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We compute the monoid V (LK(E)) of isomorphism classes of finitely generated projective modules over certain graph algebras LK(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 LK(E) and the lattice of order-ideals of V… (More)

- Pere Ara, M . Brustenga, Guillermo Cortiñas
- 2008

Let E be a row-finite quiver and let E0 be the set of vertices of E; consider the adjacency matrix N ′ E = (nij) ∈ Z(E0×E0), nij = #{ arrows from i to j}. Write N E and 1 for the matrices ∈ Z (E0×E0\Sink(E)) which result from N ′t E and from the identity matrix after removing the columns corresponding to sinks. We consider the K-theory of the Leavitt… (More)

- Pere Ara
- 1996

We prove a cancellation theorem for simple refinement monoids satisfying the weak comparability condition, first introduced by K.C. O’Meara in the context of von Neumann regular rings. This result is then applied to von Neumann regular rings and C∗-algebras of real rank zero via the monoid of isomorphism classes of finitely generated projective modules.… (More)

- Pere Ara, Martin Mathieu
- 2007

A new C*-enlargement of a C*-algebra A nested between the local multiplier algebra Mloc(A) of A and its injective envelope I(A) is introduced. Various aspects of this maximal C*-algebra of quotients, Qmax(A), are studied, notably in the setting of AW*algebras. As a by-product we obtain a new example of a type I C*-algebra A such that Mloc(Mloc(A)) 6=… (More)

We extend the notion of a purely infinite simple C*-algebra to the context of unital rings, and we study its basic properties, specially those related to K-Theory. For instance, if R is a purely infinite simple ring, then K0(R) + = K0(R), the monoid of isomorphism classes of finitely generated projective R-modules is isomorphic to the monoid obtained from… (More)

- Pere Ara
- 2006

We characterize the values of the stable rank for Leavitt path algebras, by giving concrete criteria in terms of properties of the

Replacing invertibility with quasi-invertibility in Bass’ first stable range condition we discover a new class of rings, the QB−rings. These constitute a considerable enlargement of the class of rings with stable rank one (B−rings), and include examples like EndF(V ), the ring of endomorphisms of a vector space V over some field F, and B(F), the ring of all… (More)

- Pere Ara
- 2006

A longstanding open problem in the theory of von Neumann regular rings is the question of whether every directly finite simple regular ring must be unit-regular. Recent work on this problem has been done by P. Menal, K.C . O'Meara, and the authors. To clarify some aspects of these new developments, we introduce and study the notion of almost isomorphism… (More)

We compute the Hochschild homology of Leavitt path algebras over a field k. As an application, we show that L2 and L2 ⊗ L2 have different Hochschild homologies, and so they are not Morita equivalent; in particular they are not isomorphic. Similarly, L∞ and L∞ ⊗ L∞ are distinguished by their Hochschild homologies and so they are not Morita equivalent either.… (More)

A graph monoid is a commutative monoid for which there is a particularly simple presentation, given in terms of a quiver. Such monoids are known to satisfy various nonstable K-theoretical representability properties for either von Neumann regular rings or C*-algebras. We give a characterization of graph monoids within finitely generated antisymmetric… (More)