### Citations

1278 |
Operator Algebras and Quantum Statistical Mechanics.
- Bratteli, Robinson
- 1979
(Show Context)
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... |

924 | Equilibrium States and the Ergodic theory of Anosov Di+eomorphisms. - BOWEN - 1975 |

793 |
Homological algebra
- Cartan, Eilenberg
- 1956
(Show Context)
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... |

732 |
The index of elliptic operators
- Atiyah, Singer
- 1971
(Show Context)
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... |

558 | Operator Algebras,
- Blackadar
- 2006
(Show Context)
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... |

344 |
Deformation theory and quantization
- Bayen, Flato, et al.
- 1978
(Show Context)
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 ... |

277 |
Elliptic operators, discrete groups and von Neumann algebras.
- Atiyah
- 1976
(Show Context)
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... |

235 | Non-commutative differential geometry, - Connes - 1985 |

231 |
Classifying space for proper actions and K-theory of group C∗-algebras
- Baum, Connes, et al.
- 1993
(Show Context)
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... |

170 |
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... |

168 |
C∗-algèbres et géométrie differentielle,
- 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 ... |

149 |
Intermediate spaces and interpolation, the complex method,
- Calderon
- 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 ... |

127 |
Stable isomorphism and strong Morita equivalence of C*-algebras,
- 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 ... |

125 |
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 ... |

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

118 |
Stable isomorphism of hereditary subalgebras of C∗-algebras”,
- Brown
- 1977
(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... |

115 |
Une classification des facteurs de type III
- Connes
- 1973
(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... |

112 |
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... |

107 | Elliptic operators and compact groups, - Atiyah - 1974 |

105 | Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem. - ADAMJAN, AROV, et al. - 1971 |

103 | The index of elliptic operators on compact manifolds. - Atiyah, Singer - 1963 |

102 |
Théorie bivariante de Kasparov et opérateurs non bornés dans les C ∗ -modules hilbertiens
- Baaj, Julg
- 1983
(Show Context)
Citation Context ... ∈ C. Then every normalized Kasparov A⊗C1-C-bimodule is of µ λ � � �� 0 −i the form H⊗K , F ⊗ for a unique odd Fredholm module (H, F ) i 0 over A. The proof is straightforward. The next result due to =-=[30]-=- gives a useful construction of Kasparov bimodules from unbounded operators in C ∗ -modules as well as a useful notion: Definition 14. Let B be a C ∗ -algebra and E a C ∗ -module over B; then an unbou... |

97 |
Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one
- Cowling, Haagerup
- 1989
(Show Context)
Citation Context ... if Γ2 is discrete in SL(2, R), then there does not exist any homomorphism of R(Γ1) into R(Γ2); 2) one can distinguish countably many discrete subgroups Γ of Sp(n, R) by their associated factors R(Γ) =-=[84]-=-. Problem 2. Determine the fundamental group of R(Γ) for Γ rigid.sCHAPTER 6 The metric aspect of noncommutative geometry The geometric spaces of Gauss and Riemann are defined as manifolds in which the... |

94 | From groups to groupoids: a brief survey, - Brown - 1987 |

93 |
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... |

92 |
Cyclic cohomology and the transverse fundamental class of a foliation. In: Geometric methods in operator algebras,
- 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... |

92 |
The action functional in noncommutative geometry,”Communications inMathematical
- 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-... |

91 | Outer conjugacy classes of automorphisms of factors,” Annales Scientifiques de l’Ecole Normale Superieure, - Connes - 1975 |

89 | Supersymmetry and the Atiyah-Singer index theorem,” - Alvarez-Gaume - 1983 |

73 |
Global theory of elliptic operators
- ATIYAH
- 1970
(Show Context)
Citation Context ...perties of the triple (A, H, F ), or equivalently of the representation of A in the pair (H, F ). The following notion of Fredholm representation, or equivalently of Fredholm module, is due to Atiyah =-=[18]-=-, Mishchenko [395], Brown, Douglas and Fillmore [70], and Kasparov [335]. Definition 1. Let A be an involutive algebra (over C). Then a Fredholm module over A is given by: 1) an involutive representat... |

