David P. Blecher, Benton L. Duncan
In analogy with the C*-algebra theory, we study variants appropriate to nonselfadjoint algebras of nuclearity, the local lifting property, exactness, and the weak expectation property. In addition, we study the relationships between these notions, and how they are connected with the classical C*-algebra theory through the use of C*-algebras generated by the algebra.
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$p$-nuclearity of reduced group $L^p$-operator algebras
本文证明了约化群 $L^p$-算子代数 $F^p_λ(G)$ 的 $p$-核性蕴含离散群 $G$ 的顺从性，从而解答了 N. C. Phillips 提出的一个公开问题。
We prove a relative analogue of equivalence between nuclearity and CPAP. In its proof, the notion of weak containment for C$^*$-correspondences plays an important role. As an application we prove $KK$-equivalence between full and reduced amalgamated free products of C$^*$-algebras under a strengthened variant of `relative nuclearity'.
A higher rank numerical semigroup is a positive cone whose seminormalization is isomorphic to the free abelian semigroup. The corresponding nonselfadjoint semigroup algebras are known to provide examples that answer Arveson's Dilation Problem to the negative. Here we show that these algebras share the polydisc as the character space in a canonical way. We subsequently use this feature in order to identify higher rank numerical semigroups from the corresponding nonselfadjoint algebras.
We introduce a graph theoretic property called Condition (N) for finitely separated graphs and prove that it is equivalent to both nuclearity and exactness of the associated universal tame graph C*-algebra.
We expand on some invariants used for classifying nonselfadjoint operator algebras. Specifically to nonselfadjoint operator algebras which have a conditional expectation onto a commutative diagonal we construct an edge-colored directed graph which can be used as an operator algebra invariant.
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…
In this paper, we show that if E is a Fell bundle over an amenable étale locally compact Hausdorff groupoid such that every fiber on the unit space is nuclear, then $C?^*r(E)$ is also nuclear. In order to show this result, we introduce (minimal) tensor products of Fell bundles with fixed $C^*$-algebras.
Nuclearity of Hypergraph C*-Algebras
本文通过定制的超图minor关系部分刻画了超图C*-代数的核性，证明了有限图关联的图C*-代数均为核的。
Graph products of groups were introduced by Green in her thesis. They have an operator algebraic counterpart introduced and explored by Fima and the first-named author. In this paper we prove Khintchine type inequalities for general C$^{\ast}$-algebraic graph products which generalize results by Ricard and Xu on free products of C$^{\ast}$-algebras. We apply these inequalities in the context of (right-angled) Hecke C$^{\ast}$-algebras, which are deformations of the group algebra of Coxeter groups. For these we deduce a Haagerup inequality. We further use this to study the simplicity and trace …
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.
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。
$p$-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 semigroup C*-algebra of a positive cone P of a weakly quasi-lattice ordered group. That is, P is a subsemigroup of a discrete group G with P\cap P^{-1}=\{e\} and such that any two elements of P with a common upper bound in P also have a least upper bound. We find sufficient conditions for the semigroup C*-algebra of P to be nuclear. These conditions involve the idea of a generalised length function, called a "controlled map", into an amenable group. Here we give a new definition of a controlled map and discuss examples from different sources. We apply our main result to establish …
Every unital nonselfadjoint operator algebra possesses canonical and functorial classes of faithful (even completely isometric) Hilbert space representations satisfying a double commutant theorem generalizing von Neumann's classical result. Examples and complementary results are given.
Nuclearity and CPC*-systems
该研究提出CPC*-系统概念，将可分核C*-代数表为带完全正连接映射的有限维C*-代数归纳系统极限，借映射渐增的正交保持性为算子空间极限赋乘法构为C*-代数，推广了NF系统至拟对角情形外。
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.
