19 problems

Let $E$ and $F$ be graphs and let $\mathsf{k}$ be a field. Consider the Leavitt path algebras $L_\mathsf{k}(E)$ and $L_\mathsf{k}(F)$, and the graph $C^*$-algebras $C^*(E)$ and…

Let $\Gamma$ be a graph, and let $Q_n$ and $Q(\Gamma)$ be the graph algebras considered in the paper. A graded algebra is (graded) coherent when the kernel of every homogeneous map…

Let $\Lambda$ be the graded commutative algebra formed from the antisymmetric subspace of connected tri-/univalent graphs with three labeled univalent vertices, and let $\Lambda_0$…

Let a graph have all but finitely many vertices lying on finitely many infinite paths. Fold each of these paths so that it produces countably infinitely many edges between two dist…

Fau's graph-algebra conjecture. $C^*(E)$ is nowhere scattered if and only if $C^*(E)$ is $\mathcal{Z}$-stable, if and only if $E$ has Condition (K) and distinct detours.

Diagonal-preserving classification conjecture. Two graph $C^*$-algebras are isomorphic via an isomorphism that preserves the diagonal subalgebras if and only if the associated Leav…

Graph-algebra characterization conjecture. The following conditions are equivalent:

Divisibility-and-Condition (K) conjecture. The divisibility condition and Condition (K) are equivalent to $$ -stability of the graph algebra.

Algebraic Kirchberg–Phillips conjecture. If there is a $\mathbb{Z}$-isomorphism $\phi:T_E\to T_F$ such that

Random regular Cuntz-algebra conjecture. For $p=2$, or for an odd prime $p$ dividing $r-1$,

Cuntz-algebra probability product conjecture. The asymptotic probability that the graph algebra associated to the random $r$-regular multigraph is stably isomorphic to a Cuntz alge…

Let $S_n$ be the star graph on $n$ vertices, with central vertex labeled by $1$, and let … For a vertex subset $S$, write $Y_S$ for the associated cluster-algebra element and…

Gauge-coaction normality conjecture. Every gauge coaction on a representation of $\Lambda$ is normal. Consequently, the other results in the relevant section should require only th…

Let $E$ and $F$ be finite graphs, and let $R$ be a field. Write $\mathcal{G}_E$ and $\mathcal{G}_F$ for their graph groupoids, and let $H_0^{\operatorname{gr}}$ denote graded group…

Conjecture. Theorem 4.5 is true for the Leavitt path algebras $L_S(\Gamma)$ over an arbitrary graph $\Gamma$.

The transfer-operator characterization conjecture. In general, $mathcal L$ is a transfer operator for a certain endomorphism if and only if $E$ has no infinite emitters; in that ca…

Six-vertex infinite-dimensionality conjecture. The algebra $\mathcal E_G$ is infinite-dimensional if and only if $G$ contains a subgraph isomorphic to one of the graphs shown in Fi…

Graphical-dimension conjecture. The Krull dimension of $\bar{A}(G)$ is equal to the graphical dimension of $G$.

Let $\mathcal Z$ be the Jiang--Su algebra and let … For a graph $G$, let $B(G)$ denote the associated CCR algebra. Non-realizability claim. The algebra $A_{\aleph_0\aleph_1}$ is no…