Papers

To probe a bipartite quantum system, one may use arbitrary global operators or restrict to product operators acting separately on the two subsystems. We determine the sharp universal comparison between the resulting norms. For every $z\in M_n\otimes M_m$, we prove $\|z\|_1\leq\sqrt{2}\min\{n,m\}\|z\|_\varepsilon$, where $\|\cdot\|_1$ is the trace norm and $\|\cdot\|_\varepsilon$ is the injective tensor norm associated with the trace norms on $M_n$ and $M_m$. To prove the upper bound, we establish an $L_1$ noncommutative Khintchine inequality whose random coefficients are the entries of a Haar unitary. We also show that the coefficient $\sqrt{2}$ is sharp. As applications, we show that the same sharp constant governs the gap between bipartite correlation measured in trace norm and that measured by a correlation function, and obtain an improved universal upper bound for quantum data hiding. The upper bound has also been formalized and machine-checked in Lean.

In this note, we observe that the class of unbounded subnormal operators on quaternionic Hilbert spaces is a solution of quaternionic quantum harmonic q-oscillator whose complex case is addressed by Szefraniec in [19]. We also address the connection of unbounded subnormal operators on quaternionic Hilbert spaces, quaternionic quantum harmonic q-oscillator and quaternionic quantum economics using model theory of unbounded subnormal operators.

The fundamental reliance on nonincreasing rearrangements restricts the recently introduced Wiener–Luxemburg amalgam spaces only to the rearrangement-invariant setting. To overcome this barrier, we develop a novel comprehensive framework that expands these amalgam structures to arbitrary quasi-Banach function spaces. By constructing a new quasinorm based on measure-constrained suprema and infima, we successfully separate these amalgams from rearrangement theory. We resolve the ensuing structural challenges regarding the Fatou property in this generalized setting, show consistency with the original theory, and fully characterize associate spaces, continuous embeddings, and the absolute continuity of the norm.

In this paper we first introduce and study the notions of pointwise convergence and uniform convergence of sequences of partial metric valued functions. Also introducing the notions of pointwise Cauchyness and uniform Cauchyness of sequences of partial metric valued functions, we study their relationship with pointwise convergence and uniform convergence. Further we introduce and study the notions of statistical pointwise convergence and statistical uniform convergence of sequences of partial metric valued functions including their interrelationship. Also, introducing the notion of equi-statistical convergence of sequences of partial metric valued functions we study its relationship with statistical pointwise convergence and statistical uniform convergence.

The generalized Busemann–Petty problem asks whether origin-symmetric convex bodies in $\mathbb{R}^n$ having larger volume of all $m$-dimensional sections necessarily have larger volume. When $m\geq 4$, this is known to be false, but the cases $m=2, 3$ for $n\geq 5$ have remained open since the 1990s. In this paper, we resolve these cases. Together with the known results, the generalized Busemann–Petty problem is completely solved: the answer is affirmative for $m=1, 2, 3$, and negative for $m\geq 4$.

This paper establishes essentially optimal asymptotic bounds for Kolmogorov widths under holomorphic mappings between complex Banach spaces. Given a compact parameter set whose Kolmogorov widths decay algebraically with rate s, we prove that the widths of its image under a holomorphic mapping decay algebraically with every rate t<s, thereby answering an open question raised by Cohen and DeVore. As an application, we obtain a sharp characterization of the approximability of solution manifolds associated with inf-sup stable parametrized PDEs. We also construct an explicit example showing that the arbitrarily small loss in the algebraic decay exponent is unavoidable. Finally, we provide similar characterization of asymptotic bounds for Kolmogorov widths in the exponentially decaying regime. Our analysis involves multilinear Taylor expansion in Banach spaces and a novel block dyadic expansion-truncation technique.

