Nonstable K-theory for Graph Algebras

By P. Ara, M. Moreno, Enrique Pardo. 2004

411 citations recorded by the source corpus.

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(LK(E)). When K is the field $\mathbb C$ of complex numbers, the algebra $L_{\mathbb C}(E)$ 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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c99ac1ab758d967c4c07e03ad650ecc73814feb6

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In Algebra and Number Theory

1. G. Abrams
2. M. Tomforde
3. Roozbeh Hazrat
4. Aidan Sims
5. P. Ara
6. G. Pino
7. Mercedes Siles Molina
8. K. Goodearl

In Mathematical Physics

1. G. Abrams
2. M. Tomforde
3. Roozbeh Hazrat
4. Aidan Sims
5. P. Ara
6. G. Pino
7. Mercedes Siles Molina
8. R. Exel

Citations in the corpus

• Kumjian-Pask algebras of higher-rank graphs (2011, background): “The Leavitt path algebras, which are a purely algebraic analogue of graph C∗algebras, were first studied around 2005 [1, 5].”
• Chain conditions for Leavitt path algebras (2009, neutral): “It is shown in [7] that the hereditary saturated closure of a set X is X = ⋃∞ n=0 Λn(X ), where Λ0(X ) = T (X ), and Λn(X ) = {y ∈ E0 | s−1(y) = ∅ and r(s−1(y)) ⊆ Λn−1(X )} ∪ Λn−1(X ), for n ≥ 1.”
• Irreducible representations of Leavitt path algebras (2011, background): “Q/ of Q with coefficients in k is relatively new, which is introduced in [1, 7, 8].”
• Isomorphism and Morita equivalence of graph algebras (2008, background): “Such K-algebras include many well-known classes of algebras and have been studied intensely in the algebraic community since their introduction in 2005 [6, 13].”
• Purely infinite simple Leavitt path algebras (2005, background): “furthermore, the edge ehas e∈ F1 j, where s(e) = v j. Thus we conclude that L(E) = ∪∞ i=1L(F i). (We note here that the embedding of graphs j: F i → Eis a complete graph homomorphism in the sense of [6], so that the conclusion L(E) = ∪∞ i=1L(F i) can also be achieved by invoking [6, Lemma 2.1].) Sin”
• Graded Rings and Graded Grothendieck Groups (2014, methodology): “1 K0 of Leavitt path algebras For a Leavitt path algebra LK(E), the monoid V(LK(E)) is studied in [6].”
• The graded Grothendieck group and the classification of Leavitt path algebras (2011, background): “For a Leavitt path algebra LK(E), the monoid V(LK(E)) is studied in [7].”
• Leavitt path algebras and direct limits (2007, background): “This is needed in our final result, extending a theorem of Ara-Moreno-Pardo [10], which says that V (LK(E)) is always an unperforated, separative refinement monoid.”
• Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions☆ (2012, background): “Their associated graph C*-algebras C(E,C) and Leavitt path algebras LK(E,C) (for an arbitrary field K), provide generalizations of the usual graph C*-algebras ([47]) and Leavitt path algebras ([2], [10]) associated to directed graphs, although these algebras behave quite differently from the usual g”
• Exchange Leavitt path algebras and stable rank (2005, background): “The proofs presented here significantly differ to those of the analytic setting of C∗-algebras and the arguments are necessarily different in the purely algebraic context since many of the tools used there are not available in our case.”
• Leavitt path algebras with coefficients in a commutative ring (2009, methodology): “3 uses different techniques than [3] or [26].”
• Graded Steinberg algebras and their representations (2017, methodology): “Ara, Moreno and Pardo [5] showed that for a Leavitt path algebra associated to the graph E, the monoid V(LK(E)) is entirely determined by elementary graph-theoretic data.”