arXiv subjects

Walter D. van Suijlekom

Publications and source records attributed to Walter D. van Suijlekom.

At least 19 recordsLinked to original sources

arXiv2026

K-theoretic invariants of the Toeplitz operator system and its dual, and of all graph operator systems

We compute $K$-theoretic invariants for several finite-dimensional operator systems. These include the Toeplitz operator system and its dual (the Fejér-Riesz operator system), as well as operator systems associated to tolerance relations, aka graph operator systems. A variety of techniques is applied, adapted to each of these cases, ranging from a generalization of Carathéodory's factorization result for block Toeplitz matrices, to Wiener-Hopf factorization and Schur complements. In most instances we find that the invariants coincide with their analogues for the $C^*$-envelope, except for the $K_1$-invariants of the Fejér-Riesz operator system.

math.OA↗

arXiv2026

K-theoretic invariants of four three-dimensional operator systems

We compute $K$-theoretic invariants for four three-dimensional operator systems to illustrate the general theory. We find that in some cases the invariants distinguish the operator system from its $C^*$-envelope, while in others they do not. We also explore extensions of the theory.

math.OA↗

arXiv2026

Particle Creation in a Cosmological Background in Analogy to the Schwinger Effect

We consider a gravitational analogue of the Schwinger effect in a cosmological context. While the Schwinger effect is usually attributed to a static electric background, its derivation is actually based on a switching on/off of the electric field in the infinite past/future. Motivated by this, and our previous work on particle production in a gravitational background, we consider a long pulse of the gravitational field in an FLRW-spacetime, thus simulating a static background. We rigorously derive particle production by a novel application of the Heun equation. In fact, the recently obtained connecting formulas between its local solutions can be used to determine the Bogolyubov coefficients, and subsequently the particle production probabilities, in the limit of an infinitely long pulse. The particle production in the FLRW-model is found to have a lower threshold on the late time frequencies, which can be related to the duration of the time interval of the switching on/off the background field. For large frequencies and in the spatially flat case, we find a Planckian frequency spectrum whose temperature is inversely proportional to the duration of scale change. We compare our findings to Schwinger's result on particle production for a long pulse of an electromagnetic background field, for which we also include a detailed derivation.

gr-qc↗

arXiv2026

Operator systems and positive extensions over discrete groups

The extension problem asks whether positive semi-definite functions on a symmetric unital subset of a discrete group can be extended to positive semi-definite functions on the whole group. It has been known at least since the work of Rudin in the 1960s that this is closely related to the problem of finding sums of squares factorisations of positive elements in the group C*-algebra. We give an operator system perspective at these two problems explaining their equivalence: the extension property is characterised by a certain quotient map on the Fourier--Stieltjes algebra, and the factorisation property by a certain complete order embedding into the group C*-algebra. These properties are linked to the duality of the operator systems which have recently emerged from spectral and Fourier truncations in noncommutative geometry. We exemplify how one can relate certain extension problems to operator system techniques such as nuclearity and the C*-envelope.

math.OA↗

arXiv2025

Quadratic Forms, Real Zeros and Echoes of the Spectral Action

For a real distribution $\mathcal{D}$ on the interval $[0,L]$ with $\tilde{\mathcal{ D}}$ the associated even distribution on the interval $[-L, L]$, we prove that if the associated quadratic form with Schwartz kernel $\tilde{\mathcal{D}}(x - y)$ defines a lower-bounded selfadjoint operator on $L^2([-\frac{L}{2}, \frac{L}{2}])$, whose lowest spectral value $λ$ is a simple, isolated eigenvalue with even eigenfunction $ξ$, then all the zeros of the entire function $\widehat ξ(z)$, the Fourier transform of $ξ$, lie on the real line. The proof proceeds in five steps. (1) We give a C*-algebraic proof of a corollary of Carathéodory-Fejér's 1911 structure Theorem for Toeplitz matrices: if $T \in M_n(\mathbb{C})$ is a Hermitian, positive semidefinite Toeplitz matrix of rank $n - 1$, and $ξ\in \ker T$, then the polynomial $P(z) = \sum ξ_j z^j$ has all its zeros on the unit circle. (2) We formulate and prove a continuous analogue of this result, replacing the Toeplitz matrix with a convolution operator with continuous kernel $h(x - y)$, and the polynomial $P(z)$ with the Fourier transform of the eigenfunction corresponding to the largest eigenvalue. (3) We analyze finite-dimensional truncations of the quadratic forms defined by real, even distributions $\mathcal{D}$ on $[-L, L]$, and observe that the resulting matrices exhibit a structure previously encountered in perturbative expansions of the spectral action. (4) We establish an analogue of Carathéodory-Fejér's corollary for matrices of this specific structure, thereby extending the zero localization result beyond the classical Toeplitz setting. (5) Finally, we apply a classical theorem of Hurwitz concerning the zeros of uniform limits of holomorphic functions to deduce the general result stated above.

