aa r X i v : . [ m a t h . R A ] O c t LEAVITT PATH ALGEBRAS: THE FIRST DECADE
GENE ABRAMS
Abstract.
The algebraic structures known as
Leavitt path algebras were initially developed in 2004by Ara, Moreno and Pardo, and almost simultaneously (using a different approach) by the author andAranda Pino. During the intervening decade, these algebras have attracted significant interest andattention, not only from ring theorists, but from analysts working in C ∗ -algebras, group theorists,and symbolic dynamicists as well. The goal of this article is threefold: to introduce the notion ofLeavitt path algebras to the general mathematical community; to present some of the importantresults in the subject; and to describe some of the field’s currently unresolved questions. Keywords:Leavitt path algebra; graph C ∗ -algebra Our goal in writing this article is threefold: first, to provide a history and overall viewpoint of theideas which comprise the subject of Leavitt path algebras; second, to give the reader a general senseof the results which have been achieved in the field; and finally, to give a broad picture of some of theresearch lines which are currently being pursued. The history and overall viewpoint (Section 1) arepresented with a completely general mathematical audience in mind; the writing style here will bemore chatty than formal. Our description of the results in the field has been split into two pieces: wedescribe Leavitt path algebras of row-finite graphs (Sections 2, 3, and 4), and subsequently discussvarious generalizations of these (Section 5). Our intent and hope in ordering the presentation thisway is to allow the non-expert to appreciate the key ideas of the subject, without getting ensnarledin the at-first-glance formidable constructs which drive the generalizations. We close with Section6, in which we describe some of the current lines of investigation in the subject. In part, our hopehere is to attract mathematicians from a wide variety of fields to join in the research effort.The exhilarating increase in the level of interest in Leavitt path algebras during the first decadesince their introduction has resulted in the publication of scores of articles on these and relatedstructures. Certainly it is not the goal of the current article to review the entirety of the literaturein the subject. Rather, we have tried to strike a balance between presenting enough information tomake clear the beauty and diversity of the subject on the one hand, while avoiding “informationoverload” on the other. Apologies are issued in advance to those authors whose work in the fieldhas consequently not been included herein.In keeping with our goal of making this article accessible to a broad audience, we will offer either acomplete, formal
Proof , or an intuitive, informal
Sketch of Proof , only for specific results for whichsuch proofs are particularly illuminating. In other situations we will simply present statementswithout proof. Appropriate references are provided for all key results.
Acknowledgments : This work was partially supported by Simons Foundation Collaboration Grant History and overview
The fundamental examples of rings that are encountered during one’s algebraic pubescence (e.g.,fields K , Z , K [ x ], K [ x, x − ], M n ( K )) all have the following property. Definition 1.1.
The unital ring R has the Invariant Basis Number (IBN) property in case, foreach pair i, j ∈ N , if the left R -modules R i and R j are isomorphic, then i = j . A wide class of rings can easily be shown to have the IBN property, including rings possessingany sort of reasonable chain condition on one-sided ideals, as well as commutative rings. But thereare naturally occurring examples of algebras which are not IBN.Let V be a countably infinite dimensional vector space over a field K , and let R denote End K ( V ),the algebra of linear transformations from V to itself. It is not hard to see that R i ∼ = R j as left(or right) R -modules for any pair i, j ∈ N , as follows. One starts by viewing R as the K -algebraRFM N ( K ) consisting of those N × N matrices M having the property that each row of M contains atmost finitely many nonzero entries. (In this context we view transformations as acting on the right,and define composition of transformations by setting f ◦ g to mean “first f , then g ”. Of course,depending on the reader’s tastes, the same analysis can be performed by considering the analogousalgebra CFM N ( K ) of column-finite matrices.) Then a left-module isomorphism R R → R R is easyto establish, by considering the map that associates with any row-finite matrix M the pair of row-finite matrices ( M , M ), where M is built from the odd-indexed columns of M , and M fromthe even-indexed columns. Once such an isomorphism R R ∼ = R R is guaranteed, then by using theobvious generalization of the observation R R ∼ = R R ⊕ R R ∼ = R R ⊕ R R ∼ = R R, we see that R R i ∼ = R R j for all i, j ∈ N .The following is easy to prove. Lemma 1.2.
Let R be a unital ring, and let n ∈ N . Then R R ∼ = R R n as left R -modules if and onlyif there exist elements x , . . . , x n , y , . . . , y n of R for which ( † ) y i x j = δ i,j , and n X i =1 x i y i = 1 . In effect, the ring R = End K ( V ) lies on the complete opposite end of the spectrum from theIBN property, in that every pair of finitely generated free left R -modules are isomorphic; such aring is said to have the Single Basis Number (SBN) property. The question posed (and answeredcompletely) by William G. Leavitt in the early 1960’s regards the existence of a middle groundbetween IBN and SBN: do there exist rings for which R R i ∼ = R R j for some, but not all, pairs i, j ∈ N ? If we assume an isomorphism exists between R R i and R R j for some pair i = j , thenclearly by appending k copies of R to this isomorphism we get that R R i + k ∼ = R R j + k . With this ideain mind, and using only basic properties of the semigroup N × N , it is easy to prove the following. Lemma 1.3.
Let R be a unital ring. Assume R is not IBN, i.e., that there exist i = j ∈ N with R R i ∼ = R R j . Let m be the least integer for which R R m ∼ = R R j for some j = m . For this m , let n denote be the least integer for which R R m ∼ = R R n and n > m . Let k denote n − m ∈ N . Then forany pair i, j ∈ N , R R i ∼ = R R j ⇐⇒ i, j ≥ m and i ≡ j ( modk ) . EAVITT PATH ALGEBRAS: THE FIRST DECADE 3
We call the pair ( m, n ) the module type of R . (We caution that some authors, including Leavitt,instead use the phrase module type to denote the pair ( m, k ).) In particular, the ring R = End K ( V )has module type (1 , can -happen-actually- does -happen” result. Theorem 1.4. ( Leavitt’s Theorem ) [76, Theorem 8]
Let m, n ∈ N with n > m , and let K beany field. Then there exists a K -algebra L K ( m, n ) having module type ( m, n ) . Additionally:(1) L K ( m, n ) is universal, in the sense that if S is any K -algebra having module type ( m, n ) , thenthere exists a nonzero K -algebra homomorphism ϕ : L K ( m, n ) → S .(2) L K ( m, n ) is simple (i.e., has no nontrivial two-sided ideals) if and only if m = 1 . (Thiswas shown in [77, Theorems 2 and 3] .) In this case, for each = x ∈ L K (1 , n ) , there exists a, b ∈ L K (1 , n ) for which axb = 1 .(3) L K ( m, n ) is explicitly described in terms of generators and relations. We will refer to L K ( m, n ) as the Leavitt algebra of type ( m, n ).Since the ring R = RFM N ( K ) ∼ = End K ( V ) has module type (1 , R which satisfy the appropriate relations; for clarity, we note thatone such set is given by Y = ( δ i, i − ) , Y = ( δ i, i ) , X = ( δ i − ,i ) , and X = ( δ i,i ) . Moreover, the subalgebra of RFM N ( K ) generated by these four matrices is isomorphic to L K (1 , m = 1, the explicit description of the algebra L K (1 , n ) given in [76]yields that L K (1 , n ) is the free associative K -algebra K h x , . . . , x n , y , . . . , y n i modulo the relationsgiven in ( † ). So, with Lemma 1.2 in mind, we may view L K (1 , n ) as essentially the “smallest”algebra of type (1 , n ).In the following subsection we will rediscover the algebras L K (1 , n ) from a different point of view.We begin this subsection with a well-known idea, perhaps cast more formally than is typical.Let P be a finitely generated projective left R -module. (The notions of “finitely generated” and“projective” are categorical, and make sense even in case R is nonunital.) Definition 1.5.
For a ring R , let V ( R ) denote the set of isomorphism classes of finitely generatedprojective left R -modules, and define the obvious binary operation ⊕ on V ( R ) by setting [ P ] ⊕ [ Q ] =[ P ⊕ Q ] . Then ( V ( R ) , ⊕ ) is easily seen to be a commutative monoid (with neutral element [0] ). For any idempotent e ∈ R , [ Re ] ∈ V ( R ); indeed, elements of V ( R ) of this form will play a centralrole in the subject. If R is a division ring, then V ( R ) ∼ = Z + ; the same is true for R = Z , as well asfor various additional classes of rings. The wide range of monoids which can arise as V ( R ) will bedemonstrated in Theorem 1.6. For an arbitrary ring R , it’s fair to say that an explicit descriptionof V ( R ) is typically hard to come by. The well-studied Grothendieck group K ( R ) of R is preciselythe universal abelian group corresponding to the commutative monoid V ( R ).When R is unital then [ R ] ∈ V ( R ). Key information about R may be provided by the pair( V ( R ) , [ R ]). If R and S are isomorphic as rings then there exists an isomorphism of monoids ϕ : V ( R ) → V ( S ) for which ϕ ([ R ]) = [ S ]. (In this situation we write ( V ( R ) , [ R ]) ∼ = ( V ( S ) , [ S ]).) Moregenerally, if R and S are Morita equivalent (denoted R ∼ M S ), then there exists an isomorphism of GENE ABRAMS monoids ψ : V ( R ) → V ( S ). However, such an induced isomorphism ψ need not have the property ψ ([ R ]) = [ S ]. For instance, if S = M ( K ) for a field K , then S ∼ M K , and V ( S ) ∼ = V ( K ) ∼ = Z + .But ( V ( K ) , [ K ]) = ( Z + , V ( S ) , [ S ]) = ( Z + , Z + can take 1 to 2.In addition to the commutativity of ( V ( R ) , ⊕ ), the monoid V ( R ) has the following two easy-to-seeproperties. First, V ( R ) is conical : if x, y ∈ V ( R ) have x ⊕ y = 0, then x = y = 0. Second (for R unital), V ( R ) contains a distinguished element d : for each x ∈ V ( R ), there exists y ∈ V ( R ) and n ∈ N having x ⊕ y = nd (specifically, d = [ R ]). In 1974, George Bergman established the followingremarkable result. Theorem 1.6. ( Bergman’s Theorem ) ([49, Theorem 6.2]) Let M be a finitely generated com-mutative conical monoid with distinguished element d = 0, and let K be any field. Then thereexists a K -algebra B = B ( M, d ) for which ( V ( B ) , [ B ]) ∼ = ( M, d ) . Additionally:1) B = B ( M, d ) is universal, in the sense that if C is any unital K -algebra for which there exists amonoid homomorphism ψ : V ( B ) → V ( C ) having ψ ([ B ]) = [ C ], then there exists a (not necessarilyunique) K -algebra homomorphism Ψ : B → C for which the induced map Ψ : C ⊗ B : V ( B ) →V ( C ) is precisely ψ .2) B = B ( M, d ) is left and right hereditary (i.e., every left ideal and every right ideal of B isprojective).3) The construction of B ( M, d ) depends on the specific representation of M as F / hRi , where F is a finitely generated free abelian monoid, and R is a given (finite) set of relations in F . With F and R viewed as starting data, the algebra B = B ( M, d ) = B ( F / hRi , d ) is constructed explicitlyvia a finite sequence of steps, where each step consists of adjoining elements satisfying explicitlyspecified relations (provided by R ) to an explicitly described algebra.We will refer to B = B ( M, d ) = B ( F / hRi , d ) as the Bergman algebra of ( F / hRi , d ). Example 1.7. Three important examples. (1) Perhaps not surprisingly, when (
M, d ) = ( Z + ,
1) = ( Z + / h∅i , B ( Z + / h∅i ,
1) = K . (Inthis situation we view Z + as the free abelian monoid on one generator.)(2) Somewhat more subtly, we consider the same pair ( Z + , Z + as Z + / h i . Then B ( Z + / h i ,
1) = K [ x, x − ], the Laurent polynomial algebra withcoefficients in K .(3) Let n ∈ N , n ≥
