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## The Equivariant Noncommutative Atiyah-Patodi-Singer Index Theorem ∗ (2006)

Citations: | 1 - 1 self |

### Citations

620 | Spectral asymmetry in Riemannian geometry, I, II and III,
- Atiyah, Patodi, et al.
- 1975
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Citation Context ... this paper, we extend this theorem to the equivariant case. Keywords: Equivariant total eta invariants; Clifford asymptotics; C(1)-Fredholm module; superconnection. MSC: 58J20, 19K 1 Introduction In =-=[APS]-=-, Atiyah-Patodi-Singer proved their famous Atiyah-Patodi-Singer index theorem for manifolds with boundary. In [D], Donnelly extended this theorem to the equivariant case by modifying the Atiyah-Patodi... |

422 |
Trace ideals and their applications
- Simon
- 1979
(Show Context)
Citation Context ...al case by using the method in [CH] and [F]. Firstly, recall some Lemmas in [CH] and [F]. Let H be a Hilbert space. For q ≥ 0, denote by ||.||q Schatten p-norm on Schatten ideal L p (for details, see =-=[S]-=-). L(H) denotes the Banach algebra of bounded operators on H. Lemma 2.2 ([CH],[F]) (i) Tr(AB) = Tr(BA), for A, B ∈ L(H) and AB, BA ∈ L1 . (ii) For A ∈ L1 , we have |Tr(A)| ≤ ||A||1, ||A|| ≤ ||A||1. (i... |

109 |
Elements of noncommutative geometry. Birkhäuser Adv. Texts, Birkhäuser,
- Gracia-Bondía, Várilly, et al.
- 2001
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Citation Context ...perator and ˜ dg is a bounded operator. Let A = C ∞ (N), then the data (A,H,D,G) defines a finitely (hence θ-summable) equivariant unbounded Fredholm module in the sense of [KL] (for details see [CH],=-=[FGV]-=- and [KL]). Similar to [CH] or [W], for equivariant θ-summable Fredholm module (A,H,D,G), we can define the equivariant cochain ˜ ch G k (tD,D) (k is even) by the formula: ˜ ch G k (tD,D)(f 0 , · · · ... |

79 |
The analysis of elliptic families
- Bismut, Freed
- 1986
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Citation Context ...gher eta invariant in [Wu]) which is the generalization of the classical Atiyah-Patodi-Singer eta invariants [APS] , then proved its regularity by using the Getzler symbol calculus [G1] as adopted in =-=[BF]-=- and computed its radius of convergence. Subsequently, he proved the variation formula of eta cochains, using which he got the noncommutative Atiyah-Patodi-Singer index theorem. In [G2], using superco... |

51 |
Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem.
- Getzler
- 1983
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Citation Context ...iant (called the higher eta invariant in [Wu]) which is the generalization of the classical Atiyah-Patodi-Singer eta invariants [APS] , then proved its regularity by using the Getzler symbol calculus =-=[G1]-=- as adopted in [BF] and computed its radius of convergence. Subsequently, he proved the variation formula of eta cochains, using which he got the noncommutative Atiyah-Patodi-Singer index theorem. In ... |

27 |
Eta invariants for G-spaces
- Donnelly
(Show Context)
Citation Context ...symptotics; C(1)-Fredholm module; superconnection. MSC: 58J20, 19K 1 Introduction In [APS], Atiyah-Patodi-Singer proved their famous Atiyah-Patodi-Singer index theorem for manifolds with boundary. In =-=[D]-=-, Donnelly extended this theorem to the equivariant case by modifying the Atiyah-Patodi-Singer original method. In [Z], Zhang got this equivariant Atiyah-Patodi-Singer index theorem by using a direct ... |

25 |
On the Chern character of a theta-summable Fredholm module
- Getzler, Szenes
- 1989
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Citation Context ... σ[D,p], · · · ,σ[D,p] } {{ } i times ,σ(p − 1 ), σ[D,p], · · · ,σ[D,p] 2 } {{ } (2l − i − j) times , 〉 √ t (g) ⎤⎫ ⎪⎬ ⎥ ⎦ ⎪⎭ . (3.6) The following equality is the equivariant case of Lemma 2.2 (2) in =-=[GS]-=-. Assume gD = Dg and gAi = Aig for 0 ≤ i ≤ n, then n∑ 〈A0, · · · ,An〉D(g) = (−1) i=0 (|A0|+···+|Ai|)(|Ai+1+···+|An|) 〈1,Ai+1, · · · ,An,A0, · · · ,Ai〉D(g). [ ] (3.7) 0 −D By (3.7), Lemma 3.2 in [G2] a... |

16 |
Equivariant Chern character for the invariant Dirac operator.
- Chern, Hu
- 1997
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Citation Context ...rac operator and ˜ dg is a bounded operator. Let A = C ∞ (N), then the data (A,H,D,G) defines a finitely (hence θ-summable) equivariant unbounded Fredholm module in the sense of [KL] (for details see =-=[CH]-=-,[FGV] and [KL]). Similar to [CH] or [W], for equivariant θ-summable Fredholm module (A,H,D,G), we can define the equivariant cochain ˜ ch G k (tD,D) (k is even) by the formula: ˜ ch G k (tD,D)(f 0 , ... |

