Nikolay A Ivanov
We find a necessary and sufficient conditions for the simplicity and uniqueness of trace for reduced free products of finite families of finite dimensional $C^*$-algebras with specified traces on them.
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In the paper, we prove that reduced free products of unital AH algebras with respect to given faithful tracial states, in the sense of Voiculescu, are Blackadar and Kirhcberg's MF algebras. We also show that the reduced free products of unital AH algebras with respect to given faithful tracial states, under mild conditions, are not quasidiagonal. Therefore we conclude, for a large class of AH algebras, the Brown-Douglas-Fillmore extension semigroups of the reduced free products of these AH algebras with respect to given faithful tracial states are not groups. Our result is based on Haagerup an…
In this paper, we concentrate on the MF property of reduced free products of unital C*-algebras with amalgamation over finite dimensional C*-algebras. More specifically, we give a necessary and sufficient condition for a reduced free product of two UHF algebras amalgamated over a finite-dimensional C*-algebra with respect to trace preserving conditional expectations to be MF.
Some completely positive maps on reduced amalgamated free products of C*-algebras are constructed; these allow a proof that the class of exact unital C*-algebras is closed under taking reduced amalgamated free products. Consequently, the class of exact discrete groups is closed under taking amalgamated free products.
We construct a weak conditional expectation from the section C*-algebra of a Fell bundle over a unital inverse semigroup to its unit fibre. We use this to define the reduced C*-algebra of the Fell bundle. We study when the reduced C*-algebra for an inverse semigroup action on a groupoid by partial equivalences coincides with the reduced groupoid C*-algebra of the transformation groupoid, giving both positive results and counterexamples.
We prove that the crossed product $C^*$-algebra $C^*_r(Γ,\partialΓ)$ of a free group $Γ$ with its boundary $\partialΓ$ naturally sits between the reduced group $C^*$-algebra $C^*_rΓ$ and its injective envelope $I(C^*_rΓ)$. In other words, we have natural inclusion $C^*_rΓ\subset C^*_r(Γ,\partialΓ) \subset I(C^*_rΓ)$ of $C^*$-algebras.
Let A_i be a family of unital C*-algebras, respectively, of von Neumann algebras and phi: N_0 \to C. We show that if a Hankel matrix related to phi is trace-class, then there exists a unique completely bounded map M_phi on the reduced free product of the A_i, which acts as an radial multiplier. Hereby we generalize a result of Wysoczański for Herz-Schur multipliers on reduced group C*-algebras for free products of groups.
Given reduced amalgamated free products of C$^*$-algebras, $(A,phi)=*_i(A_i,phi_i)$ and $(D,psi)=*_i(D_i,psi_i)$, an embedding $A\to D$ is shown to exist assuming there are conditional expectation preserving embeddings $A_i\to D_i$. This result is extended to show the existance of the reduced amalgamated free product of certain classes of unital completely positive maps. Analogues of the above mentioned results are proved for von Neumann algebras.
We show that if E is an equivalence of upper semicontinuous Fell bundles B and C over groupoids, then there is a linking bundle L(E) over the linking groupoid L such that the full cross-sectional algebra of L(E) contains those of B and C as complementary full corners, and likewise for reduced cross-sectional algebras. We show how our results generalise to groupoid crossed-products the fact, proved by Quigg and Spielberg, that Raeburn's symmetric imprimitivity theorem passes through the quotient map to reduced crossed products.
Given a separated graph $(E,C)$, there are two different C*-algebras associated to it, the full graph C*-algebra $C^*(E,C)$, and the reduced one $C^*_{\text{red}} (E,C)$. For a large class of separated graphs $(E,C)$, we prove that $C^*_{\text{red}} (E,C)$ either is purely infinite simple or admits a faithful tracial state. The main tool we use to show pure infiniteness of reduced graph C*-algebras is a generalization to the amalgamated case of a result on purely infinite simple free products due to Dykema.
We will study some modifications to the notion of an exact C*-algebra by replacing the minimal tensor product with the reduced free product. First we will demonstrate how the reduced free product of a short exact sequence of C*-algebras with another C*-algebra may be taken. It will then be demonstrated that this operation preserves exact sequences. We will also establish that adjoining arbitrary k-tuples of operators in a free way behaves well with respect to taking ultrapowers.
Certain classes of automorphisms of recued amalgamated free products of C*-algebras are shown to have Brown-Voiculescu topological entropy zero. Also, for automorphisms of exact C*-algebras, the Connes-Narnhofer-Thirring entropy is shown to be bounded above by the Brown-Voiculescu entropy. These facts are applied to generalize Stormer's result about entropy of automorphisms of the II_1-factor of a free group.
