Huaxin Lin
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 $$ (K_0(A), K_0(A)_+, [1_A], K_1(A), T(A)) \cong (K_0(B), K_0(B)_+, [1_B], K_1(B), T(B)). $$
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We give two characterizations of tracially nuclear C*-algebras. The first is that the finite summand of the second dual is hyperfinite. The second is in terms of a variant of the weak* uniqueness property. The necessary condition holds for all tracially nuclear C*-algebras. When the algebra is separable, we prove the sufficiency.
From Stable Rank One to Real Rank Zero: A Note on Tracial Approximate Oscillation Zero
本文通过迹振荡方法证明了稳定秩一的C*-代数具有实秩零，并建立了迹近似振荡零与实秩零之间的等价关系。
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.
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.
The real and stable rank of tracially complete C*-algebras
本文证明可追溯完备C*-代数在满足CPoU条件下具有零真实秩和一稳定秩，从而完整描述了这些代数的Cuntz半群。
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.
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.
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。
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 $A\cong B.$
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。
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…
The inheritance tracial topological rank zero for centrally large subalgebra
本文证明若无限维单位单纯 C*-代数满足 μ-ozLLP 性质，且其中心大型子代数具有迹拓扑秩零，则该代数本身也具有迹拓扑秩零。
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…
Let $A$ be a unital separable simple \CA with $\tr(A)\le 1$ and $α$ be an automorphism. We show that if $α$ satisfies the tracially cyclic Rokhlin property then $\tr(A\rtimes_α\Z)\le 1.$ We also show that whenever $A$ has a unique tracial state and $α^m$ is uniformly outer for each $m (\not= 0)$ and $α^r$ is approximately inner for some $r>0,$ $α$ satisfies the tracial cyclic Rokhlin property. By applying the classification theory of nuclear \CA s, we use the above result to prove a conjecture of Kishimoto: if $A$ is a unital simple $A{\mathbb T}$-algebra of real rank zero and $α\in \Aut(A)$ w…
We show by construction that when $G$ is an elementary amenable group and $A$ is a unital simple nuclear and tracially approximately divisible $C^*$-algebra, there exists an action $ω$ of $G$ on $A$ with the tracial Rokhlin property in the sense of Matui and Sato. In particular, group actions with this Matui-Sato tracial Rokhlin property always exist for unital simple separable nuclear $C^*$-algebras with tracial rank at most one. If $A$ is simple with rational tracial rank at most one, then the crossed product $A\rtimes_ωG$ is also simple with rational tracial rank at most one.
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。
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 Rokhlin property was introduced by Phillips to prove various structure theorems for crossed product. But it is defined for actions on simple C*-algebras only. This paper consists of two major parts: In section 2 and 3, we study the permanence properties and give a complete classification of tracial Rokhlin property for product-type actions; In section 4 and 5, we introduce the weak tracial Rokhlin property for actions on non-simple C*-algebras. We prove that when the action has the weak tracial Rokhlin property and the crossed product is simple, the properties on $A$ of having tracial …
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.
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…
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…
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
We observe that a recent theorem of Sato, Toms-White-Winter and Kirchberg-Rordam also holds for certain nonunital C*-algebras. Namely, we show that an algebraically simple, separable, nuclear, nonelementary C*-algebra with strict comparison, whose cone of densely finite traces has as a base a Choquet simplex with compact, finite dimensional extreme boundary, and which admits a continuous rank function, tensorially absorbs the Jiang-Su algebra Z.
Certain tracially nuclear dimensional for certain crossed product ${\rm C^*}$-algebras
该研究证明渐近迹属于某类迹核维数有界的${\rm C^*}$-代数类的单代数，其迹核维数仍有界，并将其应用于具有迹Rokhlin性质的有限群作用的交叉乘积代数，给出其迹核维数上界。
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 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…
Let $A$ be a stably finite simple unital $C^*$-algebra and suppose $α$ is an action of a finite group $G$ with the tracial Rokhlin property. Suppose further $A$ has real rank zero and the order on projections over $A$ is determined by traces. Then the crossed product $C^*$-algebra $C^*(G,A, α)$ also has real rank zero and order on projections over $A$ is determined by traces. Moreover, if $A$ also has stable rank one, then $C^*(G,A, α)$ also has stable rank one.
