Xuanlong Fu, Kang Li, Huaxin Lin
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.
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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。
On permanence of regularity properties II
本文研究了简单单位C*代数中正则性性质的持久性，证明了在特定条件下，某些tracial性质可以从B传递到A。
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。
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.
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…
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 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…
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。
Rokhlin property and approximate representability for inclusions of index-finite type
该研究针对有限指数型单位C*-代数包含引入弱迹Rokhlin性质及对偶的弱迹近似可表性，证明其被Jones–Watatani基本构造交换，还得到相关持久性结果。
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。
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。
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…
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 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…
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.
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.
We introduce the notion of the weak tracial approximate representability of a discrete group action on a unital $C^*$-algebra which could have no projections like the Jiang-Su algebra $\mathcal{Z}$. Then we show a duality between the weak tracial Rokhlin property and the weak tracial approximate representability. More precisely, when $G$ is a finite abelian group and $α:G\curvearrowright A$ is a group action on a unital simple infinite dimensional $C^*$-algebra, we prove that 1. $α$ has the weak tracial Rokhlin property if and only if $\hatα$ has the weak tracial approximate representability. …
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…
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…
We establish four results concerning connections between actions on separable C*-algebras with Rokhlin-type properties and absorption of the Jiang-Su algebra Z. For actions of residually finite groups or of the reals which have finite Rokhlin dimension with commuting towers, we show that if the action of any nontrivial group element is approximately inner then the C*-algebra acted upon is Z-stable. Without the assumption on approximate innerness, we show that the crossed product has good divisibility properties under mild assumptions. We also establish an analogous result for the generalized t…
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。
A finitary criterion for selfless tracial C*-algebras
本文通过引入近似无私性这一有限性条件，证明了可分迹C*-代数中无私性与近似无私性的等价性，并应用于证明具有拓扑自由极值边界的可数群是C*-无私的。
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}…
In this paper, we prove that unital homomorphisms from continuous functions on a compact metric space to matrices over a C*-algebra with tracial rank at most one are approximately diagonalizable. We also consider some generalizations of this result.
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.
We study a pair of $C^*$-algebras by associating a $*$-homomorphism from $A$ to $B$ allowing an approximate left-inverse to the sequence algebra of $A$ in a manner reminiscent of several tracial approximation properties. We are particularly interested how regularity properties in the Elliott classification program pass from $B$ to $A$. Among them, we show that the strict comparison property and $\mathcal{Z}$-stability pass from $B$ to $A$ in our setting.
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…
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。
Perfect Quantum Approximate Strategies for Imitation Games
研究模仿博弈的完美量子近似策略，证明其与两个通用C*-代数极小张量积上的双迹态等价，并探讨与von Neumann代数及顺从迹态的联系。
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 classify unital monomorphisms into certain simple Z-stable C^*-algebras up to approximate unitary equivalence. The domain algebra C is allowed to be any unital separable commutative C^*-algebra, or any unital simple separable nuclear Z-stable C^*-algebra satisfying the UCT such that C\otimes B is of tracial rank zero for a UHF algebra B. The target algebra A is allowed to be any unital simple separable Z-stable C^*-algebra such that A\otimes B has tracial rank zero for a UHF algebra B, or any unital simple separable exact Z-stable C^*-algebra whose projections separate traces and whose extr…
Let $C$ be a general unital AH-algebra and let $A$ be a unital simple $C^*$-algebra with tracial rank at most one. Suppose that $ϕ, ψ: C\to A$ are two unital monomorphisms. We show that $ϕ$ and $ψ$ are approximately unitarily equivalent if and only if \beq[ϕ]&=&[ψ] {\rm in} KL(C,A), ϕ_{\sharp}&=&ψ_{\sharp}\tand ϕ^†&=&ψ^†, \eneq where $ϕ_{\sharp}$ and $ψ_{\sharp}$ are continuous affine maps from tracial state space $T(A)$ of $A$ to faithful tracial state space $T_{\rm f}(C)$ of $C$ induced by $ϕ$ and $ψ,$ respectively, and $ϕ^‡$ and $ψ^‡$ are induced homomorphisms from $K_1(C)$ into $\Aff(T(A))…
Let $A$ be a simple separable unital locally approximately subhomogeneous C*-algebra (locally ASH algebra). It is shown that $A\otimes Q$ can be tracially approximated by unital Elliott-Thomsen algebras with trivial $\textrm{K}_1$-group, where $Q$ is the universal UHF algebra. In particular, it follows that $A$ is classifiable by the Elliott invariant if $A$ is Jiang-Su stable.