68 |
Connes, A.: Geometric K-theory for Lie groups and foliations
- Baum
(Show Context)
Citation Context ...ard techniques of algebraic topology, and a map µ, the analytic assembly map, from the geometric group to the K-group K(X) of the C ∗ -algebra associated to X µ : K∗(BX)→K(X). The general conjecture (=-=[32]-=-) that this map µ is an isomorphism is a guiding principle of great relevance. The notations BX and K∗(BX) will be explained later, but, essentially, BX stands for the homotopy type of an ordinary spa... |

64 |
A geometric construction of the discrete series for semisimple Lie groups
- Atiyah, Schmid
- 1977
(Show Context)
Citation Context ...x C) shows that the analytic assembly map is an isomorphism K ∗ top(G)→ µ K∗(C ∗ (G)).s10. THE ANALYTIC ASSEMBLY MAP AND LIE GROUPS 154 For semisimple Lie groups the results of [497], [433], [24] and =-=[25]-=- on the geometric realisation of all discrete-series representations by Dirac induction, together with [334] and [132] Section 7.5, suggested that µr should be an isomorphism µr : K ∗ top(G)→K∗(C ∗ r ... |

55 |
Relative Hamiltonian for faithful normal states of a von Neumann algebra
- Araki
- 1973
(Show Context)
Citation Context ... the Radon–Nikod´ym derivative (dψ/dϕ) ; 2) in the case of statistical mechanics, with the difference of the Hamiltonians corresponding to two equilibrium states, or the relative Hamiltonian of Araki =-=[9]-=-. It follows that, given a von Neumann algebra M , there exists a canonical homomorphism δ of R into the group OutM = AutM/InnM (the quotient of the automorphism group by the normal subgroup of inner ... |

55 |
Explicit functional determinants in four dimensions
- Branson, Oersted
(Show Context)
Citation Context ...d in local terms, and defines an elliptic differential operator P of order 4 on Σ such that � (4.30) I(X) = P (X) X dv. Σ This operator P is (up to normalization) equal to the Paneitz operator P (cf. =-=[72]-=-) already known to be the analogue of the scalar Laplacian in 4-dimensional conformal geometry. Equation 7 uses the volume element dv so that P itself is not conformally invariant. Its principal symbo... |

51 |
Almost periodic states and factors of type III1
- Connes
- 1974
(Show Context)
Citation Context ...ple, for the hyperfinite factor R, Inn R = AutR. More precisely, assuming M finite, in order that InnM be closed in AutM it is necessary and sufficient 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 predua... |

50 |
K-Theory of C ∗ -algebras in Solid State Physics
- Bellissard
- 1986
(Show Context)
Citation Context ...observables appropriate for the computation of the Hall conductivity, besides containing f(H), f ∈ C0(R) and f(J), f ∈ C0(R2 ), should be invariant under the automorphism group αs of W . In fact (cf. =-=[43]-=-), it is not difficult to see that the C∗-algebra A generated by the αs(f(H)) does contain the functions of the current, and is thus the natural algebra of observables for our problem. On A⊂W we have ... |

50 |
A factor of type II1 with countable fundamental group,
- Connes
- 1980
(Show Context)
Citation Context ...N) = {Modθ; θ ∈ AutÑ} is obviously an algebraic invariant of N. In fact, an example of a type II1 factor not anti-isomorphic to itself was not obtained until long after ([87]), nor was the existence (=-=[94]-=-) of a type II1 factor whose group F is distinct from R ∗ + (the only calculable examples always gave F = R ∗ +). Finally, to conclude this review of the results of Murray and von Neumann, we mention ... |

49 |
A survey of foliations and operator algebras”, Operator algebras and applications
- Connes
- 1982
(Show Context)
Citation Context ...n 5.3, but we shall describe it in detail. We shall assume for notational convenience that the manifold G is Hausdorff, but as this fails to be the case in very interesting examples we shall refer to =-=[96]-=- for the removal of this hypothesis. The basic elements of C ∗ r (V, F ) are smooth half-densities with compact supports on G, f ∈ C∞ c (G, Ω1/2 ), where Ω 1/2 γ for γ ∈ G is the one-dimensional compl... |

48 |
Quantum Mechanics as a Deformation of Classical Mechanics.
- Bayen, Flato, et al.
- 1977
(Show Context)
Citation Context ...e 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 water ... |

47 | Cantor spectrum for the almost Mathieu equation, J.Funct.Anal - Béllissard, Simon - 1982 |

