Simple C*-algebras with a unique tracial state

On ultraproduct embeddings and amenability for tracial von Neumann algebras

We define the notion of self-tracial stability for tracial von Neumann algebras and show that a tracial von Neumann algebra satisfying the Connes Embedding Problem is self-tracially stable if and only if it is amenable. We then generalize a result of Jung by showing that a separable tracial von Neumann algebra that satisfies the Connes Embedding Problem is amenable if and only if any two embeddings into $R^\mathcal{U}$ are ucp-conjugate. Moreover we show that for a II$_1$ factor $N$ satisfying CEP, the space $\mathbb{H}$om$(N, \prod_{k\to \mathcal{U}}M_k)$ of unitary equivalence classes of emb…

The Rohlin property for automorphisms on simple C*-algebras

We study a general Kishimoto's problem for automorphisms on simple C*-algebras with tracial rank zero. Let $A$ be a unital separable simple C*-algebra with tracial rank zero and let $α$ be an automorphism. Under the assumption that $α$ has certain Rokhlin property, we present a proof that $A\rtimes_α\Z$ has tracial rank zero. We also show that if the induced map $α_{*0}$ on $K_0(A)$ fixes a "dense" subgroup of $K_0(A)$ then the tracial Rokhlin property implies a stronger Rokhlin property. Consequently, the induced crossed product C*-algebras have tracial rank zero.

Tracial algebras and an embedding theorem

We prove that every positive trace on a countably generated *-algebra can be approximated by positive traces on algebras of generic matrices. This implies that every countably generated tracial *-algebra can be embedded into a metric ultraproduct of generic matrix algebras. As a particular consequence, every finite von Neumann algebra with separable pre-dual can be embedded into an ultraproduct of tracial *-algebras, which as *-algebras embed into a matrix-ring over a commutative algebra.

Non-unital tracially approximated ${\rm C^*}$-algebras

In this paper, we introduce a class of non-unital tracial approximation ${\rm C^*}$-algebras. Consider the class of ${\rm C^*}$-algebras which are tracially $\mathcal{Z}$-absorbing (in the sense of Amint, Golestani, Jamali, Phillips's simple tracially $\mathcal{Z}$-absorbing or Castillejos, Li, Szabvo's tracial $\mathcal{Z}$-stability). Then $A$ is tracially $\mathcal{Z}$-absorbing for any simple ${\rm C^*}$-algebra $A$ in the corresponding class of non-unital tracial approximation ${\rm C^*}$-algebras.

Tracial approximation in simple C*-algebras

We revisit the notion of tracial approximation for unital simple C*-algebras. We show that a unital simple separable C*-algebra A is asymptotically tracially in the class of C*-algebras with finite nuclear dimension if and only if A is asymptotically tracially in the class of simple nuclear Z-stable C*-algebras.

Traces of $C^*$-algebras of connected solvable groups

We give an explicit description of the tracial state simplex of the $C^*$-algebra $C^*(G)$ of an arbitrary connected, second countable, locally compact, solvable group $G$. We show that every tracial state of $C^*(G)$ lifts from a tracial state of the $C^*$-algebra of the abelianized group, and the intersection of the kernels of all the tracial states of $C^*(G)$ is a proper ideal unless $G$ is abelian. As a consequence, the $C^*$-algebra of a connected solvable nonabelian Lie group cannot embed into a simple unital AF-algebra.

迹正映射的分解及其在量子信息中的应用

Decomposition of tracial positive maps and applications in quantum information

本文利用 Aron–Berner 扩张证明迹正多重线性映射可经交换值域/定义域映射分解并因而完全正，并将其用于量子力学的广义方差、协方差与不确定关系。

Generalized Tracially Approximated C*-algebras

In this paper, we introduce some classes of generalized tracial approximation ${\rm C^*}$-algebras. Consider the class of unital ${\rm C^*}$-algebras which are tracially $\mathcal{Z}$-absorbing (or have tracial nuclear dimension at most $n$, or have the property $\rm SP$, or are $m$-almost divisible). Then $A$ is tracially $\mathcal{Z}$-absorbing (respectively, has tracial nuclear dimension at most $n$, has the property $\rm SP$, is weakly ($n, m$)-almost divisible) for any simple unital ${\rm C^*}$-algebra $A$ in the corresponding class of generalized tracial approximation ${\rm C^*}$-algebra…

