18 problems

Mazur-Ulam property conjecture. Every uniform algebra satisfies the Mazur-Ulam property.

Let $A$ be a uniform algebra on a compact metric space $K$. Say that (i) the maximal ideal space of $A$ is $K$, and (ii) every point of $K$ is a peak point for $A$, meaning that fo…

Let $A_\mathbb{R}(\mathbb{D})$ denote the real disc algebra, consisting of functions analytic on $\mathbb{D}$, continuous on $\overline{\mathbb{D}}$, and satisfying the relevant re…

Non-semi-algebraic interpolation-body conjecture. For a sufficiently pathological uniform algebra, there will exist interpolation bodies that are not semi-algebraic.

Dense-invertibles proper-Shilov-boundary conjecture. No uniform algebra with dense invertible group can have a proper Shilov boundary.

Stolzenberg's conjecture. Whenever a uniform algebra has proper Shilov boundary, its maximal ideal space must contain analytic structure.

Morris's conjecture. Whenever $R(X)$ has no non-zero, bounded point derivations, $R(X)$ must be weakly amenable.

Let $K$ be a compact space and let $A$ be a closed subalgebra of $C(K)$, with the Nevanlinna–Pick property represented by $A\in NP_{\infty}$. Nevanlinna–Pick norm conjecture for un…

Izzo's higher-power ideal conjecture. For each integer $m\geq 2$, the condition

The nonexistence conjecture. There are no nontrivial strongly regular uniform algebras.

A strongly regular uniform algebra is a uniform algebra satisfying the strong regularity property at every point of its maximal ideal space. Nonexistence conjecture. No nontrivial…

Let $R(K)$ denote the uniform algebra of rational functions with poles off a compact set $K\subset\mathbb{C}$. For $x\in K$, write $M_x$ for the maximal ideal of functions in…

Let $L\subset K\subset\mathbb{C}$ be compact sets, let $s\in\mathbb{Z}_+$, and let $x\in L$. For a compact set $E$ and $z\in E$, write $R(E)$ for the uniform algebra of rational fu…

Let $\bar M$ be a domain with boundary in a complex manifold $N$. For every uniform algebra $A$ on $X$, every totally disconnected peak-interpolation set $S\subset X$ for $A$, and…

Unital uniform-algebra conjecture. If $\|\theta_\alpha\|\leq 1$, then

Let $A$ be a uniform algebra on a compact Hausdorff space $K$, and let $\mathfrak{M}_A$ denote its maximal ideal space. An analytic disc is a nonconstant holomorphic map from the u…

Let $X$ be a compact metric space, let $C(X)$ be the algebra of continuous complex-valued functions on $X$, and let $A$ be a uniform algebra on $X$. Suppose that the maximal ideal…

Let $E$ be a compact set in $\mathbb C^n$, and let $\widehat E$ denote its polynomially convex hull. Analytic-disc conjecture. If $\widehat E$ is strictly larger than $E$, then…