Papers

We prove $K_G^{\mathbb{C}}>1.35584631827168$ for the classical complex Grothendieck constant, closing more than one quarter of the gap between Davie's lower bound and Haagerup's upper bound. The numerical part of the proof is rigorously verified by interval arithmetic. The lower bound and the proof were discovered by the Odin Automatic AI Research Agent.

We introduce the notion of perfect regularity as an appropriate weakening of perfect normality, both for spaces and locales. Various characterizations are given, using Dedekind-MacNeille completions, injective hulls, and sublocales. We place the new class of perfectly regular frames among various well-studied classes of frames. We also introduce the construction of perfect regularization of a completely regular frame, compare it to Isbell's well-known booleanization construction, and argue that it is at least as important as the latter.

We determine the spectrum of the predual generator of an irreducible Gaussian quantum Markov semigroup on trace-class operators, the natural space for density operators. The drift matrix governs the linear part of the Gaussian evolution and organizes the spectral classification. When the linear flow generated by this matrix is not periodic, the spectrum is the closed left half-plane. If all drift eigenvalues have negative real part, every interior point is an eigenvalue. There are no trace-class eigenvectors if some drift eigenvalue has positive real part or all have zero real part. For periodic flow, the spectrum consists of unions of horizontal half-lines or filled parabolic regions. Examples show that irreducibility cannot in general be omitted. We apply the results to oscillators, free particles, and boson chains.

Based on the representation theory of the $q$-CCR algebra, we develop a representation-theoretic construction of the $q$-deformed polyanalytic Fock spaces, their reproducing kernels, and the associated $q$-deformed polyanalytic Bargmann transforms. The associated determinantal point processes provide $q$-deformations of the pure and full polyanalytic Ginibre ensembles. Moreover, we determine their limiting density and establish central limit theorems for the fluctuations of their linear statistics as the number of particles tends to infinity and the deformation parameter $q$ tends to 1 simultaneously. Remarkably, although the $q$-deformation changes the macroscopic density and the size of the droplet, the limiting covariance structures coincide, after spatial rescaling, with those of the classical polyanalytic Ginibre ensembles.

We solve four questions raised by Bayart concerning coefficient summability for multilinear forms with prescribed supports. For every $m\ge 2$ and $d\in[1,m]$, we determine the product-summability exponent and the multilinear summability invariant: \[ \mathrm{prod}(m,d) =\min\left\{\frac{m}{d},\,m-\lceil d\rceil+1\right\}, \qquad \gamma_{\mathrm{mult}}(m,d) =\min\left\{m-\lceil d\rceil+1,\frac{2m}{d+1}\right\}. \] In particular, $\gamma_{\mathrm{mult}}(4,2)=8/3$, showing that the multilinear invariant need not be an integer. We also prove that, for every $d\in[1,m]$, there is a single infinite support of exact combinatorial dimension $d$ on which the dimensional Hardy–Littlewood bound is attained over both scalar fields for every anisotropic parameter $\mathbf p=(p_1,\ldots,p_m)$ with $\sum_j 1/p_j<1$, simultaneously across the two regimes separated by $\sum_j 1/p_j=1/2$.

We record two results in the theory of vector and Banach lattices related to order adherence. First, we give a negative answer to Gao and Leung's question on whether the $uo$-adherence of sublattices must be order closed. Second, we present a self-contained counterexample showing that a Banach lattice with a weakly Fatou norm need not admit any equivalent lattice norm with the Fatou property.

For every $2<p<\infty$ and every integer $r\ge2$, we construct a finite set $T\subseteq S_{L^p[0,1]}$ such that $|T|\le2^{Cr^2}$, $\gamma_2(T)\le Cr$, and $\gamma_2(\conv T)\ge c r^{3/2-1/p}$. Consequently, for every fixed $p>2$, both estimates in Talagrand's Research Problem~2.11.3 fail in each of the spaces $L^p[0,1]$ and $\ell_p$. The lower bound follows from a multilevel product principle applied to a scale-separated product of Euclidean simplices. The same construction gives the sharp convexification profile of $\ell_p^r(\ell_2^{2^{128r}})$ for $2<p\le\infty$ and, by finite representability, quantitative failure in every infinite-dimensional Banach space with cotype index greater than $2$.

