Sums of commutators in ideals and modules of type II factors

Generalized Berezin Transform and Commutator Ideals

For a quasi-free module over a function algebra $A(Ω)$, we define an analogue of the Berezin transform and relate this to the quotient of the C*-algebra it generates modulo the commutator ideal.

由换位子生成的环与 C*-代数

Rings and C*-algebras generated by commutators

本文刻画含单位元环由换位子理想生成的条件，并给出矩阵环、真无限或实秩零及 Jiang-Su 稳定 C*-代数的统一乘积项数界。

Commutator estimates in $W^*$-factors

Let $\mathcal{M}$ be a $W^*$-factor and let $S\left( \mathcal{M} \right) $ be the space of all measurable operators affiliated with $\mathcal{M}$. It is shown that for any self-adjoint element $a\in S(\mathcal{M})$ there exists a scalar $λ_0\in\mathbb{R}$, such that for all $\varepsilon > 0$, there exists a unitary element $u_\varepsilon$ from $\mathcal{M}$, satisfying $|[a,u_\varepsilon]| \geq (1-\varepsilon)|a-λ_0\mathbf{1}|$. A corollary of this result is that for any derivation $δ$ on $\mathcal{M}$ with the range in an ideal $I\subseteq\mathcal{M}$, the derivation $δ$ is inner, that is $δ(…

Commutator estimates in $W^*$-algebras

Let $\mathcal{M}$ be a $W^*$-algebra and let $LS(\mathcal{M})$ be the algebra of all locally measurable operators affiliated with $\mathcal{M}$. It is shown that for any self-adjoint element $a\in LS(\mathcal{M})$ there exists a self-adjoint element $c_{_{0}}$ from the center of $LS(\mathcal{M})$, such that for any $ε>0$ there exists a unitary element $ u_ε$ from $\mathcal{M}$, satisfying $|[a,u_ε]| \geq (1-ε)|a-c_{_{0}}|$. A corollary of this result is that for any derivation $δ$ on $\mathcal{M}$ with the range in a (not necessarily norm-closed) ideal $I\subseteq\mathcal{M}$, the derivation $…

环中完全非中心Lie理想与不变加法子群

Fully noncentral Lie ideals and invariant additive subgroups in rings

本文证明了环中完全非中心子群包含所有加法换位子的条件，并揭示了在特定代数中完全非中心子空间成为Lie理想与内自同构不变性的等价关系。

A survey on the interplay between arithmetic mean ideals, traces, lattices of operator ideals, and an infinite Schur-Horn majorization theorem

The work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators showed that arithmetic means play an important role in the study of operator ideals, and we explored their role in a multipaper project which we survey in this article. We start by presenting the notions of arithmetic mean ideals and arithmetic mean at infinity ideals. Then we explore their connections with commutator spaces, traces, elementary operators, lattice and sublattice structure of ideals, arithmetic mean ideal cancellation properties of first and second order, and softness properties - a term that we intr…

布尔二次型与正切律

The Boolean quadratic forms and tangent law

本文研究布尔交换子与反交换子加权和的极限，证明平移广义正切函数出现在极限定理中，并给出二次型的任意累积量公式，应用于布尔概率论。

具有消失阿贝尔化的齐次对称迹正六次多项式是埃尔米特平方与交换子的和

Homogeneous symmetric trace-positive sextics with vanishing abelianization are sums of Hermitian squares and commutators

本文在阿贝尔化消失的双变量对称齐次六次多项式空间中，证明迹正性等价于属于埃尔米特平方和的锥，给出了该空间中两类锥的完整显式描述。

交换子的端点Schatten类性质

Endpoint Schatten class properties of commutators

该研究探讨拉普拉斯幂与乘法算子交换子的迹理想性质，给出其属于弱Schatten类的充要条件及奇异值渐近性，证明用到双算子积分工具。

来自分数阶拉普拉斯算子的理想量子度量

Ideal quantum metrics from fractional Laplacians

本文利用分数阶拉普拉斯算子的 Schatten 理想与交换子构造 Monge--Kantorovič 量子度量，给出基于谱的闭式公式，并推广到动力系统与非交换几何。

Essentially Reductive Hilbert Modules II

Many Hilbert modules over the polynomial ring in m variables are essentially reductive, that is, have commutators which are compact. Arveson has raised the question of whether the closure of homogeneous ideals inherit this property and provided motivation to seek an affirmative answer. Positive results have been obtained by Arveson, Guo, Wang and the author. More recently, Guo and Wang extended the results to quasi-homogeneous ideals in two variables. Building on their techniques, in this note the author extends this result to Hilbert modules over certain Reinhardt domains such as ellipsoids i…

Finite sums of projections in von Neumann algebras

We first prove that in a sigma-finite von Neumann factor M, a positive element $a$ with properly infinite range projection R_a is a linear combination of projections with positive coefficients if and only if the essential norm ||a||_e with respect to the closed two-sided ideal J(M) generated by the finite projections of M does not vanish. Then we show that if ||a||_e>1, then a is a finite sum of projections. Both these results are extended to general properly infinite von Neumann algebras in terms of central essential spectra. Secondly, we provide a necessary condition for a positive operator …

Spectra of Abelian C*-Subalgebra Sums

Let $C_b(X)$ be the C*-algebra of bounded continuous functions on some non-compact, but locally compact Hausdorff space $X$. Moreover, let $A_0$ be some ideal and $A_1$ be some unital C*-subalgebra of $C_b(X)$. For $A_0$ and $A_1$ having trivial intersection, we show that the spectrum of their vector space sum equals the disjoint union of their individual spectra, whereas their topologies are nontrivially interwoven. Indeed, they form a so-called twisted-sum topology which we will investigate before. Within the whole framework, e.g., the one-point compactification of $X$ and the spectrum of th…