arXiv subjects

A. Rennie

Publications and source records attributed to A. Rennie.

18 recordsLinked to original sources

arXiv2026

Analytic spectral flow formula for unitaries and Levinson's theorem

We prove an integral formula for the spectral flow of differentiable loops of unitaries of the form ${\rm Id}+$Schatten. Our formula is in terms of a regularised winding number, expressed in terms of exact differential forms, and we show how the formula extends to non-closed paths. Applying these ideas to the scattering operator of Schrödinger scattering systems yields explicit formulae for the number of bound states, possibly modified by the presence of resonances, of the system in terms of the potential. We finish by briefly considering the paths of unbounded operators obtained from unitary loops via the Cayley transform. These include cases of moving domain as well as paths with non-constant Hilbert space.

math.FA↗

arXiv2024

Topological Levinson's theorem in presence of embedded thresholds and discontinuities of the scattering matrix

A family of discrete Schroedinger operators is investigated through scattering theory. The continuous spectrum of these operators exhibit changes of multiplicity, and some of these operators possess resonances at thresholds. It is shown that the corresponding wave operators belong to an explicitly constructed C*-algebra, whose K-theory is carefully analysed. An index theorem is deduced from these investigations, which corresponds to a topological version of Levinson's theorem in presence of embedded thresholds, resonances, and changes of multiplicity of the scattering matrices. In the second half of the paper, very detailed computations for the simplest realisation of this family of operators are provided. In particular, a surface of resonances is exhibited, probably for the first time. For Levinson's theorem, it is shown that contributions due to resonances at the lowest value and at the highest value of the continuous spectrum play an essential role.

math-ph↗

arXiv2023

Levinson's theorem for two-dimensional scattering systems: it was a surprise, it is now topological!

We prove a general Levinson's theorem for Schrödinger operators in two dimensions with threshold obstructions at zero energy. Our results confirm and simplify earlier seminal results of Bollé, Gesztesy et al., while providing an explicit topological interpretation. We also derive explicit formulas for the wave operators, and so show that they are elements of a $C^*$-algebra introduced by Cordes. As a consequence of our approach, we provide an evaluation of the spectral shift function at zero in the presence of $p$-resonances.

math.SP↗

arXiv2018

The non-unital version of Connes' theorem on the Hochschild class of the Chern character

We offer a short proof of Connes' Hochschild class of the Chern character formula for non-unital semifinite spectral triples. The proof is simple due to its reliance on the authors' extensive work on a refined version of the local index formula, and the consequent understanding of the passage from generalised residues of zeta functions to representations in terms of singular traces.

math.KT↗

arXiv2018

Poincare duality for Cuntz-Pimsner algebras of bimodules

We present a new approach to Poincare duality for Cuntz-Pimsner algebras. We provide sufficient conditions under which Poincare self-duality for the coefficient algebra of a Hilbert bimodule lifts to Poincare self-duality for the associated Cuntz-Pimsner algebra. With these conditions in hand, we can constructively produce fundamental classes in K-theory for a wide range of examples. We can also produce K-homology fundamental classes for the important examples of Cuntz-Krieger algebras (following Kaminker-Putnam) and crossed products of manifolds by isometries, and their non-commutative analogues.

math.KT↗

arXiv2014

Spectral flow for nonunital spectral triples

We prove two results about nonunital index theory left open by [CGRS2]. The first is that the spectral triple arising from an action of the reals on a C*-algebra with invariant trace satisfies 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.

math.KT↗

arXiv2014

Universal measurability and the Hochschild class of the Chern character

We study notions of measurability for singular traces, and characterise universal measurability for operators in Dixmier ideals. This measurability result is then applied to improve on the various proofs of Connes' identification of the Hochschild class of the Chern character of Dixmier summable spectral triples. The measurability results show that the identification of the Hochschild class is independent of the choice of singular trace. As a corollary we obtain strong information on the asymptotics of the eigenvalues of operators naturally associated to spectral triples (A,H,D) and Hochschild cycles for A.

math.FA↗

arXiv2013

Dense domains, symmetric operators and spectral triples

This article is about erroneous attempts to weaken the standard definition of unbounded Kasparov module (or spectral triple). We present counterexamples to claims in the literature that Fredholm modules can be obtained from these weaker variations of spectral triple. Our counterexamples are constructed using self-adjoint extensions of symmetric operators.

math.KT↗

arXiv2013

Index theory for locally compact noncommutative geometries

Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, we prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in our algebra. This formula has been successfully used to calculate index pairings in numerous noncommutative examples. The absence of any other effective method of investigating index problems in geometries that are genuinely noncommutative, particularly in the nonunital situation, was a primary motivation for this study and we illustrate this point with two examples in the text. In order to understand what is new in our approach in the commutative setting we prove an analogue of the Gromov-Lawson relative index formula (for Dirac type operators) for even dimensional manifolds with bounded geometry, without invoking compact supports. For odd dimensional manifolds our index formula appears to be completely new. As we prove our local index formula in the framework of semifinite noncommutative geometry we are also able to prove, for manifolds of bounded geometry, a version of Atiyah's L^2-index Theorem for covering spaces. We also explain how to interpret the McKean-Singer formula in the nonunital case. In order to prove the local index formula, we develop an integration theory compatible with a refinement of the existing pseudodifferential calculus for spectral triples. We also clarify some aspects of index theory for nonunital algebras.

math.OA↗

arXiv2010

Families of Type {\rm III KMS} States on a Class of $C^*$-Algebras containing $O_n$ and $\mathcal{Q}_\N$

