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Leavitt Path Algebras Gene Abrams, Pere Ara, Mercedes Siles Molina Leavitt path algebras June 14, 2016 Springer vi Preface The great challenge in writing a book about a topic of ongoing mathematical research interest lies in determining who and what. Who are the readers for whom the book is intended? What pieces of the research should be included? The topic of Leavitt path algebras presents both of these challenges, in the extreme. Indeed, much of the beauty inherent in this topic stems from the fact that it may be approached from many different directions, and on many different levels. The topic encompasses classical ring theory at its finest. While at first glance these Leavitt path algebras may seem somewhat exotic, in fact many standard, well-understood algebras arise in this context: matrix rings and Laurent polynomial rings, to name just two. Many of the fundamental, classical ring-theoretic concepts have been and continue to be explored here, including the ideal structure, Z-grading, and structure of finitely generated projective modules, to name just a few. The topic continues a long tradition of associating an algebra with an appropriate combinatorial structure (here, a directed graph), the subsequent goal being to establish relationships between the algebra and the associated structures. In this particular setting, the topic allows for (and is enhanced by) visual, pictorial representation via directed graphs. Many readers are no doubt familiar with the by-now classical way of associating an algebra over a field with a directed graph, the standard path algebra. The construction of the Leavitt path algebra provides another such connection. The path algebra and Leavitt path algebra constructions are indeed related, via algebras of quotients. However, one may understand Leavitt path algebras without any prior knowledge of the path algebra construction. The topic has significant, deep connections with other branches of mathematics. For instance, many of the initial results in Leavitt path algebras were guided and motivated by results previously known about their analytic cousins, the graph C*-algebras. The study of Leavitt path algebras quickly matured to ado- lescence (when it became clear that the algebraic results are not implied by the C* results), and almost immediately thereafter to adulthood (when in fact some C* results, including some new C* results, were shown to follow from the algebraic results). Indeed, a number of longstanding questions in algebra have recently been resolved using Leavitt path algebras as a tool, thus further establishing the maturity of the subject. The topic continues a deep tradition evident in many branches of mathematics in which K-theory plays an important role. Indeed, in retrospect, one can view Leavitt path algebras as precisely those algebras constructed to produce specified K-theoretic data in a universal way, data arising naturally from directed graphs. Much of the current work in the field is focused on better understanding just how large a role the K-theoretic data plays in determining the structure of these algebras. Our goal in writing this book, the Why? of this book, simultaneously addresses both the Who? and What? questions. We provide here a self-contained presentation of the topic of Leavitt path algebras, a presentation which will allow readers having different backgrounds and different topical interests to understand and appreciate these structures. In particular, graduate students having only a first year course in ring theory should find most of the material in this book quite accessible. Similarly, researchers who don’t self-identify as algebraists (e.g., people working in C*-algebras or symbolic dynamics) will be able to understand how these Leavitt path algebras stem from, or apply to, their own research interests. While most of the results contained here have appeared elsewhere in the literature, a few of the central results appear here for the first time. The style will be relatively informal. We will often provide historical motivation and overview, both to increase the reader’s understanding of the subject and to play up the connections with other areas of mathematics. Although space considerations clearly require us to eliminate some otherwise interesting and important topics from inclusion, we provide an extensive bibliography for those readers who seek additional information about various topics which arise herein. More candidly, our real Why? for writing this book is to share what we know about Leavitt path algebras in such a way that others might become prepared, and subsequently inspired, to join in the game. GA PA MSM Contents 1 The basics of Leavitt path algebras: motivations, definitions and examples ............... 1 1.1 A motivating construction: the Leavitt algebras.....................................2 1.2 Leavitt path algebras...........................................................4 1.3 The three fundamental examples of Leavitt path algebras............................7 1.4 Connections and motivations: the algebras of Bergman, and graph C*-algebras..........9 1.5 The Cohn path algebras and connections to Leavitt path algebras...................... 10 1.6 Direct limits in the context of Leavitt path algebras................................. 16 1.7 A brief restrospective on the history of Leavitt path algebras.......................... 21 2 Two-sided ideals .................................................................. 23 2.1 The Z-grading................................................................ 25 2.2 The Reduction Theorem and the Uniqueness Theorems.............................. 32 2.3 Additional consequences of the Reduction Theorem................................ 36 2.4 Graded ideals: basic properties and quotient graphs................................. 38 2.5 The Structure Theorem for Graded Ideals, and the internal structure of graded ideals..... 44 2.6 The socle.................................................................... 52 2.7 The ideal generated by the vertices in cycles without exits........................... 57 2.8 The Structure Theorem for Ideals, and the internal structure of ideals.................. 61 2.9 Additional consequences of the Structure Theorem for Ideals. The Simplicity Theorem... 66 3 Idempotents, and finitely generated projective modules ................................ 71 3.1 Purely infinite simplicity, and the Dichotomy Principle.............................. 72 3.2 Finitely generated projective modules: the V -monoid............................... 74 3.3 The exchange property......................................................... 79 3.4 Von Neumann regularity........................................................ 83 3.5 Primitive non-minimal idempotents.............................................. 86 3.6 Structural properties of the V -monoid............................................ 88 3.7 Extreme cycles................................................................ 94 3.8 Purely infinite without simplicity................................................ 96 4 General ring-theoretic results....................................................... 105 4.1 Prime and primitive ideals in Leavitt path algebras of row-finite graphs................ 105 4.2 Chain conditions on one-sided ideals............................................. 110 4.3 Self-injectivity................................................................ 117 4.4 The center.................................................................... 121 4.5 The stable rank............................................................... 133 4.6 Historical remarks............................................................. 141 ix x Contents ∗ 5 Graph C*-algebras, and the relationship between LC(E) and C (E) . 143 5.1 Brief overview of C∗-algebras................................................... 143 5.2 Embedding Leavitt path algebras in graph C∗-algebras.............................. 146 5.3 Projections in graph C∗-algebras................................................. 151 5.4 Closed ideals of C∗(E) containing no vertices...................................... 154 5.5 Structure of closed ideals of C∗(E) for row-finite graphs............................. 158 5.6 Stable rank of graph C∗-algebras................................................. 159 5.7 Non-parallel results............................................................ 160 5.8 From Leavitt path algebras to graph C∗-algebras.................................... 160 6 K-theory ......................................................................... 161 6.1 The Grothendieck group K0(LK(E)) ............................................. 161 6.2 The Whitehead group K1(LK(E)) ................................................ 166 6.3 The Algebraic Kirchberg Phillips Question........................................ 169 6.4 The Isomorphism Conjecture, and the Morita Equivalence Conjecture................. 181 6.5 Tensor products and Hochschild homology........................................ 189 7 Generalizations, applications, and current lines of research ............................ 197 7.1 Generalizations of Leavitt path algebras........................................... 197 7.2 Applications of Leavitt path algebras............................................. 199 7.3 Current lines of research in Leavitt path algebras................................... 202 References........................................................................ 207 Index ...............................................................................
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