Direct limits of graded matrix algebras

The direct limit of finite-dimensional semisimple associative algebras arises as a purely algebraic counterpart to important $C^\ast$-algebras. In this paper, we classify direct limits of matrix algebras endowed with a grading by a finite abelian group over an algebraically closed field. In particular, we give an explicit description of the graded $K_0$ group of the direct limit of matrix algebras, and we provide conditions under which this limit absorbs graded-division algebras.