𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Serre–Swan theorem for non-commutative C∗-algebras

✍ Scribed by Katsunori Kawamura

Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
185 KB
Volume
48
Category
Article
ISSN
0393-0440

No coin nor oath required. For personal study only.

✦ Synopsis

For a Hilbert C * -module X over a C * -algebra A, we introduce a vector bundle E X associated to X. We prove that E X has an hermitian metric and a flat connection. We introduce a vector space Γ X of holomorphic sections of E X with the following properties: (i) Γ X is a Hilbert A-module, (ii) the action of A on Γ X is defined by means of the connection of A, (iii) the C * -inner product of Γ X is induced by the hermitian metric of E X .

We prove that the Hilbert C * -module Γ X is isomorphic to X. This sectional representation is a generalization of the Serre-Swan theorem to non-commutative C * -algebras. We show that E X is isomorphic to an associated bundle of an infinite dimensional Hopf bundle with the structure group U(1).

📜 SIMILAR VOLUMES

A Serre-Swan Theorem for Bundles of Topo
A Serre-Swan Theorem for Bundles of Topological Modules
✍ Maria H. Papatriantafillou 📂 Article 📅 2006 🏛 John Wiley and Sons 🌐 English ⚖ 436 KB

## Abstract Let __R__ be a unital topological ring whose set of invertible elements is open and inversion is continuous, and let __X__ be a compact Hausdorff space admitting continuous __R__‐valued partitions of unity. Considering bundles over __X__ of fibre type a projective finitely generated __R

Existence and Density Theorems for Stoch
Existence and Density Theorems for Stochastic Maps on Commutative C*-algebras
✍ Peter M. Alberti; Armin Uhlmann 📂 Article 📅 1980 🏛 John Wiley and Sons 🌐 English ⚖ 876 KB

## Abstract This paper presents theoremes on the structure of stochastic and normalized positive linear maps over commutative __C__\*‐algebras. We show how strongly the solution of the __n__‐tupel problem for stochastic maps relates to the fact that stochastic maps of finite rank are weakly dense w