Kenneth R. Davidson, Ronald G. Douglas
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.
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For an arbitrary operator ideal I, every nilpotent element of I is a single commutator of operators from I^t, for an exponent t that depends on the degree of nilpotency.
Let M be a factor of type II_\infty or II_1 having separable predual and let M-bar be the algebra of affiliated τ-measureable operators. We characterize the commutator space [I,J] for sub-(M,M)-bimodules I and J of M-bar.
Toeplitz algebra and Symbol map via Berezin transform on $H^2(\mathbb{D}^n)$
本文利用Berezin变换研究开单位多圆盘上Hardy空间算子的类Douglas定理，得到一类更大的$C^*$-代数仍满足符号映射类似结论。
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…
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。
We show that many invariant subspaces M for d-shifts (S_1,...,S_d) of finite rank have the property that the projection P onto M almost commutes with the S_k in the sense that the commutators PS_k - S_kP belong to the Schatten-von Neumann class L^p for every p > d. In such cases the d-tuple of operators (T_1,...,T_d) obtained by compressing (S_1,...,S_d) to the orthocomplement of M generates a *-algebra whose commutator ideal is contained in L^p, p > d. It follows that the C*-algebra generated by T_1,...,T_d is commutative modulo compact operators, the associated Dirac operator is Fredholm, an…
Various questions on Lie ideals of C*-algebras are investigated. They fall roughly under the following topics: relation of Lie ideals to closed two-sided ideals; Lie ideals spanned by special classes of elements such as commutators, nilpotents, and the range of polynomials; characterization of Lie ideals as similarity invariant subspaces.
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。
For each unital $C^*$-algebra $A$, we denote $cel_{CU}(A)=\sup\{cel(u):u\in CU(A)\}$, where $cel(u)$ is the exponential length of $u$ and $CU(A)$ is the closure of the commutator subgroup of $U_0(A)$. In this paper, we prove that $cel_{CU}(A)=2π$ provided that $A$ is an $AH$ algebras with slow dimension growth whose real rank is not zero. On the other hand, we prove that $cel_{CU}(A)\leq 2π$ when $A$ is an $AH$ algebra with ideal property and of no dimension growth (if we further assume $A$ is not of real rank zero, we have $cel_{CU}(A)= 2π$).
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。
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 …
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。
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 $…
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…
Prime ideals in C*-algebras and applications to Lie theory
本文证明C*-代数中每个真稠密理想都包含于某个素理想，从而将素环的李理论结果转移到C*-代数，并揭示平方零元素生成子空间与换位子空间的新联系。
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。
Lie ideals in properly infinite C*-algebras
该研究证明含单位元的真无限C*-代数中李理想与唯一双边理想交换子等价，得到实秩零无特征标C*-代数的类似结果，并应用于von Neumann代数解决了相关公开问题。
We use operator algebras and operator theory to obtain new result concerning Berezin quantization of compact Kähler manifolds. Our main tool is the notion of subproduct systems of finite-dimensional Hilbert spaces, which enables all involved objects, such as the Toeplitz operators, to be very conveniently expressed in terms of shift operators compressed to a subspace of full Fock space. This subspace is not required to be contained in the symmetric Fock space, so from finite-dimensional matrix algebras we can construct noncommutative manifolds with extra structure generalizing that of a projec…
Let $X$, $Y$ be Banach spaces and let $\mathcal{L}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. We develop the theory of double operator integrals on $\mathcal{L}(X,Y)$ and apply this theory to obtain commutator estimates of the form $\|f(B)S-Sf(A)\|_{\mathcal{L}(X,Y)}\leq \textrm{const} \|BS-SA\|_{\mathcal{L}(X,Y)}$ for a large class of functions $f$, where $A\in\mathcal{L}(X)$, $B\in \mathcal{L}(Y)$ are scalar type operators and $S\in \mathcal{L}(X,Y)$. In particular, we establish this estimate for $f(t):=|t|$ and for diagonalizable operators on $X=\ell_{p}$ and $Y=\ell_{q…
In this paper we define B-Fredholm elements in a Banach algebra $A$ modulo an ideal $J$ of $A.$ When a trace function is given on the ideal $J,$ it generate an index for B-Fredholm elements. In the case of a B-Fredholm operator $T$ acting on a Banach space, we prove that its usual index $ind(T)$ is equal to the trace of the commutator $ [T, T_0],$ where $T_0$ is a Drazin inverse of $T$ modulo the ideal of finite rank operators, extending a Fedosov's trace formula for Fredholm operators. In the case of a semi-simple Banach algebra, we prove a punctured neighborhood theorem for the index.
Let $A$ be an $AH$ algebra $A=\lim\limits_{n\to \infty}(A_{n}=\bigoplus\limits_{i=1}\limits^{t_{n}}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}, ϕ_{n,m})$, where $X_{n,i}$ are compact metric spaces, $t_{n}$ and $[n,i]$ are positive integers, and $P_{n,i}\in M_{[n,i]}(C(X_{n,i}))$ are projections. Suppose that $A$ has the ideal property: each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two sided ideal. In this article, we will classify all $AH$ algebras with ideal property of no dimension growth---that is, $sup_{n,i}dim(X_{n,i})<+\infty$. This result gener…
Quantum Gromov-Hausdorff Convergence for Extensions of $C^*$-Algebras
研究由稳定理想对幺正\(C^*\) - 代数的托普利兹型扩张，利用谱度量空间构造，通过引入幺正2 - 压缩逼近等工具，证明在一定条件下，稳定理想基础代数的完备子算子系统序列收敛时，扩张中的相应序列也收敛，反之亦然。
Let A denote the C*-algebra of bounded operators on L^2(RxS^1) generated by: (a) multiplications by smooth functions on S^1; (b) multiplications by continuous functions on the two point compactification of R; (c) multiplications by 2π-periodic continuous functions; (d) the operator L given by the inverse of the square root of the identity operator minus the Laplacian operator on RxS^1; and (e) operators of the form DL, where D is either the differencial operator on R or a first order differential operator on S^1 with smooth coefficients. Let σbe the complex-valued symbol on A that arises from …
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