Some properties for certain generalized tracial approximated ${\rm C^*}$-algebras

In this paper, we introduce a class of generalized tracial approximation ${\rm C^*}$-algebras. Let $\mathcal{P}$ be a class of unital ${\rm C^*}$-algebras which have tracially $\mathcal{Z}$-absorbing (tracial nuclear dimension at most $n$, $\rm SP$ property, $m$-almost divisible, weakly $(m, n)$-divisible). Then $A$ has tracially $\mathcal{Z}$-absorbing (tracial nuclear dimension at most $n$, $\rm SP$ property, weakly $m$-almost divisible, secondly weakly $(m, n)$-divisible) for any simple unital ${\rm C^*}$-algebra $A$ in the class of this generalized tracial approximation ${\rm C^*}$-algebra…

Faithful tracial states on quotients of C*-algebras

We study the existence of faithful tracial states on C*-algebras as well as the stronger proerty that all quotients admit faithful tracial states. We provide a sufficient and necessary condition for when C*-algebras admit faithful tracial states in terms of the Cuntz semigroup and use this to give an equivalent formulation for all quotients to admit faithful tracial states. We relate this to the notion of strong quasidiagonality, and show that any amenable discrete group with faithful tracial states on all quotients of the corresponding group-C*-algebra is strongly quasidiagonal under the cond…

Inductive limits, unique traces and tracial rank zero

In the program to classify C$^*$-algebras, it is very important to find abstract conditions which are sufficient to imply that a given algebra has tracial rank zero, in the sense of Huaxin Lin. Even in the presence of a unique trace, we show that the union of the known necessary conditions is not enough.

On Local AH algebras

We show that every unital amenable separable simple $C^*$-algebra with finite tracial rank which satisfies the UCT has in fact tracial rank at most one. We also show that unital separable simple $C^*$-algebrass which are "tracially" locally AH with slow dimension growth are ${\cal Z}$-stable. As a consequence, unital separable simple $C^*$-algebras which are locally AH with no dimension growth are isomorphic to a unital simple AH-algebra with no dimension growth.

The tracial Rokhlin property for actions of finite groups on C*-algebras

We define "tracial" analogs of the Rokhlin property for actions of finite groups, approximate representability of actions of finite abelian groups, and of approximate innerness. We prove four analogs of related "nontracial" results. First, the crossed product of an infinite dimensional simple separable unital C*-algebra with tracial rank zero by an action of a finite group with the tracial Rokhlin property again has tracial rank zero. Second, an outer action of a finite abelian group on an infinite dimensional simple separable unital C*-algebra has the tracial Rokhlin property if and only if i…

迹反射的C*-代数

Tracially reflexive C*-algebras

本文研究了迹反射C*-代数的概念，证明了交换C*-代数和具有分离拓扑维数零的C*-代数都是迹反射的，并探讨了迹反射性质的持久性及两个判定准则。

Simple tracially $\mathcal{Z}$-absorbing C*-algebras

We define a notion of tracial $\mathcal{Z}$-absorption for simple not necessarily unital C*-algebras, study it systematically, and prove its permanence properties. This extends the notion defined by Hirshberg and Orovitz for unital C*-algebras. The Razak-Jacelon algebra, simple C*-algebras with tracial rank zero, and simple purely infinite C*-algebras are tracially $\mathcal{Z}$-absorbing. We obtain the first purely infinite examples of tracially $\mathcal{Z}$-absorbing C*-algebras which are not $\mathcal{Z}$-absorbing. We use techniques from reduced free products of von~Neumann algebras to co…

A tracially AF algebra which is not $\mathcal Z$-absorbing

We show that there is a simple separable unital (non-nuclear) tracially AF algebra $A$ which does not absorb the Jiang-Su algebra $\mathcal Z$ tensorially, i.e., $A \ncong A\otimes\mathcal Z$.

