On the homogeneous zero components of Leavitt algebras

We prove that the zero component $L(m,n{)}_0$ of a Leavitt algebra $L(m,n)$ with respect to the canonical grading is a direct limit ${\mathrm{}}_{z}L(m,n{)}_{0,z}$, where each algebra $L(m,n{)}_{0,z}$ is a free product of two Bergman algebras. For the special case $m=1,n>1$, one recovers the known result that the zero component $L(1,n{)}_0$ is a direct limit of matrix algebras. Moreover, we show that $L(m,n{)}_0$ has the IBN property.