Leavitt Path Algebras by Gene Abrams Pere Ara & Mercedes Siles Molina

Author:Gene Abrams, Pere Ara & Mercedes Siles Molina
Language: eng
Format: epub
Publisher: Springer London, London

We note that although I lce is an essential ideal of L K (E) when E 0 is finite, I lce need not equal all of L K (E). We see this behavior in L K (E T ), where E T is the Toeplitz graph as discussed in Example 3.7.3. Here we have P ec (E T ) = ∅ = P c (E T ), and P l (E T ) is the sink v. So I lce (E T ) = I({v}); but I({v}) ≠ L K (E T ), since {v} is hereditary saturated.

3.8 Purely Infinite Without Simplicity

We conclude Chap. 3 by presenting a description of the purely infinite (but not necessarily simple) Leavitt path algebras arising from row-finite graphs. As happened in the purely infinite simple case (Sect. 3.1), an in-depth analysis of the idempotent structure of L K (E) will be required. Roughly speaking, the first half of this section (through Lemma 3.8.10) will be a discussion of the purely infinite notion for general rings, while the second half will be taken up in considering this notion in the specific context of Leavitt path algebras. Many of the fundamental ideas in this section can be found in the seminal paper [39].

The general theory of purely infinite rings works smoothly for s-unital rings, defined here.

Definition 3.8.1

A ring R is said to be s-unital if for each a ∈ R there exists a b ∈ R such that a = ab = ba. By Ara [18, Lemma 2.2], if R is s-unital then for each finite subset F ⊆ R there is an element u ∈ R such that ux = x = xu for all x ∈ F.

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