C*-代数渐近迹逼近的若干性质

some properties for asymptotically tracially approximation of C*-algebras

本文研究C*-代数的渐近迹逼近性质，证明三类C*-代数的相关性质可被TΩ类单含单位元C*-代数继承，并应用于广义迹秩、迹拓扑秩相关的C*-代数分类研究。

将映射分类到一致迹序列代数中

Classifying maps into uniform tracial sequence algebras

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。

An L^2-Kunneth formula for tracial algebras

We prove a Kunneth formula computing the Connes-Shlyakhtenko L^2-Betti numbers of the algebraic tensor product of two tracial *-algebras in terms of the L^2-Betti numbers of the two original algebras. As an application, we construct examples of non-finite, non-cocommutative, compact quantum groups with a non-vanishing first L^2-Betti number.

本质迹秩至多为一的C*-代数

C*-algebras of Essentially Tracially Rank One

本文引入单C*-代数的本质迹秩至多为一的概念，刻画了可分单有限顺从𝒵稳定C*-代数，完成了长期研究项目，还证明了相关拟对角性及分解秩的结论，且不依赖UCT。

C$^*$-代数上的迹态空间

The Space of Tracial States on a C$^*$-Algebra

本文利用有限von Neumann代数上的中心值迹，给出了unital C$^*$-代数的迹态空间是Choquet单纯形的简单初等证明。

Tracially sequentially-split ${}^*$-homomorphisms between $C^*$-algebras

We define a tracial analogue of the sequentially split $*$-homomorphism between $C^*$-algebras of Barlak and Szabó and show that several important approximation properties related to the classification theory of $C^*$-algebras pass from the target algebra to the domain algebra. Then we show that the tracial Rokhlin property of the finite group $G$ action on a $C^*$-algebra $A$ gives rise to a tracial version of sequentially split $*$-homomorphism from $A\rtimes_αG$ to $M_{|G|}(A)$ and the tracial Rokhlin property of an inclusion $C^*$-algebras $A\subset P$ with a conditional expectation $E:A \…

拓扑图代数上的迹权重

Tracial weights on topological graph algebras

本文通过刻画第二可数拓扑图边界路径空间上的正则不变测度，描述了图C*-代数上的非规范不变极值迹权重，并证明了无环拓扑图的C*-代数上所有迹权重均规范不变。

Approximated by finite-dimensional homomorphisms into Simple C*-Algebras with Tracial Rank One

We discuss when a unital homomorphism ϕ : C(X) \rightarrow A can be approximated by finite-dimensional homomorphisms, where X is a compact metric space and A is unital simple C*-algebra with tracial rank one. In this paper, we will give a necessary and sufficient condition.

关于有限群作用于简单单位C*-代数的弱迹Rokhlin性质的一个注记

A note on the weak tracial Rokhlin property for finite group actions on simple unital C*-algebras

本文证明弱迹Rokhlin性质定义中的一个条件可被更弱或更强的条件替代，并在非纯无限情形下冗余，同时给出弱迹Rokhlin性质推出迹Rokhlin性质的充分条件。

Tracially Z-absorbing C*-algebras

We study a tracial notion of Z-absorption for simple, unital C*-algebras. We show that if A is a C*-algebra for which this property holds then A has almost unperforated Cuntz semigroup, and if in addition A is nuclear and separable we show this property is equivalent to having A being Z-absorbing. We furthermore show that this property is preserved under forming certain crossed products by actions satisfying a tracial Rokhlin type property.

Classification of simple $C^*$-algebras of tracial topological rank zero

We give a classification theorem for unital separable simple nuclear $C^*$-algebras with tracial topological rank zero which satisfy the Universal Coefficient Theorem. We prove that if $A$ and $B$ are two such $C^*$-algebras and $$ (K_0(A), K_0(A)_+, [1_A], K_1(A)) $$ $$ \cong (K_0(B), K_0(B)_+, [1_B], K_1(B)), $$ then $A\cong B.$

Homotopy of unitaries in simple C*-algebras with tracial rank one

Let $ε>0$ be a positive number. Is there a number $δ>0$ satisfying the following? Given any pair of unitaries $u$ and $v$ in a unital simple $C^*$-algebra $A$ with $[v]=0$ in $K_1(A)$ for which $$ \|uv-vu\|<\dt, $$ there is a continuous path of unitaries $\{v(t): t\in [0,1]\}\subset A$ such that $$ v(0)=v, v(1)=1 \and \|uv(t)-v(t)u\|<ε\forall t\in [0,1]. $$ An answer is given to this question when $A$ is assumed to be a unital simple $C^*$-algebra with tracial rank no more than one. Let $C$ be a unital separable amenable simple $C^*$-algebra with tracial rank no more than one which also satisf…

$\mathcal Z$-stability and finite dimensional tracial boundaries

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.