Let $A$ be a unital separable simple $C^*$-algebra with tracial rank zero and let $x, \, y\in A$ be two normal elements. We show that $x$ is in the closure of the convex full of the unitary obit of $y$ if and only if there exists a sequence of unital completely positive linear maps $ϕ_n$ from $A$ to $A$ such that the sequence $ϕ_n(y)$ convergent to $x$ in norm and also approximately preserves the trace values. A purely measure theoretical description for normal elements in the closure of convex hull of unitary orbit of $y$ is also given. In the case that $A$ has a unique tracial state some cla…
Tracial oscillation zero and stable rank one
本文证明在具有严格比较的可分单 $C^*$-代数中，迹近似振荡为零等价于稳定秩一且 Cuntz 半群到仿射函数空间的典范映射 $\Gamma$ 满射，并推出近似稳定秩一在相应条件下蕴含稳定秩一。
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., $A \ncong A\otimes\mathcal Z$.
Stable rank for crossed products by finite group actions with the weak tracial Rokhlin property
本文研究具有弱迹Rokhlin性质的有限群作用下C*-代数交叉积的性质，证明若原C*-代数有性质(TM)则交叉积也有该性质，并推出满足条件的交叉积具有稳定秩一。
States on coupled quantum system whose restrictions to each subsystems are normalized traces are called marginal tracial states. We investigate extremal marginal tracial states and maximal rank of such states. Diagonal marginal tracial states are also considered.
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.
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…
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 $$ \af={\rm Ad}\, u_n\circ σ_n\circ \bt\circ σ_n^{-1}\andeqn \lim_{n\to\infty}…
Let $A$ be a simple infinite dimensional stably finite unital C*-algebra, and let $B$ be a centrally large subalgebra of $A$. We prove that if $A$ is tracially ${\mathcal{Z}}$-absorbing if and only if $B$ is tracially ${\mathcal{Z}}$-absorbing. If $A$ and $B$ are also separable and nuclear, we prove that $A$ is ${\mathcal{Z}}$-absorbing if and only if $B$ is ${\mathcal{Z}}$-absorbing.
We study a tracial notion of Z-absorption for simple, unital C*-algebras. We show that if A is a C*-algebra for which this property holds then A has almost unperforated Cuntz semigroup, and if in addition A is nuclear and separable we show this property is equivalent to having A being Z-absorbing. We furthermore show that this property is preserved under forming certain crossed products by actions satisfying a tracial Rokhlin type property.
Let $A$ be a unital separable simple infinite dimensional \CA with tracial rank no more than one and with the tracial state space $T(A)$ and let $U(A)$ be the unitary group of $A.$ Suppose that $u\in U_0(A),$ the connected component of $U(A)$ containing the identity. We show that, for any $\ep>0,$ there exists a selfadjoint element $h\in A_{s.a}$ such that $$ \|u-\exp(ih)\|<\ep. $$ We also study the problem when $u$ can be approximated by unitaries in $A$ with finite spectrum. Denote by $CU(A)$ the closure of the subgroup of unitary group of $U(A)$ generated by its commutators. It is known tha…
Let $G$ be a finite group, $A$ a unital separable finite simple nuclear C*-algebra, and $α$ an action of $G$ on $A$. Assume that $A$ absorbs the Jiang-Su algebra $\mathcal{Z}$, the extremal boundary of the trace space of $A$ is compact and finite dimensional and that $α$ fixes any tracial state of $A$. Then tsr$(A \rtimes_αG) = 1$. In particular, when $A$ has a unique tracial state, we conclude it without above conditons on a tracial state space of $A$.
