Operator algebras
19 families · 15 verified by lean
This page gathers the 19 result families that the openai/math repository catalogues under Operator algebras. Each claim comes from a manuscript; 15 of them have a Lean formalization checked by software, and the other 4 are still to be verified. Open any card to read a plain-language explainer, the original statement and its sources.
Counterexamples to Baum–Connes and Kadison–Kaplansky
Disproves the coefficient-free reduced Baum–Connes conjecture through a failure of rational injectivity and a separate failure of surjectivity of assembly. A finitely generated torsion-free example witnesses the injectivity failure. Separately, a torsion-free group has a nontrivial projection in its reduced group C∗-algebra, disproving the Kadison–Kaplansky conjecture.
Rigidity and arithmetic of lattice von Neumann algebras
Classifies finite-index bimodules between scalar-twisted group factors of ICC groups commensurable with property- lattices over characteristic-zero local fields and arbitrary ICC group factors. Every such bimodule is a summand of finite sums of models arising from finite-index subgroup isomorphisms and finite-dimensional projective representations. The classification also recovers the group, scalar cocycle and amplification scale up to the stated stable equivalence.
Isomorphism of the free group factors
Resolves the free group factor isomorphism problem: , and hence all interpolated free group factors, including , are isomorphic. Their common factor has fundamental group .
Kadison's similarity conjecture
Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C∗-algebra to operators on a Hilbert space becomes a -homomorphism after conjugation by a bounded invertible operator.
Strong Kadison–Kastler stability and its spatial boundaries
Proves that sufficiently close unital von Neumann algebras on the same Hilbert space are conjugate by a unitary arbitrarily close to the identity, with a universal tolerance in the operator-norm distance between unit balls. Counterexamples show that near-identity conjugacy fails for one-sided near inclusions, and that arbitrarily close norm-separable C∗-algebras need not be ambiently unitarily conjugate.
Relative bicentralizers and modular spectral recovery
Proves Connes' bicentralizer conjecture for every type III1 factor with separable predual and every faithful normal state. More generally, for every inclusion of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra with , resolving the relative bicentralizer conjecture.
Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability
Proves equivariant Jiang–Su stability for every countable discrete amenable group action on a simple separable unital infinite-dimensional nuclear stably finite Jiang–Su-stable C∗-algebra, resolving this case of Szabó's conjecture without restrictions on trace dynamics. The family also proves the unital Toms–Winter conjecture, equating strict comparison, finite nuclear dimension and Jiang–Su stability in the simple separable unital infinite-dimensional nuclear setting.
Kirchberg's norm-ultrapower embedding problem
Constructs an explicit separable unital full group C∗-algebra that cannot embed unitally into the norm ultrapower of any fixed nonzero unital nuclear C∗-algebra, for any free ultrafilter on the natural numbers. Taking the target to be answers Kirchberg's norm-ultrapower embedding problem negatively.
Invariant projections, hyperinvariant subspaces, and transitive algebras
Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator. The construction also gives operators with no nontrivial invariant projection in the hyperfinite type II1 factor.
Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness
Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra admitting normalized 2-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite C∗-algebras can have a properly infinite minimal tensor product, with one factor .
The Kadison–Ringrose cohomology conjecture
Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive. Equivalently, all higher bounded Hochschild cohomology groups vanish, resolving the Kadison–Ringrose conjecture.
The generator problem for finite factors
Proves that every type II1 factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion of such factors, the unitaries u with form a dense Gδ subset in the trace 2-norm topology.
A ZFC counterexample to Naimark's problem
Gives an alternative to [Tanaka's ZFC construction](https://arxiv.org/abs/2609.26930v1) of a unital infinite-dimensional simple complex C∗-algebra with a faithful tracial state and exactly one nonzero irreducible representation up to unitary equivalence. Thus the unrestricted compact-operator characterization fails without additional set-theoretic assumptions; the counterexample is nonseparable.
Two notions of free entropy differ even when both are finite
Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy . This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite.
The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability
A nonzero unital separable complex C∗-algebra is Jiang–Su stable exactly when its norm central-sequence algebra has no characters, for every free ultrafilter. This answers the Kirchberg–Rørdam character question. Also, the infinite minimal tensor power of every such algebra without characters is Jiang–Su stable, answering the Dadarlat–Toms question.
Approximation and quadratic strong-operator paving
Proves that every self-adjoint element of a complex von Neumann algebra admits strong-operator paving relative to any maximal abelian subalgebra with blocks. The norm bound holds after compression by a projection arbitrarily close to the identity in the strong topology, resolving the Popa–Vaes quadratic paving conjecture.
Trace cones and Razak–Jacelon stabilization
Classifies separable nuclear complex C∗-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their full topological cones of extended lower-semicontinuous tracial weights. This answers Robert’s trace-cone question, including algebras with arbitrary ideal structure and both finite and infinite subquotients.
Radius of comparison equals half the mean dimension
For every minimal homeomorphism h of an infinite compact metrizable space X, the radius of comparison of equals , including infinite values. Zero mean dimension is equivalent to the small boundary property, Jiang–Su stability and finite nuclear dimension; in this case nuclear dimension is at most one.
Weak pure infiniteness and Cuntz-algebra absorption
Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex C∗-algebras. For exact algebras, proper infiniteness of one fixed finite amplification of every positive element also suffices. Consequently, every separable nuclear algebra with this property absorbs , without unitality or simplicity assumptions.