14 | Cyclic homology and the Atiyah-Patodi-Singer index theorem
- Getzler
- 1993
(Show Context)
Citation Context ... as adopted in [BF] and computed its radius of convergence. Subsequently, he proved the variation formula of eta cochains, using which he got the noncommutative Atiyah-Patodi-Singer index theorem. In =-=[G2]-=-, using superconnection, Getzler gave another proof of the noncommutative Atiyah-PatodiSinger index theorem, which was more difficult, but avoided mention of the operators b and B of cyclic cohomology... |

11 |
character in equivariant entire cyclic cohomology, K-Theory 4
- Klimek, Lesniewski, et al.
- 1991
(Show Context)
Citation Context ...g commutes with the Dirac operator and ˜ dg is a bounded operator. Let A = C ∞ (N), then the data (A,H,D,G) defines a finitely (hence θ-summable) equivariant unbounded Fredholm module in the sense of =-=[KL]-=- (for details see [CH],[FGV] and [KL]). Similar to [CH] or [W], for equivariant θ-summable Fredholm module (A,H,D,G), we can define the equivariant cochain ˜ ch G k (tD,D) (k is even) by the formula: ... |

9 |
Transgression and Chern character of finite dimensional
- Connes, Moscovici
(Show Context)
Citation Context ...[GS]. Let (N) (0 ≤ i ≤ 2q + 1), we have gDN = DNg for g ∈ G and f i ∈ C ∞ G − d dt chG 2q+1 (tDN)(f 0 , · · · ,f 2q+1 )(g) = b ˜ ch G 2q (tDN,DN)(f 0 , · · · ,f 2q+1 )(g) Similar to the discussion in =-=[CM]-=-, we have By (4.3) and (4.4), then +B ˜ ch G 2q+2(tDN,DN)(f 0 , · · · ,f 2q+1 )(g). (4.3) limt→∞ch G 2q+1(tDN)(f 0 , · · · ,f 2q+1 )(g) = 0. (4.4) 1 Γ( 1 2 )limt→0ch G 2q+1(tDN)(f 0 , · · · ,f 2q+1 )(... |

7 |
Local index theorem for Dirac operators
- Yu
(Show Context)
Citation Context ...need only to prove that ∫ ∫ limt→0t −1 2 | F Nξ(ε) k |λ|+ t 2 Tr{(D λ i )x[Dexp(−tD 2 )(x,g · x)]}dNξdξ| ≤ C, (2.3.7) 9for some constant C > 0. Here Nξ(ε) = {v ∈ Nξ(F)| ||v|| < ε }. Similar to [LYZ],=-=[Y]-=-, for ξ ∈ F, we choose an open neighborhood U of ξ and the orthogonal frame over U (see [LYZ], pp.574). Consider the oriented orthonormal frame field E g·x defined over the patch U by requiring that E... |

5 | A note on the noncommutative Chern character (in - Feng |

5 |
A note on equivariant eta invariants
- Zhang
(Show Context)
Citation Context ...d their famous Atiyah-Patodi-Singer index theorem for manifolds with boundary. In [D], Donnelly extended this theorem to the equivariant case by modifying the Atiyah-Patodi-Singer original method. In =-=[Z]-=-, Zhang got this equivariant Atiyah-Patodi-Singer index theorem by using a direct geometric method [LYZ] . In [Wu], Wu proved the Atiyah-Patodi-Singer index theorem in the framework of noncommutative ... |

4 |
The Chern-Connes character for the Dirac operators on manifolds with boundary. K-Theory 7
- Wu
- 1993
(Show Context)
Citation Context ...609061v1 [math.DG] 2 Sep 2006 The Equivariant Noncommutative Atiyah-Patodi-Singer Index Theorem ∗ Yong Wang Nankai Institute of Mathematics Tianjin 300071, P.R.China; wangy581@nenu.edu.cn Abstract In =-=[Wu]-=-, the noncommutative Atiyah-Patodi-Singer index theorem was proved. In this paper, we extend this theorem to the equivariant case. Keywords: Equivariant total eta invariants; Clifford asymptotics; C(1... |

1 |
Heat kernels and Dirac operators, Spring-Verlag, Berline
- Berline, Getzler, et al.
- 1992
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Citation Context ...re H ⊗ C r ⊕ H ⊗ C r . By Duhamel principle and gAt = Atg, then ∫ +∞ 0 Str(ge A2 ) = = ∫ +∞ ∫ 1 0 ∫ +∞ 0 0 Str [ Str ge s0A 2 t 1 [ g dAt dt eA2 t 2 t−1 ] dt. By gAt = Atg, similar to Theorem 9.23 in =-=[BGV]-=-, we have, ∫ +∞ d Str(ge 0 A2 ∫ +∞ [ ) = d Str g 0 dAt dt eA2 t 2dtDue (1−s0)A 2 t ] dt = limt→+∞Chg(At) − limt→0Chg(At). Using Duhamel principle and similar to the proof of Lemma 1.1 in [Wu], we have... |