Integral Formula for Spectral Flow for -Summable Operators

2019 ◽  
Vol 71 (2) ◽  
pp. 337-379
Author(s):  
Magdalena Cecilia Georgescu
Keyword(s):  
Fredholm Operator ◽  
Unbounded Operator ◽  
Von Neumann Algebra ◽  
Integral Formula ◽  
Direct Proof ◽  
Spectral Flow ◽  
Fredholm Operators ◽  
Von Neumann ◽  
Bounded Perturbations ◽  
The Ideal

AbstractFix a von Neumann algebra ${\mathcal{N}}$ equipped with a suitable trace $\unicode[STIX]{x1D70F}$. For a path of self-adjoint Breuer–Fredholm operators, the spectral flow measures the net amount of spectrum that moves from negative to non-negative. We consider specifically the case of paths of bounded perturbations of a fixed unbounded self-adjoint Breuer–Fredholm operator affiliated with ${\mathcal{N}}$. If the unbounded operator is $p$-summable (that is, its resolvents are contained in the ideal $L^{p}$), then it is possible to obtain an integral formula that calculates spectral flow. This integral formula was first proved by Carey and Phillips, building on earlier approaches of Phillips. Their proof was based on first obtaining a formula for the larger class of $\unicode[STIX]{x1D703}$-summable operators, and then using Laplace transforms to obtain a $p$-summable formula. In this paper, we present a direct proof of the $p$-summable formula that is both shorter and simpler than theirs.

scholarly journals SPECTRAL FLOW, INDEX AND THE SIGNATURE OPERATOR

2011 ◽  
Vol 03 (01) ◽  
pp. 37-67 ◽  
Author(s):  
SARA AZZALI ◽  
CHARLOTTE WAHL
Keyword(s):  
Von Neumann Algebra ◽  
Flow Index ◽  
Spectral Flow ◽  
Bounded Operators ◽  
Fredholm Operators ◽  
Foliated Manifold ◽  
Von Neumann ◽  
Signature Operator ◽  
Integral Formulas ◽  
Manifold With Boundary

We relate the spectral flow to the index for paths of selfadjoint Breuer–Fredholm operators affiliated to a semifinite von Neumann algebra, generalizing results of Robbin–Salamon and Pushnitski. Then we prove the vanishing of the von Neumann spectral flow for the tangential signature operator of a foliated manifold when the metric is varied. We conclude that the tangential signature of a foliated manifold with boundary does not depend on the metric. In the Appendix we reconsider integral formulas for the spectral flow of paths of bounded operators.

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.

scholarly journals INTEGRATION WITH RESPECT TO A WEIGHT

1991 ◽  
Vol 02 (02) ◽  
pp. 177-182 ◽  
Author(s):  
MICHAEL LEINERT
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Interpolation Space ◽  
Integral Formula ◽  
Adjoint Operator ◽  
Simple Approach ◽  
Complex Interpolation ◽  
Linear Combinations ◽  
Von Neumann ◽  
Positive Form

A simple approach to non-commutative integration for weights is described, following the lines of [7] i.e., using a natural upper integral (which is in fact an integral) and interpolation. If [Formula: see text] is a von Neumann algebra on the Hilbert space H and φ is a faithful normal semifinite weight on [Formula: see text], the space D of all φ-bounded vectors in H is contained in the domain of every closed positive form coming from a positive self-adjoint operator T affiliated to [Formula: see text] with finite upper integral [Formula: see text]. The (classes of) linear combinations of such forms constitute [Formula: see text]. In an obvious sense, [Formula: see text] consists of forms, too (bounded ones). [Formula: see text] is the complex interpolation space [Formula: see text]. It is checked that [Formula: see text] is isometrically isomorphic to Vp in [10], so [Formula: see text] is what it ought to be.

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 A Trace Formula for the Index of B-Fredholm Operators

2018 ◽  
Vol 61 (4) ◽  
pp. 1063-1068 ◽  
Author(s):  
Mohammed Berkani
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Fredholm Operator ◽  
Trace Formula ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Drazin Inverse ◽  
Trace Function ◽  
Fredholm Operators ◽  
The Ideal

AbstractIn this paper we define B-Fredholm elements in a Banach algebraAmodulo an idealJofA. When a trace function is given on the idealJ, it generates an index for B-Fredholm elements. In the case of a B-Fredholm operatorTacting on a Banach space, we prove that its usual index ind(T) is equal to the trace of the commutator [T, T0], whereT0is a Drazin inverse ofTmodulo the ideal of finite rank operators, extending Fedosov's trace formula for Fredholm operators (see Böttcher and Silbermann [Analysis of Toeplitz operators, 2nd edn (Springer, 2006)]. In the case of a primitive Banach algebra, we prove a punctured neighbourhood theorem for the index.

scholarly journals On Compact and Fredholm Operators over C*-algebras and a New Topology in the Space of Compact Operators

2008 ◽  
Vol 2 (2) ◽  
pp. 329-351 ◽  
Author(s):  
Anwar A. Irmatov ◽  
Alexandr S. Mishchenko
Keyword(s):  
Fredholm Operator ◽  
Main Property ◽  
Compact Operators ◽  
Continuous Functions ◽  
Bounded Operators ◽  
Fredholm Operators ◽  
C Algebras ◽  
Compact Topological Space ◽  
Space Of Compact Operators ◽  
The Ideal

AbstractIt is well-known that bounded operators in Hilbert C*-modules over C*-algebras may not be adjointable and the same is true for compact operators. So, there are two analogs for classical compact operators in Hilbert C*-modules: adjointable compact operators and all compact operators, i.e. those not necessarily having an adjoint.Classical Fredholm operators are those that are invertible modulo compact operators. When the notion of a Fredholm operator in a Hilbert C*-module was developed in [6], the first analog was used: Fredholm operators were defined as operators that are invertible modulo adjointable compact operators.In this paper we use the second analog and develop a more general version of Fredholm operators over C*-algebras. Such operators are defined as bounded operators that are invertible modulo the ideal of all compact operators. The main property of this new class is that a Fredholm operator still has a decomposition into a direct sum of an isomorphism and a finitely generated operator.The special case of Fredholm operators (in the sense of [6]) over the commutative C*-algebra C(K) of continuous functions on a compact topological space K was also considered in [2]. In order to describe general Fredholm operators (invertible modulo all compact operators over C(K)) we construct a new IM-topology on the space of compact operators on a Hilbert space such that continuous families of compact operators generate the ideal of all compact operators over C(K).

scholarly journals Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

2015 ◽  
Vol 26 (08) ◽  
pp. 1550064
Author(s):  
Bachir Bekka
Keyword(s):  
Von Neumann Algebra ◽  
Regular Representation ◽  
Von Neumann ◽  
Lp Spaces ◽  
Local Rigidity ◽  
Left Regular Representation ◽  
Left Regular ◽  
Property T ◽  
Finite Factor ◽  
Finite Index Subgroup

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

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