Crossed products by finite cyclic group actions with the tracial Rokhlin property

We define the tracial Rokhlin property for actions of finite cyclic groups on stably finite simple unital C*-algebras. We prove that when the algebra is in addition simple and has tracial rank zero, then the crossed product again has tracial rank zero. Under a kind of weak approximate innerness assumption and one other technical condition, we prove that if the action has the the tracial Rokhlin property and the crossed product has tracial rank zero, then the original algebra has tracial rank zero. We give examples showing how the tracial Rokhlin property differs from the Rokhlin property of Iz…

Compact Group Actions with the Tracial Rokhlin Property

We define a "tracial" analog of the Rokhlin property for actions of second countable compact groups on infinite dimensional simple separable unital C*-algebras. We prove that fixed point algebras under such actions (and, in the appropriate cases, crossed products by such actions) preserve simplicity, Property (SP), tracial rank zero, tracial rank at most one, the Popa property, tracial Jiang-Su stability, Jiang-Su stability when the algebra is nuclear, infiniteness, and pure infiniteness. We also show that the radius of comparison of the fixed point algebra is no larger than that of the origin…

Tracial invariants, classification and II_1 factor representations of Popa algebras

Using various finite dimensional approximation properties, four convex subsets of the tracial space of a unital C*-algebra are defined. Applications of these tracial invariants include: (1) An analogue of Szego's limit theorem for arbitrary self adjoint operators. (2) A McDuff factor embeds into the ultrapower of the hyperfinite II_1 factor if and only if it contains a weakly dense operator system which is injective. (3) There exists a simple, quasidiagonal, real rank zero C*-algebra with non-hyperfinite II_1 factor representations and which is not tracially AF. This answers negatively questio…

A classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras, I: $C^*$-algebras with generalized tracial rank one

A class of $C^*$-algebras, to be called those of generalized tracial rank one, is introduced, and classified by the Elliott invariant. A second class of unital simple separable amenable $C^*$-algebras, those whose tensor products with UHF-algebras of infinite type are in the first class, to be referred to as those of rational generalized tracial rank one, is proved to exhaust all possible values of the Elliott invariant for unital finite simple separable amenable ${\cal Z}$-stable $C^*$-algebras. An isomorphism theorem for a special sub-class of those $C^*$-algebras are presented. This provide…

Tracial stability for C*-algebras

We consider tracial stability, which requires that tuples of elements of a C*-algebra with a trace that nearly satisfy the relation are close to tuples that actually satisfy the relation. Here both "near" and "close" are in terms of the associated 2-norm from the trace, e.g., the Hilbert-Schmidt norm for matrices. Precise definitions are stated in terms of liftings from tracial ultraproducts of C*-algebras. We completely characterize matricial tracial stability for nuclear C*-algebras in terms of certain approximation properties for traces. For non-nuclear $C^{\ast}$-algebras we find new obstr…

渐近迹逼近 C*-代数的可除性质

Divisible properties for asymptotically tracially approximation ${\rm C^*}$-algebras

本文研究渐近迹属于类 P 的单含单位元 C*-代数，证明其继承原类代数的 m-几乎可除与弱 (m,n)-可除性质。

关于非单单位元 C*-代数的迹 $\mathcal Z$-稳定性

On tracial $\mathcal Z$-stability of simple non-unital C*-algebras

本文研究了非单位元 C*-代数的迹 $\mathcal Z$-稳定性，并证明在可分简单核 C*-代数类中该概念与 $\mathcal Z$-稳定性等价。

Some analysis on amalgamated free products of von Neumann algebras in non-tracial setting

Several techniques together with some partial answers are given to the questions of factoriality, type classification and fullness for amalgamated free product von Neumann algebras.

