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Andrej Novikov

Publications and source records attributed to Andrej Novikov.

6 recordsLinked to original sources
arXiv2026

Upper and lower limits of scales and the interchange of quantifiers

We define lower and upper limits for a family of structured subobjects of a fixed ambient object, as the direct limit of the meets over tails and the inverse limit of the joins over tails, and construct the canonical comparison morphism between them. The family need not be a diagram: its members are related only through the ambient object. We distinguish two senses in which the comparison may be invertible, before and after forgetting the structure, and prove that they coincide exactly when the forgetful functor reflects isomorphisms; they therefore agree for sets, for vector spaces and for Banach spaces, and separate for locally convex spaces. For countable scales of Banach spaces the separation cannot occur when the upper limit is ultrabornological, so under those hypotheses the obstruction is localised in the topology of that limit. Intermediate conditions correspond to factorisations of the forgetful functor, of which bornological convergence is one. In the cases that matter the comparison is an interchange of quantifiers: membership in the lower limit means that one value of the second index serves every value of the first, and membership in the upper limit that every value of the first is served by some value of the second. We exhibit this for spaces of test functions, for weighted Sobolev spaces, for Laurent series and for the adeles, and discuss three further conditions that suggest themselves.

math.OA↗
arXiv2026

Operator Michelson Contrast: Logarithmic Parametrisation, Subadditivity, and a Sharp Noncommutativity Threshold

We study the operator Michelson contrast Delta(A) = (kappa(A)-1)/(kappa(A)+1), with kappa(A)=||A|| ||A^{-1}||, for positive invertible operators, and introduce its logarithmic parametrization via arctanh(Delta(A)) = (1/2) ln kappa(A). This defines the Operator Logarithmic Contrast (OLC), extended to all invertible operators via polar decomposition. We prove OLC is subadditive under multiplication: OLC(AB) <= OLC(A)+OLC(B). For positive invertible A, we identify OLC(A) exactly as the projective Thompson distance from the identity ray to the ray of A. We establish a sharp dimensional threshold for equality in subadditivity: for dimensions 2 and 3, equality forces commutativity, while n=4 is the smallest dimension admitting noncommuting equality examples. Moreover, equality still forces commutativity in any total dimension if the operators factor as pure Kronecker products X = tensor X_i and Y = tensor Y_i with each factor dimension <= 3. We derive contraction inequalities, sum bounds, trace estimates for density operators, Lipschitz continuity (including singular limits), and limiting behavior for Cesaro means and infinite products. For density operators, we obtain an exact closed-form bijection between OLC and von Neumann entropy for qubits; prove this bijection does not exist for d >= 3; and establish upper/lower entropy envelopes at fixed condition number, showing the admissible range [S_min(kappa), S_max(kappa)] satisfies ln 2 <= S_max(kappa) < ln 3 for every kappa > 1. The qubit case recovers, as an operator lift, the classical Michelson-Jensen-Shannon equivalence of Bruni, Rossi, and Vitulano, while a counterexample shows this equivalence fails for d > 2 as a direct consequence of the non-bijectivity.

math.QA↗
arXiv2026

Finite-Dimensional Type I von Neumann Algebras in PyTorch: A GPU-Accelerated Framework for Random Block-Diagonal Operators

We present \texttt{torch\_vn\_algebra}, an open-source Python library built on PyTorch for numerical experiments with finite-dimensional Type I von Neumann algebras (direct sums of matrix algebras). The library provides: $\bullet$ a compact batched tensor representation $(B,C,k_{\max},k_{\max})$ that handles both Monte Carlo samples and multiple direct summands; $\bullet$ lazy evaluation of operators to avoid unnecessary memory allocation; $\bullet$ generation of random operators with arbitrary eigenvalue distributions (user-provided samplers) and various unitary ensembles (Haar, $\mathrm{SU}(n)$, COE, CSE, diagonal phases); $\bullet$ functional calculus via SVD (absolute value, square root, inverse, entropy) and a hybrid method for extreme eigenvalues (exact diagonalisation for $k_{\max}\le256$, otherwise power iteration); $\bullet$ three trace functionals (blunt, normalised subspace trace, and the von Neumann tracial state); $\bullet$ GPU-accelerated batched linear algebra for moderate-scale Monte Carlo studies (e.g., $2\times10^4$ samples of $100\times100$ operators). The library is validated against analytical expectations (Haar moments, trace properties). Performance benchmarks on a Tesla P100 GPU are presented and discussed. Limitations and future work are outlined. The code is open-source.

cs.MS↗
arXiv2015

$L_1$-space for a positive operator affiliated with von Neumann algebra

In this paper we suggest an approach for constructing an L1-type space for a positive selfadjoint operator affiliated with von Neumann algebra. For such operator we intro- duce a seminorm, and prove that it is a norm if and only if the operator is injective. For this norm we construct an L1 -type space as the complition of the space of hermitian ultraweakly continuous linear functionals on von Neumann algebra, and represent L1- type space as a space of continuous linear functionals on the space of special sesquilinear forms. Also, we prove that L1 -type space is isometrically isomorphic to the predual of von Neumann algebra in a natural way. We give a small list of alternate definitions of the seminorm, and a special definition for the case of semifinite von Neumann algebra, in particular. We study order properties of L1-type space, and demonstrate the con- nection between semifinite normal weights and positive elements of this space. At last, we construct a similar L-space for the positive element of C*-algebra, and study the connection between this L-space and the L1 -type space in case when this C*-algebra is a von Neumann algebra.

math.OA↗