The convergence rate of a free energy Wasserstein gradient flow is quantified by its so-called Polyak-Lojasiewicz (PL) constant $\lambda$, which relates the objective function to its dissipation along the flow. Such a flow is the mean-field limit as $N$ goes to infinity of a system of $N$ interacting particles, whose convergence rate in relative entropy towards its Gibbs measure is quantified by its log-Sobolev constant $\lambda_N$. Different behaviours of $\lambda_N$ as $N$ goes to infinity thus describe drastically different phenomena, such as fast relaxation or metastability, with many models undergoing phase transitions between these regimes, depending typically on temperature. Under fairly general conditions, a uniform-in-$N$ log-Sobolev constant (i.e. fast exponential convergence for the particle system) is known to induce a positive PL constant (i.e. exponential convergence for the mean-field flow). A conjecture was stated by Delgadino, Gvalani, Pavliotis and Smith according to which the converse implication was true ($\lambda>0$ implies $\liminf \lambda_N >0$), even with $\lim \lambda_N =\lambda$. First, we will prove this converse implication, although without the equality $\lim \lambda_N = \lambda$, for a general class of mean-field models. Second, we also notice that this implication fails if the free energy minimiser is not unique, and provide an explicit counter-example. Third, we also consider the same question of relating $N$-particle and mean-field inequalities in the context of more general Lojasiewicz inequalities, which correspond to polynomial (instead of exponential) convergence rates, and can describe the situation exactly at a phase transition.

We study the variation index of a continuous path along a refining partition sequence, defined as the infimum of exponents $p\ge1$ for which the corresponding $p$-th variation sums are uniformly bounded. Under natural geometric assumptions on the partitions, we prove that this index coincides with the classical variation index, defined using all finite partitions, for every path with positive Hölder regularity. The result extends to paths with an all-orders Dini modulus and yields a characterization of the classical variation index in terms of dyadic Faber–Schauder coefficients. Explicit counterexamples show that the partition assumptions cannot in general be omitted. Even under these assumptions, however, membership in the corresponding function spaces at the critical exponent may still depend on the partition sequence. We further obtain compact subcritical embeddings into the space of paths with vanishing classical $p$-variation and establish a strict hierarchy of coefficient and variation spaces. Finally, we show that the continuous embedding of the classical $p$-variation space into the space of continuous paths with uniformly bounded dyadic $p$-th variation sums has nonclosed range.

We characterize locally finite sets \(T\subset[0,\infty)\) that preserve exact observability from nonnegative integer times for every bounded positive operator \(A\) and bounded observation operator \(C\). The Stein identity reduces universality to uniform upper and lower bounds on radial Stein–Pick cones over all finite spectral models. Writing \(N_T(R)=\#(T\cap[0,R])\), we prove that universal Bessel preservation is equivalent to \(N_T(R)=O(R)\). The conditions \(0\in T\) and \(N_T(R)\asymp R\) characterize universal observability preservation under \([A,C^*C]=0\), as well as uniform bound preservation for each fixed number of spectral values. However, we construct a clustered set satisfying these conditions whose optimal sampled lower bounds tend to zero as the spectral count grows, despite fixed integer observability bounds. The same set destroys the frame property of a singly generated positive diagonal Carleson frame while preserving its Bessel property. When natural density exists, universality is equivalent to \(0\in T\) and \(0<d(T)<\infty\). More generally, containing zero and a subset of positive finite natural density suffices under linear upper counting. For sets containing zero, universality is invariant under sublinear differences of counting functions. For general normal operators, universal Bessel preservation is instead equivalent to uniformly bounded counts on unit intervals; under a fixed logarithmic sector condition, it is again characterized by linear upper counting.

We show that there is a Hadamard space in which the closed convex hull of a set containing merely three points is noncompact, by providing an AI-generated construction. The seemingly innocuous problem of finding a compact or finite set with this property, or showing that none exists, was recorded by Gromov and periodically revisited by researchers in metric geometry, functional analysis, fixed-point theory, and geometric group theory. Nevertheless, it remained unanswered for over three decades. The construction itself is based on building an inductive sequence of CAT(1) metric graphs and taking the Euclidean cone over the limiting space. We discuss the history of this problem, the main ideas and details of the construction, and its relationship with existing techniques.

