Spectral flow in Fredholm modules, eta invariants and the JLO cocycle

被引:47
作者
Carey, A [1 ]
Phillips, J
机构
[1] Australian Natl Univ, Inst Math Sci, Canberra, ACT, Australia
[2] Univ Victoria, Dept Math & Stat, Victoria, BC V8W 3P4, Canada
关键词
spectral flow; theta-summable Fredholm module; eta invariant; index;
D O I
10.1023/B:KTHE.0000022922.68170.61
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We give a comprehensive account of an analytic approach to spectral flow along paths of self-adjoint Breuer-Fredholm operators in a type I-infinity or IIinfinity von Neumann algebra N. The framework is that of odd unbounded theta-summable Breuer-Fredholm modules for a unital Banach *-algebra, A. In the type IIinfinity case spectral flow is real-valued, has no topological definition as an intersection number and our formulae encompass all that is known. We borrow Ezra Getzler's idea (suggested by I. M. Singer) of considering spectral flow (and eta invariants) as the integral of a closed one-form on an affine space. Applications in both the types I and II cases include a general formula for the relative index of two projections, representing truncated eta functions as integrals of one forms and expressing spectral flow in terms of the JLO cocycle to give the pairing of the K-homology and K-theory of A.
引用
收藏
页码:135 / 194
页数:60
相关论文
共 34 条
[1]  
[Anonymous], 1997, CYCLIC COHOMOLOGY NO
[2]  
Atiyah M. F., 1969, PUBL MATH-PARIS, V37, P5
[3]   SPECTRAL ASYMMETRY AND RIEMANNIAN GEOMETRY .3. [J].
ATIYAH, MF ;
PATODI, VK ;
SINGER, IM .
MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 1976, 79 (JAN) :71-99
[4]   SPECTRAL ASYMMETRY AND RIEMANNIAN GEOMETRY .1. [J].
ATIYAH, MF ;
PATODI, VK ;
SINGER, IM .
MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 1975, 77 (JAN) :43-69
[5]   THE INDEX OF A PAIR OF PROJECTIONS [J].
AVRON, J ;
SEILER, R ;
SIMON, B .
JOURNAL OF FUNCTIONAL ANALYSIS, 1994, 120 (01) :220-237
[6]  
BAAJ S, 1983, CR ACAD SCI I-MATH, V296, P875
[7]  
BOOSSBAVNBEK B, 1998, TOKYO J MATH, V21, P1
[8]   FREDHOLM THEORIES IN VON NEUMANN ALGEBRAS .I. [J].
BREUER, M .
MATHEMATISCHE ANNALEN, 1968, 178 (03) :243-&
[9]   FREDHOLM THEORIES IN VON NEUMANN ALGEBRAS .2. [J].
BREUER, M .
MATHEMATISCHE ANNALEN, 1969, 180 (04) :313-&
[10]   Spectral flow and Dixmier traces [J].
Carey, A ;
Phillips, J ;
Sukochev, F .
ADVANCES IN MATHEMATICS, 2003, 173 (01) :68-113