15 problems

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…

Let $A$ and $B$ be C$^*$-algebras equipped with faithful tracial states, and let $A*B$ denote their reduced free product with respect to these states. The stable-rank-one conjectur…

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

Let $\Lambda$ be a finite-dimensional algebra over a field $\mathcal{K}$, let $\mathrm{st}(\Lambda)$ be the stable rank at which the descending powers of the module-category radica…

Let $\Lambda$ be a special biserial algebra, and let $\mathrm{st}(\Lambda)$ denote its stable rank, the ordinal at which the descending powers of the radical stabilize. A special b…

Let $\Lambda$ be a special biserial algebra over an algebraically closed field, and let $\mathrm{st}(\Lambda)$ be the least ordinal at which the decreasing sequence of powers of th…

Let $X$ be a variety, let $H\subset X$ be a nonsingular hypersurface, let $\mathfrak a$ be an ideal on $X$, and let $P\in H$. Assume that $\mathfrak a\cdot\mathcal O_H$ is nonzero.…

Let $\Lambda$ be a special biserial algebra with at least one band, and let $\mathrm{st}(\Lambda)$ denote its stable rank. Stable-rank conjecture. The stable rank satisfies … This…

Let $\mathcal{E}'(\mathbb{R})$ be the commutative unital convolution algebra of compactly supported distributions on $\mathbb{R}$. Its Bass stable rank…

Let $X$ be an infinite, compact, metrizable space, and let $\alpha\colon X\to X$ be a minimal homeomorphism. Set … Then $A$ is a separable, unital, simple, non-elementary, nuclear…

Let $X$ be a compact Hausdorff space, and let $\tau$ be a topological involution of $X$. Denote the set of fixed points of $\tau$ by $E$. The algebra $C(X,\tau)$ is the algebra of…

Stable rank one conjecture. The crossed product always has stable rank one:

Quantitative stable-rank conjecture. There exist $h_1,h_2,g_1,g_2\in H^\infty_\mathbb{R}(\mathbb{D})$ such that

Let $H^\infty_\mathbb{R}(\mathbb{D})$ denote the algebra of bounded holomorphic functions on the unit disk that are real on the real interval. The Bass stable rank conjecture. The…

Uniform corona conjecture. There exist $g_1,\ldots,g_n\in H^\infty_B$ such that