2. Let V n denote the free abelian monoid having a single generator x ,subject to the relation nx = x . So V n = { , x, x, . . . , ( n − x } , | M n | = n , and ( M n , x ) clearlysatisfies the hypotheses of Bergman’s Theorem. (In [49], the semigroup V n is denoted V ,n .) In thissituation, Bergman’s explicit construction yields that B ( V n , x ) = K h x , . . . , x n , y , . . . , y n i , withrelations given by exactly the same defining relations as given in ( † ) above, namely, y i x j = δ i,j , and n X i =1 x i y i = 1 . Consequently, as observed in [49, Theorem 6.1], the Bergman algebra B ( V n , x ) is precisely theLeavitt algebra L K (1 , n ). EAVITT PATH ALGEBRAS: THE FIRST DECADE 5
Because of the central role they play in both the genesis and the ongoing development Leavittpath algebras, no history of the subject would be complete without a discussion of graph C ∗ -algebras .We present here only the most basic description of these algebras, just enough so that even thereader who is completely unfamiliar with them can get a sense of their connection to Leavitt pathalgebras.Throughout this subsection all algebras are assumed to be unital algebras over the complexnumbers C (but most of these ideas can be cast significantly more generally). The algebra A is a ∗ -algebra in case there is a map ∗ : A → A which has: ( x + y ) ∗ = x ∗ + y ∗ ; ( xy ) ∗ = y ∗ x ∗ ; 1 ∗ = 1;( x ∗ ) ∗ = x ; and ( α · x ) ∗ = αx ∗ for all x, y ∈ A and α ∈ C , where α denotes the complex conjugate of α . Standard examples of ∗ -algebras include matrix rings M n ( C ) (where ∗ is ‘conjugate transpose’),and the ring C ( T ) of continuous functions from the unit circle T = { z ∈ C | | z | = 1 } to C (where ∗ is defined by setting f ∗ ( z ) = f ( z ) for z ∈ T ).A C ∗ - norm on a ∗ -algebra A is a function k · k : A → R + for which: k ab k ≤ k a k·k b k ; k a + b k ≤ k a k + k b k ; k aa ∗ k = k a k = k a ∗ k ; k a k = 0 ⇔ a = 0; and k λa k = | λ |k a k for all a, b ∈ A and λ ∈ C . For A = M n ( C ), a C ∗ -norm on A is given by operator norm, where weview elements of M n ( C ) as operators C n → C n , with the Euclidean norm on C n . (This operatornorm assigns to M ∈ M n ( C ) the square root of the largest eigenvalue of the matrix M ∗ M .) AC ∗ -norm on C ( T ) is also given by an operator norm.A C ∗ -norm on a ∗ -algebra A induces a topology on A in the usual way, by defining the ǫ -ballaround an element a ∈ A to be { b ∈ A | k b − a k < ǫ } . Definition 1.8. A C ∗ -algebra is a ∗ -algebra A endowed with a C ∗ -norm k·k , for which A is completewith respect to the topology induced by k · k .A second description of a C ∗ -algebra, from an operator-theoretic point of view, is given here. Let H be a Hilbert space, and let B ( H ) denote the continuous linear operators on H . A C ∗ -algebra isan adjoint-closed subalgebra of B ( H ) which is closed with respect to the norm topology on B ( H ).In general, and especially relevant in the current context, one often builds a C ∗ -algebra by startingwith a given set of elements in B ( H ), and then forming the smallest C ∗ -subalgebra of B ( H ) whichcontains that set.A partial isometry is an element x in a C ∗ -algebra A for which y = x ∗ x is a self-adjoint idempotent;that is, in case y ∗ = y and y = y . Such elements are characterized as those elements z of A forwhich zz ∗ z = z in A . For instance, in M n ( C ), any element which is the sum of distinct matrixunits e i,i (1 ≤ i ≤ n ) is a partial isometry (indeed, a projection); there are other partial isometriesin M n ( C ) as well. Since the only idempotents in C ( T ) are the constant functions 0 and 1, it is nothard to show that the set of partial isometries in C ( T ) consists of { } ∪ { f ∈ C ( T ) | f ( T ) ⊆ T } .The study of C ∗ -algebras has its roots in the early development of quantum mechanics; thesewere used to model algebras of physical observables. Various questions about the structure of C ∗ -algebras arose over the years. One of the most important of these questions, the explicit descriptionof a separable simple infinite C ∗ -algebra, was resolved in 1977 by Cuntz ([56, Theorem 1.12]). AC ∗ -algebra is simple in case it contains no nontrivial closed two-sided ideals. (It can be shownthat this is equivalent to the algebra containing no nontrivial two-sided ideals, closed or not.) AC ∗ -algebra is infinite in case it contains an element x for which xx ∗ = 1 and x ∗ x = 1. Cuntz’ Theorem
Let n ∈ N . Consider a Hilbert space H , and a set { S i } ni =1 of isometries (i.e., S ∗ i S i = 1) on H . Assume that P ni =1 S i S ∗ i = 1. Let O n denote C ∗ ( S , . . . , S n ), the C ∗ -algebragenerated by { S i } ni =1 . Then the infinite separable C ∗ -algebra O n is simple. GENE ABRAMS
Indeed, Cuntz proves much more in [56, Theorem 1.12] than we have stated here. Additionally, itis shown in [56, Theorem 1.13] that if X is any nonzero element in O n , then there exist A, B ∈ O n for which AXB = 1.Cuntz notes that the condition P ni =1 S i S ∗ i = 1 implies that the S i S ∗ i are pairwise orthogonal. Sothe C ∗ -algebra O n is the C ∗ -completion of a C -subalgebra T n of B ( H ), where T n as a C -algebra isgenerated by isometries { S i } ni =1 , for which P ni =1 S i S ∗ i = 1. Since a C ∗ -algebra is adjoint-closed, wesee that O n may also be viewed as the C ∗ -completion of a C -subalgebra L n of B ( H ) generated byisometries { S i } ni =1 together with { S ∗ i } ni =1 , for which P ni =1 S i S ∗ i = 1.In retrospect, such a C -algebra L n is seen to be isomorphic to L C (1 , n ).Subsequent to the appearance of [56], a number of researchers in operator algebras investigatednatural generalizations of the Cuntz C ∗ -algebras O n ; see especially [59]. In the early 1980’s, variousconstructions of C ∗ -algebras corresponding to directed graphs were studied by Watatani and others(e.g., [100]). Even though, via this approach, the Cuntz algebra O n could be realized as the C ∗ -algebra corresponding to the graph R n (see Example 2.13 below), this methodology did not gainmuch traction at the time. Instead, the study of these C ∗ -algebras from a different point of view(arising from matrices with non-negative integer entries, or arising from groupoids) became morethe vogue. But then, in the fundamental article [75] (in which groupoids are still in the picture,and the corresponding graphs could not have sinks), and the subsequent followup articles [74] and[47], the power of constructing a C ∗ -algebra based on the data provided by a directed graph becameclear. Definition 1.9.
A (directed) graph is a quadruple E = ( E , E , s, r ), where E and E are sets(the vertices and edges of E , respectively), and s and r are functions from E to E (the source and range functions of E , respectively). A sink is an element v ∈ E for which s − ( v ) = ∅ . E is finite in case both E and E are finite sets. Definition 1.10.
Let E be a finite graph. Let C ∗ ( E ) denote the universal C ∗ -algebra generatedby a collection of mutually orthogonal projections { p v | v ∈ E } together with partial isometries { s e | e ∈ E } which satisfy the Cuntz-Krieger relations :(CK1) s ∗ e s e = p r ( e ) for all e ∈ E , and(CK2) p v = P { e ∈ E | s ( e )= v } s e s ∗ e for each non-sink v ∈ E .For example, in [74] the authors were able to identify those finite graphs E for which C ∗ ( E ) issimple, and those for which C ∗ ( E ) is purely infinite simple. (The germane graph-theoretic termswill be described in Notations 2.7 and 3.2 below. A unital C ∗ -algebra A is purely infinite simple incase A = C , and for each 0 = x ∈ A there exist a, b ∈ A with axb = 1.) Theorem 1.11. (Simplicity and Purely Infinite Simplicity Theorems for graph C ∗ -algebras) Let E be a finite graph. Then C ∗ ( E ) is simple if and only if the only hereditary saturatedsubsets of E are trivial, and every cycle in E has an exit. Moreover, C ∗ ( E ) is purely infinite simpleif and only if C ∗ ( E ) is simple, and E contains at least one cycle. Subsequently, in [47], results were clarified, sharpened, and extended; and the groupoid techniqueswere eliminated from the arguments.
EAVITT PATH ALGEBRAS: THE FIRST DECADE 7
During the same timeframe, Kirchberg (unpublished) and Phillips ([85]) independently proveda beautiful, deep result which classifies up to isomorphism a class of C ∗ -algebras satisfying var-ious properties. Although the now-so-called Kirchberg Phillips Theorem covers a wide class ofC ∗ -algebras, it manifests in the particular case of purely infinite simple graph C ∗ -algebras as fol-lows. Theorem 1.12. (The Kirchberg Phillips Theorem for graph C ∗ -algebras) Let E and F befinite graphs. Suppose C ∗ ( E ) and C ∗ ( F ) are purely infinite simple. Suppose there is an isomorphism K ( C ∗ ( E )) ∼ = K ( C ∗ ( F )) for which [1 C ∗ ( E ) ] [1 C ∗ ( F ) ] . Then C ∗ ( E ) ∼ = C ∗ ( F ) . The work described in [47] became the basis of a newly-energized research program in the C ∗ -algebra community, a program which continues to flourish to this day. For additional informationabout graph C ∗ -algebras, see [86]; for a more complete description of the history of graph C ∗ -algebras, see [99, Appendix B].With the overview of Leavitt algebras, Bergman algebras, and graph C ∗ -algebras now in place,we are in position to describe the genesis of Leavitt path algebras.There are two plot lines to the history.The first historical plot line begins with an investigation into the algebraic notion of purely infinitesimple rings, begun by Ara, Goodearl, and Pardo (each of whom has significant expertise in bothring theory and C ∗ -algebras) in [35]. In it, the authors “... extend the notion of a purely infinitesimple C ∗ -algebra to the context of unital rings, and study its basic properties, especially thoserelated to K -theory”.The authors note in the introduction of [35] that “The Cuntz algebra O n is the C ∗ -completion ofthe Leavitt algebra V ,n over the field of complex numbers.” Although this connection between theCuntz and Leavitt algebras is now viewed as almost obvious, it was not until the early 2000’s thatsuch a connection was first noted in the literature. (A somewhat earlier mention of this connectionappears in [3]; the observation in [3] was included at the request of an anonymous referee.)With the notion of purely infinite simple rings so introduced, the same three authors (togetherwith Gonz´alez-Barroso) set out to find large classes of explicit examples of such rings. With thepurely infinite simple graph C ∗ -algebras as motivation, the four authors in [29] introduced the“algebraic Cuntz-Krieger (CK) algebras.” (Retrospectively, these are seen to be the Leavitt pathalgebras corresponding to finite graphs having neither sources nor sinks, and which do not consistof a disjoint union of cycles.) These algebraic Cuntz-Krieger algebras arose as specific examplesof fractional skew monoid rings , and the germane ones were shown to be purely infinite simple byusing techniques which applied to the more general class.With the K -theory of the corresponding graph C ∗ -algebras in mind, it was then natural to askanalogous K -theoretic questions about the algebraic CK algebras. In addition, earlier work by Ara,Goodearl, O’Meara and Pardo [33] regarding semigroup-theoretic properties of V (e.g., separativityand refinement) for various classes of rings provided the motivation to ask similar questions about V ( A ) for these algebras.Once various specific examples had been completely worked out, it became clear to Ara andPardo that much of the information about the V -monoid of the algebraic CK algebras could be seen GENE ABRAMS directly in terms of relations between vertices and edges in an associated graph E . Indeed, theserelations between vertices and edges could be codified as information which could then be used togenerate a monoid in a natural way, defined here. Definition 1.13.
Let E be a finite graph, with E = { v , v , . . . , v n } . The graph monoid M E of E is the free abelian monoid on a generating set { a v , a v , . . . , a v n } , modulo the relations a v i = X { e ∈ E | s ( e )= v i } a r ( e ) for each non-sink v i . In a private communication to the author, Enrique Pardo wrote that, with all this informationand background as context, ... at some moment [early in 2004] one of us suggested that prob-ably Bergman’s coproduct construction would be a good manner ofsolving the computation and prove that both monoids coincide.