math.OA↗

arXiv2025

An upper limit to the lifetime of stellar remnants from gravitational pair production

Black holes are assumed to decay via Hawking radiation. Recently we found evidence that spacetime curvature alone without the need for an event horizon leads to black hole evaporation. Here we investigate the evaporation rate and decay time of a non-rotating star of constant density due to spacetime curvature-induced pair production and apply this to compact stellar remnants such as neutron stars and white dwarfs. We calculate the creation of virtual pairs of massless scalar particles in spherically symmetric asymptotically flat curved spacetimes. This calculation is based on covariant perturbation theory with the quantum field representing, e.g.,\ gravitons or photons. We find that in this picture the evaporation timescale, $τ$, of massive objects scales with the average mass density, $ρ$, as $τ\proptoρ^{-3/2}$. The maximum age of neutron stars, $τ\sim 10^{68}\,\text{yr}$, is comparable to that of low-mass stellar black holes. White dwarfs, supermassive black holes, and dark matter supercluster halos evaporate on longer, but also finite timescales. Neutron stars and white dwarfs decay similarly to black holes, ending in an explosive event when they become unstable. This sets a general upper limit for the lifetime of matter in the universe, which in general is much longer than the Hubble--Lemaître time, although primordial objects with densities above $ρ_\text{max} \approx 3\times 10^{53}\,\text{g/}\text{cm}^3$ should have dissolved by now. As a consequence, fossil stellar remnants from a previous universe could be present in our current universe only if the recurrence time of star forming universes is smaller than about $\sim 10^{68}\,\text{years}$.

gr-qc↗

arXiv2025

Reply to "Comment on 'Gravitational Pair Production and Black Hole Evaporation'"

In a recent letter, the authors presented a unified derivation of the electric Schwinger effect and a generalized Hawking effect with an additional radiation component. The approach discloses a radial profile of black hole pair production and traces the emission back to local tidal forces which are independent of the black hole event horizon. It uses an effective action valid to second order in curvature and arbitrary order in proper time. A comment on the letter supposed two inconsistencies when applying the central formula to the Schwinger effect in the presence of magnetic fields. The present letter points out that the partially flawed argumentation does not cast doubt on the results.

gr-qc↗

arXiv2024

Higher K-groups for operator systems

We extend our previous definition of K-theoretic invariants for operator systems based on hermitian forms to higher K-theoretical invariants. We realize the need for a positive parameter $δ$ as a measure for the spectral gap of the representatives for the K-theory classes. For each $δ$ and integer $p \geq 0$ this gives operator system invariants $\mathcal V_p^δ(-,n)$, indexed by the corresponding matrix size. The corresponding direct system of these invariants has a direct limit that possesses a semigroup structure, and we define the $K_p^δ$-groups as the corresponding Grothendieck groups. This is an invariant of unital operator systems, and, more generally, an invariant up to Morita equivalence of operator systems. Moreover, there is a formal periodicity that reduces all these groups to either $K_0^δ$ or $K_1^δ$. We illustrate our invariants by means of the spectral localizer.

