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The Basics of Leavitt Path Algebras: Motivations, Definitions and Examples

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Leavitt Path Algebras

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2191))

Abstract

We introduce the central idea, that of a Leavitt path algebra. We start by describing the classical Leavitt algebras. We then proceed to give the definition of the Leavitt path algebra L K (E) for an arbitrary directed graph E and field K. After providing some basic examples, we show how Leavitt path algebras are related to the monoid realization algebras of Bergman, as well as to graph C ∗-algebras. We then introduce the more general construction of relative Cohn path algebras C K X(E), and show how these are related to Leavitt path algebras. We finish by describing how any Cohn (specifically, Leavitt) path algebra may be constructed as a direct limit of Cohn (specifically, Leavitt) path algebras corresponding to finite graphs. We conclude the chapter with an historical overview of the subject.

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References

  1. Gene Abrams. Leavitt path algebras: the first decade. Bull. Math. Sci., 5(1):59–120, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  2. Gene Abrams and Pham N. Ánh. Some ultramatricial algebras which arise as intersections of Leavitt algebras. J. Algebra Appl., 1(4):357–363, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  3. Gene Abrams, Pham N. Ánh, Adel Louly, and Enrique Pardo. The classification question for Leavitt path algebras. J. Algebra, 320(5):1983–2026, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  4. Gene Abrams, Pham N. Ánh, and Enrique Pardo. Isomorphisms between Leavitt algebras and their matrix rings. J. Reine Angew. Math., 624:103–132, 2008.

    MathSciNet  MATH  Google Scholar 

  5. Gene Abrams and Gonzalo Aranda Pino. The Leavitt path algebra of a graph. J. Algebra, 293(2):319–334, 2005.

    Google Scholar 

  6. Gene Abrams and Gonzalo Aranda Pino. Purely infinite simple Leavitt path algebras. J. Pure Appl. Algebra, 207(3):553–563, 2006.

    Google Scholar 

  7. Gene Abrams, Gonzalo Aranda Pino, Francesc Perera, and Mercedes Siles Molina. Chain conditions for Leavitt path algebras. Forum Math., 22(1):95–114, 2010.

    Google Scholar 

  8. Gene Abrams, Gonzalo Aranda Pino, and Mercedes Siles Molina. Locally finite Leavitt path algebras. Israel J. Math., 165:329–348, 2008.

    Google Scholar 

  9. Gene Abrams, Jason P. Bell, Pinar Colak, and Kulumani M. Rangaswamy. Two-sided chain conditions in Leavitt path algebras over arbitrary graphs. J. Algebra Appl., 11(3):1250044, 23, 2012.

    Google Scholar 

  10. Gene Abrams, Jason P. Bell, and Kulumani M. Rangaswamy. On prime nonprimitive von Neumann regular algebras. Trans. Amer. Math. Soc., 366(5):2375–2392, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  11. Gene Abrams, Adel Louly, Enrique Pardo, and Christopher Smith. Flow invariants in the classification of Leavitt path algebras. J. Algebra, 333:202–231, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  12. Gene Abrams and Kulumani M. Rangaswamy. Regularity conditions for arbitrary Leavitt path algebras. Algebr. Represent. Theory, 13(3):319–334, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  13. Gene Abrams and Mark Tomforde. Isomorphism and Morita equivalence of graph algebras. Trans. Amer. Math. Soc., 363(7):3733–3767, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  14. Gene Abrams and Mark Tomforde. A class of C ∗-algebras that are prime but not primitive. Münster J. Math., 7(2):489–514, 2014.

    MathSciNet  MATH  Google Scholar 

  15. Pere Ara. Extensions of exchange rings. J. Algebra, 197(2):409–423, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  16. Pere Ara. Morita equivalence for rings with involution. Algebr. Represent. Theory, 2(3):227–247, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  17. Pere Ara. The exchange property for purely infinite simple rings. Proc. Amer. Math. Soc., 132(9):2543–2547 (electronic), 2004.

    Google Scholar 

  18. Pere Ara. Rings without identity which are Morita equivalent to regular rings. Algebra Colloq., 11(4):533–540, 2004.

    MathSciNet  MATH  Google Scholar 

  19. Pere Ara. The realization problem for von Neumann regular rings. In Ring theory 2007, pages 21–37. World Sci. Publ., Hackensack, NJ, 2009.

