Uniqueness theorems and ideal structure for Leavitt path algebras

By M. Tomforde. 2006

184 citations recorded by the source corpus.

Abstract We prove Leavitt path algebra versions of the two uniqueness theorems of graph C ∗ -algebras. We use these uniqueness theorems to analyze the ideal structure of Leavitt path algebras and give necessary and sufficient conditions for their simplicity. We also use these results to give a proof of the fact that for any graph E the Leavitt path algebra L C ( E ) embeds as a dense ∗-subalgebra of the graph C ∗ -algebra C ∗ ( E ) . This embedding has consequences for graph C ∗ -algebras, and we discuss how we obtain new information concerning the construction of C ∗ ( E ) .

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/08632e0b1d8499ff05e25f44b3b910befe424578

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In Algebra and Number Theory

1. G. Abrams
2. Roozbeh Hazrat
3. Aidan Sims
4. M. Tomforde
5. P. Ara
6. G. Pino
7. Mercedes Siles Molina
8. Xiao-Wu Chen

In Mathematical Physics

1. G. Abrams
2. Aidan Sims
3. Roozbeh Hazrat
4. M. Tomforde
5. P. Ara
6. G. Pino
7. Mercedes Siles Molina
8. Xiao-Wu Chen

Citations in the corpus

• Graded Steinberg algebras and their representations (2017, methodology): “Tomforde [2007, Theorem 5.7] used the admissible pairs ( H , S ) of vertices to parametrise the graded ideals of L K ( E ) for a graph E which is not row-finite.”
• Isomorphism and Morita equivalence of graph algebras (2008, methodology): “We make this precise and give a proof of the injectivity of ιE in the following proposition. Proposition 3.1 ([ 40, Theorem 7.3 ]).”
• Graph inverse semigroups: their characterization and completion (2011, background): “Although Paterson makes an explicit connection in [41, 42], they are not mentioned in [45] or [49].”
• Leavitt Path Algebras as Partial Skew Group Rings (2012, methodology): “To further illustrate how the interaction between the theories of Leavitt path algebras and partial skew group rings can be fruitful, in Section 4 we provide new proofs of the Cuntz–Krieger uniqueness theorem and the simplicity criteria for Leavitt Path Algebras (originally proved in [1, 21]).”
• Irreducible representations of Leavitt path algebras (2011, background): “The references for quivers are [11, Chapter III] and [1], and for Leavitt path algebras are [1, 2, 8, 26].”
• Socle theory for Leavitt path algebras of arbitrary graphs (2008, methodology): “1 we can see how i t reduces considerably in length the proofs given in [30] of the so-called Uniqueness T heorems.”
• A groupoid generalisation of Leavitt path algebras (2014, background): “Moreover, the graded uniqueness theorem for Leavitt path algebras implies that the C∗-algebra of a directed graph is a norm completion of its Leavitt path algebra [4], [25].”
• Analogs of Cuntz algebras on Lp spaces (2012, background): “4 of [25], where explicit formulas, valid for any graph algebra, are given, and Remark 1.”
• Leavitt path algebras and direct limits (2007, background): “A number of key results about the ideal theory of Leavitt path algebras of countable graphs are established in [24] and [3].”
• Kumjian-Pask algebras of higher-rank graphs (2011, methodology): “The first versions for Leavitt algebras were in [1, 29, 37].”
• Leavitt path algebras: the first decade (2014, background): “It is established in [97] that the expected map from LC(E) to C (E) is in fact injective.”
• Leavitt path algebras with coefficients in a commutative ring (2009, methodology): “3 uses different techniques than [3] or [26].”
• Chain conditions for Leavitt path algebras (2009, methodology): “[16] or [22]), and we have the decomposition L(E) = ⊕ v∈E0 L(E)v as left L(E)-modules.”
• Leavitt path algebras of separated graphs (2010, background): “Leavitt path algebras LK(E), natural algebraic versions of graph C -algebras, were introduced and studied firstly in [1] and [9] for row-finite graphs, and in [2] and [39] for general graphs; see also [4], [5], [10], [11], [26].”