Once some additional necessary machinery was included (the notion of a complete subgraph), thenAra and Pardo, together with Pardo’s colleague Mariangeles Moreno-Fr´ıas, had all the ingredientsin hand to make the following definition, and prove the subsequent theorem, in [36]. (We state thedefinition and theorem here only for finite graphs; these results were established for more generalgraphs in [36], and the general version will be discussed below.)
Definition 1.14. ( [36, p. 161] ) Let E be a finite graph, and let K be a field. We define the graph K -algebra L K ( E ) associated with E as the K -algebra generated by a set { p v | v ∈ E } together witha set { x e , y e | e ∈ E } , which satisfy the following relations:(1) p v p v ′ = δ v,v ′ p v for all v, v ′ ∈ E . (2) p s ( e ) x e = x e p r ( e ) = x e for all e ∈ E . (3) p r ( e ) y e = y e p s ( e ) = y e for all e ∈ E . (4) y e x e ′ = δ e,e ′ p r ( e ) for all e, e ′ ∈ E .(5) p v = P { e ∈ E | s ( e )= v } x e y e for every v ∈ E that emits edges. We note that both the terminology used in this definition (“graph algebra”), as well as thenotation, is quite similar to the terminology and notation which was already being employed in thecontext of graph C ∗ -algebras. Theorem 1.15. (The Ara / Moreno / Pardo Realization Theorem) ([36, Theorem 3.5])
Let E be a finite graph and K any field. Then there is a natural monoid isomorphism V ( L K ( E )) ∼ = M E . By examining the proof of [36, Theorem 3.5], and using Bergman’s Theorem, we can in factrestate this fundamental result as follows.
Theorem 1.15 ′ (The Ara / Moreno / Pardo Realization Theorem, restated) Let E bea finite graph and K any field. Let M E be the monoid given by the specific set of generators andrelations presented in Definition 1.13. Let d denote the element P v ∈ E a v of M E . Then L K ( E ) ∼ = B ( M E , d ) . Consequently, V ( L K ( E )) ∼ = M E . Moreover, L K ( E ) is hereditary. EAVITT PATH ALGEBRAS: THE FIRST DECADE 9
In the same groundbreaking article [36], Ara, Moreno, and Pardo were also able to establish aconnection between the V -monoids of L K ( E ) and C ∗ ( E ). Theorem 1.16. (The Ara / Moreno / Pardo Monoid Isomorphism Theorem) ([36, The-orem 7.1])
Let E be a finite graph. Then there is a natural monoid isomorphism V ( L C ( E )) ∼ = V ( C ∗ ( E )) . We conclude our discussion of Historical Plot Line
For us the motivation was to give an algebraic framework to allthese families of (purely infinite simple) C ∗ -algebras associated tocombinatorial objects, say Cuntz-Krieger algebras and graph C ∗ -algebras. For this reason we always looked at properties that wereknown in C ∗ case and were related to combinatorial information:we wanted to know which part of these results relies in algebraic in-formation, and which ones in analytic information. So, we lookedat K-Theory, stable rank, exchange property (in C ∗ -algebras this isreal rank zero property), prime and primitive ideals, the classifica-tion problem and Kirchberg-Phillips Theorem... We will visit each of these topics later in the article.The second historical plot line begins with the author’s interest in Leavitt’s algebras, specificallythe algebras L K (1 , n ). For instance, these algebras were used in [1] to produce non-IBN ringshaving unexpected isomorphisms between their matrix rings; were used again in [2] to solve aquestion (posed in [79]) about strongly graded rings; and were subsequently investigated yet againin [3], in joint work with P.N. ´Anh of the R´enyi Institute of Mathematics (Hungarian Academy ofSciences, Budapest).During a Spring 2001 visit to the University of Iowa, ´Anh met the analyst Paul Muhly. Subse-quently, ´Anh invited Muhly to give a talk at the R´enyi Institute (during a 2003 trip that Muhlyand his wife were making to Budapest anyway, to visit their son); it was during this visit thatthe two mathematicians began to consider the potential for connections between various topics.Muhly was one of the organizers of the May/June 2004 NSF - CBMS conference “Graph Algebras:Operator Algebras We Can See”, delivered by Iain Raeburn, held at the University of Iowa. Muhlyconsequently extended invitations to attend that conference to the author, to ´Anh, and to a handfulof other ring theorists. During conference coffee break discussions, the algebraists began to realize The ´Anh / Muhly meeting was quite fortuitous. ´Anh was a visiting research guest of Kent Fuller at the Universityof Iowa during Spring Semester 2001. Fuller regularly went to lunch at various Iowa City restaurants with a group ofhis departmental colleagues, an excursion in which Paul Muhly was a frequent participant; Fuller of course invited´Anh to join in. National Science Foundation - Conference Board in Mathematical Sciences. The NSF-CBMS Regional ResearchConferences in the Mathematical Sciences are a series of five-day conferences, each of which features a distinguishedlecturer delivering ten lectures on a topic of important current research in one sharply focused area of the mathe-matical sciences. V. Camillo, L. M´arki, and E. Ortega also attended. that when one considered the “pre-completion” version of the graph C ∗ -algebras, the remainingalgebraic structure looked quite familiar, specifically, as some sort of modification of the well-knownnotion of a quiver algebra or path algebra . Definition 1.17.
Let F = ( F , F , s, r ) be a graph and K any field. The path K -algebra of F (alsoknown as the quiver K -algebra of F ), denoted KF , is the K -vector space having basis F ⊔ F , withmultiplication given by the K -linear extension of p · q = ( pq if r ( p ) = s ( q )0 otherwise . (If v ∈ F we denote s ( v ) = v = r ( v ) .) Gonzalo Aranda Pino visited the author’s home institution for the period July 2004 throughDecember 2004. Early in Aranda Pino’s visit, the author shared with him some of the ideas whichhad been discussed in Iowa City during the previous month. A few weeks of collaborative effortsubsequently led to the following.
Definition 1.18.
Given a directed graph E = ( E , E , s, r ) we define the extended graph of E asthe graph b E = ( E , E ⊔ ( E ) ∗ , s ′ , r ′ ) , where ( E ) ∗ = { e ∗ i : e i ∈ E } , and the functions r ′ and s ′ aredefined by setting r ′ | E = r, s ′ | E = s, r ′ ( e ∗ i ) = s ( e i ) , and s ′ ( e ∗ i ) = r ( e i ) . Definition 1.19.
Let E be a finite graph and K any field. The Leavitt path K -algebra L K ( E ) isdefined as the path K -algebra K b E , modulo the relations:(CK1) e ∗ i e j = δ ij r ( e j ) for every e j ∈ E and e ∗ i ∈ ( E ) ∗ .(CK2) v i = P { e j ∈ E | s ( e j )= v i } e j e ∗ j for every v i ∈ E which is not a sink. Some of the notation which was developed in the C ∗ -algebra context is also used in the Leavittpath algebra world, e.g., the use of the “CK” labels to denote the two key relations. (Cf. Definition1.10).With both Leavitt’s Theorem (part 2 of Theorem 1.4) and The Simplicity Theorem for graphC ∗ -algebras (Theorem 1.11) in mind, the author and Aranda Pino focused their initial investigationon an internal, multiplicative question about the algebras L K ( E ): for which graphs E and fields K is L K ( E ) simple? Using techniques completely unlike those utilized to achieve Theorem 1.11,the following result was established. (See Notations 2.7 and 3.2 below for definitions of appropriateterms.) Theorem 1.20. (The Abrams / Aranda Pino Simplicity Theorem) [7, Theorem 3.11] ) Let E be a finite graph and K any field. Then L K ( E ) is simple if and only if the only hereditarysaturated subsets of E are trivial, and every cycle in E has an exit. As if the ´Anh / Muhly meeting (and consequent attendance of the ring theorists at the 2004 Iowa CBMSconference) was not fortuitous enough, it turned out that, many months prior to that conference, Mercedes SilesMolina had contacted the author regarding the possibility of having the author host one of her Ph.D. students for asix month visit at the University of Colorado, to commence July 2004. That having been arranged, Gonzalo ArandaPino arrived in Colorado Springs at precisely the time that this new idea was blossoming.
EAVITT PATH ALGEBRAS: THE FIRST DECADE 11
By making the obvious correspondences v ↔ p v , e ↔ x e , and e ∗ ↔ y e , we see immediately:For a finite graph E and field K ,the graph K -algebra of Definition 1.14 is the same algebraas the Leavitt path K -algebra of Definition 1.19.It is of historical interest to note that the work on [7] was started in July 2004. Subsequently, [7]was submitted for publication in September 2004, accepted for publication in June 2005, appearedonline in September 2005, and appeared in print in November 2005. On the other hand, the workon [36] was started in early 2004. Subsequently, [36] was submitted for publication in late 2004 (andposted on ArXiV at that time), and accepted for publication in early 2005, but did not appear inprint until April 2007. So even though [7] appeared in print eighteen months prior to the appearancein print of [36], in fact most the mathematical work done to produce the latter preceded that of theformer.Both [7] and [36] should be viewed as the foundational articles on the subject.2. Leavitt path algebras of row-finite graphs: general properties and examples
Section 1 of this article was meant to give the reader an overall view of the motivating ideaswhich led naturally to the construction of Leavitt path algebras. Over the next three sections wedescribe some of the key ideas and results for Leavitt path algebras arising from row-finite graphs.Subsequently, in Section 5 we relax this hypothesis on the graphs. (For those results which do notextend verbatim to the unrestricted case, we will indicate in the statement that the graph must berow-finite (or finite); otherwise, we will make no such stipulation in the statement.)
Notation 2.1.
A vertex v in a graph E = ( E , E , s, r ) is called regular in case < | s − ( v ) | < ∞ ;otherwise, v is called singular. Specifically, if s − ( v ) = ∅ then v is called a sink, while v is calledan infinite emitter in case | s − ( v ) | is infinite. E is called row-finite in case E contains no infiniteemitters. Here is the formal definition of a Leavitt path algebra arising from a row-finite graph.
Definition 2.2.
Let E = ( E , E , s, r ) be a row-finite graph and K any field. Let b E denote theextended graph of E . The Leavitt path K -algebra L K ( E ) is defined as the path K -algebra K b E ,modulo the relations:(CK1) e ∗ i e j = δ ij r ( e j ) for every e j ∈ E and e ∗ i ∈ ( E ) ∗ .(CK2) v i = P { e j ∈ E | s ( e j )= v i } e j e ∗ j for every non-sink v i ∈ E .Equivalently, we may define L K ( E ) as the free associative K -algebra on generators E ⊔ E ⊔ ( E ) ∗ ,modulo the relations(1) vv ′ = δ v,v ′ v for all v, v ′ ∈ E . (2) s ( e ) e = er ( e ) = e for all e ∈ E . (3) r ( e ) e ∗ = e ∗ s ( e ) = e ∗ for all e ∈ E . (4) e ∗ e ′ = δ e,e ′ r ( e ) for all e, e ′ ∈ E . (5) v = P { e ∈ E | s ( e )= v } ee ∗ for every non-sink v ∈ E . It is established in [97] that the expected map from L C ( E ) to C ∗ ( E ) is in fact injective. Withthis and the construction of the graph C ∗ -algebra C ∗ ( E ), we get Proposition 2.3.
For any graph E , L C ( E ) is isomorphic to a dense ∗ -subalgebra of C ∗ ( E ) . The interplay between graphs and algebras will play a major role in the theory. It is importantto note at the outset that in general, if F is a subgraph of E , then L K ( F ) need not correspond toa subalgebra of L K ( E ), because the (CK2) relation imposed at a vertex v in L K ( F ) need not bethe same as the relation imposed at v in L K ( E ). For a row-finite graph E , a subgraph F is saidto be complete in case, whenever v ∈ F , then either s − F ( v ) = ∅ , or s − F ( v ) = s − E ( v ). (In otherwords, if v ∈ F , then either v emits no edges in F , or emits the same edges in F as it does in E .) Perhaps not surprisingly, when F is a complete subgraph of E , then there is an injection ofalgebras L K ( F ) ֒ → L K ( E ). Moreover, Proposition 2.4. [36, Lemma 3.2]
The assignment E L K ( E ) can be extended to a functor L K from the category of row-finite graphs and complete graph inclusions to the category of K -algebrasand (not necessarily unital) algebra homomorphisms. The functor L K commutes with direct limits.It follows that every L K ( E ) for a row-finite graph E is the direct limit of graph algebras correspondingto finite graphs. Because of Proposition 2.4, it is often the case that a result which holds for the Leavitt pathalgebras of finite graphs can be extended to the row-finite case.
