68 problems

Let $X$ be an infinite compact Hausdorff space, and let $C(X)$ be the Banach algebra of continuous real-valued functions on $X$. Consider an algebra homomorphism from $C(X)$ into a…

Let $\mathcal{A}$ be a commutative Banach algebra, and let $D:\mathcal{A}\to\mathcal{A}$ be a derivation, meaning that $D(ab)=aD(b)+bD(a)$ for all $a,b\in\mathcal{A}$. Singer–Werme…

Let $A$ be a unital Banach algebra, and suppose that every maximal left ideal of $A$ is finitely-generated. Dales–Želazko conjecture. Then $A$ is finite-dimensional. This conjectur…

Banach-algebra linearity conjecture. The condition

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

Let $A$ be a Banach algebra, and let $C(\mathbf{S}^2,A)$ denote the algebra of continuous $A$-valued functions on the $2$-sphere. Write $Bsr(A)$ for the Bass stable rank of $A$. St…

Point-evaluation conjecture. Every nonzero compact endomorphism of $A$ is essentially a point evaluation.

Let $D$ be the open unit disk, let $\Gamma$ be the unit circle, and let $A^+$ be the algebra of absolutely convergent Taylor series. Let $I\ne\{0\}$ be a closed ideal of $A^+$, let…

Let $\mathcal{A}$ be a Banach algebra. It is super-amenable (or contractable) if every bounded derivation $d:\mathcal{A}\rightarrow E$ into every Banach $\mathcal{A}$-bimodule $E$…

Let $G$ be a locally compact group, and let $A(G)$ denote its Fourier algebra. Amenability equivalence conjecture. The group $G$ is amenable if and only if the Banach algebra…

Amenability conjecture. The Fourier algebra $A(G)$ is amenable if and only if $G$ is amenable.

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…

Converse conjecture for $B(G)$. If $B(G)$ has a normal, virtual diagonal, then $G$ has an abelian subgroup of finite index; moreover, the same converse should hold with the existen…

Derivative contraction conjecture. If $c6$ induces a compact endomorphism of $D([0,1],M)$, then

Let $F_\xi$ be a fiber of the restriction map associated with a point $\xi$ as in part (ii) of Theorem 2.3, where $F_\xi$ lies in the maximal ideal space of the relevant Banach alg…

Sparse-stencil conjecture. In models with sparse bracket stencils, the true convergence radius satisfies

Infinite upper-rank conjecture. If $R(S)=\infty$, then the Bass stable rank of $\mathscr{H}^\infty_S$ is infinite.

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…

Failure conjecture. The Cline formula may not hold for one-sided (generalized) Drazin inverses.

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 $A$…

A JB$^*$-algebra is a Jordan–Banach $^*$-algebra. An element is nilpotent if some positive power of it is zero; it is non-trivial when it is nonzero. The nilpotent characterization…

Let $A$ be a non-unital Banach algebra satisfying the von Neumann inequality, and let $H$ be a Hilbert space. The conjecture. $A$ must be isometrically isomorphic to a closed subal…