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^*(…
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。
We prove several results of the following general form: automorphisms of (or actions of ${\mathbb{Z}}^d$ on) certain kinds of simple separable unital C*-algebras $A$ which have a suitable version of the Rokhlin property are generic among all automorphisms (or actions), or in a suitable class of automorphisms. That is, the ones with the version of the Rokhlin property contain a dense $G_δ$-subset of the set of all such automorphisms (or actions). Specifically, we prove the following. If $A$ is stable under tensoring with the Jiang-Su algebra $Z,$ and has tracial rank zero, then automorphisms wi…
In this paper we will introduce the tracial Rokhlin property for an inclusion of separable simple unital C*-algebras $P \subset A$ with finite index in the sense of Watatani, and prove theorems of the following type. Suppose that $A$ belongs to a class of C*-algebras characterized by some structural property, such as tracial rank zero in the sense of Lin. Then $P$ belongs to the same class. The classes we consider include:(1) Simple C*-algebras with real rank zero or stable rank one, (2) Simple C*-algebras with tracial rank zero or tracial rank less than or equal to one, (3) Simple C*-algebras…
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…
Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple C*-algebra with tracial rank no more than one. Suppose that $ϕ, ψ: C\to A$ are two unital monomorphisms. With some restriction on $C,$ we show that $ϕ$ and $ψ$ are approximately unitarily equivalent if and only if [ϕ]=[ψ] in KL(C,A) τ\circ ϕ=τ\circ ψfor all tracial states of A and ϕ^‡=ψ^‡, here ϕ^‡ and ψ^‡ are homomorphisms from $U(C)/CU(C)\to U(A)/CU(A) induced by ϕand ψ, respectively, and where CU(C) and CU(A) are closures of the subgroup generated by commutators of the unitary groups of C and B.
Some Permanence properties for crossed products by compact group actions with the tracial Rokhlin property
本文研究紧群作用带迹Rokhlin性质的交叉积，证明不动点代数与交叉积继承原C*-代数的稳定秩一、实秩零、β-比较、Winter的n-比较、m-几乎可除及弱（m,n）-可除等持久性性质。
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。
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。
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…
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.
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…
Let A be a unital simple separable C*-algebra. If $A$ is nuclear and infinite-dimensional, it is known that strict comparison is equivalent to Z-stability if the extreme boundary of its tracial state space is non-empty, compact and of finite covering dimension. Here we will provide the first proof of this result on the case of certain non-compact extreme boundaries. Besides, if A has strict comparison of positive elements, it is known that the Cuntz semigroup of this C*-algebra is recovered functorially from the Murray-von Neumann semigroup and the tracial state state space whenever the extrem…
In this paper we show that certain simple locally recursive subhomogeneous (RSH) C*-algebras are tracially approximately interval algebras after tensoring with the universal UHF algebra. This involves a linear algebraic encoding of the structure of the local RSH algebra allowing us to find a path through the algebra which looks like a discrete version of [0,1] and exhausts most of the algebra. We produce an actual copy of the interval and use properties of C*-algebras tensored with UHF algebras to move the honest interval underneath the discrete version. It follows from our main result that su…
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…
Let A be a unital simple direct limit of recursive subhomogeneous C*-algebras with no dimension growth. We give criteria which specify exactly when A has real rank zero, and exactly when A has the Property (SP): every nonzero hereditary subalgebra of A contains a nonzero projection. Specifically, A has real rank zero if and only if the natural map from K_0 (A) to the continuous affine functions on the tracial state space has dense range, A has the Property (SP) if and only if the range of this map contains strictly positive functions with arbitrarily small norm. By comparison with results for …
Tracial approximation and ${\cal Z}$-stability
本文证明在迹态空间极端边界为$\sigma$-紧可数维的条件下，单单位可分非初等顺从简单稳定有限C*-代数的${\cal Z}$-稳定性等价于严格比较加稳定秩一，并推广至非单位情形。
Divisibility and Real Rank Zero
本文研究了简单可分精确C*-代数的可除性性质，证明了实秩零、迹近可除性、迹m-近可除性、迹近振荡零和Property (TM)等性质等价，并进一步探讨了稳定秩为一的C*-代数的迹完成性质。
Tracially Complete C*-Algebras
本文引入迹完全C*-代数类，针对满足合适性质γ的顺从该类代数得到结构分类结果，其结果介于Connes定理与单核C*-代数分类定理之间，为Toms-Winter猜想论证提供支撑。
Strict comparison and stable rank one
本文证明了具有严格比较性质的 $σ$-单位有限单 $C^*$-代数在典范映射满射或 Cuntz 半群几乎无穿孔且几乎可除的条件下，具有稳定秩一和迹近似振荡零。
Compact Group Actions with the Tracial Rokhlin Property II: Examples and Nonexistence Theorems
本文构造了三类具有限制性迹Rokhlin性质但不具有Rokhlin性质的紧群作用例子，并证明了圆周群在单AF代数上不存在具有最弱迹Rokhlin性质的直接极限作用。
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