We construct a family of purely infinite $C^*$-algebras, $\mathcal{Q}^λ$ for $λ\in (0,1)$ that are classified by their $K$-groups. There is an action of the circle $\T$ with a unique ${\rm KMS}$ state $ψ$ on each $\mathcal{Q}^λ.$ For $λ=1/n,$ $\mathcal{Q}^{1/n}\cong O_n$, with its usual $\T$ action and ${\rm KMS}$ state. For $λ=p/q,$ rational in lowest terms, $\mathcal{Q}^λ\cong O_n$ ($n=q-p+1$) with UHF fixed point algebra of type $(pq)^\infty.$ For any $n>0,$ $\mathcal{Q}^λ\cong O_n$ for infinitely many $λ$ with distinct KMS states and UHF fixed-point algebras. For any $λ\in (0,1),$ $\mathcal{Q}^λ\neq O_\infty.$ For $λ$ irrational the fixed point algebras, are NOT AF and the $\mathcal{Q}^λ$ are usually NOT Cuntz algebras. For $λ$ transcendental, $K_1\cong K_0\cong\Z^\infty$, so that $\mathcal{Q}^λ$ is Cuntz' $\mathcal Q_{\N}$, \cite{Cu1}. If $λ^{\pm 1}$ are both algebraic integers, the {\bf only} $O_n$ which appear satisfy $n\equiv 3(mod 4).$ For each $λ$, the representation of $\mathcal{Q}^λ$ defined by the KMS state $ψ$ generates a type ${\rm III}_λ$ factor. These algebras fit into the framework of modular index (twisted cyclic) theory of \cite{CPR2,CRT} and \cite{CNNR}.

math.OA↗

arXiv2009

Integration on locally compact noncommutative spaces

We present an ab initio approach to integration theory for nonunital spectral triples. This is done without reference to local units and in the full generality of semifinite noncommutative geometry. The main result is an equality between the Dixmier trace and generalised residue of the zeta function and heat kernel of suitable operators. We also examine definitions for integrable bounded elements of a spectral triple based on zeta function, heat kernel and Dixmier trace techniques. We show that zeta functions and heat kernels yield equivalent notions of integrability, which imply Dixmier traceability.

math.OA↗

arXiv2008

Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras

This paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with the Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on the Cuntz algebra. We introduce a modified $K_1$-group of the Cuntz algebra so as to pair with this twisted cocycle. As a corollary we obtain a noncommutative geometry interpretation for Araki's notion of relative entropy in this example. We also note the connection of this example to the theory of noncommutative manifolds.

math.KT↗

arXiv2008

A noncommutative Atiyah-Patodi-Singer index theorem in KK-theory

We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS boundary conditions in the setting of KK-theory, generalising the commutative theory. We find that Cuntz-Kreiger systems provide a natural class of examples for our construction and the index pairings coming from APS boundary conditions yield complete K-theoretic information about certain graph C*-algebras.

math.KT↗

arXiv2008

Spectral flow invariants and twisted cyclic theory from the Haar state on SU_q(2)

In [CPR2], we presented a K-theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely SU_q(2), and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes from the generator of the modular group of the Haar state. In contrast to the Cuntz algebras studied in [CPR2], the computations are considerably more complex and interesting, because there are nontrivial `eta' contributions to this index.

math.OA↗

arXiv2006

The Dixmier trace and asymptotics of zeta functions

We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup. We prove our results in a general semi-finite von Neumann algebra. We find for p>1 that the asymptotics of the zeta function determines an ideal strictly larger than {\mathcal L}^{p,\infty} on which the Dixmier trace may be defined. We also establish stronger versions of other results on Dixmier traces and zeta functions.

math.OA↗

arXiv2005

An Analytic Approach to Spectral Flow in von Neumann Algebras

The analytic approach to spectral flow is about ten years old. In that time it has evolved to cover an ever wider range of examples. The most critical extension was to replace Fredholm operators in the classical sense by Breuer-Fredholm operators in a semifinite von Neumann algebra. The first part of this paper gives a brief account of this theory extending and refining earlier results. It is then applied in the latter parts of the paper to a series of examples. One of the most powerful tools is an integral formula for spectral flow. This integral formula was known for Dirac operators in a variety of forms ever since the fundamental papers of Atiyah, Patodi and Singer. One of the purposes of this exposition is to make contact with this early work so that one can understand the recent developments in a proper historical context. In addition we show how to derive these spectral flow formulae in the setting of Dirac operators on (non-compact) covering spaces of a compact spin manifold using the adiabatic method. Finally we relate our work to that of Coburn, Douglas, Schaeffer and Singer on Toeplitz operators with almost periodic symbol. We generalise their work to cover the case of matrix valued almost periodic symbols on \R^N using some ideas of Shubin.

math.OA↗

arXiv2003

The Hochschild Class of the Chern Character For Semifinite Spectral Triples

We provide a proof of Connes' formula for a representative of the Hochschild class of the Chern character for (p,\infty)-summable spectral triples. Our proof is valid for all semifinite von Neumann algebras, and all integral p\geq 1. We employ the minimum possible hypotheses on the spectral triples.

math.OA↗

arXiv1999

Commutative Geometries are Spin Manifolds

In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spin^c geometry depending on whether the geometry is ''real'' or not. We attempt to flesh out the details of Connes' ideas. As an illustration we present a proof of his claim, partly extending the validity of the result to pseudo-Riemannian spin manifolds. Throughout we are as explicit and elementary as possible.

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