    Google Scholar 

  20. Pere Ara and Miquel Brustenga. \(K_{1}\) of corner skew Laurent polynomial rings and applications. Comm. Algebra, 33(7):2231–2252, 2005.

    Google Scholar 

  21. Pere Ara and Miquel Brustenga. The regular algebra of a quiver. J. Algebra, 309(1):207–235, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  22. Pere Ara and Miquel Brustenga. Module theory over Leavitt path algebras and K-theory. J. Pure Appl. Algebra, 214(7):1131–1151, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  23. Pere Ara, Miquel Brustenga, and Guillermo Cortiñas. K-theory of Leavitt path algebras. Münster J. Math., 2:5–33, 2009.

    MathSciNet  MATH  Google Scholar 

  24. Pere Ara and Alberto Facchini. Direct sum decompositions of modules, almost trace ideals, and pullbacks of monoids. Forum Math., 18(3):365–389, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  25. Pere Ara, María A. González-Barroso, Kenneth R. Goodearl, and Enrique Pardo. Fractional skew monoid rings. J. Algebra, 278(1):104–126, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  26. Pere Ara and Kenneth R. Goodearl. Stable rank of corner rings. Proc. Amer. Math. Soc., 133(2):379–386 (electronic), 2005.

    Google Scholar 

  27. Pere Ara and Kenneth R. Goodearl. Leavitt path algebras of separated graphs. J. Reine Angew. Math., 669:165–224, 2012.

    MathSciNet  MATH  Google Scholar 

  28. Pere Ara, Kenneth R. Goodearl, Kevin C. O’Meara, and Enrique Pardo. Separative cancellation for projective modules over exchange rings. Israel J. Math., 105:105–137, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  29. Pere Ara, Kenneth R. Goodearl, and Enrique Pardo. \(K_{0}\) of purely infinite simple regular rings. K-Theory, 26(1):69–100, 2002.

    Google Scholar 

  30. Pere Ara and Martin Mathieu. Local multipliers of C ∗ -algebras. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2003.

    Book  MATH  Google Scholar 

  31. Pere Ara, María A. Moreno, and Enrique Pardo. Nonstable K-theory for graph algebras. Algebr. Represent. Theory, 10(2):157–178, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  32. Pere Ara and Enrique Pardo. Stable rank of Leavitt path algebras. Proc. Amer. Math. Soc., 136(7):2375–2386, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  33. Pere Ara and Enrique Pardo. Towards a K-theoretic characterization of graded isomorphisms between Leavitt path algebras. J. K-Theory, 14(2):203–245, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  34. Pere Ara and Enrique Pardo. Representing finitely generated refinement monoids as graph monoids. J. Algebra, 480:79–123, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  35. Pere Ara, Gert K. Pedersen, and Francesc Perera. An infinite analogue of rings with stable rank one. J. Algebra, 230(2):608–655, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  36. Pere Ara and Francesc Perera. Multipliers of von Neumann regular rings. Comm. Algebra, 28(7):3359–3385, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  37. Gonzalo Aranda Pino, John Clark, Astrid an Huef, and Iain Raeburn. Kumjian-Pask algebras of higher-rank graphs. Trans. Amer. Math. Soc., 365(7):3613–3641, 2013.

    Google Scholar 

  38. Gonzalo Aranda Pino and Kathi Crow. The center of a Leavitt path algebra. Rev. Mat. Iberoam., 27(2):621–644, 2011.

    Google Scholar 

  39. Gonzalo Aranda Pino, Kenneth R. Goodearl, Francesc Perera, and Mercedes Siles Molina. Non-simple purely infinite rings. Amer. J. Math., 132(3):563–610, 2010.

    Google Scholar 

  40. Gonzalo Aranda Pino, Enrique Pardo, and Mercedes Siles Molina. Exchange Leavitt path algebras and stable rank. J. Algebra, 305(2):912–936, 2006.

    Google Scholar 

  41. Gonzalo Aranda Pino, Enrique Pardo, and Mercedes Siles Molina. Prime spectrum and primitive Leavitt path algebras. Indiana Univ. Math. J., 58(2):869–890, 2009.

    Google Scholar 

  42. Gonzalo Aranda Pino, Kulumani Rangaswamy, and Lia Vaš. ∗-regular Leavitt path algebras of arbitrary graphs. Acta Math. Sin. (Engl. Ser.), 28(5):957–968, 2012.

