## Serre-Swan theorem for non-commutative C ∗-algebras. Revised edition 1 (2006)

### BibTeX

@MISC{Kawamura06serre-swantheorem,

author = {Katsunori Kawamura},

title = {Serre-Swan theorem for non-commutative C ∗-algebras. Revised edition 1},

year = {2006}

}

### OpenURL

### Abstract

We generalize the Serre-Swan theorem to non-commutative C ∗-algebras. For a Hilbert C ∗-module X over a C ∗-algebra A, we introduce a hermitian vector bundle EX associated to X. We show that there is a linear subspace ΓX of the space of all holomorphic sections of EX and a flat connection D on EX with the following properties: (i) ΓX is a Hilbert A-module with the action of A defined by D, (ii) the C ∗-inner product of ΓX is induced by the hermitian metric of EX, (iii) EX is isomorphic to an associated bundle of an infinite dimensional Hopf bundle, (iv) ΓX is isomorphic to X.

### Citations

832 |
Operator Algebras and Quantum Statistical Mechanics I
- Bratteli, Robinson
- 1979
(Show Context)
Citation Context ...onal canonical commutation relation U(s)V (t) = e √ −1st V (t)U(s) for s,t ∈ R, its uniform Kähler bundle is (P(H),p, {1pt}). The spectrum is a one-point set {1pt} from von Neumann uniqueness theorem =-=[3]-=-. (iii) The CAR algebra A is a UHF algebra with the nest {M2 n(C)}n∈N. The uniform Kähler bundle has the base space 2 N and each fiber on 142 N is a separable infinite dimensional projective Hilbert ... |

438 |
C ∗ –algebras and their Automorphism Groups
- Pedersen
- 1979
(Show Context)
Citation Context ...fold structure induced by τ b . Furthermore the following holds. Theorem 2.3 (i) For a unital C∗-algebra A, let (P,p,B) be as in Definition 1.2 and assume that B is endowed with the Jacobson topology =-=[14]-=-. Then (P,p,B) is a uniform Kähler bundle. (ii) Let Ai be a C ∗ -algebra with the associated uniform Kähler bundle (Pi,pi,Bi) for i = 1,2. Then A1 and A2 are ∗ isomorphic if and only if (P1,p1,B1) and... |

343 |
Fundamentals of the theory of operator algebras
- Kadison, Ringrose
- 1983
(Show Context)
Citation Context ...ce class of identity representation (H,id L(H)) of L(H) on H. Since the primitive spectrum of L(H) is a two-point set, the topology of 2 [0,1] ∪ {b0} is equal to { ∅, 2 [0,1] , {b0}, 2 [0,1] ∪ {b0} } =-=[8]-=-. In this way, the base space of the uniform Kähler bundle is not always a singleton when the C ∗ -algebra is type I. (ii) For the C ∗ -algebra A generated by the Weyl form of the 1-dimensional canoni... |

148 | Non-commutative differential geometry, Publ - Connes - 1986 |

77 |
Vector bundles and projective modules
- Swan
- 1962
(Show Context)
Citation Context ...s isomorphic to X. Mathematics Subject Classifications (2000). 46L87, 46L08, 58B34. Key words. Serre-Swan theorem, Hilbert C ∗ -module, non-commutative geometry. 1 Introduction The Serre-Swan theorem =-=[9, 15, 16]-=- is described as follows: Theorem 1.1 Let Ω be a connected compact Hausdorff space and let C(Ω) be the algebra of all complex-valued continuous functions on Ω. Assume that X is a module over C(Ω). The... |

61 |
Foundations of Differential Geometry, vol II
- Kobayashi, Nomizu
- 1969
(Show Context)
Citation Context ...n commutative Lie bracket. In this sense, the function algebra is not always commutative. As a special case, for a Kähler manifold, the smooth function algebra has Kähler bracket from its Kähler form =-=[11]-=- which is the special case of symplectic form. For example, a complex projective space CP n is a Kähler manifold with a metric called Fubini-Study type. Cirelli, Manià and Pizzocchero realize non-comm... |