We study the Tsallis-to-Kullback–Leibler (KL) limit for entropy-regularized optimal transport with nonnegative bounded continuous costs. Fixing the regularization parameter $\varepsilon > 0$, we first derive an exact variational reformulation of Tsallis-regularized optimal transport in terms of the Tsallis information projection onto the set of couplings. The formula isolates an explicit correction term and thereby explains why, unlike in the KL case, the regularized transport problem and the corresponding information projection problem do not coincide exactly. We also establish existence and uniqueness for the Tsallis information projection. We then prove, with respect to the narrow topology, the $\Gamma$-convergence of the Tsallis-regularized functionals to the KL-regularized functional as $q\downarrow1$, together with narrow convergence of their unique minimizers. Finally, we obtain explicit error estimates of order $O(q-1)$ for both the regularized optimal transport values and the associated information projection values. These results quantify the passage from Tsallis regularization to the classical KL setting and clarify the relation between entropic regularization and information projection for $1 < q \leq 2$.

There is a remarkable variety of natural generalizations of the notion of scalar positivity to square matrices. The lists and tables indicate those that we consider here; they are practically exhaustive of those that arise in practice and theory. Since positive scalars are closed under taking arbitrary powers and roots, it is natural to ask which of these matrix generalizations enjoy analogous closure properties. Our purpose here is to examine and advance these generalizations with respect to their closure under integer powers and fractional matrix roots. In addition to surveying known results, we establish for several important classes of matrices that positivity is preserved under fractional powers, showing that key structural features are retained under matrix roots. The role of Pick functions in the analysis is emphasized throughout. Proofs, counterexamples, and related results are presented, and summary tables are included to provide a concise and useful reference.

In the 2004 collection Unsolved Problems in Mathematical Systems and Control Theory, Erik Verriest posed the problem of characterizing Riccati stability “without invoking additional matrices.” We give such a characterization through a scalar invariant of the resolvent family. The invariant is defined by covariance balances using at most $n^2+1$ frequency–direction pairs. A finite-dimensional separation argument shows that this covariance radius equals the optimal common ellipsoidal norm of the resolvent family. Combined with the strict bounded real lemma, this yields a necessary and sufficient condition for Riccati stability involving only the Hurwitz property of the first matrix and a single scalar inequality. The invariant reduces to the ordinary spectral radius for one matrix, lies between the pointwise spectral-radius and unscaled small-gain levels of the resolvent family, and coincides with the operator-space spectral radius of Shalit and Shamovich for the natural resolvent function space. The main results have been formally verified in the Lean proof assistant.

We study an interpolation problem in extended Gevrey classes defined by the sequences $M_n^{\tau,\sigma}=n^{\tau n^\sigma}$, $n\in\mathbb N$, $\tau>0$, $\sigma>1$. We show that, under a suitable growth condition on a divergent sequence of derivative orders, estimates of extended Gevrey type imposed only along this sequence imply corresponding estimates for all derivative orders. In particular, we explicitly quantify the change of the parameters in the resulting estimates. To this end, we first establish an equivalence between two definitions of the extended Gevrey classes, one involving the supergeometric factor $h^{n^\sigma}$ and another in which this factor is omitted. We then establish a corresponding interpolation principle for extended Gelfand–Shilov spaces, including a symmetric characterization and a formulation in terms of the associated function of the defining sequence.

A lattice Gabor frame consists of a window function and its time-frequency shifts along a lattice. We classify the lattices that admit Schwartz-class and, equivalently, Feichtinger frame windows. Except for symplectically irrational lattices of critical covolume 1,, we determine the lattice-dependent achievable window smoothness and decay using the scale of modulation spaces. For symplectically rational lattices, including the rational lattices typically used in digital applications, the classification is governed by an integer lattice parameter introduced herein, the symplectic index gap. Our results extend the classical and amalgam Balian-Low theorems. The companion paper "Tilings, Packings, and Smooth and Compactly Supported Gabor Windows" develops the associated tiling-packing theorem and compact-support constructions.

Horn conjectured that whenever $K$ is a nonempty compact convex subset of a Banach space and $F,G:K\to K$ are continuous commuting maps, there exists $x\in K$ such that $F(x)=G(x)$. We give a counterexample in a separable Hilbert space. The construction uses an exactly commuting ladder of coincidence-free maps between finite trees, due to Oversteegen and Rogers.

