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

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。

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…

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。

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 $$ (K_0(A), K_0(A)_+, [1_A], K_1(A), T(A)) \cong (K_0(B), K_0(B)_+, [1_B], K_1(B), T(B)). $$

中心大型子代数的继承迹拓扑秩零

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。

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…

The Rokhlin property and the tracial topological rank

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…

Existence of the Matui-Sato tracial Rokhlin property

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.

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 Rokhlin property and non-commutative dimensions

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 …

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.

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

Z-stability, finite dimensional tracial boundaries and continuous rank functions

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.

稳定秩一、迹局部同质性和统一性质Γ

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。

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…

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

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.

Convex hulls of unitary orbits of normal elements in $C^*$-algebras with tracial rank zero

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…

迹振荡为零与稳定秩一

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。

具有弱迹Rokhlin性质的有限群作用交叉积的稳定秩

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。

Maximal rank of extremal marginal tracial states

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.

Kishimoto's Conjugacy Theorems in simple $C^*$-algebras of tracial rank one

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}…

Traces on Topological Graph Algebras

Given a topological graph $E$, we give a complete description of tracial states on the C*-algebra $\mathrm{C}^*(E)$ which are invariant under the gauge action; there is an affine homeomorphism between the space of gauge invariant tracial states on $\mathrm{C}^*(E)$ and Radon probability measures on the vertex space $E^0$ which are, in a suitable sense, invariant under the action of the edge space $E^1$. It is shown that if $E$ has no cycles, then every tracial state on $\mathrm{C}^*(E)$ is gauge invariant. When $E^0$ is totally disconnected, the gauge invariant tracial states on $\mathrm{C}^*(…

Unitaries in a Simple C*-algebra of Tracial Rank One

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…

Stable rank for crossed products by actions of finite groups on C*-algebras

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 $A$.

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^*(…

The tracial Rokhlin property is generic

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…

The Jiang-Su absorption for inclusions of unital C*-algebras

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…

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…

Approximate Unitary Equivalence in Simple C^*-algebras of Tracial Rank One

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.

紧群作用带迹Rokhlin性质的交叉积的若干持久性性质

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。

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…

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.

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…

Real rank and property (SP) for direct limits of recursive subhomogeneous algebras

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 …

迹逼近与${\cal Z}$-稳定性

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。

迹顺从作用与纯无限交叉积

Tracially amenable actions and purely infinite crossed products

本文引入迹顺从作用概念并给出等价刻画，证明含自由群F2的群在简单、单位、Z-稳定C*-代数上的外迹顺从作用具有纯无限交叉积，并提供自由群在AF-代数上的具体例子。

可除性与实秩零

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。

严格比较与稳定秩一

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。

The Automorphism group of a simple tracially AI algebra

The structure of the automorphism group of a simple TAI algebra is studied. In particular, we show that $\frac{\bar{\mrm{Inn}} (A)}{\bar{\mrm{Inn}}_{0} (A)}$ is isomorphic (as a topological group) to an inverse limit of discrete abelian groups for a unital, simple, AH algebra with bounded dimension growth. Consequently, $\frac{\bar{\mrm{Inn}} (A)}{\bar{\mrm{Inn}}_{0} (A)}$ is totally disconnected. Another consequence of our results is the following: Suppose $A$ is the transformation group \cstar-algebra of a minimal Furstenberg transformation $(\mbb{T}^{n}, h_{n})$ with a unique $h_{n}$-invari…

Decomposition rank of UHF-absorbing C*-algebras

Let A be a unital separable simple C*-algebra with a unique tracial state. We prove that if A is nuclear and quasidiagonal, then A tensored with the universal UHF-algebra has decomposition rank at most one. Then it is proved that A is nuclear, quasidiagonal and has strict comparison if and only if A has finite decomposition rank. For such A, we also give a direct proof that A tensored with a UHF-algebra has tracial rank zero. Applying this characterization, we obtain a counter-example to the Powers-Sakai conjecture.

Tracial Rokhlin property for automorphisms on simple $A{\mathbb T}$-algebras

Let $A$ be a unital simple $A\T$-algebra of real rank zero. Given an isomorphism $γ_1: K_1(A)\to K_1(A),$ we show that there is an automorphism $\af: A\to A$ such that $\af_{*1}=γ_1$ which has the tracial Rokhlin property. Consequently, the crossed product $A\rtimes_{\af}\Z$ is a simple unital AH-algebra with real rank zero. We also show that automorphism with Rokhlin property can be constructed from minimal homeomorphisms on a connected compact metric space.

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…

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…

迹空间上的度量

Metrics on trace spaces

本文针对实秩零、稳定秩一的可分类$\text{C}^*$-代数，用最优输运方法给出了$^*$-同态酉轨道距离的迹数据计算方式，还利用迹Lipschitz结构表征了自同态的统计特征。

Crossed products by minimal homeomorphisms

Let X be an infinite compact metric space with finite covering dimension and let h be a minimal homeomorphism of X. Let A be the associated crossed product C*-algebra. We show that A has tracial rank zero whenever the image of K_0 (A) in the affine functions on the tracial state space of A is dense. As a consequence, we show that these crossed product C*-algebras are in fact simple AH algebras with real rank zero. When X is connected and h is further assumed to be uniquely ergodic, then the above happens if and only if the rotation number associated to h has irrational values. By applying the …

The Tracial Rokhlin Property for Automorphisms on Non-Simple C*-algebras

Let A be a unital AF-algebra (simple or non-simple) and let αbe an automorphism of A. Suppose that αhas certain Rokhlin property and A is α-simple. Suppose also that there is an integer J\geq1 such that α^{J}_{*0}=id_{K_{0}(A)}, we show that A\rtimes_α\mathbb{Z} has tracial rank zero.

The weak tracial Rokhlin property for finite group actions on simple C*-algebras

We develop the concept of weak tracial Rokhlin property for finite group actions on simple (not necessarily unital) C*-algebras and study its properties systematically. In particular, we show that this property is stable under restriction to invariant hereditary C*-algebras, minimal tensor products, and direct limits of actions. Some of these results are new even in the unital case and answer open questions asked by N. C. Phillips in full generality. We present several examples of finite group actions with the weak tracial Rokhlin property on simple stably projectionless C*-algebras. We prove …

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…

有限群交叉积的稳定秩与实秩

Stable and real rank for crossed products by finite groups

本文研究了有限群作用于C*-代数时交叉积的稳定秩与实秩问题，在移除对代数额外假设的条件下，证明了迹Rokhlin性质可保证交叉积与不动点代数保持原秩，并揭示了弱迹Rokhlin性质与迹Rokhlin性质的等价条件。

Rokhlin dimension: duality, tracial properties, and crossed products

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 …

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 …

Almost finiteness, comparison, and tracial $\mathcal{Z}$-stability

Inspired by Kerr's work on topological dynamics, we define tracial $\mathcal{Z}$-stability for sub-$C^*$-algebras. We prove that for a countable discrete amenable group $G$ acting freely and minimally on a compact metrizable space $X$, tracial $\mathcal{Z}$-stability for the sub-$C^*$-algebra $(C(X)\subseteq C(X)\rtimes G)$ implies that the action has dynamical comparison. Consequently, tracial $\mathcal{Z}$-stability is equivalent to almost finiteness of the action, provided that the action has the small boundary property.

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.