22 problems

Let $E$ and $F$ be finite graphs, let $\mathsf{k}$ be a field, and write $L_{\mathsf{k}}(E)$ and $L_{\mathsf{k}}(F)$ for their Leavitt path algebras. Their graded Grothendieck grou…

Let $E$ and $F$ be finite graphs, and let $K$ be a field. For a … -graded case with its natural $\mathbb{Z}[x,x^{-1}]$-module structure. Let $T_E$ denote the talented monoid of $E$…

Let $Q$ be a finite quiver, let $\widehat{Q}$ denote its Kronecker square, and let $L_{\mathbb K}(Q)$ be the Leavitt path algebra over a field $\mathbb K$. Let…

Let $Q$ be a quiver, let $\widehat{Q}$ denote its Kronecker square, and let $L_{\mathbb K}(Q)$ be the Leavitt path algebra over a field $\mathbb K$. Let…

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…

Hazrat's conjecture. The following are equivalent:

Morita–singularity classification conjecture. The following conditions are equivalent: 1. There is an isomorphism of partially ordered $\mathbb{Z}[x,x^{-1}]$-modules

Hazrat's fullness conjecture. There exists a unital $\mathbb{Z}$-graded $\mathsf{k}$-homomorphism $\psi:L(E)\rightarrow L(F)$ such that

Graded classification conjecture. There is an order-preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism

Free-subgroup conjecture. Every non-central normal subgroup of $L_K(E)^\times$ contains a non-cyclic free subgroup.

Let $\ell$ be a field. Consider the category of unital Leavitt path $\ell$-algebras with graded homomorphisms modulo inner automorphisms, and the category of preordered abelian gro…

Let $E$ and $F$ be finite graphs, let $k$ be a field, let $C^*(E)$ and $C^*(F)$ denote their graph $C^*$-algebras, let $L_k(E)$ and $L_k(F)$ denote their Leavitt path algebras, and…

Let $E_1$ and $E_2$ be finite graphs and $F$ a field. Write $M^{\operatorname{gr}}_{E_i}$ for the graded graph monoid and $K_0^{\operatorname{gr}}(L_F(E_i))$ for the graded Grothen…

Let $K$ be a field with a fixed involution, and let $E$ and $F$ be directed graphs. Write $L_K(E)$ and $L_K(F)$ for their Leavitt path algebras, equipped with the induced involutio…

Let $E$ and $F$ be directed graphs, and let $L_{\mathbb{C}}(E)$ and $L_{\mathbb{C}}(F)$ be their Leavitt path algebras over the complex numbers, while $C^*(E)$ and $C^*(F)$ are the…

Let $E$ and $F$ be directed graphs, let $L_{\mathbb{C}}(E)$ and $L_{\mathbb{C}}(F)$ be their Leavitt path algebras, and let $C^*(E)$ and $C^*(F)$ be the associated graph $C^*$-alge…

Let $E$ and $F$ be directed graphs, and let $L_{\a0\mathbb{C}}(E)$ and $L_{\a0\mathbb{C}}(F)$ denote their Leavitt path algebras over the complex numbers, with the complex-conjugat…

A ring is $^*$-regular when it is von Neumann regular and its involution is proper; it is unit-regular when for every element $a$ there is a unit $u$ such that $a=aua$. Handelman's…

Hazrat's strong classification conjecture. The functor $K_0^{\operatorname{gr}}$ is fully faithful from the category of unital Leavitt path algebras with graded homomorphisms modul…

Let $K_0^{\operatorname{gr}}$ denote the graded Grothendieck group functor. Consider the category whose objects are Leavitt path algebras and whose morphisms are graded homomorphis…

Let $E$ and $F$ be row-finite graphs over a field $K$. Write $\operatorname{\mathcal L}_K(E)$ and $\operatorname{\mathcal L}_K(F)$ for their Leavitt path algebras, and let…

Let $E$ and $F$ be row-finite graphs. Write $\operatorname{\mathcal L}(E)$ and $\operatorname{\mathcal L}(F)$ for their Leavitt path algebras, with graded isomorphisms denoted by…