Let $G$ and $K$ be convex bodies in $\mathbb{R}^d$, where $0 \in \text{int} G$ and $G \subset \text{int} K$. Given $\alpha > 0$, the Hilbert packing number $M_H(G, K; \alpha)$ is the maximum cardinality of a set of points in $G$, each pair of which is separated by a distance of at least $\alpha$ in the Hilbert geometry defined by $K$. The Hilbert convexified packing number $\widehat{M}_H(G, K; \alpha)$ is the maximum length of a sequence of points in $G$, such that each point is separated by distance at least $\alpha$ from the convex hull of its predecessors. We prove a dimension-free primal-polar bound for convexified packing in Hilbert geometry. Letting $G^\circ$ and $K^\circ$ denote the polar bodies, we show that there exist absolute constants $C, c > 0$ such that, for every $\alpha > 0$, \[ \widehat{M}_H(G, K; \alpha) ~ \leq ~ C \cdot \widehat{M}_H(K^\circ, G^\circ; c\alpha)^2 M_H(K^\circ, G^\circ; c\alpha). \] As a direct corollary, we have \[ \widehat{M}_H(G, K; \alpha) ~ \leq ~ C \cdot M_H(K^\circ, G^\circ; c\alpha)^3. \] Thus, the primal convexified packing number is bounded by a fixed polynomial in the ordinary packing number for the reversed polar bodies, with absolute constants independent of the dimension. This is motivated by the duality conjecture for packing and covering numbers, which relates covering $G$ by $K$ to covering $K^\circ$ by $G^\circ$. Our results represent a key step in extending the work of Artstein, Milman, Szarek, and Tomczak-Jaegermann from normed spaces to Hilbert geometries.

Let $C_q$ denote the group of the $q$th roots of unity. A question arising from the work of Becker, Klein, Slote, Volberg and Zhang is whether the dimension-free Bohnenblust–Hille constants for functions on $C_q^N$ grow subexponentially with the degree. We answer this question affirmatively. In fact, we prove a stronger estimate for functions whose Fourier characters involve at most $d$ coordinates. If $\BHint{d}{q}$ is the optimal constant for this larger class, then, for every fixed $q\geq2$, \[ \BHint{d}{q}\leq \exp\left(c_q\sqrt{d\log d} +O_q\left(\sqrt{\frac d{\log d}}\log\log d\right)\right), \] where $c_2=2$ and $c_q=\sqrt{2q\log(q-1)/(q-2)}$ for $q\geq3$. As an application, we obtain two-sided estimates for the Bohr radius of the Fourier layer formed by characters involving exactly $d$ coordinates, and we determine its asymptotic behaviour in natural joint regimes of $d$ and $N$.

The classical Blaschke–Lebesgue theorem identifies the Reuleaux triangle as the planar convex body of constant width with minimum area. We investigate this extremal problem under prescribed symmetry constraints. Specifically, we classify the minimum-area convex bodies of constant width that are invariant under a finite group $G$ of isometries of $\mathbb{R}^2$ fixing the origin. For the exceptional reflection group $D_1$, the minimizers are precisely the Reuleaux triangles invariant under the prescribed reflection. If $G$ contains the half-turn $\mathcal R_\pi$, the disk is the unique minimizer. For odd $n\geq 3$, the minimizers are regular Reuleaux $n$-gons, unique up to rotation in the cyclic case $C_n$, and exactly those satisfying the prescribed reflection symmetry in the dihedral case $D_n$.

We introduce a purely operator-theoretic framework for quantitative homogenization that bypasses the traditional reliance on large-scale spatial regularity and probabilistic assumptions. Inspired by Tartar's vision of a point-free theory, we derive explicit norm resolvent estimates using only the algebraic structure of multiscale operators and the abstract geometry of Hilbert spaces. In this framework, the effective macroscopic dynamics and the abstract corrector emerge naturally from an orthogonal decomposition of the state space, governed algebraically by a Schur complement. To quantify the convergence rate, we introduce a frequency-splitting technique and solve a generalized Sylvester equation that controls the commutator between the differential structure and the highly oscillatory material properties. This abstract perspective unifies stationary, non-stationary, periodic, quasi-periodic, and stochastic homogenization. We demonstrate that the physical distinctions between these media–and their respective convergence rates–are entirely captured by the behavior of the spectral measures of the microscopic and macroscopic derivative operators near zero frequency.

We show that a finite number of time-frequency shifts of a Schwartz function can be linearly dependent. This disproves the so-called HRT conjecture of Heil, Ramanathan, and Topiwala. In particular, we provide an example consisting of 12 time-frequency shifts.

We prove $a_n(S) \le e\,(n+1)\, s_n(S)$ for every s-number sequence $(s_n)$, every bounded linear operator $S$ between normed spaces, and every $n \in \mathbb{N}_0$, where $a_n$ are the approximation numbers, which are the largest s-numbers. This is sharp up to the constant and settles conjectures of Mityagin, Henkin, Carl and Pietsch dating back to 1963. We also extend it to widths of convex sets and discuss optimality there. The proof is elementary.