46 |
Gauss polynomials and the rotation algebra
- Choi, Elliott, et al.
- 1990
(Show Context)
Citation Context ...− − − − − − Figure 7. Hall current which, when θ is a Liouville number, is invertible for any λ outside a nowhere dense Cantor set K [45] on which the index changes discontinuously. We refer to [402] =-=[83]-=- [45] for more information on the operator T and gap labelling. In [83] the following q-analogue of the binomial formula is successfully used to show that the spectrum of T is a Cantor set when θ is a... |

43 |
On the cohomology of operator algebras
- Connes
- 1978
(Show Context)
Citation Context ...roof of Grothendieck’s inequality for arbitrary C ∗ -algebras (which completes the work of Grothendieck and Pisier), succeeded in showing that a C ∗ -algebra is amenable if and only if it is nuclear (=-=[93]-=-, [253], [254]). 8. The Flow of Weights: Mod(M) In this section we shall describe in detail the main invariant of type III factors, the flow of weights. This invariant was discovered for the first tim... |

42 |
The cyclic homology of the groups rings,
- Burghelea
- 1985
(Show Context)
Citation Context ...iate linearization of the nonabelian category of algebras. For group rings, i.e. for algebras over C of the form A = CΓ where Γ is a discrete (countable) group, the following theorem of D. Burghelea (=-=[76]-=-) yields a natural S 1 -space whose S 1 -equivariant cohomology (with complex coefficients) is the cyclic cohomology of A. Recall that the free loop space Y S1 of a given topological space Y is the sp... |

41 |
Cyclic homology and the algebraic K-theory of spaces II
- BURGHELEA, FIEDOROWICZ
- 1986
(Show Context)
Citation Context ... X (or more generally to cyclic topological spaces, called cyclic spaces for short) and denote by |X| the geometric realization of the underlying simplicial set (resp. space). One has Proposition 12. =-=[77]-=- [237] [371] Let X be a cyclic space. Then |X|, the geometric realization of the underlying simplicial space, admits a canonical action of S 1 . One obtains in this way a functor from the category of ... |

40 |
Sur la thèorie non commutative de l’intègration. Algèbres d’opérateurs (Sém., Les Plans-sur-Bex
- Connes
- 1978
(Show Context)
Citation Context ...y, as in [27], yields the Chern character (for simplicity, F is assumed oriented) Ch(σD) ∈ H ∗ (V, Q) as an element of the rational cohomology of the manifold V . We can now state the general result (=-=[95]-=-). Theorem 7. Let (V, F ) be a compact foliated manifold (1), D a longitudinal elliptic operator on V , and X = V/F the space of leaves. a) There exists a Borel transversal B to F such that the bundle... |

38 |
Unbounded negative definite functions.
- Akemann, Walter
- 1981
(Show Context)
Citation Context ...ve on K∗(C ∗ (G)) as soon as G has property T of Kazhdan, which is the case for any semisimple Lie group of real rank ≥ 2. First one has the following characterization of property T : Proposition 22. =-=[6]-=- Let G be a locally compact group. The following properties are equivalent: 1) G has property T of Kazhdan ([347]). 2) There exists an orthogonal projection e ∈ C ∗ (G), e �= 0, such that π(e) = 0 for... |

38 |
Principe d’Oka, K-théorie et systèmes dynamiques non commutatifs
- Bost
- 1990
(Show Context)
Citation Context ...under holomorphic functional calculus then so is Mn(A) in Mn(A) for any n ∈ N.sAPPENDIX C. STABILITY UNDER HOLOMORPHIC FUNCTIONAL CALCULUS 292 We refer to [497] for a detailed proof. One can use (cf. =-=[58]-=-) the following identity in M2(A) as a substitute for the determinant of matrices � �−1 � −1 a11 a12 a11 0 = a21 a22 0 (a22 − a21a −1 11 a12) −1 � � 1 −a12(a22 − a21a −1 11 a12) −1 � � 1 0 0 1 −a21a −... |

37 |
Hausdorff dimension of quasicircles, Inst
- Bowen
- 1979
(Show Context)
Citation Context ...nents U± of the complement of C. The discrete subgroup h(Γ) is a quasi-Fuchsian group and its limit set C is a quasicircle. It is a Jordan curve whose Hausdorff dimension is strictly bigger than one (=-=[64]-=-). Let us choose a coordinate in P1(C) = C ∪ {∞} in such a way that ∞ ∈ Σ− and use the Riemann mapping theorem to provide a conformal equivalence Z : D→Σ+⊂C where D = {z ∈ C; |z| < 1} is the unit disk... |