Definition 2.5.
Let E be any graph and A any K -algebra. A Leavitt E -family in A is a subset S = { a v | v ∈ E } ∪ { b e | e ∈ E } ∪ { c e | e ∈ E } of A for which(1) a v a v ′ = δ v,v ′ a v for all v, v ′ ∈ E . (2) a s ( e ) b e = b e a r ( e ) = b e for all e ∈ E . (3) a r ( e ) c e = c e a s ( e ) = c e for all e ∈ E . (4) c e b e ′ = δ e,e ′ a r ( e ) for all e, e ′ ∈ E .(5) a v = P { e ∈ E | s ( e )= v } b e c e for every non-sink v ∈ E . By the description of L K ( E ) as a quotient of a free associative K -algebra modulo the germanerelations given in Definition 2.2, we immediately get the following result, which often proves to bequite useful in the subject. Proposition 2.6. (Universal Homomorphism Property of Leavitt path algebras)
Let E be a graph, and suppose S is a Leavitt E -family in the K -algebra A . Then there exists a unique K -algebra homomorphism ϕ : L K ( E ) → A for which ϕ ( v ) = a v , ϕ ( e ) = b e , and ϕ ( e ∗ ) = c e for all v ∈ E and e ∈ E . Notation 2.7.
A sequence of edges α = e , e , ..., e n in a graph E for which r ( e i ) = s ( e i +1 ) for all ≤ i ≤ n − is called a path of length n . We typically denote such α more simply by e e · · · e n .Each vertex v of E is viewed as a path of length . The set of paths of length n in E is denoted by E n ; the set of all paths in E is denoted Path( E ) . So we have Path( E ) = ⊔ n ∈ Z + E n . EAVITT PATH ALGEBRAS: THE FIRST DECADE 13
For α = e e · · · e n ∈ Path( E ) , s ( α ) denotes s ( e ) , r ( α ) denotes r ( e n ) , and Vert( α ) denotes theset { s ( e ) , s ( e ) , . . . , s ( e n ) , r ( e n ) } . The path e e · · · e n is closed if s ( e ) = r ( e n ) . A closed path c = e e · · · e n is simple in case s ( e i ) = s ( e ) for all ≤ i ≤ n . Such a simple closed path c is saidto be based at v = s ( e ) . A simple closed path c = e e · · · e n is a cycle in case there are no repeatsin the list of vertices s ( e ) , s ( e ) , ..., s ( e n ) . E is called acyclic in case there are no cycles in E .An exit for a path e e · · · e n is an edge f ∈ E for which s ( f ) = s ( e i ) and f = e i for some ≤ i ≤ n .The graph E satisfies Condition (L) in case every cycle in E has an exit.The graph E satisfies Condition (K) in case no vertex in E is the base of exactly one simpleclosed path in E . If α = e e · · · e n is a path in E , then we may view α as an element of the path algebra KE ,and as an element of the Leavitt path algebra L K ( E ) as well. (In this sense, concatenation in thegraph E is interpreted as multiplication in KE or L K ( E ).) We denote by α ∗ the element e ∗ n · · · e ∗ e ∗ of L K ( E ). We often refer to a path α = e e · · · e n of E (viewed as an element of L K ( E )) as a realpath , while an element of L K ( E ) of the form α ∗ = e ∗ n · · · e ∗ e ∗ is called a ghost path . Here are someeasily verified basic properties of Leavitt path algebras. Proposition 2.8.
Let E be any graph and K any field(1) Every nonzero element r of L K ( E ) may be written (not necessarily uniquely) as r = n X i =1 k i α i β ∗ i , where k i ∈ K × , and α i , β i ∈ Path( E ) with r ( α i ) = r ( β i ) for ≤ i ≤ n .(2) For each α ∈ Path( E ) , α ∗ α = r ( α ) .(3) The natural K -algebra map KE → L K ( E ) is a one-to-one homomorphism.(4) L K ( E ) is unital (with multiplicative identity P v ∈ E v ) if and only if E is finite. In general, L K ( E ) has a set of enough idempotents, consisting of finite sums of distinct vertices.(5) The map ∗ : L K ( E ) → L K ( E ) induces an isomorphism L K ( E ) ∼ = L K ( E ) op . In particular, forLeavitt path algebras, the categories of left L K ( E ) -modules and right L K ( E ) -modules are isomorphic. Examples of familiar / “known” algebras which arise as Leavitt path algebras.
Wesaw in Section 1 how specific algebras arise from Bergman’s Theorem, starting with a specifiedmonoid. We re-examine those here, and present additional examples as well.
Example 2.9. Full matrix K -algebras. Let A n denote the graph • v e / / • v e / / · · · • v n − e n − / / • v n Then L K ( A n ) ∼ = M n ( K ) . This is not hard to see. We present two different approaches, in order toplay up the germane ideas.The first approach: consider the standard matrix units { E i,j | ≤ i, j ≤ n } in M n ( K ) . Since eachvertex (other than v n ) emits a single edge, the (CK2) relation at these vertices becomes e i e ∗ i = v i .Using this, it is straightforward to verify that the set S = { E i,i | ≤ i ≤ n } ∪ { E i,i +1 | ≤ i ≤ n − } ∪ { E i +1 ,i | ≤ i ≤ n − } is an A n -family in M n ( K ) . So the Universal Homomorphism Property ensures the existence of a K -algebra homomorphism ϕ for which ϕ ( v i ) = E i,i , ϕ ( e i ) = E i,i +1 , and ϕ ( e ∗ i ) = E i +1 ,i . That ϕ isan isomorphism is easily checked (for instance, by constructing the expected function ψ : M n ( K ) → L K ( A n ) , and verifying that ψ = ϕ − ).The second approach: we analyze the monoid M A n , define d = P ni =1 a v i , and see easily that ( M A n , d ) ∼ = ( Z + , n ) . With the relations describing M A n , it is clear that B ( M A n , d ) ∼ = M n ( K ) . NowTheorem 1.15 ′ applies. Full matrix rings over K arise as the Leavitt path algebras of graphs other than the A n graphs. InTheorem 3.1 below we will justify the isomorphisms asserted in the next two examples. These twoexamples play up the fact that non-isomorphic graphs may have isomorphic Leavitt path algebras.(This observation lies at the heart of much of the current research activity in Leavitt path algebras.) Example 2.10. Full matrix K -algebras, revisited. For n ∈ N let B n denote the graph • w e ❍❍❍❍❍❍❍❍❍ • w e (cid:15) (cid:15) • w e | | ②②②②②②②② • v o o • w n − e n − ; ; ✈✈✈✈✈✈✈✈✈ O O b b Then L K ( B n ) ∼ = M n ( K ) . Example 2.11. Full matrix K -algebras, again revisited. For n ∈ N let D n denote thegraph • v / / e ' ' e (cid:26) (cid:26) e n − E E • w Then L K ( D n ) ∼ = M n ( K ) . Proceeding in a manner similar to that utilized in Example 2.9, one can easily establish thefollowing two claims. (See Example 1.7.)
Example 2.12. The Laurent polynomial K -algebra. Let R denote the graph • v e h h Then L K ( R ) ∼ = K [ x, x − ] , the Laurent polynomial algebra. The isomorphism is clear: v , e x , and e ∗ x − . Here is the Fundamental Example of Leavitt path algebras.
Example 2.13. Leavitt K -algebras. For n ≥ , let R n denote the graph • ve (cid:19) (cid:19) e (cid:8) (cid:8) e s s e n EAVITT PATH ALGEBRAS: THE FIRST DECADE 15
Then L K ( R n ) ∼ = L K (1 , n ) , the Leavitt algebra of order n . The isomorphism is clear: using thedescription of the generators and relations for L K (1 , n ) given in ( † ) above, v , e i x i , and e ∗ i y i . Example 2.14. The Toeplitz K -algebra. For any field K , the Jacobson algebra, described in [70] , is the K -algebra A = K h x, y | xy = 1 i . This algebra was the first example appearing in the literature of an algebra which is not directlyfinite, that is, in which there are elements x, y for which xy = 1 but yx = 1 . Let T denote the“Toeplitz graph” • ve f / / • w Then L K ( T ) ∼ = A . The isomorphism is not hard to write down explicitly. First, the set S = { yx, − yx } ∪ { y x, y − y x } ∪ { yx , x − yx } is easily shown to be a T -family in A , so by the Universal Homomorphism Property of Leavittpath algebras there exists a K -algebra homomorphism ϕ : L K ( T ) → A for which ϕ ( v ) = yx , ϕ ( w ) = 1 − yx , ϕ ( e ) = y x , ϕ ( f ) = y − y x , ϕ ( e ∗ ) = yx , and ϕ ( f ∗ ) = x − yx . On the other hand,we define X = e ∗ + f ∗ , Y = e + f in A . Using (CK1) and (CK2) we get easily that XY = 1 . Thisgives a K -algebra homomorphism ψ : A → L K ( T ) , the algebra extension of x X and y Y . Itis easy to check that ϕ and ψ are inverses. Example 2.15. Full matrix K -algebras over L K ( E ) . Let E be any graph, K any field, and n ∈ N . The graph E ( n ) is defined as follows. For each v ∈ E , one adds to E the following verticesand edges • v n f n − v / / • v n − / / · · · f v / / • v f v / / , where r ( f v ) = v. Then L K ( E ( n )) ∼ = M n ( L K ( E )) . (See [18, Proposition 9.3] .) Example 2.16. Infinite matrix K -algebras. Let I be any set. We denote by M I ( K ) the setof those I × I matrices M , having entries in K , for which M i,j = 0 for at most finitely many pairs ( i, j ) . Then M I ( K ) is a K -algebra, which is unital if and only if I is finite (and in this case M I ( K ) consists of all I × I matrices having entries in K ). When I is infinite, then M I ( K ) has a set ofenough idempotents, consisting of finite sums of distinct matrix units of the form E i,i .If A N denotes the graph • v e / / • v e / / • v / / then L K ( A N ) ∼ = M N ( K ) .More generally, for any infinite set I , let B I denote the graph having vertices { v } ∪ { w i | i ∈ I } ,and edges { e i | i ∈ I } , with s ( e i ) = w i and r ( e i ) = v for all i ∈ I . Then L K ( B I ) ∼ = M I ( K ) . Internal / multiplicative properties of Leavitt path algebras
Not surprisingly, a number of the key results in the subject focus on passing structural informationfrom the directed graph E to the Leavitt path algebra L K ( E ), and vice versa; i.e., results of theform( †† ) E has graph property P ⇐⇒ L K ( E ) has ring property Q . The Simplicity Theorem (Theorem 1.20) is the quintessential result of this type. We will describe anumber of additional such results in this section and the next. In the author’s opinion, these resultsare quite interesting, some even remarkable , in their own right. Just as compellingly, some of these results have been utilized to produce heretofore unrecognized classes of algebras having interestingring-theoretic properties.Looking ahead: in contrast, in the next section, we will engage in a discussion of the equallyimportant “external / module-theoretic” properties of Leavitt path algebras. As described in Section1, the “internal / multiplicative” and “external / module-theoretic” properties form the historicalfoundations of the subject. We will see in the final section that these also drive much of the currentinvestigative energy.3.1.
Finite dimensional Leavitt path algebras.
We start by analyzing the Leavitt path algebrasof finite acyclic graphs. From a ring-theoretic point of view, these turn out to be the most basic(least interesting?) of all the Leavitt path algebras.
Theorem 3.1. Structure Theorem of Leavitt path algebras for finite acyclic graphs.
Let E be a finite acyclic graph and K any field. Let w , . . . , w t denote the sinks of E . (At least onesink must exist in any finite acyclic graph.) For each w i , let N i denote the number of elements of Path( E ) having range vertex w i . (This includes w i itself, as a path of length .) Then L K ( E ) ∼ = t M i =1 M N i ( K ) . Sketch of Proof.