    Google Scholar 

  43. Hyman Bass. K-theory and stable algebra. Inst. Hautes Études Sci. Publ. Math., 22:5–60, 1964.

    Article  MathSciNet  MATH  Google Scholar 

  44. Hyman Bass, Alex Heller, and Richard G. Swan. The Whitehead group of a polynomial extension. Inst. Hautes Études Sci. Publ. Math., 22:61–79, 1964.

    Article  MathSciNet  MATH  Google Scholar 

  45. Teresa Bates, Jeong Hee Hong, Iain Raeburn, and Wojciech Szymański. The ideal structure of the \(C^{{\ast}}\)-algebras of infinite graphs. Illinois J. Math., 46(4):1159–1176, 2002.

    Google Scholar 

  46. Teresa Bates and David Pask. Flow equivalence of graph algebras. Ergodic Theory Dynam. Systems, 24(2):367–382, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  47. Teresa Bates, David Pask, Iain Raeburn, and Wojciech Szymański. The \(C^{{\ast}}\)-algebras of row-finite graphs. New York J. Math., 6:307–324 (electronic), 2000.

    Google Scholar 

  48. Jason P. Bell and George M. Bergman. Personal communication, 2011.

    Google Scholar 

  49. Sterling K. Berberian. Baer *-rings. Springer-Verlag, New York, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 195.

    Google Scholar 

  50. George M. Bergman. On Jacobson radicals of graded rings. Unpublished. http://math.berkeley.edu/~gbergman/papers/unpub/JG.pdf, pages 1–10.

  51. George M. Bergman. Coproducts and some universal ring constructions. Trans. Amer. Math. Soc., 200:33–88, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  52. Bruce Blackadar. K-theory for operator algebras, volume 5 of Mathematical Sciences Research Institute Publications. Cambridge University Press, Cambridge, second edition, 1998.

    Google Scholar 

  53. Gary Brookfield. Cancellation in primely generated refinement monoids. Algebra Universalis, 46(3):343–371, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  54. Jonathan Brown, Lisa Orloff Clark, Cynthia Farthing, and Aidan Sims. Simplicity of algebras associated to étale groupoids. Semigroup Forum, 88(2):433–452, 2014.

    Google Scholar 

  55. Lawrence G. Brown. Homotopy of projection in C ∗-algebras of stable rank one. Astérisque, 232:115–120, 1995. Recent advances in operator algebras (Orléans, 1992).

    Google Scholar 

  56. Lawrence G. Brown and Gert K. Pedersen. C ∗-algebras of real rank zero. J. Funct. Anal., 99(1):131–149, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  57. Lawrence G. Brown and Gert K. Pedersen. Non-stable K-theory and extremally rich C ∗-algebras. J. Funct. Anal., 267(1):262–298, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  58. Nathan Brownlowe and Adam P. W. Sørensen. \(L_{2,\mathbb{Z}} \otimes L_{2,\mathbb{Z}}\) does not embed in \(L_{2,\mathbb{Z}}\). J. Algebra, 456:1–22, 2016.

    Google Scholar 

  59. Huanyin Chen. On separative refinement monoids. Bull. Korean Math. Soc., 46(3):489–498, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  60. Lisa Orloff Clark and Cain Edie-Michell. Uniqueness theorems for Steinberg algebras. Algebr. Represent. Theory, 18(4):907–916, 2015.

    Google Scholar 

  61. Lisa Orloff Clark, Cynthia Farthing, Aidan Sims, and Mark Tomforde. A groupoid generalisation of Leavitt path algebras. Semigroup Forum, 89(3):501–517, 2014.

    Google Scholar 

  62. Lisa Orloff Clark, Dolores Martin Barquero, Cándido Martin Gonzalez, and Mercedes Siles Molina. Using the Steinberg algebra model to determine the center of any Leavitt path algebra. To appear in Israel Journal of Mathematics. arXiv, 1604.01079:1–14, 2016.

    Google Scholar 

  63. Lisa Orloff Clark and Aidan Sims. Equivalent groupoids have Morita equivalent Steinberg algebras. J. Pure Appl. Algebra, 219(6):2062–2075, 2015.