34 |
On moduli of Kleinian groups
- Bers
(Show Context)
Citation Context ...r with an isomorphism π1(Σ+) = Γ = π1(Σ−) of their fundamental groups corresponding to an orientation reversing homeomorphism Σ+→Σ−. We recall the joint uniformization theorem of L. Bers: Theorem 12. =-=[47]-=- With the above notation there exists an isomorphism h : Γ→P SL(2, C) of Γ with a discrete subgroup of P SL(2, C) whose action on P1(C) = S 2 has a Jordan curve C as limit set and is proper with quoti... |

33 |
Parallelizability of proper actions, global K-slices and maximal compact subgroups
- Abels
- 1974
(Show Context)
Citation Context ...intractable. Relying on Tomita’s theory of modular Hilbert algebras and on the earlier work of Powers, Araki, Woods and Krieger, I showed in my thesis that type III is subdivided into types IIIλ, λ ∈ =-=[0, 1]-=- and that a factor of type IIIλ, λ �= 1, can be reconstructed uniquely as a crossed product of a type II von Neumann algebra by an automorphism contracting the trace. This result was then extended by ... |

33 |
Infinite Hankel matrices and generalized problems of Caratheodory-Fejer
- AROV, KREIN
- 1968
(Show Context)
Citation Context ...ous increasing maps from S 1 to S 1 , of degree 1 and such that ϕ(Z/n + 1)⊂Z/m + 1.s3 2 APPENDIX A. THE CYCLIC CATEGORY Λ 283 4 5 1 0 Figure 3. f ∈ Hom(Λ5, Λ3), ˜ f(j) = 1 ∀j ∈ Z/6Z, f is constant on =-=[4, 3]-=- and equal to 1; it makes a complete turn on [3, 4] For λ ∈ Z/n + 1 the value of ϕ(λ) ∈ Z/m + 1 is independent of the choice of ϕ in the homotopy class f and gives a well defined increasing map � f : ... |

32 | The index theorem for homogeneous differential operators - Bott - 1965 |

32 |
Sur la classification des facteurs de type
- Connes
- 1975
(Show Context)
Citation Context ...projections P ). The group F (N) = {Modθ; θ ∈ AutÑ} is obviously an algebraic invariant of N. In fact, an example of a type II1 factor not anti-isomorphic to itself was not obtained until long after (=-=[87]-=-), nor was the existence ([94]) of a type II1 factor whose group F is distinct from R ∗ + (the only calculable examples always gave F = R ∗ +). Finally, to conclude this review of the results of Murra... |

29 |
A classification of factors
- Araki, Woods
- 1973
(Show Context)
Citation Context ...n the state ϕ and the one-parameter group (σ ϕ −t) of Tomita’s theorem is exactly the KMS condition for �β = 1 . These results, as well as the work of R. Powers [453], and of H. Araki and E. J. Woods =-=[12]-=- on factors that are infinite tensor products, proved to be of considerable importance in setting in motion the classification of factors. The point of departure of my work on the classification of fa... |

27 |
K-homology and index theory’, Operator algebras and applications
- Baum, Douglas
- 1982
(Show Context)
Citation Context ...(cf. [136]) to the case of non-compact manifolds. The element σ, which we shall not use, is, however, no longer an element of K(F ). b) The Todd genus Td(F ) of a Spin c vector bundle is defined (cf. =-=[36]-=- p.136 and [366]) by the formula Td(F ) = e c/2 � A(F ), where c is the first Chern class of the line bundle ℓ associated to the Spin c structure. 6.δ Wrong-way functoriality for K-oriented maps. Let ... |

26 |
Geometry of Yang-Mills Fields
- Atiyah
- 1979
(Show Context)
Citation Context ...−1γ(W1) ⊗ 1 i−1γ(W2) ⊗ 1 −ϕ ′ 2γ5 ⊗ Md ϕ ′ 1γ5 ⊗ Mu iγ(W 2) ⊗ 1 i−1 ⎤ ⎥ , ⎥ ⎦ γ(W 1) ⊗ 1 where A = � fsdf ′ s is a C-valued 1-form on V , and W1 + W2j = W = � qsdq ′ s is an H-valued 1-form on V (cf. =-=[20]-=-). Also, ϕj and ϕ ′ j are complex-valued functions on V given by the same formulas as above for the finite geometry, namely, ϕ1 = � fs(α ′ s − f ′ s), ϕ2 = � fsβ ′ s, ϕ ′ 1 = � � αs(f ′ s − α ′ s) + β... |

