The Matrix Type of Purely Infinite Simple Leavitt Path Algebras

Let R denote the purely infinite simple unital Leavitt path algebra L(E). We completely determine the pairs of positive integers (c, d) for which there is an isomorphism of matrix rings Mc(R) ∼= Md(R), in terms of the order of [1R] in the Grothendieck group K0(R). For a row-finite directed graph E and field k, the Leavitt path algebra Lk(E) has been defined in [1] and [9], and further investigated in numerous subsequent articles. Purely infinite simple rings were introduced in [8]; the purely infinite simple Leavitt path algebras were explicitly described in [2]. All terminology used in this article can be found in these four references. We denote Lk(E) simply by L(E) throughout. In this short note we present necessary and sufficient conditions for the existence of a ring isomorphism between the matrix rings Mc(L(E)) and Md(L(E)) (thereby yielding the so-called Matrix Type of L(E)), whenever L(E) is both purely infinite simple and unital. (L(E) is unital precisely when the graph E is finite.) The sufficiency of these conditions utilizes the deep “algebraic Kirchberg Phillips Theorem” [7, Theorem 2.5] for Leavitt path algebras: If L(E) and L(F ) are Morita equivalent purely infinite simple unital rings, and there exists an isomorphism φ : K0(L(E)) → K0(L(F )) for which φ([1L(E)]) = [1L(F )], then L(E) ∼= L(F ). The following result is well-known, but we prove it here for completeness. 2000 Mathematics Subject Classification. Primary 16S50, Secondary 16E20.