14 problems

Let $d\in\mathbb{N}$, let $\mathcal{A}$ be a unital C-algebra, and let $\mathbb{MM}_{d\times d}(\mathcal{A})$ denote the $d\times d$ Manin matrices over $\mathcal{A}$. Let…

Let $A$ be a unital C-algebra, let $\mathcal H$ be a Hilbert $A$-module, and let $\{x_j:j\in\mathbb J\}$ be a standard frame with two standard alternate dual frames…

Separation conjecture. There does not exist any non-trivial bounded $A$-linear map from $\mathcal N$ to $A$ that is equal to the zero map on $\mathcal M$.

Let $\mathcal{E}$ be a Hilbert $C^*$-module over a commutative unital $C^*$-algebra $\mathcal{A}$. Let $\{\tau_j\}_{j=1}^\infty$ and $\{\omega_k\}_{k=1}^\infty$ be two Parseval fra…

Let $\mathscr A$ be a $C^*$-algebra, let $\mathscr E$ be a Hilbert $C^*$-module over $\mathscr A$, and let $\mathscr L$ be a closed $\mathscr A$-submodule of $\mathscr E$. For a st…

Let $\mathcal{A}$ be a unital C-algebra with the invariant basis number property or a W-algebra, let $(G,\mu_G)$ be a Lie group with Haar measure, and let $d\in\mathbb{N}$. A conti…

Continuous modular Zauner's conjecture. For a given Lie group $(G,\mu_G)$ and every $d\in\mathbb{N}$, there exists a $\gamma$-equiangular tight continuous modular frame…

Let $\mathcal{A}$ be a unital C-algebra with the invariant basis number property or let $\mathcal{A}$ be a W-algebra. For $d\in\mathbb{N}$, consider unit inner product frames in th…

Let $\mathcal{A}$ be a unital C-algebra, let $d\in\mathbb{N}$, and let $\mathcal{A}^d$ denote the standard Hilbert C-module. A unit inner product frame has…

Let Conjecture c:one be the assertion that pure states separate the localized image of an $\sA$-submodule from a point outside it, and let Conjecture c:two be the assertion that ev…

Let $E$ be a Hilbert $C^*$-module over a $C^*$-algebra $\sA$, let $L\subset E$ be a closed convex subset, and let $x_0\in E\setminus L$. For each state $\go$ on $\sA$, let $E^\go$…

Let $\sA$ be a $C^*$-algebra, let $E$ be a Hilbert $C^*$-module over $\sA$, and let $L$ be a proper submodule of $\sA$. For each state $\go$ on $\sA$, let $E^\go$ be the localized…

Local orthogonality-preserver conjecture. If $\theta:E\to F$ is a $\mathbb C$-linear local map preserving orthogonality, namely

Decomposition conjecture. Any orthogonality-preserving map $T$ on Hilbert $A$-modules $d4dc$ over $C^*$-algebras $A$ should be of the form