For any unital separable simple infinite-dimensional nuclear C*-algebra with finitely many extremal traces, we prove that Z-absorption, strict comparison, and property (SI) are equivalent. We also show that any unital separable simple nuclear C*-algebra with tracial rank zero is approximately divisible, and hence is Z-absorbing.
In this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple C$^*$-algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of projections with respect to tracial states. In addition, an upper bound for the number of unitary conjugates in a convex combination needed to approximate a self-adjoint are obtained.
Let $C$ be a unital AH-algebra and $A$ be a unital simple C*-algebra with tracial rank zero. It has been shown that two unital monomorphisms $ϕ, ψ: C\to A$ are approximately unitarily equivalent if and only if $$ [ϕ]=[ψ] {\rm in} KL(C,A) and τ\circ ϕ=τ\circ ψ\tforal τ\in T(A), $$ where $T(A)$ is the tracial state space of $A.$ In this paper we prove the following: Given $κ\in KL(C,A)$ with $κ(K_0(C)_+\setminus \{0\})\subset K_0(A)_+\setminus \{0\}$ and with $κ([1_C])=[1_A]$ and a continuous affine map $λ: T(A)\to T_{\mathtt{f}}(C)$ which is compatible with $κ,$ where $T_{\mathtt{f}}(C)$ is the…
Let X be an infinite compact metric space, α: X \to X a minimal homeomorphism, u the unitary implementing αin the transformation group C*-algebra, and S a class of separable nuclear C*-algebras that contains all unital hereditary C*-subalgebras of C*-algebras in S. Motivated by the success of tracial approximation by finite dimensional C*-algebras as an abstract characterization of classifiable C*-algebras and the idea that classification results for C*-algebras tensored with UHF algebras can be used to derive classification results up to tensoring with the Jiang-Su algebra Z, we prove that th…
In this paper, we prove approximate lifting results in the C$^{\ast}$-algebra and von Neumann algebra settings. In the C$^{\ast}$-algebra setting, we show that two (weakly) semiprojective unital C*-algebras, each generated by $n$ projections, can be glued together with partial isometries to define a larger (weakly) semiprojective algebra. In the von Neumann algebra setting, we prove lifting theorems for trace-preserving *-homomorphisms from abelian von Neumann algebras or hyperfinite von Neumann algebras into ultraproducts. We also extend a classical result of S. Sakai \cite{sakai} by showing …
Covering entropy for types in tracial $\mathrm{W}^*$-algebras
结合自由概率论与模型论技术，研究迹 $\mathrm{W}^*$-代数微观态空间的覆盖熵，证明若 $h(\mathcal{N}:\mathcal{M}) \geq 0$ 则存在保持熵近似的矩阵超积嵌入，并推断所有此类嵌入自同构等价时 $\mathcal{M}$ 强 $1$-有界。
We consider the reduction of problems on general noncommutative $L_p$-spaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is a unpublished result of the first named author which approximates any noncommutative $L_p$-space by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products by a locally compact abelian group or to their noncommutative $L_p$-spaces. We present applications of these results to the theory of noncommutative martingale inequalities by reducing most recent…
Optimal transport for types and convex analysis for definable predicates in tracial $\mathrm{W}^*$-algebras
该研究在迹冯·诺依曼代数框架下建立连续模型论、自由概率论与最优传输的联系，给出Monge-Kantorovich对偶的模型论类比，刻画可定义谓词的逼近性质并证明可定义闭包的相关结构结果。
Let β: S^n \to S^n, for n = 2k + 1, k \geq 1, be one of the known examples of a non-uniquely ergodic minimal diffeomorphism of an odd dimensional sphere. For every such minimal dynamical system (S^n, β) there is a Cantor minimal system (X, α) such that the corresponding product system (X x S^n, αx β) is minimal and the resulting crossed product C*-algebra C(X x S^n) \rtimes_{αx β} \mathbb{Z} is tracially approximately an interval algebra (TAI). This entails classification for such C*-algebras. Moreover, the minimal Cantor system (X, α) is such that each tracial state on C(X x S^n) \rtimes_β \m…