We continue our study of the general theory of possibly nonselfadjoint algebras of operators on a Hilbert space, and modules over such algebras, developing a little more technology to connect `nonselfadjoint operator algebra' with the C$^*-$algebraic framework. More particularly, we make use of the universal, or maximal, C$^*-$algebra generated by an operator algebra, and C$^*-$dilations. This technology is quite general, however it was developed to solve some problems arising in the theory of Morita equivalence of operator algebras, and as a result most of the applications given here (and in …
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。
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 tensor products and nuclearity-related properties of the operator system $\mathcal S_n$ generated by the Cuntz isometries. By using the nuclearity of the Cuntz algebra, we can show that $\mathcal{S}_n$ is $C^*$-nuclear, and this implies a dual row contraction version of Ando's theorem characterizing operators of numerical radius 1. On the other hand, without using the nuclearity of the Cuntz algebra, we are still able to show directly this Ando type property of dual row contractions and conclude that $\mathcal{S}_n$ is $C^*$-nuclear, which yields a new proof of the nuclearity of the C…
We introduce a new type of spectral density condition, that we call L^2-nuclearity. One formulation concerns lowest weight unitary representations of SL(2,R) and turns out to be equivalent to the existence of characters. A second formulation concerns inclusions of local observable von Neumann algebras in Quantum Field Theory. We show the two formulations to agree in chiral Conformal QFT and, starting from the trace class condition for the semigroup generated by the conformal Hamiltonian L_0, we infer and naturally estimate the Buchholz-Wichmann nuclearity condition and the (distal) split prope…
A classical result of Namioka and Phelps states that the square is a test object to verify semi-simplexity in the tensor theory of convex compact sets. By using the quantization of generalized Namioka-Phelps test spaces we formulate a nuclearity criteria for C*-algebras, which establishes a non-commutative version of their result. The proof we suggest covers the nuclearity characterization via non-commutative polyhedron outlined by Effros. Several matrix systems studied by Farenick and Paulsen are shown to be test systems for nuclearity. We also prove that the standard Namioka-Phelps test spac…
Some recent research on the tensor products of operator systems and ensuing nuclearity properties in this setting raised many stability problems. In this paper we examine the preservation of these nuclearity properties including exactness, local liftability and the double commutant expectation property under basic algebraic operations such as quotient, duality, coproducts and tensor products. We show that, in the finite dimensional case, exactness and lifting property are dual pairs, that is, an operator system $S$ is exact if and only if the dual operator system $S^d$ has the lifting property…
We consider a variant of the notion of Morita equivalence appropriate to weak* closed algebras of Hilbert space operators, which we call {\em weak Morita equivalence}. We obtain new variants, appropriate to the dual algebra setting, of the basic theory of strong Morita equivalence, and new nonselfadjoint variants of aspects of Rieffel's $W^*$-algebraic Morita equivalence.
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。
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…
We define nonselfadjoint operator algebras with generators $L_{e_1},..., L_{e_n}, L_{f_1},...,L_{f_m}$ subject to the unitary commutation relations of the form \[ L_{e_i}L_{f_j} = \sum_{k,l} u_{i,j,k,l} L_{f_l}L_{e_k}\] where $u= (u_{i,j,k,l})$ is an $nm \times nm$ unitary matrix. These algebras, which generalise the analytic Toeplitz algebras of rank 2 graphs with a single vertex, are classified up to isometric isomorphism in terms of the matrix $u$.
We initiate an investigation into how much the existing theory of (nonselfadjoint) operator algebras on a Hilbert space generalizes to algebras acting on L^p spaces. In particular we investigate the applicability of the theory of real positivity, which has recently been useful in the study of L^2-operator algebras and Banach algebras, to algebras of bounded operators on Lp spaces.
Let X be a Hilbert bimodule over a C*-algebra A and $O_X= A \rtimes_X \Z$. Using a finite section method we construct a sequence of completely positive contractions factoring through matrix algebras over A which act on $s_ξ s_η^*$ as Schur multipliers converging to the identity. This shows immediately that for a finitely generated X the algebra $O_X$ inherits any standard approximation property such as nuclearity, exactness, CBAP or OAP from A. We generalise this to certain general Pimsner algebras by proving semi-splitness of the Toeplitz extension under certain conditions and discuss some ex…
The note presents a further study of the class of Cuntz--Krieger type algebras. A necessary and sufficient condition is identified that ensures that the algebra is purely infinite, the ideal structure is studied, % and applied to semigraph algebras, and nuclearity is proved by presenting the algebra as a crossed product of an AF-algebra by an abelian group. The results are applied to examples of Cuntz--Krieger type algebras, such as higher rank semigraph $C^*$-algebras and higher rank Exel-Laca algebras.
