Banach and Hilbert spaces, linear operators, and operator algebras.
Let X and Y be real Banach spaces and F:B_X→B_Y be non-expansive on their closed unit balls. If F(S_X)⊆S_Y⊆F(B_X), then F is the restriction of a surjective real-linear isometry X→Y. Sphere coverage is required; coverage of the ball’s interior and injectivity are not assumed.
For every 0<ε<1, every self-adjoint x in any von Neumann algebra M admits strong-operator paving over any maximal abelian subalgebra A with at most 5×10⁸ε⁻² projections. The count is fixed before the inclusion, operator and strong neighborhood, without separability or conditional-expectation assumptions.
An explicit 2×2 disk-domain example meets the stated Lorist–Schwenninger Lemma 1 hypotheses with ‖T‖>1 and Re⟨E₁Tx,x⟩<0. It rules out a proposed one-step positivity shortcut.
Every action of a countable discrete amenable group on a simple, separable, unital, infinite-dimensional, nuclear, stably finite, Jiang–Su-stable complex C∗-algebra absorbs the trivial Jiang–Su action up to cocycle conjugacy, with no condition on its action on traces.
For every unital expected inclusion N⊂M of von Neumann algebras with separable preduals, there is an expected unital amenable subalgebra P⊂N with P′∩c(M)=N′∩c(M), where c(M) is the continuous core.
For separable nuclear complex C∗-algebras A and B, any isomorphism between their topological cones of all extended lower-semicontinuous tracial weights is realized by an isomorphism A⊗W⊗K ≅ B⊗W⊗K, where W is the Razak–Jacelon algebra and K the compact operators.
For an ICC group commensurable with a lattice in a finite product of the specified property-(T) algebraic groups over characteristic-zero local fields, any countably infinite ICC comparison group, and arbitrary normalized scalar cocycles at both ends, every bifinite correspondence between their twisted group factors is a closed bimodule summand of a finite sum of models from finite-index subgroup isomorphisms and finite-dimensional projective representations of the matched cocycle ratio. Conversely, every such model, sum, and closed summand is bifinite.
For every maximal abelian subalgebra of a complex von Neumann algebra, every self-adjoint operator is a strong-operator limit of self-adjoint approximants of norm at most three times its norm, each admitting norm paving to relative error ε with at most Cε⁻⁶ projections for a universal C.
A finitely generated torsion-free group has an infinite-order degree-zero K-homology class in the kernel of the coefficient-free reduced Baum–Connes assembly map.
For every irrational rotation angle θ in (0,1), there is a continuous nonnegative circle weight f with one zero and integral log f equal to negative infinity such that the weighted rotation U f(V) has no nontrivial invariant projection in its associated hyperfinite type II₁ factor.
An explicit separable unital full group C∗-algebra admits no unital embedding into the norm ultrapower B^ω of any nonzero unital nuclear C∗-algebra B for any free ultrafilter ω on the natural numbers.
There are separable real Banach spaces X and Y with a global bi-Lipschitz bijection but no bounded linear isomorphism between them; X contains an isometric copy of c₀(ℓ₂), while Y contains no bounded linear copy of that space.
Every complex C∗-algebra in which each positive element is properly infinite is strongly purely infinite, satisfying the full positive-element diagonalization property.
There is a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state for which both microstates and nonmicrostates free entropy are finite yet −∞<χ(X)≤χ*(X)−1/2<∞.
For every nonzero unital separable complex C∗-algebra A and every free ultrafilter ω on the natural numbers, its norm central-sequence algebra has no characters if and only if A absorbs the Jiang–Su algebra under minimal tensor product.
Every type II₁ factor with separable predual is generated as a von Neumann algebra by one bounded operator, equivalently two self-adjoint operators.
For every complex von Neumann algebra M and every integer k≥2, each bounded M-valued Hochschild k-cocycle has a bounded M-valued primitive, so Hᵏ_b(M,M)=0.
For every ε>0 there is a universal δ>0 such that any two unital von Neumann algebras on the same complex Hilbert space at Kadison–Kastler distance below δ are conjugate by a unitary u with norm distance from the identity below ε.
In ZFC there exists a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state whose nonzero irreducible representations are all unitarily equivalent, yet it is not an algebra of compact operators.
There is a separable unital complex C∗-algebra with normalized 2-quasitraces but no tracial state; its quasitraces fail additivity on a fixed pair of positive contractions.
For each fixed real 1 < p < infinity with p != 2 and distortion D > 1, every n-point subset of a real Lp space has a nonlinear embedding into finite-dimensional l_p with dimension at most exp(C(p,D) (log n)^gamma), where gamma = 2-p for p < 2 and 1-2/p for p > 2.