We show that, if A is a separable simple unital C*-algebra which absorbs the Jiang-Su algebra Z tensorially and which has real rank zero and finite decomposition rank, then A is tracially AF in the sense of Lin, without any restriction on the tracial state space. As a consequence, the Elliott conjecture is true for the class of C*-algebras as above which, additionally, satisfy the Universal Coefficients Theorem. In particular, such algebras are completely determined by their ordered K-theory. They are approximately homogeneous of topological dimension less than or equal to 3, approximately sub…
We study the uniform property $Γ$ for separable simple $C^*$-algebras which have quasitraces and may not be exact. We show that a stably finite separable simple $C^*$-algebra $A$ with strict comparison and uniform property $Γ$ has tracial approximate oscillation zero and stable rank one. Moreover in this case, its hereditary $C^*$-subalgebras also have a version of uniform property $Γ.$ If a separable non-elementary simple amenable $C^*$-algebra $A$ with strict comparison has this hereditary uniform property $Γ,$ then $A$ is ${\cal Z}$-stable.
We prove that a crossed product algebra arising from a minimal dynamical system on the product of the Cantor set and the circle has real rank zero if and only if that system is rigid. In the case that cocycles take values in the rotation group, it is also shown that rigidity implies tracial rank zero, and in particular, the crossed product algebra is isomorphic to a unital simple AT-algebra of real rank zero. Under the same assumption, we show that two systems are approximately $K$-conjugate if and only if there exists a sequence of isomorphisms between two associated crossed products which ap…
In this article I study a number of topological and algebraic dimension type properties of simple C*-algebras and their interplay. In particular, a simple C*-algebra is defined to be (tracially) (m,\bar{m})-pure, if it has (strong tracial) m-comparison and is (tracially) \bar{m}-almost divisible. These notions are related to each other, and to nuclear dimension. The main result says that if a separable, simple, nonelementary, unital C*-algebra A with locally finite nuclear dimension is (m,\bar{m})-pure, then it absorbs the Jiang-Su algebra Z tensorially. It follows that A is Z-stable if and on…
We present a classification theorem for amenable simple stably projectionless C*-algebras with generalized tracial rank one whose $K_0$ vanish on traces which satisfy the Universal Coefficient Theorem. One of them is denoted by ${\cal Z}_0$ which has a unique tracial state and $K_0({\cal Z}_0)=\mathbb{Z}$ and $K_1({\cal Z}_0)=\{0\}.$ Let $A$ and $B$ be two separable simple $C^*$-algebras satisfying the UCT and have finite nuclear dimension. We show that $A\otimes {\cal Z}_0\cong B\otimes {\cal Z}_0$ if and only if ${\rm Ell}(B\otimes {\cal Z}_0)={\rm Ell}(B\otimes {\cal Z}_0).$ A class of simp…
Let $A$ be a unital simple C*-algebra with tracial rank zero and $X$ be a compact metric space. Suppose that $h_1, h_2: C(X)\to A$ are two unital monomorphisms. We show that $h_1$ and $h_2$ are approximately unitarily equivalent if and only if $$ [h_1]=[h_2] {\rm in} KL(C(X),A) {\rm and} τ\circ h_1(f)=τ\circ h_2(f) $$ for every $f\in C(X)$ and every trace $τ$ of $A.$ Adopting a theorem of Tomiyama, we introduce a notion of approximate conjugacy for minimal dynamical systems. Let $X$ be a compact metric space and $α, β: X\to X$ be two minimal homeomorphisms. Using the above mentioned result, we…
Let A be a unital separable C*-algebra, and D a K_1-injective strongly self-absorbing C*-algebra. We show that if A is D-absorbing, then the crossed product of A by a compact second countable group or by Z or by R is D-absorbing as well, assuming the action satisfying a Rokhlin property. In the case of a compact Rokhlin action we prove a similar statement about approximate divisibility.
