Stable Elements and Property (S)

Inner amenability, property Gamma, McDuff II_1 factors and stable equivalence relations

We say that a countable group $G$ is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff II_1 factor. Similarly, $G$ is said to be stable if it admits such an action with the orbit equivalence relation being stable. The McDuff property, stability, inner amenability and property Gamma are subtly related and several implications and non implications were obtained in [Ef73,JS85,Va09,Ki12a,Ki12b]. We complete the picture with the remaining implications and counterexamples.

The Similarity Problem for Z-stable C*-algebras

We show that the tensor product of two unital C*-algebras, one of which is nuclear and admits a unital *-homomorphism from (the building blocks of) the Jiang-Su algebra, has Kadison's similarity property. As a consequence, we obtain that a unital C*-algebra which absorbs the Jiang-Su algebra tensorially also has this property.

Local variants of the Dixmier property and weak centrality for C*-algebras

We study variants of the Dixmier property that apply to elements of a unital C*-algebra, rather than to the C*-algebra itself. By a Dixmier element in a C*-algebra we understand one that can be averaged into a central element by means of a sequence of unitary mixing operators. Examples include all self-commutators and all quasinilpotent elements. We do a parallel study of an element-wise version of weak centrality, where the averaging to the centre is done using unital completely positive elementary operators (as in Magajna's characterization of weak centrality). We also obtain complete descri…

Stable polynomial division and essential normality of graded Hilbert modules

The purpose of this paper is to initiate a new attack on Arveson's resistant conjecture, that all graded submodules of the $d$-shift Hilbert module $H^2$ are essentially normal. We introduce the stable division property for modules (and ideals): a normed module $M$ over the ring of polynomials in $d$ variables has the stable division property if it has a generating set $\{f_1, ..., f_k\}$ such that every $h \in M$ can be written as $h = \sum_i a_i f_i$ for some polynomials $a_i$ such that $\sum \|a_i f_i\| \leq C\|h\|$. We show that certain classes of modules have this property, and that the s…

On the property IR of Friis and Rordam

In a 1997 paper Lin solved a longstanding problem as follows: For each epsilon > 0, there is delta > 0 such that if h and k are self-adjoint contractive n x n matrices and ||hk - kh|| < delta, then there are commuting self-adjoint matrices h' and k' such that ||h' - h||, ||k' - k|| < epsilon. Here delta depends only on epsilon and not on n. In a 1996 paper Friis and Rordam greatly simplified Lin's proof by using a property they called IR. They also generalized Lin's result by showing that the matrix algebras can be replaced by any C*-algebras satisfying IR. The purpose of this paper is to stud…

Stable rank of inclusion of C*-algebras of depth 2

Let $1 \in A \subset B$ be an inclusion of unital C*-algebras of index-finite type and depth 2. Suppose that $A$ is infinite dimensional simple with $tsr(A) = 1$ and SP-property. Then $tsr(B) \leq 2$. As a corollary when $A$ is a simple C*-algebra with $\tsr(A) = 1$ and SP-property and $α$ an action of a finite group $G$ on $\Aut(A)$, $\tsr(A \rtimes_αG) \leq 2$.

Unitary groups and augmented Cuntz semigroups of separable simple Z-stable C*-algebras

Let $A$ be a separable simple exact ${\cal Z}$-stable $C^*$-algebra. We show that the unitay group of ${\tilde A}$ has the cancellation property. If $A$ has continuous scale, the Cuntz semigroup of $\tilde A$ has the strict comparison property and a weak cancellation property. Let $C$ be a 1-dimensional non-commutative CW complex with $K_1(C)=\{0\}.$ Suppose that $λ: {\rm Cu}^\sim(C)\to {\rm Cu}^\sim(A)$ is a morphism in Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms $ϕ_n: C\to A$ such that $\lim_{n\to\infty}{\rm Cu}^\sim(ϕ_n)=λ.$ This result leads t…

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。

局部提升性质、FD性质和近似表示的稳定性

The Local Lifting Property, Property FD, and stability of approximate representations

本文研究了3维流形群、极限群及某些一关系群和直角Artin群的局部提升性质和FD性质，并证明这些群在归一化单位不变范数下具有高度灵活性的稳定性。

具有小边界性质的最小线束扭曲同胚的C*-代数的Z稳定性

$\mathcal{Z}$-stability for $\mathrm C^*$-algebras of minimal line-bundle-twisted homeomorphisms with the small boundary property

本文研究了具有小边界性质的最小线束扭曲同胚的C*-代数的Z稳定性，并展示了其分类及张量积的稳定性。

从局部网到Euler元

From local nets to Euler elements

该研究在含Euler元的一般李群框架下，证明广义AQFT中几何实现模群由Euler元生成，且相关条件下正则性、可局部化性成立，楔形区域von Neumann代数为III_1型因子。