    Google Scholar 

  64. P. M. Cohn. Some remarks on the invariant basis property. Topology, 5:215–228, 1966.

    Article  MathSciNet  MATH  Google Scholar 

  65. María G. Corrales García, Dolores Martín Barquero, Cándido Martín González, Mercedes Siles Molina, and José F. Solanilla Hernández. Extreme cycles. The center of a Leavitt path algebra. Publ. Mat., 60(1):235–263, 2016.

    Google Scholar 

  66. Guillermo Cortiñas and Eugenia Ellis. Isomorphism conjectures with proper coefficients. J. Pure Appl. Algebra, 218(7):1224–1263, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  67. Peter Crawley and Bjarni Jónsson. Refinements for infinite direct decompositions of algebraic systems. Pacific J. Math., 14:797–855, 1964.

    Article  MathSciNet  MATH  Google Scholar 

  68. Joachim Cuntz. Simple \(C^{{\ast}}\)-algebras generated by isometries. Comm. Math. Phys., 57(2):173–185, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  69. Joachim Cuntz. The structure of multiplication and addition in simple \(C^{{\ast}}\)-algebras. Math. Scand., 40(2):215–233, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  70. Joachim Cuntz. K-theory for certain \(C^{{\ast}}\)-algebras. Ann. of Math. (2), 113(1):181–197, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  71. Joachim Cuntz and Wolfgang Krieger. A class of \(C^{{\ast}}\)-algebras and topological Markov chains. Invent. Math., 56(3):251–268, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  72. Klaus Deicke, Jeong Hee Hong, and Wojciech Szymański. Stable rank of graph algebras. Type I graph algebras and their limits. Indiana Univ. Math. J., 52(4):963–979, 2003.

    Google Scholar 

  73. Jacques Dixmier. Sur les \(C^{{\ast}}\)-algèbres. Bull. Soc. Math. France, 88:95–112, 1960.

    Article  MathSciNet  MATH  Google Scholar 

  74. Jacques Dixmier. LesC ∗ -algèbres et leurs représentations. Deuxième édition. Cahiers Scientifiques, Fasc. XXIX. Gauthier-Villars Éditeur, Paris, 1969.

    Google Scholar 

  75. O.I. Domanov. A prime but not primitive regular ring. Uspehi Mat. Nauk, 32(6(198)):219–220, 1977.

    Google Scholar 

  76. Søren Eilers, Gunnar Restorff, Efren Ruiz, and Adam P.W. Sørensen. The complete classification of unital graph \(C^{{\ast}}\)-algebras: geometric and strong. arXiv, 1611.07120v1:1–73, 2016.

    Google Scholar 

  77. Ruy Exel. Partial Dynamical Systems, Fell Bundles and Applications, volume 224 of Mathematical Surveys and Monographs Series. American Mathematical Society, Providence, R.I., 2017.

    Google Scholar 

  78. Alberto Facchini and Franz Halter-Koch. Projective modules and divisor homomorphisms. J. Algebra Appl., 2(4):435–449, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  79. Carl Faith. Lectures on injective modules and quotient rings. Lecture Notes in Mathematics, No. 49. Springer-Verlag, Berlin-New York, 1967.

    Google Scholar 

  80. Antonio Fernández López, Eulalia García Rus, Miguel Gómez Lozano, and Mercedes Siles Molina. Goldie theorems for associative pairs. Comm. Algebra, 26(9):2987–3020, 1998.

    Google Scholar 

  81. John Franks. Flow equivalence of subshifts of finite type. Ergodic Theory Dynam. Systems, 4(1):53–66, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  82. James Gabe, Efren Ruiz, Mark Tomforde, and Tristan Whalen. K-theory for Leavitt path algebras: Computation and classification. J. Algebra, 433:35–72, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  83. Stephen M. Gersten. K-theory of free rings. Comm. Algebra, 1:39–64, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  84. Miguel Gómez Lozano and Mercedes Siles Molina. Quotient rings and Fountain-Gould left orders by the local approach. Acta Math. Hungar., 97(4):287–301, 2002.

    Google Scholar 

  85. Kenneth R. Goodearl. Notes on real and complexC ∗ -algebras, volume 5 of Shiva Mathematics Series. Shiva Publishing Ltd., Nantwich, 1982.

    Google Scholar 

  86. Kenneth R. Goodearl. von Neumann regular rings. Robert E. Krieger Publishing Co. Inc., Malabar, FL, second edition, 1991.

    Google Scholar 

  87. Kenneth R. Goodearl. Leavitt path algebras and direct limits. In Rings, Modules and Representations, volume 480 of Contemp. Math. American Mathematical Society, Providence, R.I., 2009.

