Integral mappings and the principle of local reflexivity for noncommutative L^1-spaces

几何 Elliott 不变量与映射环面的非交换刚性

A geometric Elliott invariant and noncommutative rigidity of mapping tori

本文利用非交换几何工具证明了极小拓扑动力系统映射环面的刚性，将保向叶状同伦等价提升为叶状C*-代数同构，并发展了基于拓扑与指标理论的几何Elliott不变量方法。

压缩量子平面的非交换复解析几何

Noncommutative complex analytic geometry of a contractive quantum plane

本文利用 Arens–Michael 包、Fréchet 代数预层与非交换 Harte 谱刻画 |q|≠1 时量子 q-平面的非交换复解析几何，并解决其函数演算与谱映射问题。

Analytic mappings between noncommutative pencil balls

In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. In an earlier paper we characterized NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call "NC ball maps". In this paper we turn to a more general dimension-fr…

A $C^*$-algebraic approach to the principal symbol II

We introduce an abstract theory of the principal symbol mapping for pseudodifferential operators extending the results of a preceding paper and providing a simple algebraic approach to the theory of pseudodifferential operators in settings important in noncommutative geometry. We provide a variant of Connes' trace theorem which applies to certain noncommutative settings, with a minimum of technical preliminaries. Our approach allows us to consider in a operators with non-smooth symbols, and we demonstrate the power of our approach by extending Connes' trace theorem to operators with non-smooth…

Cordes characterization for pseudodifferential operators with symbols valued on a noncommutative C*-algebra

Given a separable unital C*-algebra A, let E denote the Banach-space completion of the A-valued Schwartz space on Rn with norm induced by the A-valued inner product $<f,g>=\int f(x)^*g(x) dx$. The assignment of the pseudodifferential operator B=b(x,D) with A-valued symbol b(x,ξ) to each smooth function with bounded derivatives b defines an injective mapping O, from the set of all such symbols to the set of all operators with smooth orbit under the canonical action of the Heisenberg group on the algebra of all adjointable operators on the Hilbert module E. It is known that O is surjective if A …

p-算子逼近性质

The $p$-Operator Approximation Property

本文在算子空间非交换框架下引入p-算子逼近性质（p-OAP），建立其与算子p-紧映射理想及切片映射性质的等价刻画，研究其对偶传递性，并应用于离散群约化C*-代数中Herz-Schur乘子的cb-范数逼近。

Banach 空间上压缩投影的模理论及其在非交换 $\mathrm{L}^p$-空间中的应用

A modular theory of contractive projections on Banach spaces with applications to noncommutative $\mathrm{L}^p$-spaces

本文提出基于 Levi 形式的模理论，刻画 Banach 空间压缩投影，并应用于非交换 L^p 空间，证明 2-压缩投影等价于完全压缩，且值域具有三元结构。

Unbounded Fredholm modules and double operator integrals

In noncommutative geometry one is interested in invariants such as the Fredholm index or spectral flow and their calculation using cyclic cocycles. A variety of formulae have been established under side conditions called summability constraints. These can be formulated in two ways, either for spectral triples or for bounded Fredholm modules. We study the relationship between these by proving various properties of the map on unbounded self adjoint operators $D$ given by $f(D)=D(1+D^2)^{-1/2}$. In particular we prove commutator estimates which are needed for the bounded case. In fact our methods…

Traces of holomorphic families of operators on the noncommutative torus and on Hilbert modules

We revisit traces of holomorphic families of pseudodifferential operators on a closed manifold in view of geometric applications. We then transpose the corresponding analytic constructions to two different geometric frameworks; the noncommutative torus and Hilbert modules. These traces are meromorphic functions whose residues at the poles as well as the constant term of the Laurent expansion at zero (the latter when the family at zero is a differential operator) can be expressed in terms of Wodzicki residues and extended Wodzicki residues involving logarithmic operators. They are therefore loc…

$C(X)$-代数交叉积的大型子代数

Large subalgebras of crossed products of $C(X)$-algebras

该研究从理想出发构造非交换C*-动力系统交叉积C*-代数的大型子代数，并将其应用于映射环面等局部平凡含单位元$C(X)$-代数的研究。

Suspension spectra of matrix algebras, the rank filtration, and rational noncommutative CW-spectra

In a companion paper [ABS1] we introduced the stable $\infty$-category of noncommutative CW-spectra, which we denoted $\mathtt{NSp}$. Let $\mathcal{M}$ denote the full spectrally enriched subcategory of $\mathtt{NSp}$ whose objects are the non-commutative suspension spectra of matrix algebras. In [ABS1] we proved that $\mathtt{NSp}$ is equivalent to the $\infty$-category of spectral presheaves on $\mathcal{M}$. In this paper we investigate the structure of $\mathcal{M}$, and derive some consequences regarding the structure of $\mathtt{NSp}$. To begin with, we introduce a rank filtration of $\m…

The heat flow of the CCR algebra

Let P and Q be the canonical operators acting on the Hilbert space of all L^2 functions on the real line, defined appropriately on a common dense domain. The derivations D_P(A) = i(PA - AP) and D_Q(A) = i(QA - AQ) act on the *-algebra of all integral operators having smooth kernels of compact support, for example, and one may consider the noncommutative "Laplacian" L(A) = D_P^2(A) + D_Q^2(A), as a linear mapping of this *-algebra into itself. L generates a semigroup of normal completely positive maps on B(H), and we establish some basic properties of this semigroup and its minimal dilation to …