For each sink w i consider the ideal I ( w i ) of L K ( E ). If α, β ∈ Path( E ) have r ( α ) = r ( β ) = w i , then αβ ∗ = αw i β ∗ ∈ I ( w i ). Using the (CK1) relation with the fact that w i is asink, one shows easily that the set of N i elements { αβ ∗ | α, β ∈ Path( E ) , r ( α ) = r ( β ) = w i } is aset of matrix units, which yields that I ( w i ) ∼ = M N i ( K ) . That the sum P ti =1 I ( w i ) is direct followsby again using the hypothesis that the w i are sinks. Now let γδ ∗ be any monic monomial in L K ( E ).If v = r ( γ ) is a sink, then γδ ∗ ∈ P ti =1 I ( w i ). Otherwise, the (CK2) relation may be invoked at v ,and we may write γδ ∗ = γvδ ∗ = γ ( X e ∈ s − ( v ) ee ∗ ) δ ∗ = X e ∈ s − ( v ) ( γe )( δe ) ∗ . If r ( e ) is a sink, then the expression ( γe )( δe ) ∗ is in P ti =1 I ( w i ); if not, then in the same manner onecan use the (CK2) relation at r ( e ) to rewrite ( γe )( δe ) ∗ . Since E is finite and acyclic, the processmust terminate with expressions of the desired form. ✷ So for finite acyclic graphs, the resulting Leavitt path algebras are, among other things: unitalsemisimple; left artinian; and finite dimensional. Indeed, any of these three ring/algebra-theoreticproperties characterizes the Leavitt path algebras of finite acyclic graphs, thus yielding three exam-ples of results of type ( †† ). Perhaps more importantly, Theorem 3.1 yields a result of the followingtype: among a certain class of graphs (specifically, finite acyclic), we can determine, using easy-to-compute graph-theoretic properties, which of those graphs yield isomorphic Leavitt path algebras(specifically, those for which the number of sinks, and the corresponding N i , are equal). So Theorem3.1 may be viewed as a very basic type of Classification Theorem. Notation 3.2.
Let E be any graph, and let v, w ∈ E . We write v ≥ w in case there exists p ∈ Path( E ) for which s ( p ) = v and r ( p ) = w .Let X be a subset of E . X is called hereditary in case, whenever v ∈ X and w ∈ E and v ≥ w ,then w ∈ X . X is called saturated in case, whenever v ∈ E is regular and r ( s − ( v )) ⊆ X , then v ∈ X . (Less formally: X is saturated in case whenever v is a non-sink in E which emits finitelymany edges, and the range vertices of all of those edges are in X , then v is in X as well.) EAVITT PATH ALGEBRAS: THE FIRST DECADE 17
Clearly both E and ∅ are hereditary saturated subsets of E , and clearly the intersection of anycollection of hereditary saturated subsets of E is again hereditary saturated. If S is any subset of E , then S denotes the smallest hereditary saturated subset of E which contains S ; S is called thehereditary saturated closure of S . (Such exists by the previous observation.) The interplay between vertices E of E on the one hand (viewed as idempotent elements of L K ( E )), and ideals of L K ( E ) on the other, plays a central role in the ideal structure of L K ( E ).This connection clearly brings to light the roles of the two (CK) relations in this context. Proposition 3.3.
Let E be any graph and K any field. Let I be an ideal of L K ( E ) . Then X = E ∩ I is a hereditary saturated subset of E .Proof. Let v ∈ X . If w ∈ E for which there exists α ∈ Path( E ) with s ( α ) = v and r ( α ) = w ,then by Proposition 2.8(2) w = α ∗ α = α ∗ · v · α in L K ( E ), so that w ∈ I , and thus in X . So X is hereditary. On the other hand, suppose v ∈ E has the property that | s − ( v ) | is finite, and that r ( e ) ∈ X for each e ∈ s − ( v ). But by (CK2) v = X e ∈ s − ( v ) ee ∗ = X e ∈ s − ( v ) e · r ( e ) · e ∗ in L K ( E ), so that v ∈ I , and thus in X . So X is saturated. ✷ The Z -grading, and graded ideals. A K -algebra R is Z -graded in case R = ⊕ i ∈ Z R i as K -vector spaces, in such a way that R i · R j ⊆ R i + j for all i, j ∈ Z . The subspaces R i are calledthe homogeneous components of R . The Leavitt path algebras admit a Z -grading, as follows. Anypath K -algebra K b E of an extended graph b E is Z -graded, by setting deg( v ) = 0 for v ∈ E , anddeg( e ) = 1, deg( e ∗ ) = − e ∈ E , and extending additively and multiplicatively. Since the twosets of relations (CK1) and (CK2) consist of homogeneous elements of degree 0 with respect to thisgrading on K b E , the grading passes to the quotient algebra L K ( E ). In particular, for m ∈ Z , thehomogeneous component L K ( E ) m of degree m consists of K -linear combinations of elements of theform αβ ∗ , where r ( α ) = r ( β ), and ℓ ( α ) − ℓ ( β ) = m .A two-sided ideal I in a Z -graded ring R is called a graded ideal in case, whenever s ∈ I and s = P j ∈ Z s j is the decomposition of s into homogeneous components, then s j ∈ I for each j ∈ Z . Itis easy to show that if a two-sided ideal I in a Z -graded ring is generated by homogeneous elementsof degree 0, then I is a graded ideal. In particular, for any set of vertices X ⊂ E , the ideal I ( X )of L K ( E ) is graded. In contrast, not all ideals of a Leavitt path algebra are necessarily graded; forinstance, the ideal I (1 + x ) ⊂ K [ x, x − ] ∼ = L K ( R ) is not graded, as neither 1 nor x is in I (1 + x ).So on the one hand any ideal I of L K ( E ) gives rise to the hereditary saturated subset I ∩ E of E , while on the other, any subset X of E gives rise to the graded ideal I ( X ) of L K ( E ). Theperhaps-expected connection is the following. Proposition 3.4. [36, Theorem 5.3]
Let E be a row-finite graph. Then there is a lattice isomorphismbetween the lattice of graded ideals { I } of L K ( E ) and the lattice of hereditary saturated subsets { X } of E : I I ∩ E , X I ( X ) . In particular, every graded ideal of L K ( E ) is generated by vertices. Sketch of Proof.
It is not hard to show that I ( X ) = I ( X ). On the other hand, if v ∈ I ( X ), thenby using an explicit, iterative description of the hereditary saturated closure of a set, one can showthat v ∈ X . ✷ The connection between these two lattices does not hold verbatim in case E contains infiniteemitters, as we will see in Section 5.It was shown by Bergman [48] that if R is a Z -graded (unital) ring, then the Jacobson radical J ( R ) is necessarily a graded ideal. (See also [89, Theorem 2.5.40].) Using that J ( R ) contains nononzero idempotents in any ring R , Proposition 3.4 yields the following nice “internal” result aboutLeavitt path algebras. Corollary 3.5.
Let E be any graph and K any field. Then L K ( E ) has zero Jacobson radical. Ideals in Leavitt path algebras.
In general, loosely speaking, the two key players in thegraph E which drive the ideal structure of L K ( E ) are the vertices, and the cycles without exits.While the hereditary saturated subsets will dictate the graded structure of L K ( E ), the cycles withoutexits (when E contains such) provide additional structural nuances. The following result providessome motivation as to why this should be the case. For an element p ( x ) = P Ni = m k i x i ∈ K [ x, x − ](with k N ∈ K × ), and a cycle c in the graph E , we denote by p ( c ) the element P Ni = m k i c i of L K ( E ),where c i = ( c ∗ ) − i whenever i <
0, and c = s ( c ). Theorem 3.6. The Reduction Theorem. [42, Proposition 3.1]
Let E be any graph and K anyfield. Let = x ∈ L K ( E ) . Then there exist α, β ∈ Path( E ) for which either:(1) α ∗ xβ = kv for some v ∈ E and k ∈ K × , or(2) α ∗ xβ = p ( c ) , where = p ( x ) ∈ K [ x, x − ] and c is a cycle without exits.In other words, we can transform (via multiplication by real paths and/or ghost paths) any elementof L K ( E ) to either a nonzero multiple of a vertex, or to a nonzero polynomial in a cycle withoutexits.Sketch of Proof. The proof uses an idea similar to the one Leavitt used in his proof of theSimplicity Theorem for L K (1 , n ) ([77, Theorem 2]). Essentially, starting with x , one shows thatthere is a path γ in E for which 0 = xγ ∈ KE . This is done by finding v ∈ E for which xv = 0,then writing xv ∈ L K ( E ) in a form which minimizes the length of the ghost terms from among allpossible representations of xv , and then applying an induction argument. With this in hand, onethen modifies xγ via left multiplication by terms of the form δ ∗ to “reduce” xγ to one of the twoindicated forms. ✷ Definition 3.7.
For a hereditary saturated subset H of E , let C H denote the set of cycles c in E for which Vert( c ) ∩ H = ∅ , and for which r ( e ) ∈ H for every exit e of c .For any subset C of C H , consider a set P = P ( C ) = { p c ( x ) | c ∈ C } of noninvertible, nonzeroelements of K [ x ] . Let P C denote the subset { p c ( c ) | c ∈ C } of L K ( E ) . For instance, for the Toeplitz graph T described in Example 2.14, let H be the (only nontrivial)hereditary saturated subset { w } . Then the cycle e is in C H . For any polynomial p ( x ) ∈ K [ x ], wemay form the ideal I ( w, p ( e )) of L K ( T ) generated by the two elements w and p ( e ).In a similar way, for general graphs, using the data provided by a hereditary saturated subset H of E , a set C = C ( H ) of cycles which miss H but all of whose exits land in H , and nontrivialpolynomials P = P ( C ) = P ( C ( H )) in K [ x ] (one for each element of C ), we can build an idealin L K ( E ), namely, the ideal generated by H together with elements of L K ( E ) of the form p c ( c ). EAVITT PATH ALGEBRAS: THE FIRST DECADE 19
Rephrased, starting with such (
H, C ( H ) , P ( C ( H )), we can build the ideal I ( H, { p c ( c ) } c ∈ C ). Indeed,this process gives all the ideals of L K ( E ). Theorem 3.8. Structure Theorem of Ideals [6, Theorem 2.8.10]
Let E be a row-finite graph.Then every ideal of L K ( E ) is of the form I ( H, { p c ( c ) } c ∈ C ) as described above. Indeed, with not-hard-to-anticipate order relations defined on triples of the form (
H, C, P ), thereis a stronger form of Theorem 3.8, one which gives a lattice isomorphism between the set of appro-priate triples and the lattice of two-sided ideals of L K ( E ).There are some immediate consequences of Theorem 3.8. The most noteworthy of these is theSimplicity Theorem (Theorem 1.20): that L K ( E ) is simple if and only if the only hereditary satu-rated subsets of E are ∅ and E , and every cycle in E has an exit. (Of course the chronology hereis reversed: the historically-significant Simplicity Theorem precedes the establishment of Theorem3.8 by almost a decade.) This is seen quite readily. By Theorem 3.8, any ideal of L K ( E ) looks like I ( H, { p c ( c ) } c ∈ C ). By hypothesis there are only two possibilities for H . When H = E then C ( H ),and therefore P ( C ( H )), is empty, so that the only ideal of this form is I ( E ) = L K ( E ). On theother hand, when H = ∅ , then, as by hypothesis every cycle in E has an exit, we get that C ( H ),and therefore P ( C ( H )), is empty here as well. So the only ideal of this second form is I ( ∅ ) = { } ,and the Simplicity Theorem follows.Returning yet again to the Toeplitz graph T of Example 2.14, we see as a consequence of Theorem3.8 that the complete set of ideals of L K ( T ) consists of the three graded ideals I ( ∅ ) = { } , I ( w ),and I ( E ) = L K ( T ), together with the nongraded ideals of the form I ( w, p ( e )), where p ( x ) ∈ K [ x ]is a polynomial of degree at least 1 for which p (0) = 0.Considering the stronger (admittedly unstated) form of Theorem 3.8, a second consequence (alsoa statement of type ( †† )) is the following description of the Leavitt path algebras satisfying thechain conditions on two-sided ideals. Proposition 3.9.