38 |
Foundations of differential geometry. Vol I. Interscience Publishers, a division of
- Kobayashi, Nomizu
- 1963
(Show Context)
Citation Context ...tiability of s ∈ Γ(EX) at each B-fiber in the sense of Fréchet differentiability of Hilbert manifolds. Denote Γ∞(EX) the set of all B-fiberwise smooth sections in Γ(EX). Define the hermitian metric H =-=[12]-=- on Γ∞(EX) by Hρ(s,s ′ ) ≡ 〈s(ρ) |s ′ (ρ) 〉ρ (ρ ∈ P, s,s ′ ∈ Γ∞(EX)). (1.8) By these preparations, we state the following theorem which is a version of the Serre-Swan theorem generalized to non-commut... |

36 |
Connes’ noncommutative differential geometry and the standard
- Várilly, Gracia-Bondía
- 1993
(Show Context)
Citation Context ...rem 1.1, finitely generated projective modules over the commutative C ∗ -algebra C(Ω) and complex vector bundles over Ω are in one-to-one correspondence up to isomorphism. In non-commutative geometry =-=[6, 17]-=-, a certain module over a non-commutative C ∗ -algebra A is treated as a noncommutative vector bundle over the non-commutative space A, generalizing Theorem 1.1 in a sense of point-less geometry. Ther... |

25 |
Modules projectifs et espaces fibrés à fibre vectorielle”, Séminaire P. Dubreil, année 1957/1958
- Serre
(Show Context)
Citation Context ...s isomorphic to X. Mathematics Subject Classifications (2000). 46L87, 46L08, 58B34. Key words. Serre-Swan theorem, Hilbert C ∗ -module, non-commutative geometry. 1 Introduction The Serre-Swan theorem =-=[9, 15, 16]-=- is described as follows: Theorem 1.1 Let Ω be a connected compact Hausdorff space and let C(Ω) be the algebra of all complex-valued continuous functions on Ω. Assume that X is a module over C(Ω). The... |

24 |
An Introduction (Springer-Verlag
- Karoubi, K-Theory
- 1979
(Show Context)
Citation Context ...s isomorphic to X. Mathematics Subject Classifications (2000). 46L87, 46L08, 58B34. Key words. Serre-Swan theorem, Hilbert C ∗ -module, non-commutative geometry. 1 Introduction The Serre-Swan theorem =-=[9, 15, 16]-=- is described as follows: Theorem 1.1 Let Ω be a connected compact Hausdorff space and let C(Ω) be the algebra of all complex-valued continuous functions on Ω. Assume that X is a module over C(Ω). The... |

18 |
Elements of Mathematics. General Topology, Part 1, Hermann
- Bourbaki
- 1966
(Show Context)
Citation Context ...µ −1 (m) is a Kähler manifold. The local triviality of uniform Kähler bundle is not assumed. In general, the topological space M is neither compact nor Hausdorff. For uniform spaces, see Chapter 2 in =-=[2]-=-. Two uniform Kähler bundles (E,µ,M) and (E ′ ,µ ′ ,M ′ ) are isomorphic if there is a pair (β,φ) of a uniform homeomorphism β from E to E ′ and a homeomorphism φ from M to M ′ , such that µ ′ ◦β = φ◦... |

8 |
Pure states of general quantum mechanical systems as Kähler bundle, Nuovo Cimento
- Abbati, Cirelli, et al.
- 1984
(Show Context)
Citation Context ...ra with the associated uniform Kähler bundle (Pi,pi,Bi) for i = 1,2. Then A1 and A2 are ∗ isomorphic if and only if (P1,p1,B1) and (P2,p2,B2) are isomorphic as uniform Kähler bundle. 8Proof. (i) See =-=[1, 4]-=-. (ii) See Corollary 3.3 in [4]. By Theorem 2.3 (ii), the uniform Kähler bundle (P,p,B) associated with A is uniquely determined up to uniform Kähler isomorphism. By the above results, we obtain a fun... |

