谱小量子群表示的巴拿赫流形

Norm-Controlled Inversion in Smooth Banach Algebras, I

Every differential subalgebra of a unital $C^*$-algebra is spectrally invariant. We derive a quantitative version of this well-known fact and show that a minimal amount of smoothness, as given by a differential norm, already implies norm control. We obtain an explicit estimate for the differential norm of an invertible element $a$. This estimate depends only on the condition number of $a$ and the ratio of two norms.

Integrability of almost complex structures on Banach manifolds

We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. As an application, we extend some known results concerning existence of invariant complex structures on homogeneous spaces of Banach-Lie groups.

Infinitesimal aspects of idempotents in Banach algebras

We investigate infinitesimal properties of sets of ordered $n$-uples of idempotents in a symmetric Banach $*$-algebra. These sets are called flag manifolds and carry several interesting bundles that hold an important role in some areas of operator theory. In this direction, we introduce and study Stiefel bundles on flag manifolds, which are extensions of the well known Stiefel bundles on Grassmannians. The main ingredient of our investigation is the notion of connection on an infinite-dimensional bundle, and we survey some equivalent ocurrences of such a notion in the literature.

Manifolds of semi-negative curvature

The notion of nonpositive curvature in Alexandrov's sense is extended to include p-uniformly convex Banach spaces. Infinite dimensional manifolds of semi-negative curvature with a p-uniformly convex tangent norm fall in this class on nonpositively curved spaces, and several well-known results, such as existence and uniqueness of best approximations from convex closed sets, or the Bruhat-Tits fixed point theorem, are shown to hold in this setting, without dimension restrictions. Homogeneous spaces G/K of Banach-Lie groups of semi-negative curvature are also studied, explicit estimates on the ge…

曲率、Dolbeault-Dirac算子以及紧Kähler流形上的$\mathrm{L}^p$-指标定理

Curvature, Dolbeault-Dirac operators, and an $\mathrm{L}^p$-index theorem on compact Kähler manifolds

本文在紧Kähler流形上建立了Dolbeault-Dirac算子的$\mathrm{L}^p$-Banach非交换几何框架，证明了其闭实现是双扇形的且具有有界$\mathrm{H}^\infty$泛函演算，并得到$\mathrm{L}^p$-Hodge分解和Fredholm算子的指标等于全纯Euler示性数。

$C^{*}(G,A)$ 的逆闭子代数

The inverse-closed subalgebra of $C^{*}(G,A)$

本文研究系数为非交换 $C^*$-代数的 Roe 代数 $C^*(G,A)$ 的逆闭子代数，通过 Sobolev 型代数取交集构造 $W_a^{\infty}(G,A)$，证明其为谱不变稠密子代数，充分条件为群作用具有多项式增长。