    Google Scholar 

  88. Kenneth R. Goodearl and Robert B. Warfield, Jr. Algebras over zero-dimensional rings. Math. Ann., 223(2):157–168, 1976.

    Google Scholar 

  89. Frederick M. Goodman, Pierre de la Harpe, and Vaughan F. R. Jones. Coxeter graphs and towers of algebras, volume 14 of Mathematical Sciences Research Institute Publications. Springer-Verlag, New York, 1989.

    Google Scholar 

  90. David Handelman. Rings with involution as partially ordered abelian groups. Rocky Mountain J. Math., 11(3):337–381, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  91. Damon Hay, Marissa Loving, Martin Montgomery, Efren Ruiz, and Katherine Todd. Non-stable K-theory for Leavitt path algebras. Rocky Mountain J. Math., 44(6):1817–1850, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  92. Roozbeh Hazrat. The graded Grothendieck group and the classification of Leavitt path algebras. Math. Ann., 355(1):273–325, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  93. Roozbeh Hazrat. A note on the isomorphism conjectures for Leavitt path algebras. J. Algebra, 375:33–40, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  94. Roozbeh Hazrat. Graded rings and graded Grothendieck groups, volume 435 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2016.

    Book  Google Scholar 

  95. Richard H. Herman and Leonid N. Vaserstein. The stable range of C ∗-algebras. Invent. Math., 77(3):553–555, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  96. Danrun Huang. Flow equivalence of reducible shifts of finite type. Ergodic Theory Dynam. Systems, 14(4):695–720, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  97. Danrun Huang. Automorphisms of Bowen–Franks groups of shifts of finite type. Ergodic Theory Dynam. Systems, 21(4):1113–1137, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  98. Nathan Jacobson. Some remarks on one-sided inverses. Proc. Amer. Math. Soc., 1:352–355, 1950.

    Article  MathSciNet  MATH  Google Scholar 

  99. Nathan Jacobson. Structure of rings. American Mathematical Society, Colloquium Publications, vol. 37. American Mathematical Society, 190 Hope Street, Prov., R. I., 1956.

    Google Scholar 

  100. Ja A. Jeong and Gi Hyun Park. Graph C ∗-algebras with real rank zero. J. Funct. Anal., 188(1):216–226, 2002.

    Google Scholar 

  101. Richard V. Kadison and John R. Ringrose. Fundamentals of the theory of operator algebras. Vol. I, volume 15 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1997. Elementary theory, Reprint of the 1983 original.

    Google Scholar 

  102. Irving Kaplansky. Algebraic and analytic aspects of operator algebras. American Mathematical Society, Providence, R.I., 1970. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 1.

    Google Scholar 

  103. Eberhard Kirchberg. The classification of purely infinite C ∗-algebras using Kasparov theory. Unpublished, 3rd draft, pages 1–37, 1994.

    Google Scholar 

  104. Alex Kumjian and David Pask. Higher rank graph C ∗-algebras. New York J. Math., 6:1–20, 2000.

    MathSciNet  MATH  Google Scholar 

  105. Alex Kumjian, David Pask, and Iain Raeburn. Cuntz–Krieger algebras of directed graphs. Pacific J. Math., 184(1):161–174, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  106. Alex Kumjian, David Pask, Iain Raeburn, and Jean Renault. Graphs, groupoids, and Cuntz–Krieger algebras. J. Funct. Anal., 144(2):505–541, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  107. Tsit-Yen Lam. A first course in noncommutative rings, volume 131 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2001.

    Google Scholar 

  108. Tsit-Yuen Lam. Lectures on modules and rings, volume 189 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1999.

    Book  Google Scholar 

  109. Joachim Lambek. Lectures on rings and modules. Chelsea Publishing Co., New York, second edition, 1976.

    Google Scholar 

  110. Charles Lanski, Richard Resco, and Lance Small. On the primitivity of prime rings. J. Algebra, 59(2):395–398, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  111. Hossein Larki and Abdolhamid Riazi. Stable rank of Leavitt path algebras of arbitrary graphs. Bull. Aust. Math. Soc., 88(2):206–217, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  112. William G. Leavitt. The module type of a ring. Trans. Amer. Math. Soc., 103:113–130, 1962.

