35 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 $K_0^{\mathrm{gr}}$ denote the graded Grothendieck group of a graded algebra. For Leavitt path algebras and graph $C^*$-algebras, the relevant graded Grothendieck group is inte…

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…

Ara-Goodearl conjecture. This induced homomorphism is an isomorphism:

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 $E$ be a directed graph and let $L_K(E)$ be its Leavitt path algebra with its natural grading. The talented monoid of $L_K(E)$ is the graded monoid associated with the graded f…

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…

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…

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

Hazrat's conjecture. The following are equivalent:

Talented monoid classification conjecture. The following are equivalent: $T_E$ and $T_F$ are $\mathbb Z$-isomorphic; $L_\mathsf{k}(E)$ and $L_\mathsf{k}(F)$ are graded Morita equiv…

Graded classification conjecture. For every order-preserving $\mathbb Z[x,x^{-1}]$-module homomorphism

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…

Strong Graded Classification Conjecture. The functor

Graded Classification Conjecture. The following conditions are equivalent:

Let $E$ be a finite graph, let $\mathsf{k}$ be a field, and let $L_\mathsf{k}(E)$ be its associated Leavitt path algebra. For $n\in\mathbb{N}$, write $L_n$ for the classical Leavit…

Let the relevant graded K-theoretical invariant of a Leavitt path algebra be identified with Krieger's dimension group triple, and augment this dimension triple by the class of the…

Let $E$ be a row-finite directed graph, let $M_E^\Gamma$ be its cancellative commutative monoid with a $\Gamma$-action, and let $G_E^\Gamma$ be its Grothendieck group, equipped wit…

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…

Ara–Goodearl's conjecture. This inclusion should induce an isomorphism on non-stable $K$-theory. This problem concerns whether the algebraic and analytic separated graph constructi…

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