Let E be a row-finite graph and K any field.(1) L K ( E ) has the descending chain condition on two-sided ideals if and only if E satisfies Con-dition (K), and the descending chain condition holds in the lattice of hereditary saturated subsets of E ( [11, Theorem 3.9] ).(2) L K ( E ) has the ascending chain condition on two-sided ideals if and only if L K ( E ) has theascending chain condition on graded two-sided ideals, if and only if the ascending chain conditionholds in the lattice of hereditary saturated subsets of E . In particular, the Leavitt path algebra L K ( E ) for every finite graph E has the a.c.c. on two-sided ideals ( [11, Theorem 3.6] ). Discussion:
The Rosetta Stone.
Of great interest in the study of Leavitt path algebras is the observation that many of the resultsin the subject seem to (quite mysteriously) mimic corresponding results for graph C ∗ -algebras. Forexample, comparing the Simplicity Theorem for Leavitt path algebras (Theorem 1.20) with theSimplicity Theorem for graph C ∗ -algebras (Theorem 1.11), we see that the conditions on E whichyield simplicity of the associated graph algebra are identical in both cases. Suffice it to say that theproofs of the two Simplicity Theorems utilize significantly different tools one from the other. Moreto the point, even with the close relationship between L C ( E ) and C ∗ ( E ) in mind (cf. Proposition2.3), it is currently not understood as to whether either one of the Simplicity Theorems should“directly” imply the other. We provide in Appendix 1 a list of additional situations in which an algebraic property of L C ( E )is analogous to a topological property of C ∗ ( E ), and for which the necessary and sufficient graph-theoretic property of E is identical in each case. A systematic reason which would explain theexistence of so many such examples is usually referred to as the “Rosetta Stone of Graph Algebras”.A good reference which contains in one place a discussion of both Leavitt path algebra and graphC ∗ -algebra properties is [45]. We note that even the seemingly most basic of questions, “if L C ( E ) ∼ = L C ( F ) as rings, is C ∗ ( E ) ∼ = C ∗ ( F ) as C ∗ -algebras?” (and its converse), has only been answered (inthe affirmative) for restricted classes of graphs; the question in general remains open (see [18]). Thesearch for the Rosetta Stone comprises one of the many current lines of research in the field.3.4. Matrix rings over the Leavitt algebras.
There are too many additional “internal / mul-tiplicative” properties of Leavitt path algebras to include them all in this article. For a numberof reasons (its connection to the Rosetta Stone and its important consequences outside of Leavittpath algebras, to name two), we spend some space here describing the Isomorphism Question forMatrix Rings over Leavitt algebras.We reconsider the Leavitt algebras L K (1 , n ) for n ≥
2, the motivating examples of Leavitt pathalgebras. Fix n and K , and let R denote L K (1 , n ). By construction we have R R ∼ = R R n asleft R -modules; so by taking endomorphism rings and using the standard representation of theseendomorphism rings as matrix rings, we get R ∼ = M n ( R ) as K -algebras. Indeed, since R R ∼ = R R j ( n − for all j ∈ N , we similarly get R ∼ = M j ( n − ( R ) as K -algebras for all j ∈ N . Nowstarting from a different point of view: once we have established a ring isomorphism S ∼ = M ℓ ( S ) forsome ring S and some ℓ ∈ N , by taking ℓ × ℓ matrix rings of both sides t times, we get S ∼ = M ℓ t ( S ) forany t ∈ N . In particular, we have R ∼ = M n t ( R ) for all t ∈ N ; indeed, using the previous observation,we have more generally that R ∼ = M (1+ j ( n − t ( R ) as K -algebras for all j, t ∈ N .The question arises: if R = L K (1 , n ) is isomorphic as K -algebras to some p × p matrix ring overitself, must p be an integer of the form (1 + j ( n − t ? It is not hard to give an example where theanswer is negative: one can show (by explicitly writing down matrices which multiply correctly)that R = L K (1 ,
4) has R ∼ = M ( R ), and 2 is clearly not of the indicated form when n = 4. But ananalysis of this particular case leads easily to the observation that if d | n t for some t ∈ N , then R ∼ = M d ( R ) (by an explicitly described isomorphism).The upshot of the previous observations is the natural question:Given n ∈ N , for which d ∈ N is L K (1 , n ) ∼ = M d ( L K (1 , n )) as K -algebras?The analogous question was posed for matrix rings over the Cuntz algebras O n in [84]: given n ∈ N ,for which d ∈ N is O n ∼ = M d ( O n ) as C ∗ -algebras? The resolution of this analogous question requiredmany years of effort. In the end, the solution may be obtained as a consequence of the KirchbergPhillips Theorem: O n ∼ = M d ( O n ) if and only if g . c . d . ( d, n −
1) = 1. So while the C ∗ -algebraquestion was resolved for matrices over the Cuntz algebras, the solution did not shed any light onthe analogous Leavitt algebra question, both because the C ∗ -solution required analytic tools, andbecause it did not produce an explicit isomorphism between the germane algebras.An easy consequence of [76, Theorem 5] is that, when g . c . d . ( d, n − >
1, then L K (1 , n ) =M d ( L K (1 , n )). With this and the Cuntz algebra result in hand, it made sense to conjecture that L K (1 , n ) ∼ = M d ( L K (1 , n )) if and only if g . c . d . ( d, n −
1) = 1. Clearly if d | n t for some t ∈ N theng . c . d . ( d, n −
1) = 1, so that the conjecture is validated in this situation. The key idea was toexplicitly produce an isomorphism in situations more general than this. The method of attack wasclear: one reaches the desired conclusion by finding a subset of M d ( L K (1 , n )) of size 2 n which bothbehaves as in ( † ), and generates M d ( L K (1 , n )) as a K -algebra. EAVITT PATH ALGEBRAS: THE FIRST DECADE 21
The smallest pair d, n for which g . c . d . ( d, n −
1) = 1 but d n t for any t ∈ N is the case d = 3 , n = 5. Finding a subset of M ( L K (1 , · † ) is not hard;for instance, by (somewhat) mimicking the process used in the d | n t case, one is led to considerthese five matrices in M ( L K (1 , x x x , x x , x x x x x , x x x x x , x x x , together with their “duals” y y y , y y
00 0 10 0 0 , y y y y y , y y y y y , y y y . Although these ten matrices satisfy ( † ), they do not generate all of M ( L K (1 , e , , for example).The breakthrough came from a process which involves viewing matrices over Leavitt algebras asLeavitt path algebras for various graphs, and then manipulating the underlying graphs appropri-ately. This process led to the consideration of the following (very similar, yet) different set of fivematrices in M ( L K (1 , x x x , x x , x x x x x , x x x x x , x x x , together with their duals y y y , y y
00 0 10 0 0 , y y y y y , y y y y y , y y y . The only differences between the two sets of ten matrices lie in the fifth and tenth matrices, wheretwo of the entries have been interchanged. It is now not hard to show that this second set often matrices satisfies ( † ), and generates M ( L K (1 , K -algebra. The underlying idea whichprompted the interchange of entries is purely number-theoretic, and is fully described in Appendix2. In short, the integer 3 is used to partition the set { , , , , } into the subsets { , } ⊔ { , , } ;then, in order to build the first five matrices of this second set, one inserts monomials having left-most factor x t into row i in such a way that i and t are in the same subset with respect to thispartition. So putting the term x in row 1 and x in row 2 (as is done in the fifth matrix of the firstdisplayed set) will not work; on the other hand, putting x in row 1 and x in row 2 is consistentwith this partition, and leads to a collection with the desired properties. Once this observation wasmade, the generalization to arbitrary d, n was not overly difficult. Theorem 3.10. [5, Theorems 4.14 and 5.12]
Let n ∈ N and let K be any field. Then L K (1 , n ) ∼ = M d ( L K (1 , n )) ⇐⇒ g . c . d . ( d, n −
1) = 1 . More generally, M d ( L K (1 , n )) ∼ = M d ′ ( L K (1 , n )) ⇐⇒ g . c . d . ( d, n −
1) = g . c . d . ( d ′ , n − . Moreover, when g . c . d . ( d, n −
1) = g . c . d . ( d ′ , n − , an isomorphism M d ( L K (1 , n )) → M d ′ ( L K (1 , n )) can be explicitly described. There are two historically important consequences of the explicit construction of the isomorphismswhich yield Theorem 3.10. First, this context is one of the few places where a result from oneside of the graph algebra universe yields a result in the other. Specifically, when K = C andg . c . d . ( d, n −
1) = 1, the explicit nature of an isomorphism L C (1 , n ) ∼ = M d ( L C (1 , n )) constructedin the proof of Theorem 3.10 allows (by a straightforward completion process) for the explicit construction of an isomorphism O n ∼ = M d ( O n ). (The description of such an explicit isomorphismcame as more than a bit of a surprise to some researchers in the C ∗ -community.) Second, the explicitconstruction led to the resolution of a longstanding question in group theory. In the mid 1970’s, G.Higman produced, for each pair r, n ∈ N with n ≥
2, an infinite, finitely presented simple group,denoted G + n,r . (The groups G + n,r are called the Higman-Thompson groups.) Higman was able toestablish some sufficient conditions regarding isomorphisms between these groups, but did not havea complete classification. However, in 2011, Enrique Pardo showed how the construction given inthe proof of Theorem 3.10 could be brought to bear in this regard.
Theorem 3.11. [82, Theorem 3.6] G + n,r ∼ = G + m,s ⇐⇒ m = n and g . c . d . ( r, n −
1) = g . c . d . ( s, n − . Sketch of Proof.
The (= ⇒ ) direction was already known by Higman. Conversely, one first showsthat G + n,ℓ can be realized as an appropriate subgroup of the invertible elements of M ℓ ( L C (1 , n )) forany ℓ ∈ N . Then one verifies that the explicit isomorphism from M r ( L C (1 , n )) to M s ( L C (1 , n ))provided in the proof of Theorem 3.10 takes G + n,r onto G + n,s . ✷ For any three positive integers t, n, r (with n ≥ tV n,r )which can be viewed as a t -dimensional analog of the Higman-Thompson group, in that 1 V n,r ∼ = G + n,r .(The groups tV n,r are called the Brin-Higman-Thompson groups.) On the other hand, for t a positiveinteger and n ≥
2, one may consider the t -fold tensor product algebra L K (1 , n ) ⊗ t of L K (1 , n ) withitself t times. (We will more fully consider such tensor products in the following subsection.) In[60], Dicks and Mart´ınez-P´erez beautifully generalize Pardo’s Theorem 3.11 by showing that tV n,r is isomorphic to an appropriate subgroup of the invertible elements of M r ( L K (1 , n ) ⊗ t ) (specifically,the positive unitaries), and subsequently use this isomorphism to establish that tV n,r ∼ = t ′ V n ′ ,r ′ if and only if t = t ′ , n = n ′ , and g . c . d . ( r, n −
1) = g . c . d . ( r ′ , n ′ − . Along the way, Dicks andMart´ınez-P´erez present a streamlined, somewhat more intuitive proof of Theorem 3.10.3.5.
Tensor products of Leavitt path algebras.