On closed Lie ideals of certain tensor products of $C^*$-algebras
研究了某些张量积C*-代数的闭合李理想，证明了在特定条件下这些理想均为乘积理想，并探讨了不同范数下的结构特性。
We construct a simple, nuclear, stably projectionless C*-algebra W which has trivial K-theory and a unique tracial state, and we investigate the extent to which W might fit into the hierarchy of strongly self-absorbing C*-algebras as an analogue of the Cuntz algebra O_2. In this context, we show that every nondegenerate endomorphism of W is approximately inner and we construct a trace-preserving embedding of W into the central sequences algebra M(W)_\infty \cap W'. We conjecture that W\otimes W is isomorphic to W and we note some implications of this, for example that W would be absorbed tenso…
We give an example of an exact, stably finite, simple. separable C*-algebra D which is not isomorphic to its opposite algebra. Moreover, D has the following additional properties. It is stably finite, approximately divisible, has real rank zero and stable rank one, has a unique tracial state, and the order on projections over D is determined by traces. It also absorbs the Jiang-Su algebra Z, and in fact absorbs the 3^{\infty} UHF algebra. We can also explicitly compute the K-theory of D, namely K_0 (D) = Z[1/3] with the standard order, and K_1 (D) = 0, as well as the Cuntz semigroup of D.
We show that unital simple C*-algebras with tracial topological rank zero which are locally approximated by subhomogeneous C^-algebras can be classified by their ordered $K$-theory. We apply this classification result to show that certain simple crossed products are isomorphic if they have the same ordered $K$-theory. In particular, irrational higher dimensional non-commutative tori of the form $C({\mathbb T}^k)\times_θ{\mathbb Z}$ are in fact inductive limits of circle algebras.
Ranks of operators and pureness of C*-algebras
本文证明单非初等C*-代数中算子秩函数的均匀稠密性，由此推出严格比较蕴含几乎可除性及纯性，并应用于证明严格比较蕴含Z-稳定性，从而解决Toms-Winter猜想于近似次齐次情形，并给出精确C*-代数纯性的等价刻画。
We consider various statements that characterize the hyperfinite II$_1$ factors amongst embeddable II$_1$ factors in the non-embeddable situation. In particular, we show that "generically" a II$_1$ factor has the Jung property (which states that every embedding of itself into its ultrapower is unitarily conjugate to the diagonal embedding) if and only if it is self-tracially stable (which says that every such embedding has an approximate lifting). We prove that the enforceable factor, should it exist, has these equivalent properties. Our techniques are model-theoretic in nature. We also show h…
We examine the ranks of operators in semi-finite C*-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple C*-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray-von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are s…
We say that a unital C*-algrebra A has the approximate positive factorization property (APFP) if every element of A is a norm limit of products of positive elements of A. (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of C*-algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial K_1, and stable rank 1. In the unital case, the K_0 group separates the tr…
Let $X$ be a finite CW complex and let $h_1, h_2: C(X)\to A$ be two unital \hm s, where $A$ is a unital C*-algebra. We study the problem when $h_1$ and $h_2$ are approximately homotopic. We present a $K$-theoretical necessary and sufficient condition for them to be approximately homotopic under the assumption that $A$ is a unital separable simple C*-algebra of tracial rank zero, or $A$ is a unital purely infinite simple C*-algebra. When they are approximately homotopic, we also give a bound for the length of the homotopy. Suppose that $h: C(X)\to A$ is a monomorphism and $u\in A$ is a unitary …
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