Mauricio Achigar
We prove equality between the Topological Stable Rank and the Bass Stable Rank for finitely generated projective left modules over a unital C*-algebra. In order to do so, the concept of Stable Rank of a Hilbert module is introduced.
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Rank-preserving module maps
本文刻画了Hilbert $C^\ast-$模上的秩1保持模映射，并研究了它在自由概率论中的应用.
Additive preserving rank one maps on Hilbert $C^\ast$-modules
本文刻画了Hilbert $C^\ast$ -模上的一类可加映射，它将一个秩为1的可伴算子映射到另一个秩为1的可伴算子上.
We construct a Fredhom module representing the Kasparov gamma element in G-equivariant KK-theory for G a semisimple Lie group of real rank one. This is the main step of our proof of the Baum-Connes conjecture for such groups.
The connected stable rank and the general stable rank are homotopy invariants for Banach algebras, whereas the Bass stable rank and the topological stable rank should be thought of as dimensional invariants. This paper studies the two homotopical stable ranks, viz. their general properties as well as specific examples and computations. The picture that emerges is that of a strong affinity between the homotopical stable ranks, and a marked contrast with the dimensional ones.
Suppose that $A,B$ are nuclear, separable ${\rm C}^*$-algebras of stable rank one and real rank zero, $A$ is unital simple, $B$ is stable and $({\rm K}_0(B),{\rm K}_0^+(B))$ is weakly unperforated in the sense of Elliott \cite{Ell}. We show that any unital extension with trivial index maps of $A$ by $B$ is absorbing.
On a Question of Arveson about Ranks of Hilbert modules
众所周知，在核函数为$K(z,w)=\frac{1}{1-z_{1}w_{1}-... -z_{d}w_{d}}$的$B_{d} \in C^d$中，单位球上的泛函希尔伯特空间具有自然的$A(B_{d})$ -模结构。我们证明了一个非零子模的秩是无穷当且仅当该子模是无穷余维的。结合Arveson的膨胀理论，我们的结果表明Hilbert模与代数模的Hilbert基定理形成了鲜明的对比。这个结果回答了Arveson的一个问题。
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 study the structure and compute the stable rank of C*-algebras of finite higher-rank graphs. We completely determine the stable rank of the C*-algebra when the k-graph either contains no cycle with an entrance, or is cofinal. We also determine exactly which finite, locally convex k-graphs yield unital stably finite C*-algebras. We give several examples to illustrate our results.
Let $A$ be a separable, unital, simple C*-algebra with stable rank one. We show that every strictly positive, lower semicontinuous, affine function on the simplex of normalized quasitraces of $A$ is realized as the rank of an operator in the stabilization of $A$. Assuming moreover that $A$ has locally finite nuclear dimension, we deduce that $A$ is $\mathcal{Z}$-stable if and only if it has strict comparison of positive elements. In particular, the Toms-Winter conjecture holds for separable, unital, simple, approximately subhomogeneous C*-algebras with stable rank one.
We study the general and connected stable ranks for C*-algebras. We estimate these ranks for pullbacks of C*-algebras, and for tensor products by commutative C*-algebras. Finally, we apply these results to determine these ranks for certain commutative C*-algebras, and non-commutative CW-complexes.
We give a brief account of the interplay between spectral morphisms, K-theory, and stable ranks in the context of Banach algebras.
We introduce a class of recursive subhomogeneous algebras that we call diagonal subhomogeneous and we give a notion of diagonal maps between these algebras. We show that any simple limit of diagonal subhomogeneous algebras with diagonal maps has stable rank one. As an application we show that for any minimal homeomorphism of a compact Hausdorff space the associated crossed product has stable rank one.
On the stable rank of algebras of operator fields over N-cubes
设A是一个底空间为k-立方[0，1]^k，纤维代数为{A_t}_{t \in [0，1]^k}的算子域的单位极大满代数.则A的稳定秩由稳定秩的上确界sr（C（[0，1]^k）\otimes A_t）（t \in [0，1]^k）上有界.利用这个估计，我们计算了离散海森堡群的泛C^*-代数的稳定秩。
We study the general and connected stable ranks for $C^{\ast}$-algebras. We estimate these ranks for certain $C(X)$-algebras, and use that to do the same for certain group $C^{\ast}$-algebras. Furthermore, we also give estimates for the ranks of crossed product $C^{\ast}$-algebras by finite group actions with the Rokhlin property.
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。
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。
Rigged modules over an operator algebra are a generalization of Hilbert modules over a $C^{\star}$-algebra. We characterize the rigged modules over an operator algebra $\mathcal A$ which are orthogonally complemented in $C_\infty(\mathcal A),$ the space of infinite columns with entries in $\mathcal A.$ We show that every such rigged module `restricts' to a bimodule of Morita equivalence between appropriate stably isomorphic operator algebras.
