The functor $K_0^{\operatorname{gr}}$ is full and only weakly faithful

TL;DR

This paper investigates the properties of the graded K_0 functor for Leavitt path algebras, showing it is full in some cases but only weakly faithful in others, impacting classification efforts.

Contribution

It proves the functor is full for unital Leavitt path algebras of countable graphs and clarifies its limited faithfulness under certain conditions.

Findings

The functor $K_0^{ ext{gr}}$ is full for unital Leavitt path algebras of countable graphs.

The functor $K_0^{ ext{gr}}$ is only weakly faithful under specific conjugation conditions.

Supports the graded classification conjecture in particular cases.

Abstract

The Graded Classification Conjecture states that the pointed $K_0^{\operatorname{gr}}$-group is a complete invariant of the Leavitt path algebras of finite graphs when these algebras are considered with their natural grading by $\mathbb Z.$ The strong version of this conjecture states that the functor $K_0^{\operatorname{gr}}$ is full and faithful when considered on the category of Leavitt path algebras of finite graphs and their graded homomorphisms modulo conjugations by invertible elements of the zero components. We show that the functor $K_0^{\operatorname{gr}}$ is full for the unital Leavitt path algebras of countable graphs and that it is faithful (modulo specified conjugations) only in a certain weaker sense.