通过 Hilbert C*-模的泛范畴性质刻画紧算子的 C*-代数

Frobenius structures over Hilbert C*-modules

We study the monoidal dagger category of Hilbert C*-modules over a commutative C*-algebra from the perspective of categorical quantum mechanics. The dual objects are the finitely presented projective Hilbert C*-modules. Special dagger Frobenius structures correspond to bundles of uniformly finite-dimensional C*-algebras. A monoid is dagger Frobenius over the base if and only if it is dagger Frobenius over its centre and the centre is dagger Frobenius over the base. We characterise the commutative dagger Frobenius structures as finite coverings, and give nontrivial examples of both commutative …

希尔伯特C*-模中的B样条插值问题

B-spline interpolation problem in Hilbert C*-modules

本文在希尔伯特C*-模中引入B样条插值问题，研究其解的存在性与唯一性，给出充要条件，并推广至W*-模及C*-理想情形，辅以实例说明。

希尔伯特模与希尔伯特 W*-模的比较及其在 L^2 不变量等不变量中的应用

Hilbertian versus Hilbert W*-modules, and applications to $L^2$- and other invariants

本文比较有限冯·诺依曼代数上的希尔伯特模与希尔伯特 W*-模并建立范畴等价，借此为 L^2 不变量理论提供新证明，并提出有限生成投射 B-模的新不变量。

Hilbert $C^*$-模上的块正性与最优混合施瓦茨不等式

Block Positivity and Optimal Mixed-Schwarz Inequalities on Hilbert $C^*$-Modules

本文研究Hilbert $C^*$-模上可伴随算子理论，给出$n\ imes n$块算子矩阵正性的等价可验证条件，推导紧的带参数混合施瓦茨不等式及极值算子刻画，二者互为支撑并给出相关应用。

Hilbert C*-模与C*-代数中的框架

Frames in Hilbert C*-modules and C*-algebras

本文利用几何膨胀与投影重构方法建立C*-代数及Hilbert C*-模的模框架理论，得到表示、分解、相似性与等价性结果，并应用于Cuntz-Krieger-Pimsner代数、条件期望、C*-代数秩及小波与Gabor框架等领域。

Injective and projective Hilbert C*-modules, and C*-algebras of compact operators

We consider projectivity and injectivity of Hilbert C*-modules in the categories of Hilbert C*-(bi-)modules over a fixed C*-algebra of coefficients (and another fixed C*-algebra represented as bounded module operators) and bounded (bi-)module morphisms, either necessarily adjointable or arbitrary ones. As a consequence of these investigations, we obtain a set of equivalent conditions characterizing C*-subalgebras of C*-algebras of compact operators on Hilbert spaces in terms of general properties of Hilbert C*-modules over them. Our results complement results recently obtained by B. Magajna, J…

Complete C*-categories and a topos theoretic Green-Julg theorem

We investigate what would be a correct definition of categorical completeness for C*-categories and propose several variants of such a definition that make the category of Hilbert modules over a C*-algebra a free (co)completion. We extend results about generators and comparison theory known for W*-categories with direct sums and splitting of symmetric projections to our "complete C*-categories" and we give an abstract characterization of categories of Hilbert modules over a C*-algebra or a C*-category as "complete C*-category having enough absolutely compact morphisms (and a generator)". We th…

Hilbert $C^*$-模上斜积动力学的 Koopman 谱分析

Koopman spectral analysis of skew-product dynamics on Hilbert $C^*$-modules

针对斜积动力系统，本文在 Hilbert $C^*$-模上构造由复合与乘法定义的线性算子，提出推广特征值分解的特征算子分解以表示 Koopman 算子连续谱，并通过二维数值算例展示其相干结构刻画能力。

Characterizing projections among positive operators in the unit sphere

Let $E$ and $P$ be subsets of a Banach space $X$, and let us define the unit sphere around $E$ in $P$ as the set $$Sph(E;P) :=\left\{ x\in P : \|x-b\|=1 \hbox{ for all } b\in E \right\}.$$ Given a C$^*$-algebra $A$, and a subset $E\subset A,$ we shall write $Sph^+ (E)$ or $Sph_A^+ (E)$ for the set $Sph(E;S(A^+)),$ where $S(A^+)$ stands for the set of all positive operators in the unit sphere of $A$. We prove that, for an arbitrary complex Hilbert space $H$, then a positive element $a$ in the unit sphere of $B(H)$ is a projection if and only if $Sph^+_{B(H)} \left( Sph^+_{B(H)}(\{a\}) \right) =…

An Algebraic Duality Theory for Multiplicative Unitaries

Multiplicative Unitaries are described in terms of a pair of commuting shifts of relative depth two. They can be generated from ambidextrous Hilbert spaces in a tensor C*-category. The algebraic analogue of the Takesaki-Tatsuuma Duality Theorem characterizes abstractly C*-algebras acted on by unital endomorphisms that are intrinsically related to the regular representation of a multiplicative unitary. The relevant C*-algebras turn out to be simple and indeed separable if the corresponding multiplicative unitaries act on a separable Hilbert space. A categorical analogue provides internal charac…

更高希尔伯特空间的多面性

The many faces of higher Hilbert spaces

本文通过G- dagger范畴统一了有限维算子代数作为C*, W*, H*代数时的模范畴与对应2-范畴差异，引入G- Hermitian 2-向量空间并定义正性条件，为高维希尔伯特空间提供归纳定义框架。

Bicommutant categories from fusion categories

Bicommutant categories are higher categorical analogs of von Neumann algebras that were recently introduced by the first author. In this article, we prove that every unitary fusion category gives an example of a bicommutant category. This theorem categorifies the well known result according to which a finite dimensional *-algebra that can be faithfully represented on a Hilbert space is in fact a von Neumann algebra.

Relative tensor products for modules over von Neumann algebras

We give an overview of relative tensor products (RTPs) for von Neumann algebra modules. For background, we start with the categorical definition and go on to examine its algebraic formulation, which is applied to Morita equivalence and index. Then we consider the analytic construction, with particular emphasis on explaining why the RTP is not generally defined for every pair of vectors. We also look at recent work justifying a representation of RTPs as composition of unbounded operators, noting that these ideas work equally well for L^p modules. Finally, we prove some new results characterizin…