Of fundamental importance in the theoryof graph C ∗ -algebras is the fact that O ⊗ O ∼ = O (homeomorphically). This isomorphism isnot explicitly described; rather, it follows (originally) from some deep work done by Elliott (andstreamlined in [88]). The isomorphism O ⊗ O ∼ = O is utilized in the proof of the KirchbergPhillips Theorem. (The C ∗ -algebra O is nuclear, so that there is no ambiguity in forming thistensor product.)In the context of the previous paragraph, together with the Rosetta Stone discussion, it is thennatural to ask: is L K (1 , ⊗ K L K (1 , ∼ = L K (1 , in the negative came in early 2011, in the form of three different approaches by threedifferent investigative teams.The first proof (unpublished), offered by Warren Dicks, utilized a classical result of Cartan andEilenberg [53, Theorem X1.3.1], which yields that the flat dimension of a tensor product is atleast the sum of the flat dimensions of the two algebras. By Theorem 1.15 ′ , the global dimensionof L K (1 ,
2) (indeed, of any Leavitt path algebra) is at most 1. (
Global dimension at most
EAVITT PATH ALGEBRAS: THE FIRST DECADE 23 equivalent to hereditary .) Consequently, the flat dimension of a Leavitt path algebra L K ( E ) equals1 precisely when L K ( E ) is not von Neumann regular (i.e., when there are L K ( E )-modules whichare not flat). But it had been shown in [16] that if E is a graph containing at least one cycle,then L K ( E ) is not von Neumann regular, so, in particular, L K (1 , ∼ = L K ( R ) is not von Neumannregular. So the flat dimension, and therefore also the global dimension, of L K (1 , ⊗ K L K (1 , L K (1 , ⊗ L K (1 ,
2) cannot be a Leavitt path algebra (again using Theorem1.15 ′ ), and so can’t be isomorphic to L K (1 , L K (1 , M (involving functions on [0 , ⊆ R having finite support in Q of the form n/ j ), and showed that the left L K (1 , ⊗ L K (1 , M ⊗ M has projective dimension 2, so that L K (1 , ⊗ L K (1 ,
2) has global dimension at least 2, andthus (arguing as did Dicks) cannot be isomorphic to any Leavitt path algebra.The third approach to verifying that L K (1 , ⊗ L K (1 , = L K (1 ,
2) is the most general of thethree. Utilizing Hochschild homology, Ara and Corti˜nas in [28] showed (among many other things)the following, from which the result of interest follows immediately.
Theorem 3.12.
Suppose { E i } mi =1 and { F j } nj =1 are finite graphs, each containing at least one cycle,and let K be any field. If ⊗ mi =1 L K ( E i ) is Morita equivalent to ⊗ nj =1 L K ( F j ) , then m = n . Two currently unresolved questions about the tensor products of Leavitt path algebras will begiven in Section 6.3.6.
Some additional internal / multiplicative properties of Leavitt path algebras.
Weconclude the section by presenting five additional multiplicative properties of Leavitt path algebras:primeness; the center; Gelfand Kirillov dimension; wreath products; and the simplicity of thecorresponding bracket Lie algebra.A ring R is prime in case for any two-sided ideals I, J of R , if IJ = { } then I = { } or J = { } .A graph E is called downward directed if, for any two vertices v, w ∈ E , there exists a vertex u ∈ E for which v ≥ u and w ≥ u . Theorem 3.13. [44]
Let E be any graph and K any field. Then L K ( E ) is prime if and only if E is downward directed.Sketch of Proof. ( ⇒ ) If R denotes L K ( E ), and v, w ∈ E , then the ideals RvR and
RwR areeach nonzero, so that
RvRRwR = { } , so that vRw = { } , which yields a nonzero element of theform αβ ∗ with s ( α ) = v , and r ( β ∗ ) = s ( β ) = w , so that u = r ( α ) has the desired property.( ⇐ ) The converse can be proved ‘elementwise’, but it is easier to invoke [80, Proposition 5.2.6(1)],which implies that for a Z -graded ring, primeness is equivalent to graded primeness. So we needonly check that if I, J are nonzero graded ideals, then IJ = { } . But by Proposition 3.4 (or itsgeneralization Theorem 5.4 given below), any nonzero graded ideal contains a vertex; so if v ∈ E ∩ I and w ∈ E ∩ J , and u ∈ E with v ≥ u and w ≥ u , then 0 = u = u ∈ IJ . ✷ For a ring R , the center Z ( R ) = { r ∈ R | rx = xr for all x ∈ R } . It is well-known that Z (M n ( K )) = K · I n (where I n denotes the identity matrix in M n ( K )). Additionally, this easilyyields that the center of M N ( K ) is { } . The following result includes these observations as specificcases. Theorem 3.14. ( [41] ) Let E be a row-finite graph. Suppose L K ( E ) is simple (see Theorem 1.20).If E is finite, then Z ( L K ( E )) = K · L K ( E ) . If E is infinite, then Z ( L K ( E )) = { } . For a K -algebra A , the Gelfand-Kirillov Dimension GK K ( A ) is an algebraic invariant of A which,loosely speaking, measures how far A is from being finite dimensional. (Finite dimensional algebrashave GK dimension 0. On the other hand, the free associative K -algebra on two generators hasGK dimension ∞ . Such an algebra is said to have exponential growth ; otherwise, the algebra has polynomially bounded growth . See e.g. [72] for a full description.) If C and C ′ are two disjoint cycles(i.e., Vert( C ) ∩ Vert( C ′ ) = ∅ ), the symbol C ⇒ C ′ indicates that there is a path which starts inVert( C ) and ends in Vert( C ′ ). A sequence of disjoint cycles C , ..., C k is a chain of length k in case C ⇒ · · · ⇒ C k . Let d denote the maximal length of a chain of cycles in E , and let d denote themaximal length of a chain of cycles each of which has an exit. Theorem 3.15. [21, Theorem 5]
Let E be a finite graph and K any field.(1) L K ( E ) has exponential growth if and only if there exist a pair of distinct cycles in E whichare not disjoint.(2) In case L K ( E ) has polynomially bounded growth, then the GK dimension of L K ( E ) is max(2 d − , d ) . Further results regarding Leavitt path algebras of polynomially bounded growth, and of theautomorphism groups of some specific such algebras, are presented in [24].For a countable dimensional K -algebra C and ring-theoretic property P , an affinization of A withrespect to P is an embedding of C in a finitely generated (i.e., affine) K -algebra D , for which, if C has P , then so does D .Let E be a row-finite graph and A any associative K -algebra. In [20], the authors present theconstruction of the wreath product , denoted A wr L K ( E ). In case W is a hereditary saturatedsubset of E , then the wreath product construction allows for the realization of L K ( E ) as thewreath product of two Leavitt path algebras, namely, as L K ( W ) wr L K ( E/W ). Furthermore, let T be the Toeplitz graph of Example 2.14. Then the wreath product A wr L ( T ) is isomorphic to a K -algebra of the form K [ x, x − ] + M N ( A ) (with multiplication explicitly described). This algebracan then be embedded in an algebra of the form K [ x, x − ] + RCFM N ( A ), where RCFM N ( A ) is the(unital) ring of those N × N matrices with entries in A , for which each row and each column containsat most finitely many nonzero entries. One may then build in a natural way an affine K -algebra B , generated by four elements, for which K [ x, x − ] + M N ( A ) ⊂ B ⊂ K [ x, x − ] + RCFM N ( A ). Theorem 3.16. [20]
For an associative K -algebra A let B be the affine K -algebra described above.(1) There exists a unital algebra A for which B is an affinization of K [ x, x − ] + M N ( A ) withrespect to the property non-nil Jacobson radical .(2) There exists a unital algebra A for which B is an affinization of K [ x, x − ] + M N ( A ) withrespect to the property non-nilpotent locally nilpotent radical . Both of the constructs mentioned in Theorem 3.16 give a systematic approach to what hadbeen previously longstanding ring-theoretic questions.For a K -algebra R , the corresponding bracket Lie algebra [ R, R ] consists of K -linear combinationsof elements of the form xy − yx with x, y ∈ R . [ R, R ] is a Lie algebra, with the usual bracketoperation. A Lie algebra L is called simple in case [ L, L ] = { } , and the only Lie ideals of L are { } and L . Let E be a finite graph, and write E = { v i | ≤ i ≤ m } . If v i is a not a sink, foreach 1 ≤ j ≤ m let a ij denote the number of edges e ∈ E such that s ( e ) = v i and r ( e ) = v j . Inthis situation, define B i = ( a ij ) − ǫ i ∈ Z m (where ǫ i is the element of Z m which is 1 in the i -thcoordinate, and zero elsewhere). On the other hand, if v i is a sink, let B i = (0) ∈ Z m . EAVITT PATH ALGEBRAS: THE FIRST DECADE 25
Theorem 3.17. [15, Theorem 23]
Let K be a field, and let E be a finite graph having at leasttwo vertices for which L K ( E ) is simple. Write E = { v , . . . , v m } , and for each ≤ i ≤ m let B i be as above. Then the Lie K -algebra [ L K ( E ) , L K ( E )] is simple if and only if (1 , . . . , span K { B , . . . , B m } . As it turns out, the condition given in Theorem 3.17 for the simplicity of [ L K ( E ) , L K ( E )] dependsnot only on the structure of E but also on the characteristic of K (see [15, Examples 28 and 29]).The K -dependence of a result about Leavitt path algebras is very much the exception. But for oneintriguing additional example, see Theorem 6.2 and the subsequent discussion.By introducing and utilizing the notion of a balloon over a subset of E , Alahmedi and Alsulamiare able to extend Theorem 3.17 to all row-finite graphs (specifically, the simplicity of L K ( E ) is notrequired); see [23, Theorem 2]. For instance, it is shown in [23] that the graph E given here • / / • (cid:5) (cid:5) E E has the property that the Lie algebra [ L K ( E ) , L K ( E )] is simple, even though the Leavitt pathalgebra L K ( E ) is not simple.In related work [22], the same two authors analyze the simplicity of the Lie algebra of ∗ -skew-symmetric elements of a Leavitt path algebra.4. Module-theoretic properties of Leavitt path algebras
The module theory of Leavitt path algebras has for the most part been focused on the structureof the finitely generated projective L K ( E )-modules, owing to the Ara / Moreno / Pardo RealizationTheorem (Theorem 1.15) describing V ( L K ( E )). In this section we take a closer look at the structureof these projectives, specifically, the purely infinite modules. Of central interest here is the questionof whether or not the analog of the Kirchberg Phillips Theorem (Theorem 1.12) holds for Leavittpath algebras; we present in Theorem 4.8 the Restricted Algebraic KP Theorem. We next look atthe structure of some simple (non-projective) L K ( E )-modules. We conclude by considering somemonoid-theoretic properties of V ( L K ( E )).4.1. Purely infinite simplicity.
We have seen that the cycle structure of the graph E , and theexistence of exits for those cycles, is a significant factor driving the algebraic structure of the Leavittpath algebra L K ( E ). We have also seen behavior in the Leavitt algebras L K (1 , n ) that at first glanceseems somewhat exotic: R R ∼ = R R n as left R -modules. Specifically, the module R R has the propertythat R R ∼ = R R ⊕ P where P = { } ; i.e., that R R has a nontrivial direct summand which is isomorphicto itself. ( Nontrivial here means that the complement of the direct summand is nonzero.) Thissame sort of behavior is manifest in L K ( E ) when E has cycles with exits. Remark 4.1.
Let α ∈ Path( E ) , and let r ( α ) = w . Then L K ( E ) αα ∗ ∼ = L K ( E ) w as left L K ( E ) -modules, since, if ϕ = ρ α : L K ( E ) αα ∗ → L K ( E ) w denotes right multiplication by α , then it is easyto show that ϕ − = ρ α ∗ . Proposition 4.2.