    Article  MathSciNet  MATH  Google Scholar 

  113. William G. Leavitt. The module type of homomorphic images. Duke Math. J., 32:305–311, 1965.

    Article  MathSciNet  MATH  Google Scholar 

  114. Douglas Lind and Brian Marcus. An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 1995.

    Book  MATH  Google Scholar 

  115. Martin Lorenz. On the homology of graded algebras. Comm. Algebra, 20(2):489–507, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  116. John C. McConnell and J. Chris Robson. Noncommutative Noetherian rings, volume 30 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, revised edition, 2001. With the cooperation of L.W. Small.

    Google Scholar 

  117. Pere Menal and Jaume Moncasi. Lifting units in self-injective rings and an index theory for Rickart C ∗-algebras. Pacific J. Math., 126(2):295–329, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  118. Gerard J. Murphy. C ∗ -algebras and operator theory. Academic Press, Inc., Boston, MA, 1990.

    MATH  Google Scholar 

  119. Constantin Năstăsescu and Freddy van Oystaeyen. Graded ring theory, volume 28 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1982.

    Google Scholar 

  120. Constantin Năstăsescu and Freddy Van Oystaeyen. Methods of graded rings, volume 1836 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2004.

    Google Scholar 

  121. W. Keith Nicholson. Lifting idempotents and exchange rings. Trans. Amer. Math. Soc., 229:269–278, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  122. W.K. Nicholson. I-rings. Trans. Amer. Math. Soc., 207:361–373, 1975.

    MathSciNet  MATH  Google Scholar 

  123. Narutaka Ozawa. About the Connes embedding conjecture: algebraic approaches. Jpn. J. Math., 8(1):147–183, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  124. Andrei V. Pajitnov and Andrew A. Ranicki. The Whitehead group of the Novikov ring. K-Theory, 21(4):325–365, 2000. Special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part V.

    Google Scholar 

  125. Enrique Pardo. The isomorphism problem for Higman–Thompson groups. J. Algebra, 344:172–183, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  126. Bill Parry and Dennis Sullivan. A topological invariant of flows on 1-dimensional spaces. Topology, 14(4):297–299, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  127. William L. Paschke and Norberto Salinas. Matrix algebras over \(\mathscr{O}_{n}\). Michigan Math. J., 26(1):3–12, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  128. N. Christopher Phillips. A classification theorem for nuclear purely infinite simple C ∗-algebras. Doc. Math., 5:49–114 (electronic), 2000.

    Google Scholar 

  129. Iain Raeburn. Graph algebras, volume 103 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2005.

    Google Scholar 

  130. Iain Raeburn and Wojciech Szymański. Cuntz–Krieger algebras of infinite graphs and matrices. Trans. Amer. Math. Soc., 356(1):39–59 (electronic), 2004.

    Google Scholar 

  131. Kulumani M. Rangaswamy. The theory of prime ideals of Leavitt path algebras over arbitrary graphs. J. Algebra, 375:73–96, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  132. Kulumani M. Rangaswamy. On generators of two-sided ideals of Leavitt path algebras over arbitrary graphs. Comm. Algebra, 42(7):2859–2868, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  133. Marc A. Rieffel. Dimension and stable rank in the K-theory of \(C^{{\ast}}\)-algebras. Proc. London Math. Soc. (3), 46(2):301–333, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  134. Mikael Rørdam. A short proof of Elliott’s theorem: \(\mathscr{O}_{2} \otimes \mathscr{O}_{2}\cong \mathscr{O}_{2}\). C. R. Math. Rep. Acad. Sci. Canada, 16(1):31–36, 1994.

    MathSciNet  Google Scholar 

  135. Mikael Rørdam. Classification of Cuntz–Krieger algebras. K-Theory, 9(1):31–58, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  136. Mikael Rørdam, Flemming Larsen, and Niels Laustsen. An introduction toK-theory forC ∗ -algebras, volume 49 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2000.

    Google Scholar 

  137. Joseph J. Rotman. An introduction to homological algebra. Universitext. Springer, New York, second edition, 2009.

    Google Scholar 

  138. Efren Ruiz and Mark Tomforde. Classification of unital simple Leavitt path algebras of infinite graphs. J. Algebra, 384:45–83, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  139. Konrad Schmüdgen. Unbounded operator algebras and representation theory, volume 37 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1990.