math.OA↗

arXiv2024

A generalization of K-theory to operator systems

We propose a generalization of K-theory to operator systems. Motivated by spectral truncations of noncommutative spaces described by $C^*$-algebras and inspired by the realization of the K-theory of a $C^*$-algebra as the Witt group of hermitian forms, we introduce new operator system invariants indexed by the corresponding matrix size. A direct system is constructed whose direct limit possesses a semigroup structure, and we define the $K_0$-group as the corresponding Grothendieck group. This is an invariant of unital operator systems, and, more generally, an invariant up to Morita equivalence of operator systems. For $C^*$-algebras it reduces to the usual definition. We illustrate our invariant by means of the spectral localizer.

math.OA↗

arXiv2024

Gromov--Hausdorff Convergence of Spectral Truncations for Tori

We consider operator systems associated to spectral truncations of tori. We show that their state spaces, when equipped with the Connes distance function, converge in the Gromov--Hausdorff sense to the space of all Borel probability measures on the torus equipped with the Monge--Kantorovich distance. A crucial role will be played by the relationship between Schur and Fourier multipliers. Along the way, we introduce the spectral Fejér kernel and show that it is a good kernel. This allows to make the estimates sufficient to prove the desired convergence of state spaces. We conclude with some structure analysis of the pertinent operator systems, including the C*-envelope and the propagation number, and with an observation about the dual operator system.

math.OA↗

arXiv2023

Convergence of spectral truncations for compact metric groups

We consider Gromov-Hausdorff convergence of state spaces for spectral truncations of a compact metric group $G$. We work in the context of order-unit spaces and consider orthogonal projections $P_Λ$ in $L^2(G)$ corresponding to finite subsets of irreducible representations $Λ\subseteq \widehat G$. We then prove that the sequence of truncated state spaces $\{ S(P_ΛC(G) P_Λ)\}_Λ$ Gromov-Hausdorff converges to the original state space $S(C(G))$, when these are equipped with a metric associated to a Lip-norm which in turn is induced by the action of $G$.

math.OA↗

arXiv2023

Gravitational Pair Production and Black Hole Evaporation

We present a new avenue to black hole evaporation using a heat-kernel approach analogous as for the Schwinger effect. Applying this method to an uncharged massless scalar field in a Schwarzschild spacetime, we show that spacetime curvature takes a similar role as the electric field strength in the Schwinger effect. We interpret our results as local pair production in a gravitational field and derive a radial production profile. The resulting emission peaks near the unstable photon orbit. Comparing the particle number and energy flux to the Hawking case, we find both effects to be of similar order. However, our pair production mechanism itself does not explicitly make use of the presence of a black hole event horizon.

gr-qc↗

arXiv2022

Riemannian embeddings in codimension one as unbounded $KK$-cycles

Given a codimension one Riemannian embedding of Riemannian spin$^c$-manifolds $\imath:X \to Y$ we construct a family $\{\imath_!^ ε\}_{0< ε< ε_0}$ of unbounded $KK$-cycles from $C(X)$ to $C_0(Y)$, each equipped with a connection $\nabla^ε$ and each representing the shriek class $\imath_! \in KK(C(X), C_0(Y))$. We compute the unbounded product of $\imath_!^ε$ with the Dirac operator $D_Y$ on $Y$ and show that this represents the $KK$-theoretic factorization of the fundamental class $[X] = \imath_! \otimes [Y]$ for all $ε$. In the limit $ε\to 0$ the product operator admits an asymptotic expansion of the form $\frac{1}ε T + D_X + \mathcal{O}(ε)$ where the ``divergent'' part $T$ is an index cycle representing the unit in $KK(\mathbb{C}, \mathbb{C})$ and the constant ``renormalized'' term is the Dirac operator $D_X$ on $X$. The curvature of $(\imath_!^ε, \nabla^ε)$ is further shown to converge to the square of the mean curvature of $\imath$ as $ε\to 0$.

