## Noncommutative Geometry (1994)

Citations: | 2100 - 33 self |

### BibTeX

@MISC{Connes94noncommutativegeometry,

author = {Alain Connes},

title = {Noncommutative Geometry},

year = {1994}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

832 |
Operator Algebras and Quantum Statistical Mechanics I
- Bratteli, Robinson
- 1979
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Citation Context ...l fact of the C ∗ -algebraic formulation of quantum statistical mechanics that for a given β every KMSβ state decomposes uniquely as a statistical superposition of extreme KMSβ states: Proposition 2. =-=[66]-=- [249] Let (A, σt) be a C ∗ -dynamical system and β ∈ [0, ∞[. Then the space of KMSβ states is a compact convex Choquet simplex. For a careful discussion of the link between extreme KMSβ states and th... |

810 |
Quantum groups
- Drinfeld
- 1986
(Show Context)
Citation Context ...the indication of the existence of a deformation with one parameter ( h here) of the algebra of functions into a noncommutative algebra. I refer the reader to the literature ([13], [37], [38], [169], =-=[182]-=-, [220], [368], [474], [570]) for a description of the results of this theory. 2. Statistical State of a Macroscopic System and Quantum Statistical Mechanics A cubic centimeter of water contains on th... |

517 | Equilibrium states and the ergodic theory of Anosov diffeomorphisms - Bowen - 1975 |

485 |
Homological Algebra
- Cartan, Eilenberg
- 1956
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Citation Context ...ty follows from � a n+1 (a 0 da 1 · · · da n � ) = (a 0 da 1 · · · da n )a n+1 . Let us now recall the definition of the Hochschild cohomology groups Hn (A, M) of A with coefficients in a bimodule M (=-=[80]-=-). Let Ae = A⊗Ao be the tensor product of A by the opposite algebra. Then any bimodule M over A becomes a left Ae-module and, by definition, Hn (A, M) = Ext n Ae(A, M), where A is viewed as a bimodule... |

463 |
The index of elliptic operators
- Atiyah, Singer
- 1971
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Citation Context ...he Euler characteristic of F evaluated on the Ruelle-Sullivan cycle, is a special case of a general theorem which extends to measured foliations the Atiyah-Singer index theorems for compact manifolds =-=[26]-=- and for covering spaces [21] [523]. As we have already seen above, a typical feature of the leaves of foliations is that they fail to be compact even if the ambient manifold V is compact. However, th... |

438 |
C ∗ –algebras and their Automorphism Groups
- Pedersen
- 1979
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Citation Context ...ical extension to a state still noted ' on the multiplier algebra M (A) of A. The canonical extension of a KMS state still satisfies the KMS condition on B. Proof. For the first statement we refer to =-=[42]-=-. The proof of the second statement illustrates a general density argument, where the continuity of t 7! oet(x) is used to control the uniform continuity in the closed strip, in order to apply the Mon... |

333 |
lectures on theta
- Mumford, “Tata
- 1984
(Show Context)
Citation Context ...=x Det(m)\Gamma fi be the limit of the above expression, with f the characteristic function of the subset fxg ae M2(Z=N Z). We want to show that h(x) = N \Gamma 4 ; 8x 2 M2(Z=N Z) : (1.105) 3cf. e.g. =-=[39]-=- for the standard identification of the set of invertible Q-lattices with the above double quotient 53sSince pN is a surjection SL2(Z) ! SL2(Z=N Z) and \Gamma (N ) a normal subgroup of \Gamma , one ge... |

332 |
Noncommutative geometry and matrix theory: compactification on tori
- Connes, Douglas, et al.
- 1998
(Show Context)
Citation Context ...tative boundary of modular curves We shall explain in this section how to combine the dual of the GL2-system described above with the idea, originally developed in the work of Connes- Douglas-Schwarz =-=[12]-=- and Manin-Marcolli [37], of enlarging the boundary of modular curves with a noncommutative space that accounts for the degeneration of elliptic curves to noncommutative tori. The GL2-system described... |

312 | K–Theory for Operator Algebras
- Blackadar
- 1986
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Citation Context ...is then given by the formula ||a|| = (Spectral radius(a ∗ a)) 1/2 where the spectral radius is taken inside A (or � A if A is not unital). For such pre-C ∗ - algebras A (called local C ∗ -algebras in =-=[51]-=-) the following holds: Proposition 7. Let A be a pre-C ∗ -algebra; then: 1) Any Fredholm module (H, F ) over A extends by continuity to a Fredholm module over the associated C ∗ -algebra A. 2) The inc... |