OpenAI reports that every bounded complex-linear unital homomorphism from a unital complex C*-algebra to operators on a Hilbert space is conjugate by a bounded invertible operator to a *-homomorphism.
For every countable discrete group, the manuscript claims amenability is equivalent to strong Ulam stability: sufficiently accurate unitary almost representations on any complex Hilbert space are uniformly close in operator norm to genuine representations on the same space. It gives a fixed-distance obstruction for nonamenable groups.
For every epsilon>0, the OpenAI-authored manuscript constructs unital norm-separable C*-algebras on a common separable Hilbert space, with common identity and Kadison-Kastler distance below epsilon, which share their von Neumann closure but are not conjugate by any unitary. It does not assert abstract nonisomorphism or a von Neumann counterexample.
The OpenAI-authored manuscript constructs unital von Neumann algebra pairs on separable Hilbert spaces with one-sided near-inclusion gaps tending to zero, yet every unitary embedding the source into the target stays a fixed distance from the identity. Spatial embeddings do exist; this does not refute two-sided Kadison-Kastler stability.
On every separable infinite-dimensional complex Hilbert space, the paper constructs a nonzero bounded norm-quasinilpotent operator S whose commutant has no nonzero proper closed invariant subspace. The operator S itself has ordinary invariant subspaces.
On every separable infinite-dimensional complex Hilbert space, the commutant of the constructed operator is a proper unital complex algebra, closed in the strong operator topology and transitive.
A separable real Banach space Z has no closed linear copy of c₀ yet is globally bi-Lipschitz equivalent to Z×c₀. The construction also yields bi-Lipschitz universality for separable metric spaces.
OpenAI reports the sharp constant-two numerical-range inequality for matrix-valued polynomials, uniformly over all finite base and coefficient dimensions.
For every minimal homeomorphism h of an infinite compact metrizable space X, the manuscript claims rc(C(X)⋊ₕZ)=mdim(X,h)/2, including infinite values. The new part is the lower bound for every finite open cover.
A finitely generated group G and matrix A over Z[G] are constructed with A invertible over Q[G] and Fuglede–Kadison determinant strictly between zero and one. This contradicts the unrestricted group-ring Determinant Conjecture.
For simple separable unital infinite-dimensional nuclear C*-algebras, the companion manuscripts claim strict comparison implies Jiang–Su absorption, completing the unital Toms–Winter equivalence with finite nuclear dimension. A separate extended-functional theorem also treats nonunital algebras under its own assumptions.
For a simple separable unital infinite-dimensional nuclear stably finite C*-algebra with traces, real rank zero of the uniform tracial ultrapower of its uniform tracial completion implies a central projection halving every specified trace on every coefficient. This yields uniform property Gamma for each specified free ultrafilter.
For separable nuclear complex C*-algebras with no nonzero elementary ideal subquotients, the manuscript claims finite nuclear dimension, Jiang–Su stability and nuclear dimension at most one are equivalent. It separately claims dim_nuc(A₀⊗Z)≤1 for every separable nuclear A₀.
A finitely generated torsion-free discrete group has a scalar projection e in its reduced group C*-algebra whose canonical trace lies strictly between zero and one-half.
OpenAI claims a normal trace-preserving *-isomorphism between the free group factors L(F₂) and L(F₃), giving the affirmative branch of the classical isomorphism alternative.
For sufficiently large n=2ᵏ, Wu’s Walsh–Hadamard norm and identity map Uₙ:Xₙ→ℓ∞ⁿ make C₂ʳ(Uₙ)/max(C₂ᵍ(Uₙ),‖Uₙ‖(2,1)) grow at least as c√log(n+1). Thus the proposed universal operator-cotype bound fails already at q=2.
Sauers constructs a separable unital C*-algebra with compact nonempty trace space whose uniform tracial completion has a trace discontinuous in the uniform 2-norm, and a factorial tracially complete pair with the same failure.
Lorist and van Neerven construct two explicit 2^n-dimensional Banach-space families at p=2. One has Hilbert-transform constant of order n and UMD constant of order √n; the other reverses them, making both classical quadratic comparison exponents sharp over real and complex spaces.
Guo, Fang and Lu prove the strict bound K_G^C>1.35584631827168 using a Gaussian Hermite multiplier, exact rational parameters and a weighted operator-norm criterion.
Yoshito Ishiki constructs real Banach spaces X and Y and a bijection U from X to Y that preserves distances between every pair of points in each closed ball of radius 1/4, but is not a global isometry. This answers the unrestricted Scottish Book Problem 155 negatively.
From a property-(T) group Γ with an injective non-surjective endomorphism, Eckhardt forms its ascending HNN extension G and generalized wreath product W=(⊕_(G/Γ) Z/2Z)⋊G. His Theorem 4.3 gives a specified nonidentity element killed by every homomorphism of W to a matrix-ultraproduct quotient, showing W is not MF.