math.DG↗

arXiv2022

Cyclic cocycles and one-loop corrections in the spectral action

We present an intelligible review of recent results concerning cyclic cocycles in the spectral action and one-loop quantization. We show that the spectral action, when perturbed by a gauge potential, can be written as a series of Chern-Simons actions and Yang-Mills actions of all orders. In the odd orders, generalized Chern-Simons forms are integrated against an odd $(b,B)$-cocycle, whereas, in the even orders, powers of the curvature are integrated against $(b,B)$-cocycles that are Hochschild cocycles as well. In both cases, the Hochschild cochains are derived from the Taylor series expansion of the spectral action Tr$(f(D+V))$ in powers of $V=π_D(A)$, but unlike the Taylor expansion we expand in increasing order of the forms in $A$. We then analyze the perturbative quantization of the spectral action in noncommutative geometry and establish its one-loop renormalizability as a gauge theory. We show that the one-loop counterterms are of the same Chern-Simons-Yang-Mills form so that they can be safely subtracted from the spectral action. A crucial role will be played by the appropriate Ward identities, allowing for a fully spectral formulation of the quantum theory at one loop.

hep-th↗

arXiv2022

Cyclic cocycles in the spectral action

We show that the spectral action, when perturbed by a gauge potential, can be written as a series of Chern--Simons actions and Yang--Mills actions of all orders. In the odd orders, generalized Chern--Simons forms are integrated against an odd $(b,B)$-cocycle, whereas, in the even orders, powers of the curvature are integrated against $(b,B)$-cocycles that are Hochschild cocycles as well. In both cases, the Hochschild cochains are derived from the Taylor series expansion of the spectral action Tr$(f(D+V))$ in powers of $V=π_D(A)$, but unlike the Taylor expansion we expand in increasing order of the forms in $A$. This extends [Connes--Chamseddine 2006], which computes only the scale-invariant part of the spectral action, works in dimension at most 4, and assumes the vanishing tadpole hypothesis. In our situation, we obtain a truly infinite odd $(b,B)$-cocycle. The analysis involved draws from recent results in multiple operator integration, which also allows us to give conditions under which this cocycle is entire, and under which our expansion is absolutely convergent. As a consequence of our expansion and of the gauge invariance of the spectral action, we show that the odd $(b,B)$-cocycle pairs trivially with $K_1$.

math.QA↗

arXiv2022

Noncommutativity and Physics: A non-technical review

We give an overview of the applications of noncommutative geometry to physics. Our focus is entirely on the conceptual ideas, rather than on the underlying technicalities. Starting historically from the Heisenberg relations, we will explain how in general noncommutativity yields a canonical time evolution, while at the same time allowing for the coexistence of discrete and continuous variables. The spectral approach to geometry is then explained to encompass two natural ingredients: the line element and the algebra. The relation between these two is dictated by so-called higher Heisenberg relations, from which both spin geometry and non-abelian gauge theory emerges. Our exposition indicates some of the applications in physics, including Pati--Salam unification beyond the Standard Model, the criticality of dimension 4, second quantization and entropy.

hep-th↗

arXiv2022

Operator systems for tolerance relations on finite sets

We study the duals of a certain class of finite-dimensional operator systems, namely the class of operator systems associated to tolerance relations on finite sets or equivalently the class of operator systems that are associated with graphs. In the case where the graphs associated with these operator systems are chordal we are able to find concrete realizations of their duals as sitting inside of finite-dimensional $C^*$-algebras. We then use these concrete realizations to compute the $C^*$-envelopes, propagation numbers and extremal rays of these duals in the chordal case. Finally, we exemplify our results by applying them to operator systems of band matrices.

math.OA↗

arXiv2021

Tolerance relations and operator systems

We extend the scope of noncommutative geometry by generalizing the construction of the noncommutative algebra of a quotient space to situations in which one is no longer dealing with an equivalence relation. For these so-called tolerance relations, passing to the associated equivalence relation looses crucial information as is clear from the examples such as coarse graining in physics or the relation $d(x,y)< \varepsilon$ on a metric space. Fortunately, thanks to the formalism of operator systems such an extension is possible and provides new invariants, such as the $C^*$-envelope and the propagation number. After a thorough investigation of the structure of the (non-unital) operator systems associated to tolerance relations, we analyze the corresponding state spaces. In particular, we determine the pure state space associated to the operator system for the relation $d(x,y)< \varepsilon$ on a path metric measure space.

math.OA↗