271 |
Local Quantum Physics
- Haag
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Citation Context ...oup of GL2 by the multiplicative group Q\Lambdasacts as symmetries of the system, and the action is implemented by endomorphisms, as in the theory of superselection sectors of Doplicher-Haag-Roberts (=-=[16]-=-). It is this symmetry which is spontaneously broken below the critical temperature T = 12. The partition function of the GL2 system is i(fi)i(fi \Gammas1), for fi = 1=T , and the system exhibits thre... |

250 |
Deformation Theory and Quantization
- Bayen, Flato, et al.
- 1978
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Citation Context ...is none other than the indication of the existence of a deformation with one parameter ( h here) of the algebra of functions into a noncommutative algebra. I refer the reader to the literature ([13], =-=[37]-=-, [38], [169], [182], [220], [368], [474], [570]) for a description of the results of this theory. 2. Statistical State of a Macroscopic System and Quantum Statistical Mechanics A cubic centimeter of ... |

239 | Simple C ∗ -algebras generated by isometries - Cuntz |

226 |
Optictû Coherence and Quantum Optics
- Mandel, Wolf
- 1995
(Show Context)
Citation Context ... hand side implements this averaging as a renormalization group action. Passing to the limit N ! 1 for the phase states is a delicate process. It is known in the theory of optical coherence (cf. e.g. =-=[33]-=- x10) that one can take such limit only after expectation values have been calculated. The analogy between the laser and the ferromagnet suggests that this limiting procedure should be treated as a ca... |

192 |
Elliptic operators, discrete groups and von Neumann algebras. In Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan
- Atiyah
- 1974
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Citation Context ... the noncompact leaves L of a measured foliation (V, F ), the Atiyah-Singer index theorem for compact manifolds, and is directly along the lines of the index theorem for covering spaces due to Atiyah =-=[21]-=- and Singer [523]. 5.α Transverse measures for foliations. To get acquainted with the notion of transverse measure for a foliation, we shall first describe it in the simplest case: dim F = 1, i.e. whe... |

171 | Non-commutative differential geometry - Connes - 1985 |

163 | A class of c* algebras and topological markov chains - Krieger - 1980 |

144 |
Classifying space for proper actions and K-theory of group C ∗ -algebras
- Baum, Connes, et al.
- 1993
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Citation Context ...eometric group as K∗,τ(BG) when the holonomy groups G x x, x ∈ V , are torsion-free. As in the case of discrete groups, the general case requires more care and will be treated in Section 10 (cf. also =-=[35]-=-). b) Exactly as in Section 7, the Chern character Ch∗ is a rational isomorphism of K∗,τ(BG) with H∗(BG, Q), (if we assume to simplify that τ is oriented, i.e. that the foliation is transversally orie... |

136 |
1970] Tomita’s Theory of Modular Hilbert Algebras and its
- Takesaki
(Show Context)
Citation Context ...iquely defined so that ss(a\Lambda ) = ss(a)\Lambdas(the Hilbert space adjoint), namely f\Lambda (fl) := f(fl\Gamma 1) 8 fl 2 \Gamma 0n\Gamma =\Gamma 0: (1.27) It happens that the time evolution (cf. =-=[54]-=-) of the von Neumann algebra generated by A in the regular representation restricts to the dense subalgebra A. This implies that there is a uniquely determined time evolution oet 2 Aut(A), such that t... |

131 |
algèbres d’opérateurs dans lespace hilbertien (algèbres de von
- Dixmier, Les
- 1969
(Show Context)
Citation Context ... state on the algebra M is given by a linear form ϕ(A) = 〈Aξ, ξ〉 , where ξ is a vector of length 1 in the Hilbert space. Tomita’s theory, which has as an ancestor the notion of quasi-Hilbert algebra (=-=[172]-=-), consists in analyzing, given a von Neumann algebra M on the Hilbert space H and a vector ξ ∈ H such that Mξ and M ′ ξ are dense in H , the following unbounded operator S : Sxξ = x ∗ ξ ∀x ∈ M. This ... |

128 |
Existence of Star-Products and of Formal Deformations of the Poisson Lie Algebra o Arbitrary Symplectic Manifolds
- Wilde, Lecomte
- 1983
(Show Context)
Citation Context ...r than the indication of the existence of a deformation with one parameter ( h here) of the algebra of functions into a noncommutative algebra. I refer the reader to the literature ([13], [37], [38], =-=[169]-=-, [182], [220], [368], [474], [570]) for a description of the results of this theory. 2. Statistical State of a Macroscopic System and Quantum Statistical Mechanics A cubic centimeter of water contain... |

