Kaplansky's Quasitrace Conjecture, Naimark's Problem and 7 More in Operator Algebras

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. (Lean)

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. (Lean)

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 𝒪₂ answers Kirchberg's norm-ultrapower embedding problem negatively. (Lean)

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 C_r^*(𝔽₂). (Lean)

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 P ⊂ M of such factors, the unitaries u with M=W^*(P,u) form a dense Gδ subset in the trace 2-norm topology. (Lean)

Gives an alternative to Tanaka's ZFC construction 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. (Lean)

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. (Lean)

Proves that every self-adjoint element of a complex von Neumann algebra admits strong-operator paving relative to any maximal abelian subalgebra with O(ε^-2) 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.

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. (Lean)