Non-Commutative T-Duality

Yetter-Drinfeld C*-algebras 的范畴双子结构：超越辫对易情况

Categorical dualtiy for Yetter-Drinfeld C*-algebras. Beyond the braided-commutative case

本文研究了Yetter-Drinfeld C*-代数在紧量子群下的范畴双子结构，提出了一种超越辫对易情况的双子理论。

Doob's inequality for non-commutative martingales

Let $1\le p<\8$ and $(x_n)_{\nen}$ be a sequence of positive elements in a non-commutative $L_p$ space and $(E_n)_{\nen}$ be an increasing sequence of conditional expectations, then the $L_p$ norm of \sum_n E_n(x_n) can be estimated by c_p times the $L_p$ norm of \sum_n x_n. This inequality is due to Burkholder, Davis and Gundy in the commutative case. By duality, we obtain a version of Doob's maximal inequality for $1<p\le \8$.

A Horizontal Categorification of Gelfand Duality

In the setting of C*-categories, we provide a definition of "spectrum" of a commutative full C*-category as a one-dimensional unital saturated Fell bundle over a suitable groupoid (equivalence relation) and prove a categorical Gelfand duality theorem generalizing the usual Gelfand duality between the categories of commutative unital C*-algebras and compact Hausdorff spaces. Although many of the individual ingredients that appear along the way are well-known, the somehow unconventional way we "glue" them together seems to shed some new light on the subject.

Non-Commutative Geometry, Categories and Quantum Physics

After an introduction to some basic issues in non-commutative geometry (Gel'fand duality, spectral triples), we present a "panoramic view" of the status of our current research program on the use of categorical methods in the setting of A.Connes' non-commutative geometry: morphisms/categories of spectral triples, categorification of Gel'fand duality. We conclude with a summary of the expected applications of "categorical non-commutative geometry" to structural questions in relativistic quantum physics: (hyper)covariance, quantum space-time, (algebraic) quantum gravity.

A duality of locally compact groups which does not involve the Haar measure

We present a simple and intuitive framework for duality of locally compacts groups, which is not based on the Haar measure. This is a map, functorial on a non-degenerate subcategory, on the category of coinvolutive Hopf \cst-algebras, and a similar map on the category of coinvolutive Hopf-von Neumann algebras. In the \cst-version, this functor sends $C_0(G)$ to $C^*(G)$ and vice versa, for every locally compact group $G$. As opposed to preceding approaches, there is an explicit description of commutative and co-commutative algebras in the range of this map (without assumption of being isomorph…

Noncommmutative Gelfand Duality for not necessarily unital $C^*$-algebras, Jordan Canonical form, and the existence of invariant subspaces

Gelfand-Naimark duality (Commutative $C^*$-algebras $\equiv$ Locally compact Hausdorff spaces) is extended to $C^*$-algebras $\equiv$ Quotient maps on locally compact Hausdorff spaces. Using this duality, we give for an \emph{arbitrary} bounded operator on a complex Hilbert space of several dimensions, a functional calculus and the existence theorem for nontrivial invariant subspace.

Noncommutative Gelfand Duality and Applications I: The Existence of invariant subspaces

Gelfand duality between unital commutative C*-algebras and Compact Hausdorff spaces is extended to all unital C*-algebras, where the dual objects are what we call compact Hausdorff quantum spaces. We apply this result to obtain, a characterization of unitary groups of C*-algebras, and, for arbitrary bounded Hilbert space operators, (i) A spectral theorem cum continuous functional calculus, and (ii) A proof of the general Invariant Subspace Theorem. Also described is a nonabelian generalization of Pontryagin duality of abelian locally compact groups.

K-theoretic duality for shifts of finite type

C*-algebras generalizing Cuntz-Krieger algebras can be associated to hyperbolic homeomorphisms of compact metric spaces. They satisfy a non-commutative form of Spanier-Whitehead duality with respect to K-theory. We prove this for the case of subshifts of finite type. The special feature of the present situation is that the constructions are all done on the full Fock space and are very explicit, while the general theorem requires much more abstract machinery.

非交换 $L_p$-空间上的量子双随机算子

Quantum Doubly Stochastic Operators on Non-commutative $L_p$-Spaces

本文研究了非交换 $L_p$-空间上与半有限von Neumann代数相关的正迹保持映射，即量子双随机算子，探讨了其基本性质、严格范数不等式、Schatten理想意义下的紧性条件以及有限和无限维中的新例子，并讨论了其在量子主次性和插值稳定性中的应用。

Yetter-Drinfeld代数的范畴对偶性

Categorical duality for Yetter-Drinfeld algebras

该文研究紧量子群G作用下(Rep G)-模范畴的张量结构，证明其等价于辫子交换Yetter-Drinfeld G-C*-代数范畴与生成酉张量函子范畴的对偶，并通过实例验证等价性。

A groupoid approach to noncommutative T-duality

Topological T-duality is a transformation taking a gerbe on a principal torus bundle to a gerbe on a principal dual-torus bundle. We give a new geometric construction of T-dualization, which allows the duality to be extended in following two directions. First, bundles of groups other than tori, even bundles of some nonabelian groups, can be dualized. Second, bundles whose duals are families of noncommutative groups (in the sense of noncommutative geometry) can be treated, though in this case the base space of the bundles is best viewed as a topological stack. Some methods developed for the con…