124 |
Algebra extensions and nonsingularity
- Cuntz, Quillen
- 1995
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Citation Context ...Rham homology of S 1 and interpret the formulas as deformations of the algebra � QεA. We obtained these results by direct calculations but they also follow from general results on quasifree algebras (=-=[155]-=- [157]). � .s7. ENTIRE CYCLIC COHOMOLOGY 391 Let us first recall that given an algebra A, the standard bar resolution of the bimodule A over B = A⊗A o is given by the acyclic chain complex (Mk, b) Mk ... |

123 |
C ∗ algèbres et géométrie différentielle
- Connes
- 1980
(Show Context)
Citation Context ...ction, including the construction of the finite projective C ∗ -module E = Ep,q over Aθ which achieves the strong Morita equivalence with Aθ ′ (i.e. Aθ ′∼EndAθ (E)) are due to the author of this book =-=[98]-=-, [96] and were later extended to higher dimensional tori by Rieffel. One first determines the manifold Ep,q = {γ ∈ G ; r(γ) ∈ Np,q , s(γ) ∈ N0,1} . Let us assume that p > 0 to avoid the trivial case ... |

103 |
An amenable equivalence relation is generated by a single transformation. Ergodic Theory Dynamical Systems 1
- Connes, Feldman, et al.
- 1981
(Show Context)
Citation Context ...owed [127] that if Γ is solvable then R is of the form RT . The case that Γ is an amenable group was treated by D. Ornstein and B. Weiss [427] and the definitive answer to the problem was obtained in =-=[120]-=- by J. Feldman, B. Weiss and myself. Theorem 4. [120] R is of the form RT if and only if R is amenable in the sense of Zimmer [597]. R. Zimmer’s definition translates the amenability of the von Neuman... |

103 | Operator Algebras and Quantum - Bratteli, Robinson - 1987 |

102 | On the classification of inductive limits of sequences of semi-simple finite-dimensional algebras - Elliott - 1976 |

101 |
Connes, Hecke algebras, Type III factors and phase transitions with spontaneous symmetry breaking
- Bost, A
- 1995
(Show Context)
Citation Context ...ction In this chapter we shall start by giving a geometric interpretation in terms of the space of commensurability classes of 1-dimensional Q-lattices of the quantum statistical dynamical system (BC =-=[5]-=-). This system exhibits the relation between the phenomenon of spontaneous symmetry breaking and number theory. Its dual system obtained by taking the cross product by the time evolution is basic in t... |

97 | Cyclic cohomology, the Novikov conjecture, and hyperbolic groups, Topology 29 No.3
- Connes, Moscovici
- 1990
(Show Context)
Citation Context ...nding KK-theory was first proved by N. Higson [280]. In our joint work with N. Higson [123], it was shown that, using the theory of deformations and the notion of asymptotic morphisms, originating in =-=[129]-=-, one can define the intersection product of extensions. The resulting theory has a number of technical advantages; in particular, the complete proofs of the main properties of the intersection produc... |

97 | Local observables and particle statistics I - Doplicher, Haag, et al. - 1971 |

93 |
Classification of injective factors
- Connes
- 1976
(Show Context)
Citation Context ...ent that M not satisfy the property Γ of Section 1 ([88], [491]). When M is a factor of type II1, the approximately inner automorphisms of M are characterized by the following equivalence: Theorem 9. =-=[90]-=- Let N be a factor of type II1 with separable predual, acting on the Hilbert space H = L 2 (N, τ), where τ denotes the normalized trace of N. For an automorphism θ ∈ AutN, the following conditions are... |

90 |
Monotone Matrix Functions and Analytic Continuation
- Donoghue
- 1974
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Citation Context ...f(T ) = exp(−T −1 ). (We take, of course, f(0) = 0.) The inverse transformation f −1 is defined for 0 ≤ T ≤ λ < 1 by f −1 (T ) = −(log T ) −1 . This nonlinear transformation is operator monotone (cf. =-=[177]-=-), i.e. 0 ≤ T1 ≤ T2 ≤ 1⇒f −1 (T1) ≤ f −1 (T2), since it is a composition of the operator monotone functions log and −1/x.s8. THE CHERN CHARACTER OF θ-SUMMABLE FREDHOLM MODULES 403 For any T ∈ L(H) , T... |

