arXiv subjects

Magnus Goffeng

Publications and source records attributed to Magnus Goffeng.

At least 37 records · Page 2Linked to original sources

arXiv2022

Exotic cyclic cohomology classes and Lipschitz algebras

We study the noncommutative geometry of algebras of Lipschitz continuous and Hölder continuous functions where non-classical and novel differential geometric invariants arise. Indeed, we introduce a new class of Hochschild and cyclic cohomology classes that pair non-trivially with higher algebraic $K$-theory yet vanish when restricted to the algebra of smooth functions. Said cohomology classes provide additional methods to extract numerical invariants from Connes-Karoubi's relative sequence in $K$-theory.

math.KT↗

arXiv2022

Fell algebras, groupoids, and projections

Examples of Fell algebras with compact spectrum and trivial Dixmier-Douady invariant are constructed to illustrate differences with the case of continuous trace $C^*$-algebras. At the level of the spectrum, this translates to only assuming the spectrum is locally Hausdorff (rather than Hausdorff). The existence of (full) projections is the fundamental question considered. The class of Fell algebras studied here arise naturally in the study of Wieler solenoids and applications to dynamical systems will be discussed in a separate paper.

math.OA↗

arXiv2021

Realisations of elliptic operators on compact manifolds with boundary

This paper investigates realisations of elliptic differential operators of general order on manifolds with boundary following the approach of Bär-Ballmann to first order elliptic operators. The space of possible boundary values of elements in the maximal domain is described as a Hilbert space densely sandwiched between two mixed order Sobolev spaces. The description uses Calderón projectors which, in the first order case, is equivalent to results of Bär-Bandara using spectral projectors of an adapted boundary operator. Boundary conditions that induce Fredholm as well as regular realisations, and those that admit higher order regularity, are characterised. In addition, results concerning spectral theory, homotopy invariance of the Fredholm index, and well-posedness for higher order elliptic boundary value problems are proven.

math.AP↗

arXiv2020

On the magnitude function of domains in Euclidean space

We study Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain $X\subset \mathbb{R}^{2m-1}$, we find geometric significance in the function $\mathcal{M}_X(R) = \mathrm{mag}(R\cdot X)$. The function $\mathcal{M}_X$ extends from the positive half-line to a meromorphic function in the complex plane. Its poles are generalized scattering resonances. In the semiclassical limit $R \to \infty$, $\mathcal{M}_X$ admits an asymptotic expansion. The three leading terms of $\mathcal{M}_X$ at $R=+\infty$ are proportional to the volume, surface area and integral of the mean curvature. In particular, for convex $X$ the leading terms are proportional to the intrinsic volumes, and we obtain an asymptotic variant of the convex magnitude conjecture by Leinster and Willerton, with corrected coefficients.

math.MG↗

arXiv2019

Untwisting twisted spectral triples

We examine the index data associated to twisted spectral triples and higher order spectral triples. In particular, we show that a Lipschitz regular twisted spectral triple can always be `logarithmically dampened' through functional calculus, to obtain an ordinary (i.e. untwisted) spectral triple. The same procedure turns higher order spectral triples into spectral triples. We provide examples of highly regular twisted spectral triples with nontrivial index data for which Moscovici's ansatz for a twisted local index formula is identically zero.

math.KT↗

arXiv2019

Smale space C*-algebras have nonzero projections

The main result of the present paper is that the stable and unstable C*-algebras associated to a mixing Smale space always contain nonzero projections. This gives a positive answer to a question of the first listed author and Karen Strung and has implications for the structure of these algebras in light of the Elliott program for simple C*-algebras. Using our main result, we also show that the homoclinic, stable, and unstable algebras each have real rank zero.

math.OA↗

arXiv2019

Estimating Dixmier traces of Hankel operators in Lorentz ideals

In this paper we study Dixmier traces of powers of Hankel operators in Lorentz ideals. We extend results of Engliš-Zhang to the case of powers $p\geq 1$ and general Lorentz ideals starting from abstract extrapolation results of Gayral-Sukochev. In the special case $p=2,4,6$ we give an exact formula for the Dixmier trace. For general $p$, we give upper and lower bounds on the Dixmier trace. We also construct, for any $p$ and any Lorentz ideal, examples of non-measurable Hankel operators.

