On Hastings' approach to Lin's Theorem for Almost Commuting Matrices

Almost commuting matrices, cohomology, and dimension

We investigate which relations for families of commuting matrices are stable under small perturbations, or in other words, which commutative $C^*$-algebras $C(X)$ are matricially semiprojective. Extending the works of Davidson, Eilers-Loring-Pedersen, Lin and Voiculescu on almost commuting matrices, we identify the precise dimensional and cohomological restrictions for finite-dimensional spaces $X$ and thus obtain a complete characterization: $C(X)$ is matricially semiprojective if and only if $\dim(X)\leq 2$ and $H^2(X;\mathbb{Q})=0$. We give several applications to lifting problems for commu…

Connecting Commuting Normal Matrices

In this document we study the local path connectivity of sets of $m$-tuples of commuting normal matrices with some additional geometric constraints in their joint spectra. In particular, given $\varepsilon>0$ and any fixed but arbitrary $m$-tuple $\mathbf{X}\in {M_n(\mathbb{C})}^m$ in the set of $m$-tuples of pairwise commuting normal matrix contractions, we prove the existence of paths between arbitrary $m$-tuples in the intersection of the previously mentioned sets of $m$-tuples in ${M_n(\mathbb{C})}^m$ and the $δ$-ball $B_ð(\mathbf{X},δ)$ centered at $\mathbf{X}$ for some $δ>0$, with respec…

Almost commuting functions of almost commuting self-adjoint operators

Let $A$ and $B$ be almost commuting (i.e, $AB-BA\in\bS_1$) self-adjoint operators. We construct a functional calculus $\f\mapsto\f(A,B)$ for $\f$ in the Besov class $B_{\be,1}^1(\R^2)$. This functional calculus is linear, the operators $\f(A,B)$ and $ψ(A,B)$ almost commute for $\f,\,ψ\in B_{\be,1}^1(\R^2)$, $\f(A,B)=u(A)v(B)$ whenever $\f(s,t)=u(s)v(t)$, and the Helton--Howe trace formula holds. The main tool is triple operator integrals.

冯·诺依曼代数中邻近交换元素的投影刻画与对称性提升

A Projection Characterization and Symmetry Bootstrap for Elements of a von Neumann Algebra that are Nearby Commuting Elements

本文研究 C*-代数中邻近交换算子的对称性提升问题，通过投影刻画证明带对称性的 Lin 定理，并解决拓扑绝缘体相关的一个猜想。

Analytic Model of Doubly Commuting Contractions

An n-tuple (n \geq 2), T = (T_1, \ldots, T_n), of commuting bounded linear operators on a Hilbert space \mathcal{H} is doubly commuting if T_i T_j^* = T_j^* T_i for all $1 \leq i < j \leq n$. If in addition, each T_i \in C_{\cdot 0}, then we say that T is a doubly commuting pure tuple. In this paper we prove that a doubly commuting pure tuple $T$ can be dilated to a tuple of shift operators on some suitable vector-valued Hardy space H^2_{\mathcal{D}_{T^*}}(\mathbb{D}^n). As a consequence of the dilation theorem, we prove that there exists a closed subspace \mathcal{S}_T of the form \[\mathcal{…

Functions of almost commuting operators and an extension of the Helton-Howe trace formula

Let $A$ and $B$ be almost commuting (i.e., the commutator $AB-BA$ belongs to trace class) self-adjoint operators. We construct a functional calculus $φ\mapstoφ(A,B)$ for functions $φ$ in the Besov class $B_{\infty,1}^1({\Bbb R}^2)$. This functional calculus is linear, the operators $φ(A,B)$ and $ψ(A,B)$ almost commute for $φ,\,ψ\in B_{\infty,1}^1({\Bbb R}^2)$, and $φ(A,B)=u(A)v(B)$ whenever $φ(s,t)=u(s)v(t)$. We extend the Helton--Howe trace formula for arbitrary functions in $B_{\infty,1}^1({\Bbb R}^2)$. The main tool is triple operator integrals with integrands in Haagerup-like tensor produc…