For any countable discrete group $G$ with a reduced abelian subgroup of finite index, we construct an action $α$ of $G$ on the universal UHF algebra $\Qq$ using an infinite tensor product of permutation representations of $G$ and show that these actions possess some sort of Rokhlin property. The crossed product $\qag$ is then deduced to be tracially AF with a unique tracial state. We also compute the Elliott invariants in the case that $G$ is abelian.
Selfless Reduced Crossed Product $C^{*}$-Algebras Arising from Almost Periodic Actions
该研究证明了一类特定殆周期作用对应的约化交叉积 $C^*$-代数的性质，并结合两种方法完成证明。
Selfless reduced free product $C^*$-algebras
本文建立任意 $C^*$-概率空间滤子的快速衰减理论，并结合近似正交的 von Neumann 代数技术，证明一般约化自由积 $C^*$-代数的无私性，涵盖自由半圆系统代数并给出纯无限新例。
Von Neumann algebras as reduced twisted groupoid $C^*$-algebras
本文刻画了同构于约化扭曲群胚$C^*$-代数的冯·诺依曼代数，证明其为次齐次代数，并引入受控传播障碍，排除$B(H)$及Calkin代数的嵌入。
In this note we start the study of whether the reduced C*-algebra of an inverse semigroup is quasi-diagonal, making explicit use of the inner structure of this class of semigroups in order to produce quasi-diagonal approximations. Given a discrete inverse semigroup, we detail the relationship between its isolated subgroups and the quasi-diagonality of its reduced C*-algebra, and prove that such subgroups must be amenable. Moreover, we give a direct characterization of the quasi-diagonality of inverse semigroup whose universal groupoid is minimal. Lastly, we also study the relevance of Green's …
We survey the results required to pass between full and reduced coactions of locally compact groups on C*-algebras, which say, roughly speaking, that one can always do so without changing the crossed-product C*-algebra. Wherever possible we use definitions and constructions that are well-documented and accessible to non-experts, and otherwise we provide full details. We then give a series of applications to illustrate the use of these techniques. We obtain in particular a new version of Mansfield's imprimitivity theorem for full coactions, and prove that it gives a natural isomorphism between …
Reduced $C^{*}$-algebras of Product Systems -- an $E_0$-semigroup and a Groupoid perspective
本文建立 Ore 半群上 $E_0$-半群与反向积系统的双射，并据此证明真积系统约化代数与半群交叉积 Morita 等价，给出核性、正合性及 $K$-理论同伦不变性结论。
Covariant representations of actions of inverse semigroups: a new approach to the reduced and essential crossed-product C*-algebras
本文通过逆半群作用构造胚芽群胚及具体协变表示族，提出约化与本质交叉积C*-代数的新定义，避免双重换位子与局部乘子代数，并证明与已有定义同构。
We prove that the unitary equivalence classes of extensions of C*_r(G) by any sigma-unital stable C"-algebra, taken modulo extensions which split via an asymptotic homomorphism, form a group which can be calculated from the universal coefficient theorem of KK-theory when G is a free product of a countable collection of amenable groups.
We use a recent result by Cuntz, Echterhoff and Li about the K-theory of certain reduced C*-crossed products to describe the K-theory of C*_r(S) when S is an inverse semigroup satisfying certain requirements. A result of Milan and Steinberg allows us to show that C*_r(S) is Morita equivalent to a crossed product of the type handled by Cuntz, Echterhoff and Li. We apply our result to graph inverse semigroups and the inverse semigroups of one-dimensional tilings.
Nonamenable groups whose reduced group C*-algebras are not pure
本文构造非 amenable 群，其约化群C*-代数不纯，通过限制性圈积的理想-商同构于紧算子代数，否定 Thiel 的纯性问题。
Let a discrete group $G$ act on a unital simple C$^*$-algebra $A$ by outer automorphisms. We establish a Galois correspondence $H\mapsto A\rtimes_{α,r}H$ between subgroups of $G$ and C$^*$-algebras $B$ satisfying $A\subseteq B \subseteq A\rtimes_{α,r}G$, where $A\rtimes_{α,r}G$ denotes the reduced crossed product. For a twisted dynamical system $(A,G,α,σ)$, we also prove the corresponding result for the reduced twisted crossed product $A\rtimes^σ_{α,r}G$.
