14 problems

For $0<r_2<r_1<1$, let … be the Banach ring constructed as the non-expanding colimit of the rings $\mathbb{Z}\{(x/r)^{1/n}\}$ under the morphisms $\alpha_{n,m}$. Nuclearity conject…

Quotient-space decay conjecture. The spaces $\mathcal L^{1'}_v$ and $\mathcal L'_v$ possess the same decay properties as $\mathcal L^1_v$ and $\mathcal L_v$, respectively. The sour…

Weighted decay conjecture. There exists $\delta>0$ such that $h_L\in\mathcal L^1_{\tau_\delta}$. If true, this would support identifying $\mathcal L^1$ with a union of the weighted…

Let $A$ be a unital Banach algebra, and let $A_{\mathrm{MP}}$ denote the set of Moore–Penrose invertible elements of $A$. For $a\in A_{\mathrm{MP}}$, write $a^\dag$ for its Moore–P…

Let $V$ be a Banach algebra, let $Y$ be a Banach space, and let $\varphi\!:V\to Y$ be a bounded linear map. For $z\in V$, let ${\mathfrak{S}}_\cap^\varphi(z)$ and…

Let $V$ be a Banach algebra, and let $X$ be a Banach space. Let $\varphi\!:V\to X$ be a bounded linear map. For $z\in V$, write ${\mathfrak{C}}_{-1}(z)$ for the set of correspondin…

Let $(A,\|\cdot\|)$ be a non-unital, semisimple Banach algebra, and let $A_e$ denote its unitization. Say that $A$ has the spectral extension property (SEP) when every norm on $A$…

Let $A$ be an infinite-dimensional unital Banach algebra. Dales–Zelazko conjecture. Then $A$ has a maximal left ideal which is not finitely generated. The conjecture concerns the e…

Banach-algebra linearity conjecture. The condition

Hardy-algebra elementary-generation conjecture.

Gelfand-transform criterion. $F\in SL_n(R)$ belongs to $E_n(R)$ if and only if

Lau–Ülger–Pym conjecture. There is no faithful infinite-dimensional non-unital weakly sequentially complete Banach algebra $A$ with a BAI such that

Let $A$ be a real or complex commutative Banach algebra with involution, let $\varepsilon$ denote the hermitian sign, and let $m$ be the coefficient modulus. The algebraic-to-topol…

Amenability–similarity conjecture. A commutative Banach subalgebra of $\mathfrak{B}(\mathfrak{H})$ is amenable if and only if it is similar to a $C^*$-algebra.