial centre are discussed.
Remarks on C*-discrete inclusions
本文通过伽罗瓦对应刻画了不可约C*-离散包含的中间子代数，在迹条件下定义了张量范畴在C*-代数上的自由作用并证明了简化交叉积保持单性，以及在一定约束下A值半圆系统生成C*-离散包含。
A locally compact groupoid is said to have the weak containment property if its full $C^*$-algebra coincides with its reduced one. This property is strictly weaker than amenability and is known to be equivalent to amenability for transformation groupoids relative to actions of exact discrete groups. We believe that for general étale groupoids one should have the same equivalence of the two properties under some mild exactness assumption. In this paper we try to support this statement.
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