We construct a locally compact groupoid with the properties in the title. Our example is based closely on constructions used by Higson, Lafforgue, and Skandalis in their work on counterexamples to the Baum-Connes conjecture. It is a bundle of countable groups over the one point compactification of the natural numbers, and is Hausdorff, second countable and étale.
The Wave Equation in the Context of Reduced Groups $C^*$-Algebras
针对可数无限群，本文在约化群 $C^*$-代数框架下，利用单参数 $*$-自同构群构造 Laplacian 类似物，证明了波动方程解的存在性与唯一性。
We prove that the Cowling-Haagerup constant of a reduced free product of weakly amenable discrete quantum groups with Cowling-Haagerup constant equal to 1 is again equal to 1.
We study the ideal structure of reduced crossed product of topological dynamical systems of a countable discrete group. More concretely, for a compact Hausdorff space $X$ with an action of a countable discrete group $Γ$, we consider the absence of a non-zero ideals in the reduced crossed product $C(X) \rtimes_r Γ$ which has a zero intersection with $C(X)$. We characterize this condition by a property for amenable subgroups of the stabilizer subgroups of $X$ in terms of the Chabauty space of $Γ$. This generalizes Kennedy's algebraic characterization of the simplicity for a reduced group $\mathr…
We show that if $G$ is a second countable locally compact Hausdorff étale groupoid carrying a suitable cocycle $c:G\to\mathbb{Z}$, then the reduced $C^*$-algebra of $G$ can be realised naturally as the Cuntz-Pimsner algebra of a correspondence over the reduced $C^*$-algebra of the kernel $G_0$ of $c$. If the full and reduced $C^*$-algebras of $G_0$ coincide, we deduce that the full and reduced $C^*$-algebras of $G$ coincide. We obtain a six-term exact sequence describing the $K$-theory of $C^*_r(G)$ in terms of that of $C^*_r(G_0)$.
Non-isomorphism of reduced free group $C^\ast$-algebras
本文通过新方法证明了不同自由群的简化解$C^\ast$-代数非同构，恢复了Pimsner和Voiculescu的经典结果。
With a minor change made in the previous construction we observe that any reduced HNN extension is precisely a compressed algebra of a certain reduced amalgamated free product in both the von Neumann algebra and the $C^*$-algebra settings. It is also pointed out that the same fact holds true even for universal HNN extensions of C$^*$-algebras. We apply the observation to the questions of factoriality and type classification of HNN extensions of von Neumann algebras and also those of simplicity and $K$-theory of (reduced or universal) HNN extensions of $C^*$-algebras.
In this short note we prove that the reduced group C*-algebra of a locally compact group admits a non-zero trace if and only if the amenable radical of the group is open. This completely answers a question raised by Forrest, Spronk and Wiersma.
We use the technology of linking groupoids to show that equivalent groupoids have Morita equivalent reduced C*-algebras. This equivalence is compatible in a natural way in with the Equivalence Theorem for full groupoid C*-algebras.
A notion of unique ergodicity relative to the fixed-point subalgebra is defined for automorphisms of unital C*-algebras. It is proved that the free shift on any reduced amalgamated free product C*-algebra is uniquely ergodic relative to its fixed-point subalgebra, as are autormorphisms of reduced group C*-algebras arising from certain automorphisms of groups. A generalization of Haagerup's inequality, yielding bounds on the norms of certain elements in reduced amalgamated free product C*-algebras, is proved.
Fourier coefficients and rapid decay in reduced groupoid C*-algebras
该文研究局部紧豪斯多夫平展群胚上的扭所对应的约化C*-代数中，具开支撑的元素何时属于紧支撑截面的闭包，在群胚满足关于条件负定长度函数、或局部负型函数平方根的速降性时给出肯定结论。
We generalize the main result of Kamalov and show that if $G$ is an amenable discrete group with an action $α$ on a finite nuclear unital $C^*$-algebra $A$ such that the reduced crossed product $A\rtimes_{α,r} G$ has property $T$, then $G$ is finite and $A$ is finite dimensional. As an application, an infinite discrete group $H$ is non-amenable if and only if the uniform Roe algebra $C^*_u(H)$ has property $T$.
If $A_n$ is a sequence of C*-algebras, then the C*-algebra $\prod A_n / \bigoplus A_n$ is called a reduced product. We prove, assuming Todorcevic's Axiom and Martin's Axiom, that every isomorphism between two reduced products of separable, unital UHF algebras must be definable in a strong sense. As a corollary we deduce that two such reduced products $\prod A_n / \bigoplus A_n$ and $\prod B_n / \bigoplus B_n$ are isomorphic if and only if, up to an almost-permutation of $\mathbb{N}$, $A_n$ is isomorphic to $B_n$.