We introduce notions of finite presentation and co-exactness which serve as qualitative and quantitative analogues of finite-dimensionality for operator modules over completely contractive Banach algebras. With these notions we begin the development of a local theory of operator modules by introducing analogues of the local lifting property, nuclearity, and semi-discreteness. For a large class of operator modules we prove that the local lifting property is equivalent to flatness, generalizing the operator space result of Kye and Ruan. We pursue applications to abstract harmonic analysis, where…
Universal C$^{\ast}$-Algebras from Graph Products: Structure and Applications
本文引入并研究了一类由 C$^{\ast}$-代数的 reduced graph products 生成并辅以投影族扩充的泛 C$^{\ast}$-代数，利用其组合结构确立了泛性质、核性、精确性与理想结构，并借此为图积 C$^{\ast}$-代数结构提供了新见解。
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…
Approximate factorization properties for operator systems
本文证明算子系统张量积层次中的多种核性性质可表示为近似因子分解性质，推广了Choi-Effros的C*-代数完全正逼近性质及Han-Paulsen的算子系统核性性质。
Approximation properties of Fell bundles over inverse semigroups and non-Hausdorff groupoids
本文在逆半群上 Fell 丛的框架下证明 Fell 吸收技巧的类比，并利用 Exel 逼近性质证明全截面与约化截面 C*-代数同构，进而得到具有逼近性质且单位纤维为核的 Fell 丛的截面 C*-代数是核的，且结果同样适用于 étale 非豪斯多夫群胚。
We study property A for metric spaces $X$ with bounded geometry introduced by Guoliang Yu. Property A is an amenability-type condition, which is less restrictive than amenability for groups. The property has a connection with finite-dimensional approximation properties in the theory of operator algebras. It has been already known that property A of a metric space $X$ with bounded geometry is equivalent to nuclearity of the uniform Roe algebra C$^*_u(X)$. We prove that exactness and local reflexivity of C$^*_u(X)$ also characterize property A of $X$.
A distance between von Neumann algebras is introduced, depending on a further norm inducing the $w^*$-topology on bounded sets. Such notion is related both with the Gromov-Hausdorff distance for quantum metric spaces of Rieffel and with the Effros-Marechal topology on the von Neumann algebras acting on a Hilbert space. This construction is tested on the local algebras of free quantum fields endowed with norms related with the Buchholz-Wichmann nuclearity condition, showing the continuity of such algebras w.r.t. the mass parameter.
We study two classes of operator algebras associated with a unital subsemigroup $P$ of a discrete group $G$: 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 $P$ that reflect the space $\mathcal{J}$ 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 $\mat…
For a $C^*$-algebra $A$ and a set $X$ we give a Stinespring-type characterisation of the completely positive Schur $A$-multipliers on $K(\ell^2(X))\otimes A$. We then relate them to completely positive Herz-Schur multipliers on $C^*$-algebraic crossed products of the form $A\rtimes_{α,r} G$, with $G$ a discrete group, whose various versions were considered earlier by Anantharaman-Delaroche, Bédos and Conti, and Dong and Ruan. The latter maps are shown to implement approximation properties, such as nuclearity or the Haagerup property, for $A\rtimes_{α,r} G$.
We study properties of the central sequence algebra of a C*-algebra, and we present an alternative approach to a recent result of Matui and Sato. They prove that every unital separable simple nuclear C*-algebra, whose trace simplex is finite dimensional, tensorially absorbs the Jiang-Su algebra if and only if it has the strict comparison property. We extend their result to the case where the extreme boundary of the trace simplex is closed and of finite topological dimension. We are also able to relax the assumption on the C*-algebra of having the strict comparison property to a weaker property…
Characterizations of amenability for noncommutative dynamical systems and Fell bundles
本文通过引入对角最大张量积，解决了Fell束上近似性质及其持久性的关键问题，建立了BCAP与AP的等价性，并推导了AP在子群限制和正规子群部分商下的持久性性质。
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…
Kirchberg's Embedding Problem (KEP) asks whether every separable C$^*$ algebra embeds into an ultrapower of the Cuntz algebra $\mathcal{O}_2$. In this paper, we use model theory to show that this conjecture is equivalent to a local approximate nuclearity condition that we call the existence of good nuclear witnesses. In order to prove this result, we study general properties of existentially closed C$^*$ algebras. Along the way, we establish a connection between existentially closed C$^*$ algebras, the weak expectation property of Lance, and the local lifting property of Kirchberg. The paper c…
Let $P \subset A$ be an inclusion of $σ$-unital C*-algebras with a finite index in the sense of Izumi. Then we introduce the Rokhlin property for a conditional expectation $E$ from $A$ onto $P$ and show that if $A$ is simple and satisfies any of the property $(1) \sim (12)$ listed in the below, and $E$ has the Rokhlin property, then so does $P$. (1) Simplicity;(2) Nuclearity;(3) C*-algebras that absorb a given strongly self-absorbing C*-algebra $\mathcal{D}$; (4)C*-algebras of stable rank one; (5) C*-algebras of real rank zero;(6) C*-algebras of nuclear dimension at most $n$, where $n \in Z^+$…
On some $p$-approximation properties of discrete groups and $\ell^p$ uniform Roe algebras
研究$\ell^p$一致Roe代数的$p$-逼近性质及其与粗几何和群论的联系，引入$p$-不变平移逼近性质，并证明精确群与$p$-算子逼近性质的等价性。
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 …
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。
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…
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