181 problems

Connes' rigidity conjecture. If

Peterson–Thom conjecture for hyperbolic groups. If $\mathcal{A}\capmathcal{B}$ is diffuse, then the von Neumann algebra they generate is amenable:

Let $M$ be a type ${\rm III}_1$ factor and let $\varphi\in M_*$ be a faithful normal state. The bicentralizer conjecture. The bicentralizer satisfies … This is Connes' bicentralize…

Let $(M,\tau)$ be a finite von Neumann algebra with separable predual, and let $R$ be the hyperfinite $\mathrm{II}_1$-factor over ${\mathbbm k}\in\{{\mathbb C},{\mathbb R}\}$. Writ…

Let $\Gamma\curvearrowright (X,\mu)$ be a Bernoulli action of an icc group $\Gamma$ with Kazhdan's property (T). The action is W$^*$-superrigid if its isomorphism class as a group…

Let $G$ be a countable discrete group, and let $\operatorname{L}(G)$ be its group von Neumann algebra. Write $\operatorname{Char}(G)$ for the group of characters of $G$ and…

Connes's conjecture. If

Let $N \subset M$ be an inclusion of von Neumann algebras with expectation. The relative bicentralizer conjecture proposes an explicit computation of the relative bicentralizer…

Characteristic-invariant conjecture. The action $\alpha$ is classified up to cocycle conjugacy by the characteristic invariant $(H,\chi)$.

Popa's conjecture. The von Neumann algebra $L^\infty(X^\Gamma)\rtimes\Gamma$ has a unique Cartan subalgebra, up to unitary conjugacy.

Let the spectral density be $rho$ for the large $N$ theory, and consider the von Neumann algebra generated by the relevant single-trace operators. Leutheusser–Liu's initial conject…

Let $X$ be a free manifold in the sense of chapter 13, and consider an algebra of type $L^\infty(X)$. Free-manifold factor conjecture. The algebra $L^\infty(X)$ is a ${\rm II}_1$ f…

Let $\mathcal{M}$ be a von Neumann algebra, and let $H^n(\mathcal{M},\mathcal{M})$ denote its continuous Hochschild cohomology groups with coefficients in itself. Kadison–Ringrose'…

Let ${\mathcal M}$ and ${\mathcal N}$ be type II von Neumann algebras, and let $LS({\mathcal M})$ and $LS({\mathcal N})$ denote their algebras of locally measurable operators. A ri…

Let $\mathbb F_n$ act freely and ergodically by probability-measure-preserving transformations on $(X,\mu)$, where $n\geq 2$. The associated group measure space factor is…

Let $L(\mathbb{F}_n)$ be a free group factor and let $B\subset L(\mathbb{F}_n)$ be a diffuse amenable subalgebra. A maximal amenable extension of $B$ is an amenable subalgebra of…

Let $M$ be a von Neumann algebra, let $A\subset M$ be a MASA that is the range of a normal conditional expectation, and let $A^\omega$ denote its ultrapower. For $n\geq 2$ and a se…

Separable normal dilation conjecture. The dilation space $Z$ can be chosen to be separable, and the dilation homomorphism $\pi$ can be chosen to be ultraweakly continuous.

Let $\mathbb{F}_n \curvearrowright (X,\mu)$ be an action of a free group on a probability space, with no profiniteness assumption. The crossed product…

Let $W$ be a discrete group, let $L^2W$ be the Hilbert space completion of its real group algebra, and let $R_M:(L^2W)^n\rightarrow(L^2W)^m$ be right multiplication by a matrix $M$…

Let $\mathcal{F}(M)$ and $\mathcal{F}(\mathcal{R})$ denote the fundamental groups of a $\mathrm{II}_1$ factor and an orbit equivalence relation, respectively. For a free ergodic pr…

Superfiniteness conjecture. Any finite von Neumann category is superfinite.

Twisted-group-ring injectivity conjecture. If $0\ne\theta\in\mathbb{C}*\mathcal{G}_n$ and $0\ne f\in L^2(\mathbb{R}^n)$, then $\theta f\ne0$. This is the operator-theoretic formula…

Determinant-class conjecture. Every finite $CW$-complex is of determinant class.

Let $d4aa$ be a free ultrafilter on $d4a9$ and let $d53d$ be a separable $d40fd40f_1$-factor. Let $d52c$ denote the hyperfinite $d40fd40f_1$-factor, and let $d52c^{d4aa}$ be its ul…