We study some reduced free products of C*-algebras with amalgamations. We give sufficient conditions for the positive cone of the K_0 group to be the largest possible. We also give sufficient conditions for simplicity and uniqueness of trace. We use the later result to give a necessary and sufficient condition for simplicity and uniqueness of trace of the reduced C*-algebras of the Baumslag-Solitar groups BS(m,n).
We establish the MF property of the reduced group $ C^* $-algebra of an amalgamated free product of countable Abelian discrete groups. This result is then used to give a characterization of the amalgamated free products of Abelian groups for which the BDF semigroup of the reduced group $ C^* $-algebra is a group. Along the way we get a tensor product factorization of the corresponding group von Neumann algebra. We end the exposition by applying the ideas from the first part to give a few more examples of groups with a reduced group $ C^* $-algebra which is MF.
We establish conditions under which the universal and reduced norms coincide for a Fell bundle over a groupoid. Specifically, we prove that the full and reduced C*-algebras of any Fell bundle over a measurewise amenable groupoid coincide, and also that for a groupoid G whose orbit space is T_0, the full and reduced algebras of a Fell bundle over G coincide if the full and reduced algebras of the restriction of the bundle to each isotropy group coincide.
The generator problem was posed by Kadison in 1967, and it remains open until today. We provide a solution for the class of C*-algebras absorbing the Jiang-Su algebra Z tensorially. More precisely, we show that every unital, separable, Z-stable C*-algebras A is singly generated, which means that there exists an element x in A that is not contained in any proper sub-C*-algebra of A. To give applications of our result, we observe that Z can be embedded into the reduced group C*-algebra of a discrete group that contains a non-cyclic, free subgroup. It follows that certain tensor products with red…
We give an example of a non-trivial asymptotic representation of the reduced C*-algebra of a free group. This example allows to evaluate the asymptotic tensor C*-norm of some elements in tensor product C*-algebras and to show semi-invertibility of the non-invertible extension of $C^*_r(\mathbb F_2)$ considered by Haagerup and Thorbjornsen.
Some remarks on Reduced $C^*$-algebras of semigroup dynamical systems and product systems
研究半群动力系统的简化的交叉积和乘积系统的简化的$C^*$-代数的精确性，证明在合理假设下，简化的交叉积精确当且仅当原代数精确。
In this paper, we study operator algebraic properties of the reduced and von Neumann algebraic versions of the free wreath products $\mathbb G \wr_* S_N^+$, where $\mathbb G$ is a compact matrix quantum group. Based on recent result on their corepresentation theory by Lemeux and Tarrago, we prove that $\mathbb G \wr_* S_N^+$ is of Kac type whenever $\mathbb G$ is, and that the reduced version of $\mathbb G \wr_* S_N^+$ is simple with unique trace state whenever $N \geq 8$. Moreover, we prove that the reduced von Neumann algebra of $\mathbb G \wr_* S_N^+$ does not have property $Γ$.
We study C*-irreducibility of inclusions of reduced twisted group C*-algebras and of reduced group C*-algebras. We characterize C*-irreducibility in the case of an inclusion arising from a normal subgroup, and exhibit many new examples of C*-irreducible inclusions.
The purpose of this paper is to make a further contribution to the Mackey bijection for a complex reductive group G, between the tempered dual of G and the unitary dual of the associated Cartan motion group. We shall construct an embedding of the C*-algebra of the motion group into the reduced C*-algebra of G, and use it to characterize the continuous field of reduced group C*-algebras that is associated to the Mackey bijection. We shall also obtain a new characterization of the Mackey bijection using the same embedding.
To each discrete left cancellative semigroup $S$ one may associate a certain inverse semigroup $I_l(S)$, often called the left inverse hull of $S$. We show how the full and the reduced C*-algebras of $I_l(S)$ are related to the full and reduced semigroup C*-algebras for $S$ recently introduced by Xin Li, and give conditions ensuring that these algebras are isomorphic. Our picture provides an enhanced understanding of Li's algebras.
C. Schochet shows Künneth theorem for the $C^*$-algebras in the smallest class of nuclear $C^*$-algebras which contains the separable Type I algebras and is closed under some operations. We calculate the $K$-theory for the crossed product of the infinite tensor product of a unital $C^*$-algebra in this class by the shift. In particular, we calculate the $K$-theory of the lamplighter group $C^*$-algebra.