math.FA↗

arXiv2018

Constructing KMS states from infinite-dimensional spectral triples

We construct KMS-states from $\mathrm{Li}_1$-summable semifinite spectral triples and show that in several important examples the construction coincides with well-known direct constructions of KMS-states for naturally defined flows. Under further summability assumptions the constructed KMS-state can be computed in terms of Dixmier traces. For closed manifolds, we recover the ordinary Lebesgue integral. For Cuntz-Pimsner algebras with their gauge flow, the construction produces KMS-states from traces on the coefficient algebra and recovers the Laca-Neshveyev correspondence. For a discrete group acting on its Stone-Čech boundary, we recover the Patterson-Sullivan measures on the Stone-Čech boundary for a flow defined from the Radon-Nikodym cocycle.

math.OA↗

arXiv2018

Relative geometric assembly and mapping cones, Part II: Chern characters and the Novikov property

We study Chern characters and the assembly mapping for free actions using the framework of geometric $K$-homology. The focus is on the relative groups associated with a group homomorphism $ϕ:Γ_1\to Γ_2$ along with applications to Novikov type properties. In particular, we prove a relative strong Novikov property for homomorphisms of hyperbolic groups and a relative strong $\ell^1$-Novikov property for polynomially bounded homomorphisms of groups with polynomially bounded cohomology in $\C$. As a corollary, relative higher signatures on a manifold with boundary $W$, with $π_1(\partial W)\to π_1(W)$ belonging to the class above, are homotopy invariant.

math.KT↗

arXiv2018

Relative geometric assembly and mapping cones, Part I: The geometric model and applications

Inspired by an analytic construction of Chang, Weinberger and Yu, we define an assembly map in relative geometric $K$-homology. The properties of the geometric assembly map are studied using a variety of index theoretic tools (e.g., the localized index and higher Atiyah-Patodi-Singer index theory). As an application we obtain a vanishing result in the context of manifolds with boundary and positive scalar curvature; this result is also inspired and connected to work of Chang, Weinberger and Yu. Furthermore, we use results of Wahl to show that rational injectivity of the relative assembly map implies homotopy invariance of the relative higher signatures of a manifold with boundary.

math.KT↗

arXiv2017

Dixmier traces and residues on weak operator ideals

We develop the theory of modulated operators in general principal ideals of compact operators. For Laplacian modulated operators we establish Connes' trace formula in its local Euclidean model and a global version thereof. It expresses Dixmier traces in terms of the vector-valued Wodzicki residue. We demonstrate the applicability of our main results in the context of log-classical pseudo-differential operators, studied by Lesch, and a class of operators naturally appearing in noncommutative geometry.

math.FA↗

arXiv2017

Spectral triples on $O_N$

We give a construction of an odd spectral triple on the Cuntz algebra $O_{N}$, whose $K$-homology class generates the odd $K$-homology group $K^1(O_{N})$. Using a metric measure space structure on the Cuntz-Renault groupoid, we introduce a singular integral operator which is the formal analogue of the logarithm of the Laplacian on a Riemannian manifold. Assembling this operator with the infinitesimal generator of the gauge action on $O_{N}$ yields a $θ$-summable spectral triple whose phase is finitely summable. The relation to previous constructions of Fredholm modules and spectral triples on $O_{N}$ is discussed.

math.OA↗

arXiv2016

Shift tail equivalence and an unbounded representative of the Cuntz-Pimsner extension