非单位的广义迹逼近 C*-代数

Non unital generalized tracially approximated C*-algebras

本文研究了非单位的广义迹逼近C*-代数类，证明了其与弱迹逼近的等价性，并验证了该类代数具有迹Z-吸收、迹核维数至多n及性质SP等重要性质。

A classification of finite simple amenable Z-stable C*-algebras, II, --C*-algebras with rational generalized tracial rank one

A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-algebras which exhausts all possible values of the Elliott invariant for unital stably finite simple separable amenable Z-stable C*-algebras. Moreover, it contains all unital simple separable amenable C*-algebras which satisfy the UCT and have finite rational tracial rank

Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property

We introduce the tracial Rokhlin property for automorphisms of stably finite simple unital C*-algebras containing enough projections. This property is formally weaker than the various Rokhlin properties considered by Herman and Ocneanu, Kishimoto, and Izumi. Our main results are as follows. Consider a stably finite simple unital C*-algebra, and an automorphism of it which has the tracial Rokhlin property. Suppose the algebra has real rank zero and stable rank one, and suppose that the order on projections over the algebra is determined by traces. Then the crossed product by the automorphism al…

Some results on tracial stability and graph products

We establish the tracial stability of a certain class of graph products of C*-algebras. This result involves the development of the "pincushion class" of finite graphs. We then apply this result in two ways. The first application yields a selective version of Lin's Theorem for almost commuting operators. The second application addresses some approximation properties of right-angled Artin groups. In particular, we show that the full C*-algebra of any right-angled Artin group is quasidiagonal and thus has a non-trivial amenable trace, and then we apply tracial stability to show when these amenab…

Finite cyclic group actions with the tracial Rokhlin property

We give examples of actions of Z/2Z on AF algebras and AT algebras which demonstrate the differences between the (strict) Rokhlin property and the tracial Rokhlin property, and between (strict) approximate representability and tracial approximate representability. Specific results include the following. We determine exactly when a product type action of Z/2Z on a UHF algebra has the tracial Rokhlin property; in particular, unlike for the strict Rokhlin property, every UHF algebra admits such an action. We prove that Blackadar's action of Z/2Z on the 2^{\infty} UHF algebra, whose crossed produc…

Tracial Rokhlin property for actions of amenable groups on C* algebras

Tracial Rokhlin property was introduced by Chris Phillips to study the structure of crossed product of actions on simple C*-algebras. It was originally defined for actions of finite groups and group of integers. Matui and Sato generalized it to actions of amenable groups. In this paper, we give a further generalization of Matui and Sato's definition. We shall show that many known results about tracial Rokhlin property could be generalized to actions of amenable groups under this definition. We also show that some natural examples has the (weak) tracial Rokhlin property.

A proof-theoretic metatheorem for tracial von Neumann algebras

We adapt a continuous logic axiomatization of tracial von Neumann algebras due to Farah, Hart and Sherman in order to prove a metatheorem for this class of structures in the style of proof mining, a research program that aims to obtain the hidden computational content of ordinary mathematical proofs using tools from proof theory.

Simple stably projectionless C*-algebras with generalized tracial rank one

We study a class of stably projectionless simple C*-algebras which may be viewed as having generalized tracial rank one in analogy with the unital case. Some structural question concerning these simple C*-algebras are studied. The paper also serves as a technical support for the classification of separable stably projectionless simple amenable Jiang-Su stable C*-algebras.

