Operator algebras

We construct a finitely generated discrete group whose reduced group C^∗-algebra contains a projection of irrational canonical trace. Its K₀-class lies outside the image of the coefficient-free reduced Baum–Connes assembly map. This disproves the coefficient-free reduced Baum–Connes conjecture for countable discrete groups.

We construct a finitely generated torsion-free discrete group whose reduced group C^∗-algebra contains a projection other than zero and the identity. This disproves the Kadison–Kaplansky projection conjecture.

We disprove rational injectivity of the coefficient-free reduced Baum–Connes assembly map for torsion-free groups. Specifically, we construct a finitely generated torsion-free discrete group whose degree-zero assembly map has a kernel class of infinite order.

We prove stable canonical recovery for scalar-twisted group factors of ICC groups commensurable with property-\((T)\) lattices over characteristic-zero local fields. Every specified stable isomorphism to a scalar-twisted factor of any countably infinite ICC group recovers the group, the cocycle up to a normalized cochain, and equal amplification scales, with an implementing partial isometry in the given matrix corners. We compute the arithmetic correspondence subcategory for arbitrary countable ICC groups, including all bounded maps, summands, conjugates, and Connes fusion, and combine it with the companion's arithmetic exhaustion theorem. Multiplication also recovers arbitrary finite-index factor neighbors, with the exact projective condition on the matched cocycle ratio.

We prove arithmetic exhaustion for bifinite correspondences between arbitrarily scalar-twisted group factors of ICC groups commensurable with property-\((T)\) lattices over characteristic-zero local fields and twisted group factors of arbitrary countable ICC groups. Every such correspondence is a closed summand of a finite direct sum of models obtained from actual finite-index subgroup isomorphisms and finite-dimensional projective representations of the matched cocycle ratio. The proof includes reducible products, mixed real and finite places, and quaternionic and Cayley rank-one factors.

We solve the free group factor isomorphism problem affirmatively by proving that \(L(\mathbb F_2)\) and \(L(\mathbb F_3)\) are isomorphic as tracial von Neumann algebras. The classical free group factor alternative then implies that all interpolated free group factors, including \(L(\mathbb F_\infty)\), are isomorphic and have fundamental group \(\mathbb R_{\gt 0}\).

We prove that every bounded complex-linear unital algebra homomorphism from a unital complex C^∗-algebra into the bounded operators on an arbitrary Hilbert space is similar to a \(*\)-homomorphism. This resolves Kadison's similarity conjecture positively. We also obtain one universal hyperreflexivity constant for all unital von Neumann algebras on arbitrary complex Hilbert spaces.

We give a negative answer to the unrestricted small-spatial-embedding problem for one-sided near inclusions of von Neumann algebras. On separable complex Hilbert spaces, we construct pairs of unital von Neumann algebras with common identity whose one-sided gaps tend to zero. Spatial embeddings of the source into the target exist, but every implementing unitary stays a fixed positive distance from the identity. The obstruction therefore concerns small implementing unitaries, rather than the existence of spatial embeddings.

We disprove the separable C^∗-algebraic spatial form of the Kadison–Kastler conjecture. For every ε > 0, we construct unital, norm-separable C^∗-algebras on a common separable complex Hilbert space, with the same identity and Kadison–Kastler distance less than ε, that are not conjugate by any unitary. The two algebras have the same von Neumann closure.

We prove that sufficiently close unital von Neumann algebras are conjugate by a unitary arbitrarily close to the identity. The tolerance depends only on the prescribed distance of that unitary from the identity, uniformly over all algebras, representations, and Hilbert spaces. This resolves the strong Kadison–Kastler conjecture.

We prove that every inclusion \(N\subset M\) of von Neumann algebras with separable preduals and a faithful normal conditional expectation contains an expected amenable subalgebra \(P\subset N\) such that \(P'\cap c(M)=N'\cap c(M)\), where \(c(M)\) denotes the continuous core of M. This resolves the relative bicentralizer conjecture in this setting.

We prove a bounded spectral recovery theorem for a von Neumann algebra with a faithful normal state whose centralizer consists only of scalars. A positive averaged squared norm for an arbitrary bounded Hilbert-space operator on shrinking modular spectral bands can be recovered on uniformly bounded algebra elements with shrinking spectral support. We apply this theorem to spectral intertwining rigidity and obtain an alternative proof of bicentralizer triviality for type III₁ factors with separable predual.

Every action of a countable discrete amenable group on a simple, separable, unital, infinite-dimensional, nuclear, stably finite complex C^∗-algebra that is already Jiang–Su stable absorbs the trivial action on the Jiang–Su algebra up to cocycle conjugacy. No restriction is imposed on the action on the tracial-state simplex. This proves the unital, stably finite case of Szabó's Conjecture A on automatic equivariant Jiang–Su stability.

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 uniform property Γ. This answers Problem XXI of Schafhauser, Tikuisis and White affirmatively. Independently, strict comparison implies Jiang–Su absorption for simple, separable, unital, infinite-dimensional nuclear algebras, resolving the unital Toms–Winter conjecture. Under comparison tested on traces of finite target rank, we also obtain uniform property Γ and absorption for simple, separable, nuclear, stably projectionless algebras whose densely finite traces are all bounded and have a nonempty compact normalized base.

For separable nuclear C^∗-algebras with no nonzero elementary ideal subquotients, finite nuclear dimension is equivalent to Jiang–Su stability. More generally, \(\dim_{\mathrm{nuc}}(A_0\otimes\mathcal Z)\le1\) for every separable nuclear A₀. This proves Robert and Tikuisis's Conjecture (C1) and the nuclear-dimension equivalence in the nonsimple Toms–Winter regularity question.

