For a field K, the Leavitt K-algebra LK(1,n) was defined and
investigated in the early
1960's, and has enjoyed an intense revival in interest over the past few
years. When K=C (the field of
complex numbers), the Leavitt algebra is intimately connected with the
C*-algebra On, the so-called Cuntz algebra. Let R denote LK(1,n). For any ring S, let Mt(S) denote the t by t matrix ring over S. We will
first show (easily) that if t is congruent to 1 (mod n-1), then
R is isomorphic to Mt(R) as K-algebras. We will then show
(less easily) that if t divides some power of n, then we also get
that R is isomorphic to Mt(R). With these two observations as
motivation, we will then show how to prove the following: R is isomorphic to Mt(R) if and only if t and n-1 are relatively prime. The proof of this result involves some straight forward number theory (which
we will describe in detail). As a consequence of this result, we get information not only about Leavitt
K-algebras, but we answer a longstanding question about Cuntz C*-algebras as well.
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