scholarly journals The local index formula in semifinite Von Neumann algebras I: Spectral flow

2006 ◽  
Vol 202 (2) ◽  
pp. 451-516 ◽  
Author(s):  
Alan L. Carey ◽  
John Phillips ◽  
Adam Rennie ◽  
Fyodor A. Sukochev
Keyword(s):  
Von Neumann Algebras ◽  
Index Formula ◽  
Spectral Flow ◽  
Local Index ◽  
Von Neumann

scholarly journals The local index formula in semifinite von Neumann algebras II: The even case

2006 ◽  
Vol 202 (2) ◽  
pp. 517-554 ◽  
Author(s):  
Alan L. Carey ◽  
John Phillips ◽  
Adam Rennie ◽  
Fyodor A. Sukochev
Keyword(s):  
Von Neumann Algebras ◽  
Index Formula ◽  
Local Index ◽  
Von Neumann

scholarly journals Operator Integrals, Spectral Shift, and Spectral Flow

2009 ◽  
Vol 61 (2) ◽  
pp. 241-263 ◽  
Author(s):  
N. A. Azamov ◽  
A. L. Carey ◽  
P. G. Dodds ◽  
F. A. Sukochev
Keyword(s):  
Von Neumann Algebras ◽  
Spectral Shift ◽  
Shift Function ◽  
Spectral Flow ◽  
Spectral Shift Function ◽  
Spectral Measures ◽  
Unbounded Operators ◽  
Von Neumann ◽  
Operator Functions ◽  
Multiple Operator

Abstract. We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing results, and establish the Birman–Solomyak representation of the spectral shift function of M.G. Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow.

scholarly journals KK-Theory and Spectral Flow in von Neumann Algebras

2012 ◽  
Vol 10 (2) ◽  
pp. 241-277 ◽  
Author(s):  
J. Kaad ◽  
R. Nest ◽  
A. Rennie
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Closed Ideal ◽  
Spectral Triple ◽  
Spectral Flow ◽  
Von Neumann ◽  
Selfadjoint Operators ◽  
Definition Of

AbstractWe present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko(J).Given a semifinite spectral triple (A, H, D) relative to (N, τ) with A separable, we construct a class [D] ∈ KK1(A, K(N)). For a unitary u ∈ A, the von Neumann spectral flow between D and u*Du is equal to the Kasparov product [u]A[D], and is simply related to the numerical spectral flow, and a refined C*-spectral flow.

Flat bundles, von Neumann algebras andK-theory with ℝ/ℤ-coefficients

2014 ◽  
Vol 13 (2) ◽  
pp. 275-303 ◽  
Author(s):  
Paolo Antonini ◽  
Sara Azzali ◽  
Georges Skandalis
Keyword(s):  
Differential Operator ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Homology Class ◽  
Spectral Flow ◽  
Pseudo Differential Operator ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Flat Bundle ◽  
Theoretic Description

AbstractLetMbe a closed manifold andα:π1(M) →Una representation. We give a purelyK-theoretic description of the associated element in theK-theory group ofMwith ℝ/ℤ-coefficients ([α] ∈K1(M; ℝ/ℤ)). To that end, it is convenient to describe the ℝ/ℤ-K-theory as a relativeK-theory of the unital inclusion of ℂ into a finite von Neumann algebraB. We use the following fact: there is, associated withα, a finite von Neumann algebraBtogether with a flat bundleℰ→Mwith fibersB, such thatEα⊗ℰis canonically isomorphic with ℂn⊗ℰ, whereEαdenotes the flat bundle with fiber ℂnassociated withα. We also discuss the spectral flow and rho type description of the pairing of the class [α] with theK-homology class of an elliptic selfadjoint (pseudo)-differential operatorDof order 1.

scholarly journals AN ANALYTIC APPROACH TO SPECTRAL FLOW IN VON NEUMANN ALGEBRAS

2006 ◽  
pp. 297-352 ◽  
Author(s):  
MOULAY-TAHAR BENAMEUR ◽  
ALAN L. CAREY ◽  
JOHN PHILLIPS ◽  
ADAM RENNIE ◽  
FYODOR A. SUKOCHEV ◽  
...  
Keyword(s):  
Von Neumann Algebras ◽  
Analytic Approach ◽  
Spectral Flow ◽  
Von Neumann

scholarly journals Spectral Flow for Nonunital Spectral Triples

2015 ◽  
Vol 67 (4) ◽  
pp. 759-794 ◽  
Author(s):  
A. L. Carey ◽  
V. Gayral ◽  
J. Phillips ◽  
A. Rennie ◽  
F. A. Sukochev
Keyword(s):  
Index Theory ◽  
Spectral Triple ◽  
Index Formula ◽  
Spectral Flow ◽  
Spectral Triples ◽  
Local Index ◽  
C Algebra ◽  
Odd Index ◽  
Definition Of ◽  
Special Case

AbstractWe prove two results about nonunital index theory left open in a previous paper. The first is that the spectral triple arising from an action of the reals on a C*-algebra with invariant trace satisûes the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths arising from the odd index pairing for smooth spectral triples in the nonunital setting, we are able to connect with earlier approaches to the analytic definition of spectral flow

Lectures on von Neumann Algebras

2019 ◽  
Author(s):  
Serban-Valentin Stratila ◽  
Laszlo Zsido
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann
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