26 |
Ordinary quantum Hall effect and non-commutative cohomology,” in Localization in disordered systems
- Bellissard
- 1988
(Show Context)
Citation Context ...hematician, S. Novikov [418], and a physicist, D. Thouless [557]. The use of noncommutative geometry, which makes it possible to eliminate the rationality hypothesis of [418], is due to J. Bellissard =-=[44]-=-. We shall follow his work in sections β) and γ) below. 6.α Elliptic theory on T 2 θ . We have already met the noncommutative torus T2 θ in Chapter II arising from the Kronecker foliation (Section 8 β... |

26 |
Inductive limits of finite dimensional C -algebras
- Bratteli
- 1972
(Show Context)
Citation Context ...1 . The C ∗ -algebra A is then the inductive limit of the finite-dimensional algebras An, and one can calculate its invariants for the classification of Bratteli, Elliott, Effros, Shen and Handelman (=-=[65]-=-, [193], [194], [191]) of these particular C ∗ -algebras. The invariant to be calculated, due to G. Elliott, is an ordered group, namely the group K0(A), described earlier, generated by the stable iso... |

25 |
Caractérisation des espaces vectoriels ordonnés sous-jacents aux algèbres de von
- Connes
- 1974
(Show Context)
Citation Context ...er, one can show that any element of L 2 (M) admits a canonical left polar decomposition of the form (5.45) δ = u ϕ 1/2 where u is a partial isometry with initial support u ∗ u = s(ϕ), ϕ ∈ M + ∗ (cf. =-=[86]-=- [251]). The canonical involution J of L 2 (M), given by (5.46) Jδ = δ ∗ ∀δ ∈ L 2 (M) is isometric by (10), and it obviously exchanges the left and right actions of M on L 2 (M). Finally, L 2 (M) is e... |

23 | The exponent of convergence of the Poincaré series - Beardon |

20 |
Cyclic homology and equivariant theories
- Brylinski
- 1987
(Show Context)
Citation Context ...scribe the cyclic cohomology of the crossed product algebra A = C∞ c (V )⋊Γ. The analogues of the above results of D. Burghelea are due to Feigin and Tsygan [206], V. Nistor [414], and J.L. Brylinski =-=[74]-=- [75]. We shall describe an earlier result [99], namely that the periodic cyclic cohomology H∗ (A) contains as a direct factor the twisted cohomology groups H∗ τ (VΓ, C). We adopt here the notation of... |

18 |
Noncommutative differential geometry Part II, de Rham homology and noncommutative algebra, Inst. Hautes Etudes Sci., Bures-Sur-Yvette
- Connes
- 1983
(Show Context)
Citation Context ...n the right are invariant under cyclic permutations, since these do not alter the conjugacy class of the product g0 · · · gn.s3. CYCLIC COHOMOLOGY (CHAPTER III) 23 This formula was initially found in =-=[102]-=- and later extended by M. Karoubi and D. Burghelea, who computed the cyclic cohomology of group rings (cf. Chapter III Section 2). We have used in a crucial manner the above cyclic cocycles on group r... |

17 |
Poids Associe a une algebre Hilbertienne a Gauche
- COMBES
- 1971
(Show Context)
Citation Context ...utative analogue of infinite positive measures. The initial data for noncommutative integration is thus a pair (M, ϕ) consisting of a von Neumann algebra M and a weight ϕ on M in the following sense (=-=[85]-=-, [435], [252]): Definition 1. A weight on a von Neumann algebra M is an additive, positively homogeneous mapping ϕ of M+ into R+ = [0, +∞]. We say that: a) ϕ is semifinite if {x ∈ M+; ϕ(x) < ∞} is σ(... |

17 | Classification des facteurs. In Operator algebras and applications, Part 2 - Connes - 1980 |

