42 problems

Let $d4d8_p$ be the collection of symbols that are bounded Schur multipliers of the Schatten–von Neumann class $d4d4_p$, and let $d4d8_p^{\mathrm{cb}}$ be the collection of symbols…

Let $d$ be a positive integer, let $m_\delta$ be the Bochner–Riesz multiplier from the classical formulation, and let $\mathfrak{M}_p({\mathbb R}^{2d})$ denote the class of multipl…

Let $X$ be the Banach space from Proposition 3.16, which is an $M$-ideal in $X^{**}$ but admits no $M$-approximate identity in $X^{**}$. Separable intermediate-space conjecture. Th…

Let $Z_1,Z_2,\ldots$ be reflexive separable operator spaces such that $(Z_1\oplus Z_2\oplus\cdots)_{c_0}$ has the completely separable extension property (CSEP). The finite matrix…

Let $Z$ be a separable operator space. The finite matrix type conjecture. If $Z$ has the completely separable extension property (CSEP), then $Z$ itself is of finite matrix type. H…

Let $(M_{\infty,n})_{c_0}$ denote the $c_0$-sum of the spaces of matrices with $\infty$ rows and $n$ columns. An operator space is primary if, whenever it is isomorphic to a direct…

Let $G$ be a locally compact group, and let $\mathrm{S}_0(G)$ denote Feichtinger's Segal algebra equipped with the operator space structure constructed in the paper. The maximality…

Let $d$ be the chaos degree, and let the upper estimate of Theorem C have a constant depending on $d$. Polynomial growth conjecture. The growth of this constant should be polynomia…

Solel's conjecture. The same result should hold even when $\Phi$ is not weak-continuous. The paper constructs a unital, completely positive, idempotent $L^{\infty}(\mathbb{T})$-bim…

Conjecture. The answer is no to both questions: there exists a quasicentralizer that is not an associative bilinear map, and there exists an associative quasicentralizer that is no…

Amenable-group complete contraction conjecture. The identity map from $A(G)$ to $OA(G)$ is a complete contraction for each amenable $G$.

Let $n,m\geq2$ and let $\mathcal{H}\subset\mathbb{M}_{n,m}$ be a subspace of dimension $r$ that is isometric to a Hilbert space. The preceding theorem gives $r\leq n+m-2$. Hilbert-…

The direct sum axiom for operator spaces is not naturally expressed solely through the enriched category theory of modules. Enriched algebraic theory conjecture. There should exist…

Interpolation conjecture. The quasi-Banach spaces defined in the paper are complex interpolation spaces, completely isometrically, in the sense that

Let $G$ be a compact Lie group of dimension $n$ with a biinvariant Riemannian metric, and let $A$ be the relevant generator. Set … For $1<p<\infty$, let $\mathrm{L}^p_0(G)$ denote…

Let $p:\{-1,1\}^n\to\mathbb{R}$ be a polynomial of degree at most $d$. Its Fourier completely bounded $d$-norm is denoted by $\|p\|_{\operatorname{fcb},d}$, and…

Let $n\geq 3$, and let $\ell_1^n$ be the $n$-dimensional operator space. A finite-dimensional operator space is controllable when it satisfies the controllability property defined…

Let $\ell_1^3$ be the three-dimensional operator space and let $M_k(\ell_1^3)$ denote its $k\times k$ matrix amplification. Say that an operator space is controllable when it has t…

Let $\mathscr{C}=C^*(\mathbb{F}_\infty)$, and let a finite-dimensional operator subspace $E\subset\mathscr{C}$ be controllable when it is completely isometric to a subspace having…

Inner amenability–$(\kappa_\infty)$ conjecture. Inner amenability of $G$ is equivalent to property $(\kappa_\infty)$.

Failure conjecture. The converse implication fails: there exists a maximal operator space that is not strongly maximal, owing to the lack of a suitable approximation property in…

Let $X$ be a valid norm, meaning a norm on $\mathcal{S}_1^{\tilde k'\!, n}\otimes\mathcal{S}_1^{\tilde k'\!, n}\otimes\ell_2^{k\tilde k'}$ satisfying … for every $x$, together with…

Let $X$ be a norm on the algebraic tensor product $\mathcal{S}_1^{\tilde k'\!, n}\otimes\mathcal{S}_1^{\tilde k'\!, n}\otimes\ell_2^{k\tilde k'}$. Call $X$ valid if … for every $x$…

Weaker-form conjecture. There exists a dimension-independent constant $\beta<1$ such that

Let $n$, $k$, and $\tilde k'$ be the parameters in the construction, let $\mathcal{S}_1^{\tilde k'\!, n}$ denote the corresponding trace-class space, let…