We construct C*-dynamical systems for the dynamics of classical infinite particle systems describing harmonic oscillators interacting with arbitrarily many neighbors on lattices, as well on more general structures. Our approach allows particles with varying masses, varying frequencies, irregularly placed lattice sites and varying interactions subject to a simple summability constraint. A key role is played by the commutative resolvent algebra, which is a C*-algebra of bounded continuous functions on an infinite-dimensional vector space, and in a strong sense the classical limit of the Buchholz–Grundling resolvent algebra, which suggests that quantum analogs of our results are likely to exist. For a general class of Hamiltonians, we show that the commutative resolvent algebra is time-stable and admits a time-stable sub-algebra on which the dynamics is strongly continuous, therefore obtaining a C*-dynamical system.

Given H self-adjoint, V symmetric and relatively H-bounded, and f:R→C satisfying mild conditions, we show that the Gateaux derivativedndtnf(H+tV)|t=0 exists in the operator norm topology, for every natural n, and establish perturbation formulas for multiple operator integrals under relatively bounded perturbations. If the H-bound of V is less than 1, we obtain sufficient conditions on f which ensure that the Taylor expansionf(H+V)=∑n=0∞1n!dndtnf(H+tV)|t=0 exists and converges absolutely in operator norm. Assuming that V(H−i)−p∈Ss/p for p=1,…,s for some s∈N[jls-end-space/], we show that the Krein–Koplienko spectral shift functions ηk,H,V[jls-end-space/], satisfying(Formula presented) exist for every k=1,2,3,…[jls-end-space/], independently of s. The latter result (which is significantly stronger than [27]) is new also in the case that V is bounded. The proof is based on [34], combined with a generalisation of the multiple operator integral compatible with [17]. We discuss applications of our results to quantum physics and noncommutative geometry.

Buchholz and Grundling (Commun Math Phys 272:699–750, 2007) introduced a C^∗-algebra called the resolvent algebra as a canonical quantisation of a symplectic vector space and demonstrated that this algebra has several desirable features. We define an analogue of their resolvent algebra on the cotangent bundle T^∗Tⁿ of an n-torus by first generalising the classical analogue of the resolvent algebra defined by the first author of this paper in earlier work (van Nuland in J Funct Anal 277:2815–2838, 2019) and subsequently applying Weyl quantisation. We prove that this quantisation is almost strict in the sense of Rieffel and show that our resolvent algebra shares many features with the original resolvent algebra. We demonstrate that both our classical and quantised algebras are closed under the time evolutions corresponding to large classes of potentials. Finally, we discuss their relevance to lattice gauge theory.

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space Rⁿ. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden layer, for all activation functions φ that are continuous, nonpolynomial, and asymptotically polynomial at ±∞. When φ is moreover bounded, we exactly determine which functions can be uniformly approximated by neural networks, with the following unexpected results. Let N_φ^l(Rⁿ)¯ denote the vector space of functions that are uniformly approximable by neural networks with l hidden layers and n inputs. For all n and all l≥2, N_φ^l(Rⁿ)¯ turns out to be an algebra under the pointwise product. If the left limit of φ differs from its right limit (for instance, when φ is sigmoidal) the algebra N_φ^l(Rⁿ)¯ (l≥2) is independent of φ and l, and equals the closed span of products of sigmoids composed with one-dimensional projections. If the left limit of φ equals its right limit, N_φ^l(Rⁿ)¯ (l≥1) equals the (real part of the) commutative resolvent algebra, a C*-algebra which is used in mathematical approaches to quantum theory. In the latter case, the algebra is independent of l≥1, whereas in the former case N_φ²(Rⁿ)¯ is strictly bigger than N_φ¹(Rⁿ)¯.