The Reduction Theorem for Leavitt Labelled Path Algebras and Its Applications

We introduce a notion of labelled cycle for normal labelled spaces and prove a reduction theorem for Leavitt labelled path algebras. We show that every nonzero element can be reduced, by suitable left and right multiplication, either to a nonzero scalar multiple of a projection or to a polynomial supported on a labelled cycle without exits. This extends the classical reduction theorem for Leavitt path algebras of directed graphs and its analogues for ultragraph Leavitt path algebras and subshift algebras. As applications, we prove the graded uniqueness theorem and the Cuntz–Krieger uniqueness theorem for Leavitt labelled path algebras, and show that these algebras are semiprime and semiprimitive over fields.