For odd n≥3, Lu and Yang obtain a global Lipschitz family of exact hyperplane-section isometries after a positive-degree bundle pullback. Combined with their degree argument, this proves that a real Banach space whose n-dimensional subspaces are all linearly isometric is Hilbert when 2≤n<dim X.
Liehr, Taylor and Yu prove that every countable Bessel sequence of bound 1 in a complex Hilbert space has its full Gram matrix, including diagonal entries, represented by correlations of complex functions on [0,1] bounded in modulus by 1.
A nonzero Schwartz function has twelve distinct time-frequency shifts, eleven in one irrational translate of a half-integer lattice plus the origin, that are linearly dependent. The authors also prove a finite counterexample without a fixed twelve-shift count.
Das, Ferenczi and Rosendal prove that no hyperplane of the real Kalton–Peck space Z₂ admits a complex structure; hence Z₂ is not isomorphic to its hyperplanes. A symplectic operator rank-parity theorem supplies the obstruction.
Zhou constructs two nonisomorphic countable ICC property-(T) groups whose group von Neumann algebras are isomorphic. A quadratic fiber shear identifies the dual actions, while the groups have distinct module structures.
OpenAI constructs infinitely many pairwise nonisomorphic finitely generated ICC property-(T) groups whose tracial group von Neumann factors are mutually isomorphic. The family refutes Connes’s rigidity conjecture and a finite-to-one strengthening.
For finite-dimensional unitaries satisfying approximate order-two and orbit-commutator relations through 2M, with M polynomial in the target tolerance, Dogon and Vidick prove nearby exact lamplighter relations. Their argument gives polynomial upper bounds on Hilbert–Schmidt stability rate and radius.
Antonio and Pablo Acuaviva construct a unital complex Banach algebra not isomorphic to B(X)/K(X) for any Banach space X, using large simple algebras and separable-range operator obstructions.
If T:X→Y is weakly compact and Y is separable with the bounded approximation property, then T factors through a reflexive Banach space with a Schauder basis.
Every infinite-dimensional complex normed space has unit vectors separated by at least 1+ε for some ε>0 even after either vector is multiplied by any scalar of modulus one.
For every 1<p<∞, Lp(L1) has uniform primary factorization and is therefore primary. The proof includes a scalar-compression argument on its doubly cancellative product-Haar part.
For every bounded operator T:X→Y with separable Y, T is strictly cosingular if and only if its adjoint T* is strictly singular.
Ignatev uses a Claude-origin max-term asymptotic lemma twice to prove Λ(T)=[0,1] for backward shifts on ℓ^p with strictly decreasing, positive, p′-summable weights.
For every odd integer base b>1, the fair {0,2} base-b Cantor measure admits no Fourier frame. The middle-third case answers Strichartz’s question.
Theorem 6.1 gives an explicit quantitative stable-phase-retrieval bound for the closed real L² span of independent centred normalized random variables under indexed two-sided L¹ bounds on all but at most one coordinate. Its proof uses a head–tail concentration versus diffusion argument and Sperner-type anticoncentration.
Krishtal and Pfander construct a bounded invertible normal operator T on ℓ²(N) and a vector g whose normalized orbit {Tᵏg/||Tᵏg||: k≥0} is a frame.
For every M≥2 with N=4M−5, a nonempty open set of complex N×M measurement matrices has a noninjective phase-retrieval map. Hence an IID complex Gaussian matrix is noninjective with positive probability, so p_M<1.
Banach and Hilbert spaces, linear operators, and operator algebras.
Publication dates and editorial changes.
Non-expansive maps between unit balls that send the sphere onto the sphere are linear isometries
OpenAI reports bi-Lipschitz c₀ absorption without a linear c₀ copy
ChatGPT-found example refutes Talagrand’s operator-cotype bound
OpenAI reports that strong Ulam stability characterizes amenability
OpenAI reports a counterexample to the group-ring Determinant Conjecture
OpenAI reports an infinite-family counterexample to Connes rigidity
AI-assisted construction gives a distinct Connes rigidity counterexample
ChatGPT improvement yields polynomial lamplighter stability bounds
Astra-found examples make both Hilbert–UMD quadratic bounds sharp
OpenAI reports radius of comparison equals half mean dimension
ChatGPT-assisted normal orbit refutes a dynamical-sampling conjecture
OpenAI reports nonsimple Robert–Tikuisis regularity equivalence
Model-generated strategy yields quantitative stable phase retrieval
OpenAI reports close separable C*-algebras without spatial conjugacy
OpenAI reports a Hilbert-space hyperinvariant-subspace counterexample
OpenAI reports a proper transitive strongly closed operator algebra
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