Abelian self-commutators in finite factors

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 $…

Weak type commutator and Lipschitz estimates: resolution of the Nazarov-Peller conjecture

Let $\mathcal{M}$ be a semi-finite von Neumann algebra and let $f: \mathbb{R} \rightarrow \mathbb{C}$ be a Lipschitz function. If $A,B\in\mathcal{M}$ are self-adjoint operators such that $[A,B]\in L_1(\mathcal{M}),$ then $$\|[f(A),B]\|_{1,\infty}\leq c_{abs}\|f'\|_{\infty}\|[A,B]\|_1,$$ where $c_{abs}$ is an absolute constant independent of $f$, $\mathcal{M}$ and $A,B$ and $\|\cdot\|_{1,\infty}$ denotes the weak $L_1$-norm. If $X,Y\in\mathcal{M}$ are self-adjoint operators such that $X-Y\in L_1(\mathcal{M}),$ then $$\|f(X)-f(Y)\|_{1,\infty}\leq c_{abs}\|f'\|_{\infty}\|X-Y\|_1.$$ This result re…

On the relation between an operator and its self-commutator

Our main result is a theorem saying that a bounded operator $A$ on a Hilbert space belongs to a certain set associated with its self-commutator $[A^*,A]$, provided that $A-zI$ can be approximated by invertible operators for all complex numbers $z$. The theorem remains valid in a general $C^*$-algebra of real rank zero under the assumption that $A-zI$ belong to the closure of the connected component of unity in the set of invertible elements. This result implies the Brown--Douglas--Fillmore theorem and Huaxin Lin's theorem on almost commuting matrices. Moreover, it allows us to refine the forme…

由因子映射和子群胚产生的拓扑全群的最大子群

Maximal subgroups of topological full groups arising from factor maps and subgroupoids

本文通过因子映射和宽开子群胚构造拓扑全群及其换位子群的最大子群，给出最大性判据，并应用于康托尔极小系统，产生 Grigorchuk-Vorobets 猜想中的划分稳定化子及顺从群中的最大子群例子。

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(…

Commutators and linear spans of projections in certain finite C*-algebras

Assume that A is a unital separable simple C*-algebra with real rank zero, stable rank one, strict comparison of projections, and that its tracial simplex T(A) has a finite number of extremal points. We prove that every self-adjoint element a in A in the kernel of all tracial states is the sum of two commutators in A and that every positive element of A is a linear combination of projections with positive coefficients. Assume that A is as above but σ-unital. Then an element (resp. a positive element) a of A is a linear combination (resp. a linear combination with positive coefficients) of proj…

$B(H)$-Commutators: A Historical Survey II and recent advances on commutators of compact operators

A sequel to \cite{gW05}, we address again the single commutator problem \cite{PT71} of Pearcy and Topping: Is every compact operator a single commutator of compact operators? by focusing on a 35 year old test question for this posed in 1976 by the last named author and others: Are there any strictly positive operators that are single commutators of compact operators? The latter we settle here affirmatively with a modest modification of Anderson's fundamental construction \cite{jA77} constructing compact operators whose commutator is a rank one projection. Moreover we provide here a rich class …

A new kind of index theorem

Index theory has had profound impact on many branches of mathematics. In this note we discuss the context for a new kind of index theorem. We begin, however, with some operator theoretic results. In [11] Berger and Shaw established that finitely cyclic hyponormal operators have trace-class self-commutators. In [9], [31] Berger and Voiculescu extended this result to operators whose self-commutators can be expressed as the sum of a positive and a trace-class operator. In this note we show this result can't be extended to operators whose self-commutator can be expressed as the sum of a positive a…

Commutator inequalities via Schur products

For a self-adjoint unbounded operator D on a Hilbert space H, a bounded operator y on H and some complex Borel functions g(t) we establish inequalities of the type ||[g(D),y]|| \leq A|||y|| + B||[D,y]|| + ...+ X|[D, [D,...[D, y]...]]||. The proofs take place in a space of infinite matrices with operator entries, and in this setting it is possible to approximate the matrix associated to [g(D), y] by the Schur product of a matrix approximating [D,y] and a scalar matrix. A classical inequality of Bennett on the norm of Schur products may then be applied to obtain the results.