广义酉和Picard群

The generalized Picard groups for finite dimensional $C^*$-Hopf algebra coactions on unital $C^*$-algebras

We shall generalize the notion of the strong Morita equivalence for coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra and define the Picard groups with respect to the generalized strong Morita equivalence. We call them the generalized Picard groups for coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algeba. We shall investigate basic properties of the generalized Picard groups and clarify the relation between the generalized Picard groups and the Picard groups for unital inclusions of unital $C^*$-algebras.

Rigged modules I: modules over dual operator algebras and the Picard group

In a previous paper we generalized the theory of W*-modules to the setting of modules over nonselfadjoint dual operator algebras, obtaining the class of weak*-rigged modules. At that time we promised a forthcoming paper devoted to other aspects of the theory. We fulfill this promise in the present work and its sequel "Rigged modules II", giving many new results about weak*-rigged modules and their tensor products. We also discuss the Picard group of weak* closed subalgebras of a commutative algebra. For example, we compute the weak Picard group of $H^{\infty}(D)$, and prove that for a weak* cl…

Hilbert C*-模上的动力系统

Dynamical Systems on Hilbert C*-Modules

研究了广义导子，证明了单Hilbert $C^*$ -模上的广义导子是可闭的或有稠密值域.我们还将$C^*$ -代数${\mathcal A}$上的全Hilbert $C^*$ -模${\mathcal M}$上的动力系统描述为${\mathcal M}$上的酉单参数群，并证明了如果$α: \R\to U({\mathcal M})$是动力系统，其中$U({\mathcal M})$表示${\mathcal M}$上所有酉算子的集合，那么我们可以对应一个$C^*$ -动力系统$α^{'}$在${\mathcal A}$上，使得如果$δ$和$d$分别是$α$和$α^{'}$的无穷小生成元，则$δ$是$d$ -导子。

Localizing gauge theories from noncommutative geometry

We recall the emergence of a generalized gauge theory from a noncommutative Riemannian spin manifold, viz. a real spectral triple $(A,H,D;J)$. This includes a gauge group determined by the unitaries in the $*$-algebra $A$ and gauge fields arising from a so-called perturbation semigroup which is associated to $A$. Our main new result is the interpretation of this generalized gauge theory in terms of an upper semi-continuous $C^*$-bundle on a (Hausdorff) base space $X$. The gauge group acts by vertical automorphisms on this $C^*$-bundle and can (under some mild conditions) be identified with the…