We study the relation between non-anticipative functional calculus and compatible families of controlled paths. For a non-anticipative functional satisfying horizontal Lipschitz regularity, we show that the iterated vertical derivatives generate compatible higher-order controlled Taylor expansions along Hölder controls, with level-dependent remainder exponents. We derive a rough-integration criterion from these estimates and show that, for $\gamma-$Hölder controls with $\tfrac{1}{2}\geq \gamma>\sqrt{2}-1$, the first-order remainder estimate is recovered. Our main result is a converse representation theorem. We consider non-anticipative functionals $G_0,\ldots,G_p$ which satisfy compatible higher-order controlled Taylor estimates along $\gamma$-Hölder paths. Under natural continuity and compatibility assumptions, we prove that they can be represented as the iterated vertical derivatives of the base functional: $ G_j=\nabla_\omega^jG_0,\qquad j=1,\ldots,p.$ Thus the Gubinelli coefficients of a compatible controlled family are symmetric and uniquely determined by its base functional; in particular, the Gubinelli derivative is identified with the vertical derivative introduced in functional Itô calculus, giving the coefficient hierarchy an intrinsic path-space differential structure. We show that this class of compatible coefficient families is stable under admissible non-anticipative functional composition and derive the corresponding functional chain rule. As an application, we obtain well-posedness for a class of path-dependent rough differential equations with Volterra memory, and identify the Gubinelli derivative of the resulting path-dependent rough coefficient.

We develop a rigorous renormalization group (RG) formalism on Wiener space. We introduce an RG operator $\mathcal R$ acting on probability measures over continuous paths, whose fixed point is the Wiener measure. We linearize $\mathcal{R}$ around the Brownian fixed point and analyze the spectral structure of the resulting operator $\mathscr{L}$. Its eigenvectors are too singular to be realized as honest measures on path space, and we therefore develop a generalized spectral theory within the framework of white noise analysis (Hida calculus). The eigenvectors of $\mathscr{L}$ are realized as Hida distributions, satisfying $\mathscr{L} U=\lambda U$ in the weak sense. This analysis yields structural results such as a spectral gap and a diagonal-concentration property of the eigenvectors. Moreover, we identify a family $\{\mathfrak{U}_n\}_{n \geq 0}$ of eigenvectors with eigenvalues $\lambda_n = 2^{1-n/2}$, consisting of Wick polynomials of white noise formally expressed as $\mathfrak{U}_n = \int_0^1 {:}\dot W(t)^n{:}\, dt$, and rigorously constructed as Hida distributions. We then use this spectral structure to uncover a finer, second layer of universality in the Donsker invariance principle: not only is the convergence of random walks towards the Brownian scaling limit universal, but the entire hierarchy of leading corrections to the Brownian limit is universal as well, governed by the eigenpairs $(\lambda_n, \mathfrak{U}_n)$. We also show, at a formal level, that the framework developed here extends to Gibbs-type perturbations of the Brownian fixed point, in the spirit of self-interacting quantum field theories. In particular, we recover the standard irrelevant/marginal/relevant classification, usually obtained in the physics literature by power-counting, solely from the spectral analysis of $\mathscr{L}$.

We study area operators $\mathbb{A}_{\mu,l}$, $0<l<\infty$, induced by positive Borel measures on the right half-plane and acting on the Hardy spaces of Dirichlet series $\mathscr H^p$, $0<p<\infty$. We first disprove a conjecture proposed by the present authors in an earlier work by constructing a probability measure, valid for all $0<p,l<\infty$, for which the associated area operator is bounded although the measure fails the proposed Carleson conditions. We then characterize the boundedness and compactness of $\mathbb{A}_{\mu,l}$ on both $\mathscr H^p$ and the Hardy space $\mathscr H^p_0$ of Dirichlet series vanishing at $+\infty$, for every $0<p,l<\infty$, via integrability conditions for the associated cone integrals. In particular, the boundedness and compactness are equivalent for these operators. We also give direct proofs of compactness under vanishing Carleson measure and compact $H_{\mathrm i}^p$-Carleson embedding hypotheses. As an application, we establish an estimate for the essential norms of bounded Volterra operators. This leads to a complete characterization of compactness for Volterra operators on $\mathscr{H}^p$ and provides a different proof of compactness for those with Dirichlet series symbols in $\operatorname{VMOA}(\mathbb C_0)$.