We prove that the set of time-frequency shifts $\{e^{2\pi i \beta l t} g(t-\alpha k) : k,l \in \mathbb{Z}\}$ with a continuous, integrable totally positive function $g$ and lattice parameters $\alpha,\beta>0$ generates a frame for $L^2(\mathbb{R})$ if and only if $\alpha\beta<1$. This fully settles the so-called frame set problem for the class of totally positive functions. As a closely related result we prove a sharp Kadets-type theorem for every shift-invariant space generated by a continuous totally positive function. The proofs are based on Fredholm theory and limit-operator theory. A formalization of our main result in Lean 4 is also provided.

We characterize the analytic self-maps of the unit disk inducing isometric composition operators on \(\BMOA\) with respect to the Möbius-invariant \(H^p\) norm for \(p\ge1\) and the corresponding quasi-norm for \(0<p<1\). In particular, this gives a complete answer to Laitila's question on the characterization of isometric composition operators for the Möbius-invariant \(H^p\) norms, while also covering the quasi-Banach range. As a consequence, the isometric property of \(C_\varphi\) is independent of the exponent \(p\in(0,\infty)\).

Let $\varphi$ be a nonzero continuous mean-zero function on the one-dimensional torus and let $N_\varphi$ be the number of times that $\varphi$ changes signs. We prove the sharp family of oscillation inequalities of the types \begin{equation*} N_\varphi\|\varphi\|_{\dot W^{-1,s}} \gtrsim_{p,s} \frac{\|\varphi\|_1^{1+p'/s}}{\|\varphi\|_p^{p'/s}} \, \, \text{ and } \, \, (N_\varphi)^\alpha\|\phi\|_{\dot W^{-1,s}} \gtrsim_{p,q,r,s,\alpha} \frac{\|\varphi\|_p\|\varphi\|_q}{\|\varphi\|_r}. \end{equation*} This resolves an open problem posed by S. Steinerberger and strengthens the original estimate. The proof is independent of optimal transport and is based on a Gagliardo-Nirenberg-type estimate as well as a quotient-space characterization of the negative Sobolev seminorm. As applications, we derive several oscillation estimates related to Fourier projection, the uncertainty principle, and the Sturm-Hurwitz theorem.

We prove an infinitesimal circular version of Morera's theorem. Let $D\subset\mathbb{C}$ be a domain and let $f\in C(D)$. If, at every $a\in D$, $\int_{\vert{}\zeta-a\vert{}=r}f(\zeta)\,d\zeta=o(r^2)$ as $r\to0^+$, then $f$ is holomorphic in $D$. In particular, exact vanishing of all sufficiently small centered circular integrals implies holomorphicity. The proof uses a local distributional $\partial$-primitive, a circular identity for weak $\partial$-derivatives, and a pointwise asymptotic mean-value criterion for harmonicity.

We develop a Legendre–Bregman outer-approximation method for a common-solution problem in a uniformly smooth and uniformly convex Banach space. The constraint system consists of countable families of fixed-point-type mappings, equilibrium bifunctions, and monotone variational-inequality operators. Strong convergence to the Bregman projection of the initial point onto the common solution set is obtained without a family-level NST compatibility condition. A localized theorem covers negative-entropy generators, and Hilbert-space and Alber-functional cases are also derived.

The theory of Kubo-Ando means provides a fundamental correspondence between operator means and operator monotone functions on the positive half-line. In this paper, we first extend this theory to the cone of positive quaternionic Hermitian matrices, establishing quaternionic analogues of the main results of Kubo and Ando. We then develop an analogous framework for the cone of J-Hermitian positive matrices associated with an indefinite inner product. In particular, we prove that Kubo-Ando means on this cone are again in one-to-one correspondence with operator monotone functions, thereby extending the classical characterization to this indefinite setting.

This paper deals with the model theory of convolution algebras $(L^1(G),*)$ for locally compact groups $G$, seen as Banach lattices equipped with the convolution product $*$. We first prove transfer principles for elementary equivalence and elementary embeddings when the underlying group $G$ is discrete, namely, $(\ell^1(G),*) \equiv (\ell^1(H),*)$ implies $G \equiv H$, while the converse holds when $G$ and $H$ are $\omega$-saturated (likewise for elementary substructures). Without $\omega$-saturation, the converse fails. Although pure Banach lattices are model-theoretically tame, our results imply that adding convolution yields wild behavior. For example, we prove that if $G$ is any locally compact, non-discrete group, then the formula $d(x,x*y)$ is unstable with respect to $\mathrm{Th}(L^1(G),*)$. Moreover, we show that if $G$ is discrete and contains a particular configuration of amenable subgroups, then the formula $d(x*y,z)\mathbin{\dot{-}}\frac{1}{2}$ witnesses $\mathrm{TP}_2$ with respect to $\mathrm{Th}(\ell^1(G),*)$. As a consequence, if $G$ contains an infinite abelian subgroup, then $\mathrm{Th}(\ell^{1}(G),*)$ has $\mathrm{TP}_{2}$. We prove similar results in the locally compact non-discrete setting using the notion of an approximate identity. Finally, we prove a `continuous-by-discrete' approximation theorem. Namely, convolution algebras of connected abelian Lie groups admit metric embeddings into ultraproducts of convolution algebras over finite abelian groups.