16 | Expansional in Banach algebras - Araki - 1973 |

16 |
Covariance and Geometrical Invariance in ∗-Quantization.
- Arnal, Cortet, et al.
- 1983
(Show Context)
Citation Context ...space 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 centimet... |

15 | Notes on extensions of C ∗–algebras - Arveson |

15 |
Extensions of C* -algebras and K-homology
- Brown, Douglas, et al.
- 1977
(Show Context)
Citation Context ... the representation of A in the pair (H, F ). The following notion of Fredholm representation, or equivalently of Fredholm module, is due to Atiyah [18], Mishchenko [395], Brown, Douglas and Fillmore =-=[70]-=-, and Kasparov [335]. Definition 1. Let A be an involutive algebra (over C). Then a Fredholm module over A is given by: 1) an involutive representation π of A in a Hilbert space H; 293s2) an operator ... |

15 | Spectral sequence and homology of currents for operator algebras. Oberwolfach Tagungsber., 41/81, Funktionalanalysis und C∗-Algebren - Connes - 1981 |

14 | character for discrete groups. A fête of topology, 163–232 - Chern - 1988 |

13 |
Leafwise homotopy equivalence and rational Pontrjagin classes
- Baum, Connes
- 1985
(Show Context)
Citation Context ...ature operator. This question is the exact analogue of the question of the homotopy invariance of the Γ-invariant signature for covering spaces, which was proved by Mishchenko and Kasparov. One gets (=-=[33]-=- [279]): Proposition 5. Let (V, F ) be a compact foliated manifold with F even-dimensional and oriented. Let D be the leafwise signature operator. Then its analytic index Inda(D) ∈ K0(C ∗ (V, F )) is ... |

13 | la Harpe: Moyennabilité intérieure des groupes: définitions et exemples, L’Enseignement - Bédos, de - 1986 |

13 |
Uniformization by Beltrami Equations
- Bers
- 1961
(Show Context)
Citation Context ... von Neumann algebra of 2×2 matrices � � f u a = v g where f and g are measurable bounded functions on V = P1(C), and u and v are measurable bounded Beltrami differentials: u(z, z)dz/dz, v(z, z)dz/dz =-=[46]-=-. In particular, an odd element µ ∈ M, µ = µ ∗ with �µ� < 1, corresponds exactly to a single Beltrami differential v(z, z)dz/dz, with �v�∞ < 1 and v measurable, by the equality � � ∗ 0 v µ = . v 0 Now... |

12 |
On characteristic classes in the framework of Gelfand-Fuks cohomology, “Analyse et Topologie” en l’Honneur de Henri Cartan
- Bott
- 1974
(Show Context)
Citation Context ... c (V ). Since the homotopy quotient VΓ is the geometric realization of a simplicial manifold (cf. Appendix A) we can describe the twisted cohomology H ∗ τ (VΓ) as the cohomology of a double complex (=-=[61]-=- Theorem 4.5). More explicitly, we can view the space EΓ as the geometric realization of the simplicial set eΓ where (eΓ)n = Γ n+1 and (3.4) di(g0, . . . , gn) = (g0, . . . , gi ∨ , . . . , gn) ∀i = 0... |

11 |
Stable Homotopy and Generalized Homology, Univ.
- Adams
- 1974
(Show Context)
Citation Context ...e space obtained from X×Y by smashing the subspace X ∨Y to a point and is called the reduced join, or smash product, of X and Y . In particular X∧S 1 = SX is called the suspension of X. Definition 9. =-=[4]-=- A spectrum Σ is a sequence of spaces Σn and maps σn : SΣn→Σn+1 such that each Σn is a CW -complex and the maps σn are embeddings of CW -complexes. The generalized homology theory h∗ associated to a s... |

11 |
The harmonic analysis of automorphism groups. Operator algebras and applications, Part I
- Arveson
- 1980
(Show Context)
Citation Context ...∗ (Γ) is independent of the choice of the cocycle c. The construction of Ω ∗ (Γ) from the group ring A = CΓ is a special case of the universal differential algebra Ω ∗ (A) associated to an algebra A (=-=[16]-=- [325]), which we briefly recall.s1. CYCLIC COHOMOLOGY 190 Proposition 3. Let A be a not necessarily unital algebra over C. 1) Let Ω 1 (A) be the linear space � A⊗CA, where � A = A⊕C1 is the algebra o... |