    Book  Google Scholar 

  140. Konrad Schmüdgen. Noncommutative real algebraic geometry—some basic concepts and first ideas. In Emerging applications of algebraic geometry, volume 149 of IMA Vol. Math. Appl., pages 325–350. Springer, New York, 2009.

    Google Scholar 

  141. Laurence C. Siebenmann. A total Whitehead torsion obstruction to fibering over the circle. Comment. Math. Helv., 45:1–48, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  142. S. Paul Smith. Category equivalences involving graded modules over path algebras of quivers. Adv. Math., 230(4–6):1780–1810, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  143. S. Paul Smith. The space of Penrose tilings and the noncommutative curve with homogeneous coordinate ring \(k\langle x,y\rangle /(y^{2})\). J. Noncommut. Geom., 8(2):541–586, 2014.

    Google Scholar 

  144. Benjamin Steinberg. A groupoid approach to discrete inverse semigroup algebras. Adv. Math., 223(2):689–727, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  145. Benjamin Steinberg. Simplicity, primitivity and semiprimitivity of étale groupoid algebras with applications to inverse semigroup algebras. J. Pure Appl. Algebra, 220(3):1035–1054, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  146. Mark Tomforde. Extensions of graph C*-algebras. ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)–Dartmouth College.

    Google Scholar 

  147. Mark Tomforde. Uniqueness theorems and ideal structure for Leavitt path algebras. J. Algebra, 318(1):270–299, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  148. Mark Tomforde. Leavitt path algebras with coefficients in a commutative ring. J. Pure Appl. Algebra, 215(4):471–484, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  149. Leonid N. Vaseršteĭn. On the stabilization of the general linear group over a ring. Math. USSR-Sb., 8:383–400, 1969.

    Article  MathSciNet  MATH  Google Scholar 

  150. Leonid N. Vaseršteĭn. The stable range of rings and the dimension of topological spaces. Funkcional. Anal. i Priložen., 5(2):17–27, 1971.

    MathSciNet  Google Scholar 

  151. Leonid N. Vaseršteĭn. Bass’s first stable range condition. In Proceedings of the Luminy conference on algebraicK-theory (Luminy, 1983), J. Pure Appl. Algebra, volume 34, pages 319–330, 1984.

    Google Scholar 

  152. Ivan Vidav. On some ∗-regular rings. Acad. Serbe Sci. Publ. Inst. Math., 13:73–80, 1959.

    MathSciNet  MATH  Google Scholar 

  153. R.B. Warfield, Jr. Cancellation of modules and groups and stable range of endomorphism rings. Pacific J. Math., 91(2):457–485, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  154. Robert B. Warfield, Jr. Exchange rings and decompositions of modules. Math. Ann., 199:31–36, 1972.

    Article  MathSciNet  Google Scholar 

  155. Yasuo Watatani. Graph theory for \(C^{{\ast}}\)-algebras. In Operator algebras and applications, Part I (Kingston, Ont., 1980), volume 38 of Proc. Sympos. Pure Math., pages 195–197. Amer. Math. Soc., Providence, R.I., 1982.

    Google Scholar 

  156. Friedrich Wehrung. Various remarks on separativity and stable rank in refinement monoids. Unpublished manuscript.

    Google Scholar 

  157. Friedrich Wehrung. The dimension monoid of a lattice. Algebra Universalis, 40(3):247–411, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  158. Charles A. Weibel. An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994.

    Book  Google Scholar 

  159. Charles A. Weibel. TheK-book, volume 145 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2013. An introduction to algebraic K-theory.

    Google Scholar 

  160. Robert F. Williams. Classification of subshifts of finite type. Ann. of Math. (2), 98:120–153; errata, ibid. (2) 99 (1974), 380–381, 1973.

    Google Scholar 

  161. Mariusz Wodzicki. Excision in cyclic homology and in rational algebraic K-theory. Ann. of Math. (2), 129(3):591–639, 1989.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Colorado, Colorado Springs, USA

    Gene Abrams

  2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Barcelona, Spain

    Pere Ara

  3. Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, Málaga, Spain

    Mercedes Siles Molina

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Abrams, G., Ara, P., Siles Molina, M. (2017). The Basics of Leavitt Path Algebras: Motivations, Definitions and Examples. In: Leavitt Path Algebras. Lecture Notes in Mathematics, vol 2191. Springer, London. https://doi.org/10.1007/978-1-4471-7344-1_1

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