Operator Algebras
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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…
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Paper 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 C(X)\rtimesh\mathbb Z equals \tfrac12\mathrm{mdim}(X,h), including infinite values. Zero mean dimension is…
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Paper 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…
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Paper 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 O(\varepsilon^{-2}) blocks. The norm bound holds after…
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Paper 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 -\infty\lt \chi\lt \chi^\lt \infty. This answers Voiculescu’s finite-entropy…
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Paper 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,…
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Paper A ZFC counterexample to Naimark's problem
Gives an alternative to 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…
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Paper 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 N\subset M of von Neumann algebras with separable…
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Paper 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…
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Paper Rigidity and arithmetic of lattice von Neumann algebras
Classifies finite-index bimodules between scalar-twisted group factors of ICC groups commensurable with property-(T) lattices over characteristic-zero local fields and arbitrary ICC group factors. Every such…
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Paper Isomorphism of the free group factors
Resolves the free group factor isomorphism problem: L(\mathbb F2)\cong L(\mathbb F3), and hence all interpolated free group factors, including L(\mathbb F\infty), are isomorphic. Their common factor has…
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Paper 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…
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Paper 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…
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Paper Kirchberg's \mathcal O_2 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…
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Paper 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…
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Paper 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…
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