We introduce and study a one-parameter family of probability measures $\{\mu_t\}_{t>0}$ defined by the Boolean additive convolution $\mu_t=\beta_t\uplus U_t$, where $\beta_t$ is the symmetric Bernoulli distribution with variance $t$ and $U_t$ is the uniform distribution on $[-t/2,t/2]$. Exploiting the linearization property of the Boolean energy transform, we derive an explicit formula for the reciprocal Cauchy transform and obtain its Laurent expansion. As applications, we compute the Boolean cumulants and moment sequence of $\mu_t$, establish asymptotic estimates for the moments, and investigate analytic properties including support criteria and a differential equation satisfied by the Boolean energy transform. We further analyze scaling limits, prove stability under suitable normalizations, and show that the family does not form a Boolean convolution semigroup.

Let $C^1[0,1]$ be the complex linear space of all continuously differentiable complex-valued functions $f$ on the unit interval $[0,1]$ with respect to the norm $ \|f\|_{\sigma} = |f(0)| +\|f'\|_\infty$. Let $P_1, P_2: C^1[0,1] \rightarrow C^1[0,1]$ be distinct, nonzero idempotent maps, which are not necessarily linear, such that $P_1P_2 = P_2P_1 = 0$ and $P_1+P_2 = I$, where $I$ denotes the identity operator. It is proved that $\lambda_1P_1 + \lambda_2P_2$ is an isometry on $C^1[0,1]$, for some distinct unit modulus complex numbers $\lambda_1, \lambda_2$, if and only if either $\lambda_1 + \lambda_2 = 0$, or $\lambda_1P_1 + \lambda_2P_2$ is an isometry for all such complex numbers $\lambda_1, \lambda_2$. In the former case, the collection $\{P_1, P_2\}$ is called a family of generalized bi-circular idempotents; in the latter case, it is called a family of bi-circular idempotents. The structure of isometric reflection on $C^1[0,1]$, that is, an isometry $T$ such that $T^2 = I$, is characterized, and its relationship with the above class of idempotents maps is also discussed.

We show that the dual approach to Bourgain–Brezis estimates for Hodge systems is substantially more flexible than previously understood. For $1\leq l\leq n-1$, we introduce the trace-free Beurling–Ahlfors transform $S=\frac{n-l}{n}P-\frac lnP^\perp$, a canonical normalization of the generalized Beurling–Ahlfors transform on $l$-forms in $\mathbb{R}^n$. Its matrix symbol decomposes into scalar multipliers that are odd under suitable orthogonal reflections, yielding an endpoint cancellation estimate from finite measures to $L^\infty$ for $|D|^{-n}S$. This cancellation allows us to complete the Hilbertian case of the Bourgain–Brezis conjecture in every dimension and for every form degree. We then develop multilinear reflection estimates and obtain new critical Triebel–Lizorkin and Besov Bourgain–Brezis estimates. In particular, for every dimension and form degree, the Sobolev Bourgain–Brezis conjecture in $\dot W^{\frac np,p}$ holds for $p=\frac{2k}{2k-1}$, $k\geq1$, and hence for exponents arbitrarily close to $1$. We also derive endpoint Hodge decompositions and Hodge–Sobolev inequalities. Finally, except in the endpoint Besov case where the critical space already embeds into $L^\infty$, we prove that the associated bounded selections cannot be linear.

