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.

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.

M-embedded symmetric operator spaces and the derivation problem

2019 ◽  
Vol 169 (3) ◽  
pp. 607-622
Author(s):  
JINGHAO HUANG ◽  
GALINA LEVITINA ◽  
FEDOR SUKOCHEV
Keyword(s):  
Symmetric Spaces ◽  
Symmetric Operator ◽  
Von Neumann Algebra ◽  
Bounded Operators ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Unbounded Operators ◽  
Von Neumann ◽  
Symmetric Operator Spaces ◽  
Order Continuous

AbstractLet ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 < p < ∞, has a positive answer.

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 Generators of reflexive algebras

1975 ◽  
Vol 20 (2) ◽  
pp. 159-164
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Complex Hilbert Space ◽  
Bounded Operators ◽  
Finitely Generated ◽  
Separable Space ◽  
Von Neumann ◽  
Underlying Space ◽  
Weakly Closed ◽  
Reflexive Algebras

For any collection of closed subspaces of a complex Hilbert space the set of bounded operators that leave invariant all the members of the collection is a weakly-closed algebra. The class of such algebras is precisely the class of reflexive algebras as defined for example in Radjavi and Rosenthal (1969) and contains the class of von Neumann algebras.In this paper we consider the problem of when such algebras are finitely generated as weakly-closed algebras. It is to be hoped that analysis of this problem may shed some light on the famous unsolved problem of whether every von Neumann algebra on a separable Hilbert space is finitely generated. The case where the underlying space is separable and the collection of subspaces is totally ordered is dealt with in Longstaff (1974). In the present paper the result of Longstaff (1974) is generalized to the case of a direct product of countably many totally ordered collections each on a separable space. Also a method of obtaining non-finitely generated reflexive algebras is given.

Further results on the relative entropy

1987 ◽  
Vol 101 (2) ◽  
pp. 363-373 ◽  
Author(s):  
Matthew J. Donald
Keyword(s):  
Relative Entropy ◽  
Von Neumann Algebra ◽  
Measurement Problem ◽  
Bounded Operators ◽  
Many Worlds ◽  
Von Neumann ◽  
Normal States ◽  
Injective Von Neumann Algebra ◽  
Made In

Given any subset ℬ, containing the identity (1), of ℬ (ℋ) (the bounded operators on some Hilbert space ℋ), and given two states σ and ρ on ℬ(ℋ), a definition was given in [3] of entℬ (σℬ|ρ|ℬ) - ‘the entropy of σ relative to ρ given the information in ℬ’. It was shown that, for ℬ an injective von Neumann algebra, the resulting relative entropy agreed with those of Umegaki, Araki, Pusz and Woronowicz, and Uhlmann. The purpose of this paper is to explore this definition further. After some technical preliminaries in Section 2, in Section 3 a new characterization of entℬ(ℋ) (σ|ρ) for σ and ρ normal states will be given. In Section 4 it will be shown that under fairly general circumstances the relative entropy on algebras can be used for statistical inference. This is important for applications of the relative entropy. I shall given the briefest sketches of how I see these applications being made in the measurement problem in quantum theory and in a ‘many worlds’ interpretation. The vigilant reader will notice that the scheme proposed in Section 4 for modelling measurements subject to given compatibility requirements differs slightly from that proposed in the introduction to [3]. The reason for this is outlined in Section 5, where an explicit computation is made of the relative entropy for the simplest non-trivial case in which ℬ is not an algebra; when ℬ = {1, P, Q} for P and Q projections subject to certain conditions.

A Wasserstein-type Distance to Measure Deviation from Equilibrium of Quantum Markov Semigroups

2013 ◽  
Vol 20 (02) ◽  
pp. 1350009 ◽  
Author(s):  
Julián Agredo
Keyword(s):  
Von Neumann Algebra ◽  
Separable Hilbert Space ◽  
Optimal Transport ◽  
Detailed Balance ◽  
Wasserstein Distance ◽  
Markov Semigroup ◽  
Bounded Operators ◽  
Von Neumann ◽  
Detailed Balance Condition ◽  
Quantum Markov Semigroup

In this paper we define a distance W between states in the non-commutative von Neumann algebra [Formula: see text] of bounded operators on a separable Hilbert space [Formula: see text], in order to measure deviations from equilibrium using a rate ep W(·). The restriction of W to the diagonal subalgebra of [Formula: see text] coincides with the Wasserstein distance used in optimal transport. Moreover, if ρ is a normal invariant state of a quantum Markov semigroup [Formula: see text], then ep W(ρ) = 0 if and only if a detailed balance condition holds.

scholarly journals On the operator equationα+α−1=β+β−1

1986 ◽  
Vol 9 (4) ◽  
pp. 767-770 ◽  
Author(s):  
A. B. Thaheem
Keyword(s):  
Hilbert Space ◽  
Operator Equation ◽  
Von Neumann Algebra ◽  
Central Projection ◽  
Bounded Operators ◽  
New Techniques ◽  
Von Neumann ◽  
Alpha Beta

Letα,βbe∗-automorphisms of a von Neumann algebraMsatisfying the operator equationα+α−1=β+β−1. In this paper we use new techniques (which are useful in noncommutative situations as well) to provide alternate proofs of the results:- Ifα,βcommute then there is a central projectionpinMsuch thatα=βonMPandα=β−1onM(1−P); IfM=B(H), the algebra of all bounded operators on a Hilbert spaceH, thenα=βorα=β−1.

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.

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