We characterize relatively norm compact sets in the regular $C^*$-algebra of finitely generated Coxeter groups using a geometrically defined positive semigroup acting on the algebra.
For any second countable locally compact group G, we construct a simple G-C*-algebra whose full and reduced crossed product norms coincide. We then construct its G-equivariant representation on another simple G-C*-algebra without the coincidence condition. This settles two problems posed by Anantharaman-Delaroche in 2002. Some constructions involve the Baire category theorem.
A countable group is C*-simple if its reduced C*-algebra is a simple algebra. Since Powers recognised in 1975 that non-abelian free groups are C*-simple, large classes of groups which appear naturally in geometry have been identified, including non-elementary Gromov hyperbolic groups and lattices in semisimple groups. In this exposition, C*-simplicity for countable groups is shown to be an extreme case of non-amenability. The basic examples are described and several open problems are formulated.
Selfless reduced $C^{*}$-algebras of linear groups
证明非平凡线性群且可解根平凡的约化C*-代数是无我的，从而线性群的约化C*-代数中无我性与单性等价。
This paper is aimed at investigating links between Fell bundles over Morita equivalent groupoids and their corresponding reduced C*-algebras. Mainly, we review the notion of Fell pairs over a Morita equivalence of groupoids, and give the analogue of the Renault's Equivalence Theorem for the reduced C*-algebras of equivalent Fell systems. Eventually, we will use this theorem to connect the reduced C*-algebra of an S^1-central groupoid extension to that of its associated Dixmier-Douady bundle.
Let G be the p-adic group GL(n). Using C*-algebra techniques, we obtain a very explicit description of the tempered dual of G in terms of Bernstein parameters and extended quotients. We also prove that Plancherel measure (on the tempered dual of a reductive p-adic group) is rotation-invariant.
We consider reduced crossed products of twisted C*-dynamical systems over C*-simple groups. We prove there is a bijective correspondence between maximal ideals of the reduced crossed product and maximal invariant ideals of the underlying C*-algebra, and a bijective correspondence between tracial states on the reduced crossed product and invariant tracial states on the underlying C*-algebra. In particular, the reduced crossed product is simple if and only if the underlying C*-algebra has no proper non-trivial invariant ideals, and the reduced crossed product has a unique tracial state if and on…
The reduced C*-algebra of a countable linear group G is shown to be simple if and only if G has no nontrivial normal amenable subgroups. Moreover, these conditions are shown to be equivalent to the uniqueness of tracial state on the aforementioned C*-algebra.
Let $G$ be an acylindrically hyperbolic group. We prove that if $G$ has no non-trivial finite normal subgroups, then the set of invertible elements is dense in the reduced $C^\ast$-algebra of $G$. The same result is obtained for finite direct products of acylindrically hyperbolic groups.
We prove Khintchine type inequalities for words of a fixed length in a reduced free product of $C^*$-algebras (or von Neumann algebras). These inequalities imply that the natural projection from a reduced free product onto the subspace generated by the words of a fixed length $d$ is completely bounded with norm depending linearly on $d$. We then apply these results to various approximation properties on reduced free products. As a first application, we give a quick proof of Dykema's theorem on the stability of exactness under the reduced free product for $C^*$-algebras. We next study the stabi…
Let $B$ be a finite dimensional C$^\ast$-algebra equipped with its canonical trace induced by the regular representation of $B$ on itself. In this paper, we study various properties of the trace-preserving quantum automorphism group $\G$ of $B$. We prove that the discrete dual quantum group $\hG$ has the property of rapid decay, the reduced von Neumann algebra $L^\infty(\G)$ has the Haagerup property and is solid, and that $L^\infty(\G)$ is (in most cases) a prime type II$_1$-factor. As applications of these and other results, we deduce the metric approximation property, exactness, simplicity …
A characterization of simplicity of reduced groupoid C*-algebras
本文刻画了约化群胚C*-代数的简单性，证明其等价于存在余稠密单位点集具有C*-简单迷向群，并构造反例回答Christensen和Neshveyev的问题。
The induction and reduction precesses of an O*-vector space $\M$ obtained by means of a projection taken, respectively, in $\M$ itself or in its weak bounded commutant $\M'_\w$ are studied. In the case where $\M$ is a partial GW*-algebra, sufficient conditions are given for the induced and the reduced spaces to be partial GW*-algebras again.
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