We prove that strict comparison in the extended-functional sense implies Jiang–Su absorption for separable simple nuclear non-elementary C^∗-algebras. This resolves the corresponding implication of the Toms–Winter regularity problem, including nonunital algebras and allowing unbounded traces. More generally, every separable nuclear C^∗-algebra whose Cuntz semigroup is almost unperforated and fully almost divisible absorbs the Jiang–Su algebra.

We give a negative answer to Kirchberg's norm-ultrapower embedding problem. We construct an explicit separable unital full group C^∗-algebra that admits no unital embedding into B^ω, for any nonzero unital nuclear C^∗-algebra B and any free ultrafilter ω on ℕ. In particular, it does not embed unitally into \(\mathcal O_2^\omega\).

For every irrational angle, we construct a continuous nonnegative circle weight with exactly one zero and logarithmic integral \(-\infty\) whose weighted rotation has no nontrivial invariant projection in the associated hyperfinite type II₁ factor. This answers negatively the question of Zhu, Fang, and Shi, with the angle prescribed in advance. The resulting operator is nonzero and norm-quasinilpotent.

We give a negative answer to the hyperinvariant-subspace problem by constructing, on every infinite-dimensional separable complex Hilbert space, a nonzero bounded norm-quasinilpotent operator with no nonzero proper closed hyperinvariant subspace. Its commutant is a proper strongly closed unital transitive complex operator algebra.

We refute Kaplansky's quasitrace conjecture by constructing a separable unital complex C^∗-algebra that admits normalized 2-quasitraces but has no tracial state. As a consequence, we show that the minimal tensor product of two unital simple stably finite complex C^∗-algebras can be properly infinite, even when one factor is the reduced free-group algebra \(C_r^*(\mathbb F_2)\).

We prove that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with values in the algebra itself, has a bounded primitive. Together with the established degree-one inner-derivation theorem, this resolves the Kadison–Ringrose cohomology conjecture positively.

We prove that every type II₁ factor with separable predual is generated by one operator, equivalently by two self-adjoint operators. Combined with the established direct-integral reduction to this case, this gives an affirmative solution of the generator problem for von Neumann algebras with separable predual.

We construct in ZFC a unital infinite-dimensional simple complex C^∗-algebra with a faithful tracial state whose nonzero irreducible representations are all unitarily equivalent. This gives a negative answer to Naimark's problem without additional set-theoretic assumptions.

We answer the finite-entropy equality question for microstates and nonmicrostates free entropy negatively. We construct a bounded self-adjoint tuple X in a von Neumann algebra with faithful normal tracial state such that \(-\infty\lt \chi(X)\leq\chi^*(X)-\tfrac12\lt \infty\). Here χ is the original microstates entropy with an operator-norm cutoff and a limsup over matrix sizes. The counterexample uses a large but fixed number of variables.

We prove the Kirchberg–Rørdam character criterion: for every free ultrafilter on ℕ, a nonzero unital separable complex C^∗-algebra absorbs the Jiang–Su algebra if and only if its norm central-sequence algebra has no characters. We also prove that the infinite minimal tensor power of every nonzero unital separable complex C^∗-algebra without characters is \(\mathcal Z\)-stable and contains a unital copy of \(\mathcal Z\).

We prove the quadratic strong-operator paving conjecture of Popa and Vaes. For every \(0\lt \varepsilon \lt 1\), every self-adjoint element of a von Neumann algebra admits strong-operator paving over each maximal abelian subalgebra with at most \(5\times10^8\varepsilon ^{-2}\) projections. The bound is uniform over representations and requires no separability or conditional-expectation assumption.

We prove the approximation-paving conjecture of Popa and Vaes for every maximal abelian subalgebra of a complex von Neumann algebra. For each \(0\lt \varepsilon\lt 1\), every self-adjoint operator is a strong limit of self-adjoint operators of norm at most three times its norm, each admitting a norm paving with error at most ε times the approximant's own norm. The number of projections is at most \(C\varepsilon^{-6}\) for a universal constant C. No separability or conditional-expectation hypothesis is required.

We prove that the canonical topological cone of all extended lower-semicontinuous tracial weights determines a separable nuclear C^∗-algebra after tensoring with the Razak–Jacelon algebra and the compact operators, answering Robert's trace-cone classification question positively. The isomorphism realizes the prescribed cone map, with arbitrary ideal structure and without a density assumption on the finite domains of the weights.

We prove the integer-action case of the Phillips–Toms conjecture: for every minimal homeomorphism of an infinite compact metrizable space, the radius of comparison of its crossed product equals one half of its mean dimension, including equality at infinity. For these systems, zero mean dimension and the small boundary property are equivalent to Jiang–Su stability and to finite nuclear dimension of the crossed product; in this case its nuclear dimension is at most one.

We prove that, in sufficiently large smash powers, an \(MU\)-null restriction to a finite pointed subcomplex becomes stably null on the entire union of products with a fixed positive proportion of restricted factors. This coherent vanishing theorem yields boundary-preserving compression of cube-valued maps on every compact metrizable input space. When the torus slot bundle embeds continuously into a trivial bundle of rank less than twice the source rank, the compression places a positive proportion of slots on their boundaries in arbitrarily large powers and fixes every original boundary slot exactly. An example at equality shows that the strict rank inequality cannot be removed.

We prove that every complex C^∗-algebra in which each positive element is properly infinite is strongly purely infinite. This answers the ordinary-to-strong part of Kirchberg–Rørdam's comparison question. For exact algebras, we also prove that proper infiniteness of one fixed finite amplification of every positive element implies proper infiniteness of each positive element. Consequently, every separable nuclear algebra with this fixed-amplification property absorbs \(\mathcal O_\infty\), without assumptions of unitality or simplicity.