10 problems

Let $\mathbb{X}$ be a Banach space with a unique unconditional basis. For $E\in\{c_{0},\ell_{1},\ell_{2}\}$, consider the iterated copy $E(\mathbb{X})$ of $\mathbb{X}$, namely the…

Let $Y$ be a twisted Hilbert space, meaning that $Y$ has a Hilbertian subspace $X_0$ such that $Y/X_0$ is Hilbertian. Twisted Hilbert space conjecture. If $Y$ has an unconditional…

Let $X$ be a Banach space with an unconditional basis, and let $E_0$ denote eventual equality of binary sequences. The block-subspace dichotomy conjecture. Either $E_0$ is Borel re…

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…

Bourgain's conjecture. Every unconditional spreading model of $\mathbb{X}$ is isomorphic to one of

Let $X$ be a separable, non-Hilbertian, non-ergodic Banach space. The cited theorem gives a non-Hilbertian subspace $Y$ which isomorphically embeds into all of its non-Hilbertian s…

Let $X$ be a separable Banach space. A space is non-ergodic when it is not ergodic, and a Banach space is Hilbertian when it is isomorphic to a Hilbert space. Weak ergodicity conje…

Let $X$ be a Johnson space, meaning a separable Banach space with only countably many non-isomorphic subspaces. Weak Johnson conjecture. Every Johnson space has an unconditional ba…

Let $1<p<\infty$, $p\neq 2$, and let $(\Omega,\mu)$ be a finite measure space such that $L^p(\mu)$ is not separable. The Enflo–Rosenthal conjecture asserts that $L^p(\mu)$ is not a…

Dineen's conjecture. The space $P(^m E)$ has an unconditional basis if and only if $E$ is finite-dimensional.