89 |
1973] Une classification des facteurs de type III
- Connes
(Show Context)
Citation Context ...t do not depend on the choice of ϕ have real significance for M . The crucial result that allowed me to get the classification of factors going is the following analogue of the Radon–Nikod´ym theorem =-=[89]-=-:s3. MODULAR THEORY AND THE CLASSIFICATION OF FACTORS 50 For every pair ϕ, ψ of states on M , there exists a canonical 1-cocycle t→ut, ut1+t2 = ut1σ ϕ t1 (ut2) ∀t1, t2 ∈ R with values in the unitary g... |

88 |
An invitation to C ∗ -algebras
- Arveson
(Show Context)
Citation Context ...mapping. Definition 5. Let A and B be C ∗ -algebras. A linear mapping T : A → B is said to be completely positive if, for every n, the mapping 1 ⊗ T : Mn(A) → Mn(B) is positive. I refer the reader to =-=[15]-=- for further information. The principal interest of this concept is that, although the category of C ∗ -algebras and ∗-homomorphisms has relatively few morphisms, in any case not enough to connect the... |

88 |
On groups of measure preserving transformations
- Dye
- 1963
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Citation Context ...e (X, B) and if µ is a measure quasi-invariant under T , then the orbits of T in X define an equivalence relation RT ; T1 is said to be weakly equivalent to T2 if RT1 is isomorphic to RT2. Theorem 2. =-=[186]-=- Any two ergodic transformations with invariant measure are weakly equivalent. Let T be such a transformation. The state ϕ (associated with the invariant measure) on the von Neumann algebra L ∞ (RT , ... |

87 | Trace formula in noncommutative geometry and the zeros of the Riemann zeta function,” Sel - Connes - 1999 |

85 |
On the heat equation and the index theorem
- Atiyah, Bott, et al.
- 1973
(Show Context)
Citation Context ...e algebra of compact operators given by the tangent groupoid of M (Chapter II). In particular, it is integral, computes the Atiyah-Singer index map (Chapter II Section 5), and using invariant theory (=-=[22]-=-, [227]) one computes its character: Ch∗(C ∞ c (T ∗ M), ∗) = Td(T ∗ M) as the Todd genus of the symplectic manifold T ∗ M ([119]) viewed as a de Rham current on T ∗ M.sCHAPTER 5 Operator algebras The ... |

80 | modules, Topology 3 - Atiyah, Bott, et al. - 1964 |

78 |
An analogue of the Thom isomorphism for crossed products of a C∗-algebra by an action of R
- Connes
(Show Context)
Citation Context ...G)⋊ ˆ G is isomorphic to the noncommutative elementary C ∗ -algebra of compact operators. We shall see in particular how it immediately implies a far-reaching analogue of the Bott periodicity theorem =-=[105]-=- (motivated by [446]). We specialise to the case G = R, the additive group of real numbers, and our first task is to construct a natural map φα : Ki(A)→Ki+1(A⋊αR) for any C ∗ -algebra A with one-param... |

73 |
Stable isomorphism and strong Morita equivalence
- Brown, Green, et al.
- 1977
(Show Context)
Citation Context ...uivalent to the existence of a B-C equivalence bimodule. One can then take E2 = E 1. Let K be the C ∗ -algebra of compact operators on an infinite-dimensional separable Hilbert space. Theorem 8. [69] =-=[71]-=- Let B and C be two separable C ∗ -algebras (more generally, two C ∗ -algebras with countable approximate units). Then B is strongly Morita equivalent to C iff B⊗K is isomorphic to C⊗K. Of particular ... |

71 |
Cohomologie cyclique et foncteurs Ext n
- Connes
- 1983
(Show Context)
Citation Context ...rus, BS 1 = P∞(C) and H ∗ S 1(pt) is a polynomial ring in one generator of degree 2. The formal analogy between cyclic cohomology and S 1 -equivariant cohomology is well understood from the equality (=-=[111]-=-) BΛ = BS 1s2. EXAMPLES 219 where Λ is the small category introduced in [111] which governs cyclic cohomology (cf. Appendix A) through the equality HC ∗ (A) = Ext ∗ Λ(A ♮ , C ♮ ) where the functor A→A... |

70 | Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem - Adamjan, Arov, et al. - 1971 |

70 |
Cyclic cohomology and the transverse fundamental class of a foliation
- Connes
- 1983
(Show Context)
Citation Context ...to C such that: a) If n is even and e ∈ Proj Mq (Domτ) then ϕ([e]) = (τ # Tr)(e, . . . , e). b) If n is odd and u ∈ GLq ( Domτ) then ϕ([u]) = (τ # Tr)(u −1 , u, u −1 u, . . . , u −1 , u). We refer to =-=[99]-=- for the proof. Note that by [99], τ extends uniquely to an n-trace with domain the closure of Domτ under holomorphic functional calculus in B. In particular, if τ and τ ′ agree on a common dense doma... |