We prove that $c_0$ admits a coarse Lipschitz embedding into a separable dual Banach space and that the optimal coarse Lipschitz distortion is equal to two. Let $$ G=\mathbb Z^{<\omega}\subset c_0, \qquad G_R=G\cap R B_{c_0}, \quad R\in\mathbb N, $$ with the metric inherited from $c_0$. On each $G_R$ we construct a commuting family of retractions onto finite initial segments of a special ordering of $G_R$, with Lipschitz constant at most two. Associated to these retractions there is a boundedly complete Schauder basis of $\mathcal F(G_R)$ whose basis constant is at most two and which is $2R$-equivalent to the unit vector basis of $\ell_1$. Consequently, each $\mathcal F(G_R)$ is $2$-isomorphic to a separable dual Banach space, uniformly in $R$. Kalton's annular decomposition then gives an embedding of $\mathcal F(G)$ into a separable dual space with distortion at most $2(1+\varepsilon)$ for every $\varepsilon>0$. A decomposition result of Aliaga and Medina further shows that \[ \mathcal F(G) \cong \Big( \bigoplus_{n\geq0}\mathcal F(G_{2^n}) \Big)_{\ell_1}, \] and hence $\mathcal F(G)$ itself is isomorphic to a separable dual Banach space. The constant two is sharp: if $G_2$ embeds into $X^*$ with distortion strictly smaller than two, then $X$ contains an isomorphic copy of $\ell_1$. It follows that the infimum of the coarse Lipschitz distortions of embeddings of $c_0$ into separable dual Banach spaces is exactly two.

We present a novel method to recover the source intensity, $f : \mathbb{R}^n \to \mathbb{R}$, and attenuation coefficient, $\mu : \mathbb{R}^n \to \mathbb{R}$, in Compton camera imaging. We apply a non-linear model, which accounts for ray attenuation. We show that the data, $h$, can be modeled $h = \mathcal{R}(f,\mu) = R(fg)$, where $g = \exp(-G\mu)$ models attenuation, $G$ is a (linear) divergent beam transform, and $R$ is a linear operator which defines the integrals of $fg$ over cones. Commonly in the literature, $\mu$ is set to zero, and the data $h = Rf$ is linear. We address the case when $\mu \neq 0$ and the transform is non-linear. To simplify the analysis, we first transform the data into weighted line integrals, $\tilde{h} = \mathcal{D}_k(f,\mu) = D_k(fg)$, where $D_k$ is a weighted ray transform. Assuming practically reasonable geometric conditions, we show that $\tilde{h} = \exp(-X_{w_1}\mu)X_{w_2}f$, where $X_w$ is a weighted X-ray transform, and the $w_i$ are smooth weights. After which, we use the theory of conormal distributions to describe the singularities of $\tilde{h}$. We show that there are artifacts in the reconstruction, and we quantify their strength using Sobolev spaces. We combine this theory with a geometric argument to recover the edges of $f$ and ultimately prove that $f$ and $\mu$ are unique to $h$. The recovery of $f$ is notably more stable than that of $\mu$, which we also discuss. To validate our theory, we present simulated reconstructions of $f$ and $\mu$ using the proposed method.

We provide a proof of Crouzeix's conjecture, which combines the tools developed previously for weaker estimates with a simple perturbation lemma for $2$-dilations. Applying the lemma to the iterates $f^{n}$ in the double-layer potential representation yields the conjectured bound.

Let $\Omega$ be a compact Hausdorff space and $X$ be a complex Banach space such that $X^{**}$ is isometric to a $C^*$-algebra. Let $\mathcal{W}(X^*, C(\Omega))$ denote the space of weakly compact operators. In this article, we show that the collection of operators $T$ in $\mathcal{W}(X^*, C(\Omega))$ such that $T^*$ maps linear extreme points of the unit ball of $C(\Omega)^*$ to $C^*$-extreme points of $X^{**}$ is a uniformly strongly extreme set. We also investigate this phenomenon when $X^{**}$ is isometric to a space of all operators on a Banach space.