11 |
The regular representation of restricted direct product groups
- Blackadar
- 1977
(Show Context)
Citation Context ...trace. ϕβ(x) = Trω(e −H x) ∀x ∈ C ∗ (N ∗ ) d) For 0 < β ≤ 1, ϕβ is a factor state of type III1 and the associated factor is the factor R∞ of Araki-Woods. Statement d) for β = 1 is due to B. Blackadar =-=[54]-=-. We refer to Chapter IV for the definition of the Dixmier trace, whose general properties make it clear that the equality c) defines a KMS1 state.s11. HECKE ALGEBRAS, TYPE III FACTORS AND PRIME NUMBE... |

11 | Effros, Separable nuclear C ∗ -algebras and injectivity - Choi, G - 1976 |

10 |
Estimates of the spectrum of a difference of fractional powers of selfadjoint operators
- Birman, Koplienko, et al.
- 1975
(Show Context)
Citation Context ...the map A→|A| p with respect to the norms σN. We first recall that by [164] and [356] the map A→|A| is a Lipschitz map from L p,∞ to itself provided that p > 1. We shall need the following lemma (cf. =-=[50]-=-): Lemma 9. Let α ∈ ]0, 1[. There exists Cα < ∞ such that for any compact operators A and B on H one has 1 N σN (|A| α − |B| α � 1 ) ≤ Cα N σN(A �α − B) . As an immediate corollary of this lemma we ge... |

10 |
Produit eulérien et facteurs de type
- Bost, Connes
- 1992
(Show Context)
Citation Context ... In this section we shall discuss an example of a quantum statistical mechanical system, arising from the theory of prime numbers, which exhibits a phase transition with spontaneous symmetry breaking =-=[60]-=-. The original motivation for these results comes from the work of B. Julia [312] (cf. also [530]). 11.α Description of the system and its phase transition. Let us first recall our discussion of quant... |

8 | Golden-Thompson and Peierls-Bogolubov inequalities for a general von Neumann algebra - Araki - 1973 |

8 | The Hausdorff dimension of singular sets of properly discontinuous groups - Beardon - 1966 |

8 | Cyclic homology and the Selberg principle - Blanc, Brylinski - 1992 |

6 |
On induced representations of discrete groups
- Binder
- 1993
(Show Context)
Citation Context ...′ are viewed as Γ0-bi-invariant functions on Γ with finite support in Γ0\Γ/Γ0. To complete H to a C ∗ -algebra we just close it in norm in the following regular representation of H in ℓ 2 (Γ0\Γ) (cf. =-=[49]-=-).s11. HECKE ALGEBRAS, TYPE III FACTORS AND PRIME NUMBERS 527 Proposition 3. Let Γ0⊂Γ be an almost normal subgroup of the discrete group Γ. Then the following defines an (involutive) representation λ ... |

5 | Localization and Space in Quantum Physics - Bacry - 1988 |

4 |
On a tensor product C ∗ -algebra associated with the free group on two generators
- Akemann, Ostrand
- 1975
(Show Context)
Citation Context ...have property Γ ⇔ C ∗ (M, M ′ ) contains the ideal k(H) of compact operators. An example of a type II1 factor for which C ∗ (M, M ′ ) contains the ideal k(H) was given by C. Akemann and P. Ostrand in =-=[5]-=-. Thus, combined with the corollary, the theorem shows that every semi-discrete factor of type II1 has the property Γ. In fact, Effros and Lance proved in their paper [192] that every Araki–Woods fact... |

4 | The von Neumann algebra of a foliation”, Mathematical problems in theoretical physics - Connes - 1977 |

3 | A simple C ∗ -algebra with no nontrivial projections - Blackadar - 1980 |

3 | Some examples of Hochschild and cyclic homology - Brylinski - 1986 |

2 |
Localization of electrons with spin-orbit or magnetic interactions in a two-dimensional disordered crystal, Phys. Rev. B33
- Bellissard, Grempel, et al.
- 1986
(Show Context)
Citation Context ...ribute to the conductivity. This is why one observes experimentally the plateaus of the conductivity as a function of the Fermi level. This qualitative idea is put on a rigorous mathematical basis by =-=[39]-=- using [213] and [212] for realistic models. Now the meaning of the localization of the electron states with energies in a small interval ]µ − ε, µ + ε[ around the Fermi level µ is the following: Ts6.... |