Since Young's 1936 theorem, irregular integration has been organized around the threshold 1/2: above it the path determines the integral, while below it higher-order data are needed. For fractional Brownian motion, H = 1/4 is the threshold for the canonical Gaussian enhancement, although geometric rough lifts exist for every H > 0. We prove that H = 1/4 is instead the exact threshold for measurable locality. For d-dimensional fractional Brownian motion with independent components, d at least 2, if 0 < H <= 1/4, no positive-measure Borel set supports even one finite off-diagonal second-level coordinate satisfying Chen's relation and measurable interval by interval from path increments. No moment, Holder, geometricity, stationarity, or scaling assumption is imposed. If 1/4 < H <= 1/2, every full-law local rough-path lift in the standard Holder range is automatically square-integrable and hence classified without an assumed L2 condition; for H > 1/2, every finite-step enhancement with natural graded Holder bounds is uniquely the Young signature. We also introduce an exact scale calculus below classical differentiability. Matched dyadic differences recover normalized derivatives with sharp O(epsilon^2) error, exact localized inversion, and lossless reconstruction. Multiplication and smooth functional calculus transport exactly to scale coordinates, while the corona quotient yields an exact universal derivation. Reinserting the Holder amplitude and using Fourier-normal ordering produces strong geometric lifts for every positive input regularity and every lower rough exponent, with explicit ultraviolet rates and stability. Thus rough lifts exist below one quarter, but no measurable lift on a positive-measure domain can be interval-local there. The results separate existence from local recoverability and show that nondifferentiability does not destroy exact differential information.

In this article, we define classical and non-classical Besov and Triebel–Lizorkin spaces associated with the Grushin operator with or without drift. We establish various characterisations of these spaces, and study their complex interpolation, embedding properties, and fractional Leibniz rules.

Majorization inequalities encode spectral comparisons through leading partial sums, whereas complete Horn inequalities capture all admissible selected-eigenvalue constraints associated with sums of Hermitian matrices [Fulton, 2000]. We develop a general framework for lifting majorization inequalities to this complete Horn level, which is equivalent to unitary-orbit inequalities. The main ingredients are a generalized Hersch–Zwahlen variational formula for coordinatewise monotone functions of selected eigenvalues and a Schubert-geometric lifting principle that reduces the desired Horn inequalities to the concavity or convexity of suitable scalarizer functions on matrix compressions. As examples we apply the framework to three families of operator inequalities. First, we obtain a complete Horn extension of the weak-majorization inequality for convex functions established by Aujla and Silva [Aujla and Silva, 2003]. Second, we lift the concavity and convexity results for two-variable positive-power functions due to Carlen, Frank, and Lieb and Zhang [Carlen, Frank, and Lieb, 2016; Zhang, 2020]. Third, we establish multiplicative Horn inequalities for multivariable geodesic means, including the weighted Karcher mean and the two-variable geometric mean, extending determinant-level inequalities of Bourin and Hiai [Bourin and Hiai, 2014]. The resulting statements yield unitary-orbit inequalities that generalize familiar weak-majorization, trace, and determinant inequalities.

We prove that the quasi-compactness of an analytic semigroup is preserved under $\theta$-subordinate perturbations of its generator on general Banach spaces, provided the perturbations are compact along trajectories. This allows us to prove the permanence of uniform exponential stability for an analytic semigroup under similar perturbations of its generator, provided the perturbed generator generates a strongly stable semigroup. We then show how the analyticity assumption can be relaxed to the class of Crandall-Pazy semigroups under a smaller class of $\theta$-subordinate perturbations, and to immediately differentiable semigroups under bounded perturbations. As an application of the stabilization result, we consider the one-dimensional Neumann Laplacian on a non-reflexive Banach space. Finally, we present counterexamples that demonstrate the lack of uniform stabilization under the larger class of $A$-bounded perturbations.

A discrete set $\Lambda$ is called a spectrum of a Borel probability measure $\mu$ if the exponential functions $\{e^{2\pi i \langle\lambda, x\rangle}:\lambda\in\Lambda\}$ form an orthonormal basis for $L^2(\mu)$. In this work, we first give necessary and sufficient conditions for a self-replicating translation set to be a spectrum of the self-similar measure associated with a product-form Hadamard triple, which were discovered by the first-named author and Lai [Adv. Math. 431 (2023) Paper No.109257]. These results extend the studies by Łaba and Wang [J. Funct. Anal. 193 (2002) 409-420], Dutkay and Lai [J. Math. Pures Appl. 107 (2017) 183-204] in $\mathbb R$. As an application, we can explicitly construct such a self-replicating spectrum. Finally, we prove that the dual spectral set conjecture holds for a self-replicating translation set.

