100 problems

Let $X$ be a separable Banach space that is not isomorphic to a Hilbert space. Recall that $X$ is ergodic if the equivalence relation $\mathbf{E}_0$ is Borel-reducible to the isomo…

Let $A$ be the generator of a $C_0$-group on $\ell_p$, with $p>1$, whose eigenvalues, counted with multiplicity, are $\{\lambda_n\}_{n=1}^{\infty}$ and whose corresponding normaliz…

Let $X$ be the Banach space underlying a linear discrete-time system, and let $A_m^k$ denote its associated evolution operator from time $k$ to time $m$. The system is uniformly ex…

Let $X$ be a $K$-convex Banach space. The property $\mathrm{GL}_2$ is the Gordon–Lewis property with exponent $2$, and a Banach lattice has finite cotype in the usual Banach-space…

Let $X$ be a separable Banach space, let $A\subseteq X$ be Borel, and let $\mathbb{A}_{\mathsf{G}}$ be the auxiliary game referred to in the paper, with $\mathbb{A}$ denoting the c…

Let $X$ be a separable Banach space over $cmathbb{R}$, let $A\toolbox X$ be Borel and Aronszajn-null, and let $\mathcal{A}_{\mathsf{G}}$ be the game in which player $\mathbf{I}$ ch…

Let $X$ be a Banach space with a basis, respectively an unconditional basis, which is not equivalent to the canonical basis of $c_0$ or $\ell_p$ for $1\leq p<+\infty$. A normalized…

Let $X$ be a Banach space with an unconditional basis. A block-subspace is a closed subspace generated by a block-sequence of that basis, and $E_0$ is the equivalence relation of e…

A separable Banach space is a Banach space with a countable dense subset, and a Banach space is ergodic when the isomorphism relation between its infinite-dimensional subspaces is…

Let $G=\prod_{i=1}^{m}\mathbf{G}_{i}(k_{i})$, where each $k_i$ is a local field and each $\mathbf{G}_{i}(k_i)$ is the group of $k_i$-points of a Zariski connected simple $k_i$-alge…

Three-type existence claim. Every infinite-dimensional Banach space $\mathfrak X$ admits a bounded biorthogonal system $\left\{x_n,x_n^*\right\}$ that is one of the following three…

Let $X$ be a finite-dimensional Banach space, with $n=\dim X$. Finite-dimensional characterization conjecture. (a) If every one-dimensional subspace of $X$ can be represented as an…

Let $X$ be a strictly monotone Banach space with a $1$-unconditional basis, sufficiently different from a Hilbert space. Block-structure conjecture. Every norm-one projection on…

Let $X$ be a rearrangement-invariant function space on $[0,1]$, so that $X$ contains the constant functions, and let $P$ be a contractive projection on $X$ satisfying…

Embedding conjecture. The natural embedding $E\subset L_1([0,1])$ is not strictly singular if and only if $G$ embeds continuously into $E$.

Let $X$ be a Banach space, let $k\ge 2$ be fixed, and let $n\ge 2$. For $n$-point sets $Q_1,\dots,Q_n\subset X$, define $\operatorname{Tv}(X,k,n)$ as the smallest radius factor suc…

The complex cross-2 distance conjecture. For every integer $n\geq 1$ and all $1\leq q\leq 2\leq p\leq\infty$,

For $n\geq 1$, let $\ell_1^n(\mathbb{C})$ and $\ell_{\infty}^n(\mathbb{C})$ denote the $n$-dimensional complex sequence spaces, and let $d_{BM}$ denote the Banach–Mazur distance be…

Let $n\geq 1$ be an integer, and let $a_1,\ldots,a_n\in\mathbb{C}^n$ satisfy … A complex number of modulus $1$ is unimodular, and a vector in $\mathbb{C}^n$ is a unimodular vector…

Let $G$ be a Gromov hyperbolic group, and consider affine actions of $G$ on a Hilbert space whose linear parts are uniformly bounded representations. An affine action is proper whe…

Naor's lattice quotient conjecture. There exists a constant $c>0$ such that for every $r\in\mathbb{N}$ and every lattice $\Lambda\subset\mathbb{R}^r$,

For $n\to\finity$ and $p>2$, let $\mathsf{c}_2^n(L_p)$ denote the maximum, over all $n$-point metric spaces, of the least distortion with which the space embeds into $L_p$ and then…

Ferenczi–Rosendal conjecture. Every non-Hilbertian separable Banach space is ergodic.

Planar ellipse conjecture. Then $K$ or $L$ is an ellipse.

Let $X$ and $Y$ be real Banach spaces. The pair $(X,Y)$ has the Hilbert space factorization property (HFP) when every bounded operator $\phi:X\to Y$ satisfies…