We show how the fine structure in shift-tail equivalence, appearing in the noncommutative geometry of Cuntz-Krieger algebras developed by the first two authors, has an analogue in a wide range of other Cuntz-Pimsner algebras. To illustrate this structure, and where it appears, we produce an unbounded representative of the defining extension of the Cuntz-Pimsner algebra constructed from a finitely generated projective bi-Hilbertian module, extending work by the third author with Robertson and Sims. As an application, our construction yields new spectral triples for Cuntz- and Cuntz-Krieger algebras and for Cuntz-Pimsner algebras associated to vector bundles twisted by equicontinuous $*$-automorphisms.

math.KT↗

arXiv2016

The bordism group of unbounded KK-cycles

We consider Hilsum's notion of bordism as an equivalence relation on unbounded $KK$-cycles and study the equivalence classes. Upon fixing two $C^*$-algebras, and a $*$-subalgebra dense in the first $C^*$-algebra, a $\mathbb{Z}/2\mathbb{Z}$-graded abelian group is obtained; it maps to the Kasparov $KK$-group of the two $C^*$-algebras via the bounded transform. We study properties of this map both in general and in specific examples. In particular, it is an isomorphism if the first $C^*$-algebra is the complex numbers (i.e., for $K$-theory) and is a split surjection if the first $C^*$-algebra is the continuous functions on a compact manifold with boundary when one uses the Lipschitz functions as the dense $*$-subalgebra.

math.KT↗

arXiv2016

Boundaries, spectral triples and K-homology

This paper extends the notion of a spectral triple to a relative spectral triple, an unbounded analogue of a relative Fredholm module for an ideal $J\triangleleft A$. Examples include manifolds with boundary, manifolds with conical singularities, dimension drop algebras, $θ$-deformations and Cuntz-Pimsner algebras of vector bundles. The bounded transform of a relative spectral triple is a relative Fredholm module, making the image of a relative spectral triple under the boundary mapping in $K$-homology easy to compute. We introduce an additional operator called a Clifford normal with which a relative spectral triple can be doubled into a spectral triple. The Clifford normal also provides a boundary Hilbert space, a representation of the quotient algebra, a boundary Dirac operator and an analogue of the Calderon projection. In the examples this data does assemble to give a boundary spectral triple, though we can not prove this in general. When we do obtain a boundary spectral triple, we provide sufficient conditions for the boundary triple to represent the $K$-homological boundary. Thus we abstract the proof of Baum-Douglas-Taylor's "boundary of Dirac is Dirac on the boundary" theorem into the realm of non-commutative geometry.

math.KT↗

arXiv2016

Wieler solenoids, Cuntz-Pimsner algebras and K-theory

We study irreducible Smale spaces with totally disconnected stable sets and their associated $K$-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one $K$-theoretic. Using Wieler's Theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyze an explicit groupoid Morita equivalence between the groupoids of Deaconu-Renault and Putnam-Spielberg, extending results of Thomsen. The Deaconu-Renault groupoid and the explicit Morita equivalence leads to a Cuntz-Pimsner model for the stable Ruelle algebra. The $K$-theoretic invariants of Cuntz-Pimsner algebras are then studied using the Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions we characterize the KMS weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples.

math.DS↗

arXiv2016

Nonclassical spectral asymptotics and Dixmier traces: From circles to contact manifolds

We consider the spectral behavior and noncommutative geometry of commutators $[P,f]$, where $P$ is an operator of order $0$ with geometric origin and $f$ a multiplication operator by a function. When $f$ is Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudo-differential calculus is available, variations of Connes' residue trace theorem and related integral formulas continue to hold. On the circle, a large class of non-measurable Hankel operators is obtained from Hölder continuous functions $f$, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.

math.SP↗

arXiv2016

Realizing the analytic surgery group of Higson and Roe geometrically, Part II: Relative eta-invariants

We apply the geometric analog of the analytic surgery group of Higson and Roe to the relative $η$-invariant. In particular, by solving a Baum-Douglas type index problem, we give a "geometric" proof of a result of Keswani regarding the homotopy invariance of relative $η$-invariants. The starting point for this work is our previous constructions in "Realizing the analytic surgery group of Higson and Roe geometrically, Part I: The geometric model" (arXiv:1308.5990).

math.KT↗