2 | On Hilbert’s 22nd problem. Mathematical developments arising from Hilbert problems - Bers - 1976 |

2 |
A simple unital projectionless C ∗ algebra
- Blackadar
(Show Context)
Citation Context ... = He−2st shows that Gs(Λ) = e−2sΛ for all s ∈ R. Thus τ ◦θs = e−2sτ, and so τ(e) = 0 for any self-adjoint idempotent e. So C∗ r (V, F ) does not have any non-zero idempotent though it is simple (cf. =-=[52]-=- for the first example of such a C∗-algebra). We shall describe in Section 9 another example with a unital C∗-algebra. 8.γ The analytic assembly map µ : K∗,τ(BG)→K(C ∗ (V, F )). In this section we sha... |

2 |
Almost commuting algebras, K-theory and operator algebras
- Carey, Pincus
- 1977
(Show Context)
Citation Context ...high enough order belong to L 1 and hence can be integrated according to the above formulas. The above idea is directly in line with the earlier works of Helton and Howe [273] [274], Carey and Pincus =-=[79]-=- and Douglas and Voiculescu [181] in the case when A is commutative. But even in that case it improves on earlier work since the above cyclic cocycle determines all the lower-dimensional homology clas... |

1 |
A new proof of the regularity theorem for invariant eigendistributions on semisimple Lie groups
- Atiyah, Schmid
(Show Context)
Citation Context ... (Appendix C) shows that the analytic assembly map is an isomorphism K ∗ top(G)→ µ K∗(C ∗ (G)).s10. THE ANALYTIC ASSEMBLY MAP AND LIE GROUPS 154 For semisimple Lie groups the results of [497], [433], =-=[24]-=- and [25] on the geometric realisation of all discrete-series representations by Dirac induction, together with [334] and [132] Section 7.5, suggested that µr should be an isomorphism µr : K ∗ top(G)→... |

1 |
Multiplicateurs non bornés. Thèse
- Baaj
- 1980
(Show Context)
Citation Context ...d densely defined operators T and T ∗ on E commuting with the right action of B such that 1) 〈T ξ, η〉 = 〈ξ, T ∗ η〉 ∀ξ ∈ DomT , η ∈ DomT ∗ 2) 1 + T ∗ T is surjective. We refer to the thesis of S. Baaj =-=[29]-=- for this notion introduced in [30]. Theorem 15. (cf. [30]) a) Let E be an A-B C ∗ -bimodule with Z/2-grading γ and D an unbounded selfadjoint endomorphism of E anticommuting with γ and such that α) {... |

1 | A transformational approach to case based synthesis - unknown authors - 1991 |

1 |
Witt group of commutative involutive Banach algebras
- Bost
(Show Context)
Citation Context ...etion of B for the seminorm ||x|| C ∗ = Sup{||π(x)|| ; π a unitary representation of B in a Hilbert space}. This equality was proved by J.-B. Bost for arbitrary commutative involutive Banach algebras =-=[59]-=-. 2. Elementary Examples of Quotient Spaces We shall formulate the algebraic counterpart of the geometric operation of forming the quotient of a space by an equivalence relation, and show how to handl... |

1 |
Topological aspects of holonomy groupoids
- Brown
(Show Context)
Citation Context ...ology and the fibers Gx = r−1 {x}, x ∈ G (0) , of the map r, are discrete. This is what allows us to define the convolution algebra very simply by (a ∗ b)(γ) = � a(γ1)b(γ2). γ1◦γ2=γ We refer to [470] =-=[68]-=- for the general case of locally compact groupoids. Our next example of the tangent groupoid of a manifold will be easier to handle than the general case; though no longer discrete, it will be smooth ... |

1 |
Cyclic cohomology of smooth discrete groupoids
- Brylinski, Nistor
- 1993
(Show Context)
Citation Context ...e the cyclic cohomology of the crossed product algebra A = C∞ c (V )⋊Γ. The analogues of the above results of D. Burghelea are due to Feigin and Tsygan [206], V. Nistor [414], and J.L. Brylinski [74] =-=[75]-=-. We shall describe an earlier result [99], namely that the periodic cyclic cohomology H∗ (A) contains as a direct factor the twisted cohomology groups H∗ τ (VΓ, C). We adopt here the notation of Sect... |

1 | Homologie cyclique - Cartier - 1984 |