We present a result of P. Ara which establishes that the Unbounded Generating Number property is a Morita invariant for unital rings. Using this, we give necessary and sufficient conditions on a graph E so that the Leavitt path algebra associated to E has UGN. We conclude by identifying the graphs for which the Leavitt path algebra is (equivalently) directly finite; stably finite; Hermite; and has cancellation of projectives.Mathematics Subject Classifications: 16S99, 18G05, 05C25

Abstract. We compute the monoid V (L K (E)) of isomorphism classes of finitely generated projective modules over certain graph algebras L K (E), and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattice of graded ideals of L K (E) and the lattice of order-ideals of V (L K (E)). When K is the field C of complex numbers, the algebra L C (E) is a dense subalgebra of the graph C * -algebra C * (E), and we show that the inclusion map induces an isomorphism between the corresponding monoids. As a consequence, the graph C*-algebra of any row-finite graph turns out to satisfy the stable weak cancellation property.

The construction of the C*-algebra associated to a directed graph E is extended to incorporate a family C consisting of partitions of the sets of edges emanating from the vertices of E. These C*-algebras C * (E, C) are analyzed in terms of their ideal theory and K-theory, mainly in the case of partitions by finite sets. The groups K 0 (C * (E, C)) and K 1 (C * (E, C)) are completely described via a map built from an adjacency matrix associated to (E, C). One application determines the K-theory of the C*-algebras U nc m,n , confirming a conjecture of McClanahan. A reduced C*-algebra C * red (E, C) is also introduced and studied. A key tool in its construction is the existence of canonical faithful conditional expectations from the C*-algebra of any row-finite graph to the C*-subalgebra generated by its vertices. Differences between C * red (E, C) and C * (E, C), such as simplicity versus non-simplicity, are exhibited in various examples, related to some algebras studied by McClanahan.

-A separative ring is one whose finitely generated projective modules satisfy the property A EB A � A EB B � B EB B ==} A ,...., B. This condition is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. It is shown that the class of separate exchange rings is very broad, and, notably, closed under extensions of ideals by factor rings. That is, if an exchange ring R has an ideal I with I and Rf I both separative, then R is separative. SEPARATIVE CANCELLATION FOR PROJECTIVE MODULES OVER EXCHANGE RINGSP. ARA, K.R. GOODEARL, K.C. O'MEARA AND E. PARDO ABSTRACT. A separative ring is one whose finitely generated projective modules satisfy the property A EB A ~ A EB B ~ B EB B ~ A ~ B. This condition is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. It is shown that the class of separative exchange rings is very broad, and, notably, closed under extensions of ideals by factor rings. That is, if an exchange ring R has an ideal I with I and R/ I both separative, then R is separative.

No abstract