Crossed product C*-algebras by finite group actions with the projection free tracial Rokhlin property

In this paper we introduce an analog of the tracial Rokhlin property, called the {\emph {projection free tracial Rokhlin property}}, for $C^*$-algebras which may not have any nontrivial projections. Using this we show that if $A$ is an infinite dimensional stably finite simple unital $C^*$-algebra with stable rank one, with strict comparison of positive elements, with only finitely many extreme tracial states, and with the property that every 2-quasi-trace is a trace, and if $α$ is an action of a finite group $G$ with the projection free tracial Rokhlin property, then the crossed product $C^*(…

Simple nuclear $C^*$-algebras of tracial topological rank one

We give a classification theorem for unital separable nuclear simple \CA s with tracial rank no more than one. Let $A$ and $B$ be two unital separable simple nuclear \CA s with $TR(A), TR(B)\le 1$ which satisfy the universal coefficient theorem. We show that $A\cong B$ if and only if $$ (K_0(A), K_0(A)_+, [1_A], K_1(A), T(A)) \cong (K_0(B), K_0(B)_+, [1_B], K_1(B), T(B)). $$

广义迹逼近 C*-代数的若干性质

Certain properties of generalized tracially approximated C*-algebras

本文研究广义迹逼近 C*-代数，证明 WTAΩ 类中的含单位单 C*-代数可保持 Ω 类含单位 C*-代数的一致性质 Γ、迹核维数上界及弱可除性。

The Dixmier property and tracial states for C*-algebras

It is shown that a unital C*-algebra A has the Dixmier property if and only if it is weakly central and satisfies certain tracial conditions. This generalises the Haagerup-Zsido theorem for simple C*-algebras. We also study a uniform version of the Dixmier property, as satisfied for example by von Neumann algebras and the reduced C*-algebras of Powers groups, but not by all C*-algebras with the Dixmier property, and we obtain necessary and sufficient conditions for a simple unital C*-algebra with unique tracial state to have this uniform property. We give further examples of C*-algebras with t…

迹冯·诺依曼代数的张量积对初等性的保持

Preservation of elementarity by tensor products of tracial von Neumann algebras

该研究证明在迹冯·诺依曼代数范畴中与I型代数取张量积可保持初等等价性，关键方法是使用了针对度量结构直积分的新颖通用Feferman--Vaught型定理。

Closed Convex Hulls of Unitary Orbits in C$^*$-Algebras of Real Rank Zero

In this paper, we study closed convex hulls of unitary orbits in various C$^*$-algebras. For unital C$^*$-algebras with real rank zero and a faithful tracial state determining equivalence of projections, a notion of majorization describes the closed convex hulls of unitary orbits for self-adjoint operators. Other notions of majorization are examined in these C$^*$-algebras. Combining these ideas with the Dixmier property, we demonstrate unital, infinite dimensional C$^*$-algebras of real rank zero and strict comparison of projections with respect to a faithful tracial state must be simple and …

Crossed Products by Automorphisms with the Tracial Quasi-Rokhlin Property

We introduce the tracial quasi-Rokhlin property for an automorphism alpha of a unital C*-algebra A, which is not assumed to be simple. We show that under suitable hypotheses, the associated crossed product C*-algebra C*(Z,A,alpha) is simple, and there is a bijection between the space of tracial states on C*(Z,A,alpha)$ and the alpha-invariant tracial states on A. We show that, for a minimal dynamical system (X,h) and a simple, separable, unital C*-algebra A, the automorphism beta which extends the action of h on C(X) has the tracial quasi-Rokhlin property, and hence that C*(Z,C(X,A),beta) has …

Tracial approximate divisibility and stable rank one

We show that every separable simple tracially approximately divisible $C^*$-algebra has strict comparison, is either purely infinite, or has stable rank one. As a consequence, we show that every (non-unital) finite simple ${\cal Z}$-stable $C^*$-algebra has stable rank one.

Non-amenable simple C*-algebras with tracial approximation

We construct two types of unital separable simple $C^*$-alebras $A_z^{C_1}$ and $A_z^{C_2},$ one is exact but not amenable, and the other is non-exact. Both have the same Elliott invariant as the Jiang-Su algebra, namely, $A_z^{C_i}$ has a unique tracial state, $$(K_0(A_z^{C_i}), K_0(A_z^{C_i})_+, [1_{A_z^{C_i}} ])=(\mathbb Z, \mathbb Z_+,1)$$ and $K_{1}(A_z^{C_i})=\{0\}$ ($i=1,2$). We show that $A_z^{C_i}$ ($i=1,2$) is essentially tracially in the class of separable ${\cal Z}$-stable $C^*$-alebras of nuclear dimension 1. $A_z^{C_i}$ has stable rank one, strict comparison for positive elements…