11 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…

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

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 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 $E_1$ 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 $A$ be a graded $C^*$-algebra. A graded projective module is a projective module equipped with a compatible grading, and let $K^{\operatorname{gr}}_0(A)$ denote the graded…

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