4 |
of Mathematics, General topology part I, Addison-Wesley Publishing company
- Bourbaki
- 1966
(Show Context)
Citation Context ...d uniform topology in this paper. The local triviality of uniform Kähler bundle is not assumed. M is always neither compact nor Hausdorff. We simply denote (E, µ, M) by E. For a uniform topology, see =-=[2]-=-. Any metric space is a uniform space. Examples and relations with C ∗ -algebra are given later. Roughly speaking, the fiber of uniform Kähler bundle is related to non-commutativity of C ∗ -algebra. D... |

4 |
A.Manià and L.Pizzocchero, A functional representation of noncommutative C
- Cirelli
- 1994
(Show Context)
Citation Context ...n commutative vector bundle because the condition of modules is used in Theorem 1.1. On the other hand, for a unital general non commutative C ∗ -algebra A, there is a uniform Kähler bundle (P, p, B) =-=[3]-=- unique up to equivalence class of A, such that A is ∗ isomorphic onto the uniform Kähler function algebra with ∗-product on (P, p, B) which is a natural generalization of Gel’fand representation. Exa... |

4 |
Symplectic geometry, Gordon and Breach science
- Fomenko
- 1988
(Show Context)
Citation Context ...fferential manifold are generally non commutative. Furthermore, for a symplectic manifold M, the set C∞ (M) of all smooth functions on M has the Poisson bracket determined by the symplectic form of M =-=[6]-=-. It is nothing but which is non commutative Lie bracket. In this sense, the function algebra is not always commutative. As a special case, for a Kähler manifold, the smooth function algebra has Kähle... |

3 |
A functional representation for non-commutative
- Cirelli, Mania, et al.
- 1994
(Show Context)
Citation Context ...before. 2 e-mail: kawamura@kurims.kyoto-u.ac.jp. 1On the other hand, for a unital generally non-commutative C ∗ -algebra A, the functional representation on a certain geometrical space is studied by =-=[4]-=-. We review it as follows. Definition 1.2 A triplet (P,p,B) is the uniform Kähler bundle associated with A if P (= PureA) is the set of all pure states of A, endowed with the w ∗ -uniformity, i.e. the... |

3 | Infinitesimal Takesaki duality of Hamiltonian vector fields on a symplectic manifold
- Kawamura
- 2000
(Show Context)
Citation Context ... Gel’fand representation of unital commutative C ∗ -algebras. By these correspondences, we show the infinitesimal version of the Takesaki duality of Hamiltonian vector fields on a symplectic manifold =-=[10]-=-. 3 Proof of Theorem 1.5 In this section, we construct the typical fiber F b X of EX in Theorem 1.5 and show the isomorphism among vector bundles. In order to construct the typical fiber F b X of EX, ... |

2 |
theorem for non-commutative C ∗ -algebras
- Serre-Swan
- 2003
(Show Context)
Citation Context ... generalizing Theorem 1.1 in a sense of point-less geometry. Therefore both a noncommutative space and a non-commutative vector bundle are invisible even if one desires to look hard. 1 Original paper =-=[11]-=-. The essential mathematical statement is same as before. 2 e-mail: kawamura@kurims.kyoto-u.ac.jp. 1On the other hand, for a unital generally non-commutative C ∗ -algebra A, the functional representa... |

1 |
C ∗ -modules:theorems of Stinespring and Voiculescu, J.Operator Theory 4(1980
- Kasparov
(Show Context)
Citation Context ... the Hilbert C ∗ -structure is more essential than FP property according to our theorem. Here we would mention that there is a kind of FP property of Hilbert C ∗ -module by Kasparov stability theorem =-=[10]-=-. In subsection 3.2, we introduce EX in Theorem 1.2. EX is called the atomic bundle of a Hilbert C∗-module X which is a Hilbert bundle on the uniform Kähler bundle. We show its geometrical structure i... |