  It is shown that Hattori-Stallings trace induces a homomorphism of abelian
groups, called Hattori-Stallings character, from the $K_1$-group of
endomorphisms of the perfect derived category of an algebra to its zero-th
Hochschild homology, which provides a new proof of Igusa-Liu-Paquette Theorem,
i.e., the strong no loop conjecture for finite-dimensional elementary algebras,
on the level of complexes. Moreover, the Hattori-Stallings traces of projective
bimodules and one-sided projective bimodules are studied, which provides
another proof of Igusa-Liu-Paquette Theorem on the level of modules.