This paper investigates the domain $\mathbf{\Theta}_n$ from an operator-theoretic perspective. We establish several characterizations of $\mathbf{\Theta}_n$-contractions, $\mathbf{\Theta}_n$-unitaries, and $\mathbf{\Theta}_n$-isometries, and explore their connections with $\Gamma_n$- and tetrablock operator tuples, as well as with the corresponding operator classes associated with $\mathbf{\Theta}_{n+1}$. We prove that every $\mathbf{\Theta}_n$-contraction admits a canonical decomposition as the direct sum of a $\mathbf{\Theta}_n$-unitary and a completely non-unitary $\mathbf{\Theta}_n$-contraction. We further develop the conditional dilation theory for $\mathbf{\Theta}_n$-contractions and establish necessary conditions for the existence of $\mathbf{\Theta}_n$-isometric dilations. We show that the conditions in Theorem~\ref{Conditional Dilation} are not, in general, sufficient by constructing a $\mathbf{\Theta}_3$-contraction that admits a $\mathbf{\Theta}_3$-isometric dilation although condition~(2) of Theorem~\ref{Conditional Dilation} fails. We further study pure $\mathbf{\Theta}_n$-isometric dilations of $\mathbf{\Theta}_n$-contractions and establish a Hardy space characterization of such dilations. We show that the resulting $\mathbf{\Theta}_n$-isometric dilation is minimal. We also study isometric dilations of doubly commuting $\mathbf{\Theta}_n$-contractions under suitable Hardy space compatibility conditions. Finally, we identify a special class of $\mathbf{\Theta}_2$-contractions that admits a $\mathbf{\Theta}_2$-isometric dilation.

We study bounded finite-interaction-range operators $H$ on $\ell^2(\Gamma)$, where $\Gamma$ is a uniformly discrete subset of a left-invariant doubling metric measure space $(X,d,\mu)$. Our goal is to approximate spectral information of $H$ from finite sections $H_{L,x}$ supported on balls $B_L(x)\cap\Gamma$. The main technical input is a commutator estimate for Lipschitz ”tent” localisations $W_{L,x}$, which depends only on geometric properties (doubling) and a uniform interaction degree. As a consequence, we obtain explicit two-sided pseudospectral inclusion bounds of the form \[ \gamma_{L,\varepsilon}(H)\subset \sigma_\varepsilon(H)\subset \gamma_{L,\varepsilon+C_0/L}(H), \] and Hausdorff convergence of window pseudospectra to the (global) pseudospectrum as $L\to\infty$. In the self-adjoint/ normal case this yields computable gap tests and spectral sampling schemes with rigorous $O(1/L)$ error control, while in the non-normal case it leads to corresponding approximation results for pseudospectra. The framework isolates the geometric core behind earlier approaches on $\mathbb{R}^n$ and on countable Abelian groups, and covers irregular geometries arising in quasicrystal models, as well as new cases such as discrete (non-Abelian) nilpotent groups (including for example the discrete Heisenberg group).

Let $X$ and $Y$ be locally compact Hausdorff spaces and let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. We say that a bijection $T\colon C_0(X,\mathbb{K})\to C_0(Y,\mathbb{K})$ is a ring isomorphism in norm if \[ \|T(f+g)\|=\|T(f)+T(g)\|,\qquad \|T(fg)\|=\|T(f)T(g)\| \] for every $f,g\in C_0(X,\mathbb{K})$. We determine the form of such maps. When $\mathbb{K}=\mathbb{C}$, under the additional assumption that $\|T(\overline f)\|=\|T(f)\|$ for every $f\in C_0(X,\mathbb{C})$, there exist a continuous function $w\colon Y\to\{\lambda\in\mathbb{C}:|\lambda|=1\}$, a homeomorphism $\varphi\colon Y\to X$, and a closed and open subset $Y_0\subset Y$ such that \[ T(f)(y)= \begin{cases} w(y)f(\varphi(y)),& y\in Y_0,\\ w(y)\overline{f(\varphi(y))},& y\in Y\setminus Y_0, \end{cases} \] for every $f\in C_0(X,\mathbb{C})$ and $y\in Y$. When $\mathbb{K}=\mathbb{R}$, there exist a continuous function $w\colon Y\to\{\pm1\}$ and a homeomorphism $\varphi\colon Y\to X$ such that \[ T(f)(y)=w(y)f(\varphi(y)) \] for every $f\in C_0(X,\mathbb{R})$ and $y\in Y$. In particular, the real case extends the norm version of the Gelfand–Kolmogoroff theorem to the locally compact setting.

We establish a simple characterization of amenability of second countable locally compact quantum groups $\mathbb{G}$ in terms of the long-time behavior of convolution semigroups of states on $\mathbb{G}$. We then investigate the deeper problem of combining amenability of $\mathbb{G}$ with other approximation properties of the dual quantum group $\hat{\mathbb{G}}$. Specifically, we characterize the combination of ("strong") amenability of $\mathbb{G}$ with either the failure of property (T) or the presence of the Haagerup property for $\hat{\mathbb{G}}$. These results, which appear to be new even in the classical setting of locally compact groups, produce convolution semigroups on $\mathbb{G}$ whose behavior is controlled both as time goes to zero and as time goes to infinity.