70 | Von Neumann Algebras
- Jones
- 1990
(Show Context)
Citation Context ...te using Theorem 7.3. Thus, the conclusion follows from Theorem 7.6. Next, recall that a (discrete) group Γ is said to have Kazhdan’s property T if the trivial representation is isolated. Theorem 22. =-=[125]-=- Let Γ be a discrete group with infinite conjugacy classes. Then Γ has property T iff the identity correspondence of the II1 factor R(Γ) is isolated. Using the above functor it is trivial that if Γ do... |

69 |
Relative entropy of states of von Neumann algebras
- Araki
- 1975
(Show Context)
Citation Context ...ase (nonunimodular) given in [112] and developed in [134], which I shall now describe. Let M be a not necessarily finite von Neumann algebra. The functional S(ρ1, ρ2) defined above still has meaning (=-=[8]-=-). It is called the relative entropy. However, ρ1 and ρ2 are no longer positive elements of M but are positive linear forms ϕ1, ϕ2 ∈ M∗. The convexity of this functional S(ϕ1, ϕ2) remains true in full... |

67 |
C ∗ -algèbres et leurs représentations, GauthierVillars
- Dixmier, Les
- 1969
(Show Context)
Citation Context ...ose representations are exactly the unitary representations of Γ. The reduced dual � Γr of Γ is the space of irreducible representations of Γ which are weakly contained in the regular representation (=-=[173]-=-). It is the space of irreducible representations of the C ∗ -algebra C ∗ r (Γ) which is the norm closure of CΓ in ℓ 2 (Γ). To the inclusion � Γr⊂ � Γ corresponds a surjection of C ∗ -algebras C ∗ (Γ)... |

66 |
Intermediate spaces and interpolation, the complex method
- CALDERÓN
- 1964
(Show Context)
Citation Context ...ns of interpolation spaces F (B0, B1) from a pair (B0, B1) of Banach spaces that are continuously embedded in a locally convex vector space. The first functorial construction is complex interpolation =-=[78]-=-. For any θ ∈ [0, 1] one defines a Banach space B = [B0, B1]θ as the space of values f(θ), where f is a holomorphic function in the strip D = {z ∈ C; ℜz ∈ [0, 1]}, with values in B0 + B1 (viewed as a ... |

66 |
The action functional in noncommutative geometry
- Connes
- 1988
(Show Context)
Citation Context ...M) ⊂ T ∗ (M) with its induced volume element: 1 Res(T ) = n(2π) n � traceE(σ)ds. The equality of Proposition 5 can be proved using Proposition 4, or using the general properties of the Dixmier trace (=-=[101]-=-). It is important for our later purposes that the Wodzicki residue continues to make sense for pseudo-differential operators of arbitrary order [589]. It is the unique trace on the algebra of pseudo-... |

63 | Supersymmetry and the Atiyah-Singer index theorem - Alvarez-Gaumé - 1983 |

63 | Outer conjugacy classes of automorphisms of factors - Connes - 1975 |

63 |
Cours d’Arithmetique,” Presses Universitaires de
- SERRE
(Show Context)
Citation Context ...he Riemann i-function. 2) For all fi ? 2, Z\Gamma 1 'fi is a KMSfi state on C\Lambda (M2(Z)+). Proof. Any sublattice L ae Z2 is uniquely of the form L = ^ a b 0 d * Z2, where a; d * 1, 0 ^ b ! d (cf. =-=[50]-=- p. 161). Thus, the type II1 dimension of the action of \Gammasin the subspace of H spanned by the "m with Det m = N is the same as the cardinality of the quotient of fm 2 M2(Z)+; Det m = N g by \Gamm... |

61 |
Stable isomorphism of hereditary subalgebras of C
- Brown
(Show Context)
Citation Context ...is equivalent to the existence of a B-C equivalence bimodule. One can then take E2 = E 1. Let K be the C ∗ -algebra of compact operators on an infinite-dimensional separable Hilbert space. Theorem 8. =-=[69]-=- [71] Let B and C be two separable C ∗ -algebras (more generally, two C ∗ -algebras with countable approximate units). Then B is strongly Morita equivalent to C iff B⊗K is isomorphic to C⊗K. Of partic... |