On the stable rank of algebras of operator fields over metric spaces
设G是一个无挠生成的两步幂零群.设C^*（G）是G的泛C^*-代数.我们证明了acsr（C^*（G））= acsr（C（（\hat{G}）_1）），其中对单位C^*-代数A，acsr（A）是A的绝对连通稳定秩，（\hat{G}）_1是G的一维表示空间.对于稳定秩的情形，我们得到了相近的结果。在此过程中，我们给出了度量空间上算子域的极大满代数的稳定秩估计。
A remark on invariants for C*-algebras of stable rank one
证明了，对于稳定秩为1的C* 代数（即，其中可逆元是稠密的），两个著名的同构不变量，Cuntz半群和Cubersen半群，包含相同的信息。更确切地说，这两个不变量，适当地看，以自然的方式相互确定。
In this paper, we show that for unital, separable $C^*$-algebras of stable rank one and real rank zero, the unitary Cuntz semigroup functor and the functor ${\rm K}_*$ are naturallly equivalent. Then we introduce a refinement of the unitary Cuntz semigroup, say the total Cuntz semigroup, which is a new invariant for separable $C^*$-algebras of stable rank one, is a well-defined continuous functor from the category of $C^*$-algebras of stable rank one to the category ${\rm\underline{ Cu}}$. We prove that this new functor and the functor ${\rm \underline{K}}$ are naturallly equivalent for unital…
On the topological stable rank of non-selfadjoint operator algebras
我们给出了一个关于M的问题的负解。Rieffel谁问，如果右和左拓扑稳定的行列的巴拿赫代数必须始终同意。我们的例子中发现一类套代数。我们证明了对于许多其他的套代数，左和右拓扑稳定秩都是无限的。我们将后一结果推广到Popescu的非交换圆盘代数和自由半群代数。
Finitely summable Fredholm modules over higher rank groups and lattices
给出了高阶群和格上的可和Fredholm表示（Fredholm模）的一个完整分类（直到光滑同伦）。我们的结果是一个直接后果的工作巴德，弗曼，Gelander和莫诺的概括Kazhdan的性质T的局部紧群。
In the context of commutative $C^*$-algebras we solve a problem related to a question of M. Rieffel by showing that the all-units rank and the norm-one rank coincide with the topological stable rank. We also introduce the notion of unitary $M$-stable rank for an arbitrary commutative unital ring and compare it with the Bass stable rank. In case of uniform algebras, a sufficient condition for norm-one reducibility is given.
Let $A$ be a simple C*-algebra of stable rank one and let $p$ and $q$ be two $σ$-compact open projections. It is proved that there is a continuous path of unitaries in ${\tilde A}$ which connects open sub-projections of $p$ which is compactly contained in $p$ to those in $q.$ It is also shown that every Hilbert module is projective in the category whose morphisms are bounded module maps with adjoints. A discussion of projective Hilbert modules (whose morphisms are bounded module maps) is also given.
We show that the class of C*-algebras with stable rank greater than a given positive integer is axiomatizable in logic of metric structures. As a consequence we show that the stable rank is continuous with respect to forming ultrapowers of C*-algebras, and that stable rank is Kadison--Kastler stable.
We give a formula for the stable rank of inclusions of unital Banach algebras in the sense of finite Watatani index. As an application we show that the stable rank of $\ell^1$-algebras of Disk algebras by any action of finite groups is 2.
The Stable Rank of Topological Algebras and a Problem of R.G. Swan
研究了拓扑代数的稳定秩的各种概念。一些对R.G.的部分回答Swan问题（有两个Banach或好Fréchet代数在K理论中的密度定理相同的稳定秩？）获得了例如，$C^\ast$ -代数$B$的一个Fréchet稠密$\ast$ -子代数$A$，在自伴元的$C^\infty$ -谓词演算下闭，与$B$具有相同的Bass稳定秩。
Twisted group C*-algebras of acylindrically hyperbolic groups have stable rank one
该研究证明了未必具有平凡有限根的柱形双曲群的扭曲群C*-代数具有稳定秩一，拓展了相关算子代数与群论交叉领域的结论。
Simplicity and the stable rank of some free product C*-algebras
给出了有限维交换代数的C*-代数约化自由积单的一个充要条件，并证明了这类自由积的稳定秩为1.证明了C ~*-代数的其他约化自由积的相关结果。
Diagonalization of compact operators in Hilbert modules over C*-algebras of real rank zero
经典的Hilbert-Schmidt定理可以推广到有限型$W^*$ -代数上的Hilbert $A$ -模$H_A^*$中的紧算子的情形，即$H_A^*$中的紧算子在很小的限制下可以在$A$上对角化.证明了若$B$是$A$中实秩为零的弱稠密$C^*$ -子代数，且具有某些附加性质，则紧算子从$H_B$到$H_A^*\supset H_B$的自然扩张可对角化，对角元来自$C^*$ -代数$B$ .
Rank-two Milnor idempotents for the multipullback quantum complex projective plane
本文研究了多重回溯量子复射影平面的C*-代数的K₀群，证明了米尔诺模与非交换向量丛或其补集的同构关系，并展示了其中一个米尔诺模可分解为自由模与非自由投射模的直和。
Cancellation does not imply stable rank one
称有单位的C* 代数-代数A具有投影可消性，如果A上矩阵投影的Murray-von Neumann等价类半群D（A）是可消的.长期以来，人们都知道稳定秩1意味着相消，并且已经建立了一些部分相消。我们证明了对于简单的稳定有限C* 代数，消去并不意味着稳定秩1.
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