In this paper we state, prove, and discuss some new boundedness results concerning sequences of linear functionals of functions from the Lipschitz class \(Lip_1\) and general orthonormal systems. It is also proved that these results are, in a sense, the best possible.

It is known that any convex cone in a general vector space is the positive cone of a preorder compatible with the vector space, and that any convex set is an affine slice of a convex cone. We therefore treat convex sets order-theoretically, as affine slices of positive cones. We find that faces are sets of strong minima, and we generalize functionals to oriented maps valued in arbitrary ordered vector spaces. Oriented maps expose every face, and an oriented affine map $f$ separates $C$ from $D$ when $f(D) \le 0 \le f(C)$ element-wise. This generalization of hyperplane separation recovers mathematical objects that are "local", including the generated face at a point and the normal cone. The kernels of separating oriented maps form a lattice of separating affine flats whose bottom element is the joint supporting subspace (JSS), recently introduced in finite dimensions for qualification-free convex analysis results. We define the JSS in general vector spaces for two convex sets $C, D$ as the affine slice of the cross-lineality $\mathrm{lin}(K-G)$, where $K,G$ are the homogenization cones of $C, D$. We show that in finite dimensions, bilateral facial reduction corresponds to a descent in the lattice of separating affine flats, which we formalize as lexicographic products of oriented maps. The lexicographic characterization of faces follows as a corollary. By making no assumption other than convexity, our approach yields general results: we show that the face of a difference of two convex sets is a difference of faces, dropping assumptions of compactness, finite dimensionality, and exposedness. We also show that any face of an intersection of two convex sets is an intersection of faces, resolving open problem 5.10 by Weis in "A note on faces of convex sets", dropping the assumption of non-empty intrinsic core.

It is shown that a bounded linear map on a complex separable Hilbert space with non-trivial invariant probability measures doesn't have to possess eigenvalues. This resolves a question indicated by Flytzanis in 1995 and concretely asked by Grivaux–López-Martínez from 2023 resp. Grivaux–Matheron–Menet from 2021. Still, as a positive result, we prove that every bounded linear operator on a separable complex Banach space for which a non-fixing invariant probability measure exists, has to have some approximate point spectrum on the unit circle minus $1$.

We define a class of infinite dimensional linear port-Hamiltonian delay systems, give a characterization for such systems, and provide an easily verifiable sufficient condition to belong to this class, thereby extending the study by Breiten, Hinsen and Unger in [IEEE Trans. Autom. Control vol. 69, no. 12 (2024), pp. 8924-8930, doi: 10.1109/TAC.2024.3464332]. The results rely on the operator semigroup theoretic treatment of linear delay PDEs.

Klaja and Ransford exhibited a complex unital Banach algebra in which the exponential spectra of \(ab\) and \(ba\) differ away from zero, and asked whether this can occur in an algebra of bounded operators. We answer their question affirmatively. For a Bourgain–Delbaen space \(X\) obtained from Motakis' construction with Calkin algebra isomorphic as a Banach algebra to \(C(S^4)\), we construct \(S,T\in\B(X\oplus X)\) such that \[ \frac12\in\varepsilon_{\B(X\oplus X)}(ST) \quad\text{and}\quad \frac12\notin\varepsilon_{\B(X\oplus X)}(TS). \] Moreover, \(\ltsr\B(X\oplus X)=\rtsr\B(X\oplus X)=2\), which is the least possible stable rank for such an example. More generally, if \(K\) is compact metrisable and \(X\) is any space arising from Motakis' construction for \(C(K)\), then, for every \(n\geqslant1\), \[ \ltsr\B(X^n)=\rtsr\B(X^n)= \begin{cases} \left\lceil \lfloor\dim K/2\rfloor/n\right\rceil+1,&\dim K<\infty, \infty,&\dim K=\infty. \end{cases} \] This also gives counterexamples of every finite stable rank at least two and of infinite stable rank. Finally, we relate every failure of exponential-spectral commutativity to the kernel of the first matrix stabilisation map on the index group.