Suppose c is a cycle in a graph E based at a vertex v , and suppose e is an exit for c with s ( e ) = v . Then the left L K ( E ) -module L K ( E ) v has a nontrivial direct summand isomorphicto itself.Proof. Clearly L K ( E ) v = L K ( E ) cc ∗ + L K ( E )( v − cc ∗ ). But the sum is direct: if xcc ∗ = y ( v − cc ∗ )for x, y ∈ L K ( E ), then multiplying both sides on the right by cc ∗ yields xcc ∗ = y ( cc ∗ − cc ∗ ) = 0 . That L K ( E ) v ∼ = L K ( E ) cc ∗ as left L K ( E )-modules follows from the previous Remark. Since e is an exit for c we have e ∗ c = 0 by (CK1). Now to show that the complement L K ( E )( v − cc ∗ ) is nonzero,assume to the contrary that v − cc ∗ = 0. Multiplying both sides on the left by ee ∗ gives ee ∗ − ee ∗ = 0, which is impossible. ✷ A left R -module M is called infinite in case M ∼ = M ⊕ N with N = { } . An idempotent x ∈ R iscalled infinite in case Rx is infinite. The ring R is called purely infinite simple in case R is simple,and each nonzero left ideal of R contains an infinite idempotent. Purely infinite simple rings werefirst introduced in [35]; the idea was born in the context of C ∗ -algebras. Clearly a purely infinitemodule can satisfy neither of the two chain conditions, nor can it have finite uniform dimension.With the Simplicity Theorem in hand, and with Proposition 4.2 as guidance, some medium-leveleffort yields the following. Theorem 4.3. The Purely Infinite Simplicity Theorem [8]
Let E be a row-finite graph and K any field. Then L K ( E ) is purely infinite simple if and only if L K ( E ) is simple, and E containsat least one cycle. Theorems 3.1 and 4.3 together yield what is typically called the
Dichotomy for simple Leavittpath algebras : for L K ( E ) simple, either L K ( E ) is purely infinite simple, or L K ( E ) ∼ = M n ( K ) forsome n ∈ N .In the context of Leavitt path algebras, the purely infinite simple algebras play an especiallyintriguing role. For any ring S , the Grothendieck group K ( S ) is the universal group correspondingto the abelian monoid V ( S ). (Here universal means that any homomorphism from V ( S ) to anabelian group G necessarily factors through K ( S ).) When V ( S ) ∼ = Z + (as is often the case ingeneral, e.g., when S is a field or S = Z ), then one gets K ( S ) ∼ = Z , by “adding in the negatives”.As it turns out, however, if S is purely infinite simple, then V ( S ) \ { [0] } is a group, precisely K ( S ).This is perhaps counterintuitive at first glance: although V ( S ) has an identity element (namely,[0]), there still remains an identity element once [0] is eliminated. For instance, if R = L K (1 , n ),then R ⊕ R n − = R n ∼ = R . Using this, it’s trivial to conclude that [ R n − ] is an identity element for V ( R ) \ { [0] } = { [ R ] , [ R ] , . . . , [ R n − ] } . The group V ( R ) \ { [0] } is clearly isomorphic to Z / ( n − Z .Although the converse is not true for arbitrary rings, when one restricts to the class of Leavittpath algebras, then the converse is true as well [83]: L K ( E ) is purely infinite simple if and only if V ( L K ( E )) \ { [0] } is a group (necessarily K ( L K ( E ))). Moreover, this group is easy to describe inthis situation. As is standard, for a finite graph E with E = { v , v , . . . , v n } , the incidence matrix of E is the | E | × | E | Z + -valued matrix A E , where A E ( i, j ) equals the number of edges e for which s ( e ) = v i and r ( e ) = v j . By interpreting the (CK2) relation as it plays out in V ( L K ( E )), one gets Proposition 4.4.
Let E be a finite graph, with | E | = n . Suppose L K ( E ) is purely infinite simple.Then K ( L K ( E )) ∼ = Z n / ( I n − A E ) Z n , where I n denotes the n × n identity matrix. In other words, when L K ( E ) is purely infinite simple, then K ( L K ( E )) is the cokernel of thelinear transformation I n − A E : Z n → Z n induced by matrix multiplication.As an easy example of how this plays out in an already-familiar situation, suppose E = R m , thegraph with one vertex and m loops. Then A E = ( m ), so I − A E is the 1 × − m ), and K ( L K ( E )) ∼ = Z / (1 − m ) Z = Z / ( m − Z , as we’ve seen previously. EAVITT PATH ALGEBRAS: THE FIRST DECADE 27
Towards a Classification Theorem for purely infinite simple Leavitt path algebras.
In many endeavors in which an object from one class is associated to an object in another, a fun-damental question is to identify the stalks of the process; that is, determine which objects from thefirst class correspond to the same object in the second. Asked in the current context: if two graphs
E, F produce the “same” Leavitt path algebra (up to isomorphism, or up to Morita equivalence, orup to some other ring-theoretic invariant), can anything be said about the relationship between E and F ? As seen in Theorem 3.1, if E and F are finite acyclic graphs for which L K ( E ) ∼ = L K ( F ),then E and F have the same number of sinks, and the same number of directed paths ending atthose sinks. (An additional easy consequence of Theorem 3.1 is that if L K ( E ) is Morita equivalentto L K ( F ), then E and F have the same number of sinks.)We spend some time here investigating this question in the context of purely infinite simpleLeavitt path algebras. The reason is twofold: this investigation plays up an important relationshipbetween Leavitt path algebras and symbolic dynamics, and also provides the foundation for muchof the current research focus in Leavitt path algebras. The discussion here will be quite broad andintuitive; for details, the standard reference is [78].For a finite directed graph E , one defines the notion of a “flow” (essentially, “flow of information”)through the graph. Two graphs E and F are “flow equivalent” in case the collection of flows through E match up appropriately with the collection of flows through F . Two matrices with entries in Z + are called flow equivalent in case the directed graphs corresponding to the two matrices are flowequivalent. The directed graph E (or the corresponding incidence matrix A E ) is called(1) irreducible if for any pair v, w ∈ E there is a path from v to w ;(2) essential if there are neither sources nor sinks in E ; and(3) trivial if E consists of a single cycle with no other vertices or edges.A deep, fundamental result in flow dynamics is Franks’ Theorem. [64] Suppose that A and B are non-negative irreducible essential nontrivialsquare integer matrices. Then A and B are flow equivalent if and only if Z n / ( I n − A ) Z n ∼ = Z m / ( I m − B ) Z m and det( I n − A ) = det( I m − B ) . There are a number of ways to systematically modify a directed graph. As an intuitive example, expansion at v modifies the graph E to the graph E v as indicated here. E ❅❅❅❅❅❅❅ • v > > ⑦⑦⑦⑦⑦⑦⑦ ❅❅❅❅❅❅❅❅ > > ⑦⑦⑦⑦⑦⑦⑦⑦ E v (cid:31) (cid:31) ❄❄❄❄❄❄❄ • v f / / • v ∗ > > ⑥⑥⑥⑥⑥⑥⑥ ❆❆❆❆❆❆❆❆❆ ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ It can easily be shown that the graphs E and E v are flow equivalent. In a similar manner, one maydescribe five more systematic modifications of a graph (each having the property that the originalgraph is flow equivalent to the modified graph): contraction (the inverse of expansion); out-split ,as well as its inverse out-amalgamation ; and in-split , as well as its inverse in-amalgamation . Thespecific descriptions of these “graph moves” are given in Appendix 3.The second deep, fundamental theorem germane to the current discussion is The Parry / Sullivan Theorem.
Two finite directed graphs are flow equivalent if and only ifone can be gotten from the other by a sequence of transformations involving these six graph moves.Combining Franks’ Theorem with the Parry / Sullivan Theorem, we get
Theorem 4.5.
Suppose E and F are irreducible essential nontrivial graphs. Then Z n / ( I n − A E ) Z n ∼ = Z m / ( I m − A F ) Z m and det( I − A E ) = det( I − A F ) if and only if E can be obtained from F by somesequence of graph moves, with each move one of the six types described above. We are now in position to present the (miraculous?) bridge between the ideas from flow dynamicsand those of Leavitt path algebras. First, using the Purely Infinite Simplicity Theorem (Theorem4.3) and some straightforward graph theory, it is not hard to show that E is irreducible, essential,and nontrivial if and only if E has no sources and L K ( E ) is purely infinite simple. Next, Proposition 4.6.
Suppose E is a graph for which L K ( E ) is purely infinite simple. Suppose F isgotten from E by doing one of the six aforementioned graph moves. Then L K ( E ) and L K ( F ) areMorita equivalent. In particular, L K ( F ) is purely infinite simple. In addition, if v is a source in E , and F is gotten from E by eliminating v and all edges e ∈ E having s ( e ) = v , then L K ( E ) and L K ( F ) are Morita equivalent.Sketch of Proof. It is not hard to show that an isomorphic copy of L K ( E ) can be viewed as a(necessarily full, by simplicity) corner of L K ( G ) (or vice-versa), where E and G are related by oneof the graph moves. ✷ The previous discussion yields the first of two desired results.
Theorem 4.7.
Let E and F be finite graphs and K any field. Suppose L K ( E ) and L K ( F ) arepurely infinite simple. If K ( L K ( E )) ∼ = K ( L K ( F )) and det( I − A E ) = det( I − A F ) , then L K ( E ) and L K ( F ) are Morita equivalent.Sketch of Proof. Suppose E and/or F have sources; then using Proposition 4.6 we may constructgraphs E ′ and F ′ for which L K ( E ′ ) and L K ( F ′ ) are purely infinite simple, L K ( E ) is Morita equiva-lent to L K ( E ′ ), and L K ( F ) is Morita equivalent to L K ( F ′ ), where E ′ and F ′ have no sources. Butsince Morita equivalent rings have isomorphic K groups, and because (it’s straightforward to showthat) det( I − A E ) = det( I − A E ′ ) and det( I − A F ) = det( I − A F ′ ), we have that the hypotheses ofTheorem 4.5 are satisfied for E ′ and F ′ . Thus F ′ can be gotten from E ′ by a sequence of appropriategraph moves. But again invoking Proposition 4.6, each of these moves preserves Morita equivalence.So L K ( E ′ ) is Morita equivalent to L K ( F ′ ), and the result follows. ✷ The third deep, fundamental result of interest here is
Huang’s Theorem.
Suppose L K ( E ) is Morita equivalent to L K ( F ). Further, suppose there is some isomorphism ϕ : K ( L K ( E )) → K ( L K ( F )) for which ϕ ([ L K ( E )]) = [ L K ( F )]. Then there issome Morita equivalence Φ : L K ( E ) − Mod → L K ( F ) − Mod for which Φ | K ( L K ( E )) = ϕ. Consequently:
Theorem 4.8. The Restricted Algebraic Kirchberg Phillips Theorem. [13, Corollary 2.7]
Let E and F be finite graphs and K any field. Suppose L K ( E ) and L K ( F ) are purely infinite simple.If K ( L K ( E )) ∼ = K ( L K ( F )) via an isomorphism for which [ L K ( E )] [ L K ( F )] , and det( I − A E ) = det( I − A F ) , then L K ( E ) ∼ = L K ( F ) . EAVITT PATH ALGEBRAS: THE FIRST DECADE 29
Sketch of Proof.
For any Morita equivalence Φ : R − M od → S − M od , if Φ( R R ) = S S , then R ∼ = End R ( R R ) ∼ = End S (Φ( R R )) ∼ = End S ( S S ) ∼ = S as rings. Now apply Theorem 4.7 together withHuang’s Theorem. ✷ As an example of how the Restricted Algebraic KP Theorem can be implemented, let E be thegraph • (cid:31) (cid:31) ❅❅❅❅❅❅❅ • ? ? ⑧⑧⑧⑧⑧⑧⑧ ; ; • Q Q o o l l Then using the description provided in Proposition 4.4, we get K ( L K ( E )) ∼ = Z / Z ; moreover,under this isomorphism, [ L K ( E )]
1. Easily we get det( I − A E ) = − <
0. But the Leavitt pathalgebra L K ( R ) ∼ = L K (1 ,
4) has precisely the same data associated with it, so we conclude that L K ( E ) ∼ = L K (1 , . In Section 6 we describe how the Restricted Algebraic Kirchberg Phillips Theorem has beenacting as a springboard for much of the current research energy in the subject.4.3.
Simple L K ( E ) -modules. We now move our focus on L K ( E )-modules from projectives tosimples.Let p be an infinite path in E ; that is, p is a sequence e e e · · · , where e i ∈ E for all i ∈ N , andfor which s ( e i +1 ) = r ( e i ) for all i ∈ N . (N.b.: an infinite path in E is not an element of Path( E ),nor of the Leavitt path algebra L K ( E ).) The set of infinite paths in E is denoted by E ∞ . For p = e e e · · · ∈ E ∞ and n ∈ N , p >n denotes the infinite path e n +1 e n +2 · · · .Let c be a closed path in E . Then ccc · · · is an infinite path in E , denoted by c ∞ , and called a cyclic infinite path. A closed path d is irreducible in case d cannot be written as e j for any closedpath e and j >
1. For any closed path c there exists an irreducible d for which c = d n ; then c ∞ = d ∞ as elements of E ∞ .For p, q ∈ E ∞ , p and q are tail equivalent (written p ∼ q ) in case there exist integers m, n forwhich p >m = q >n (i.e., in case p and q eventually become the same infinite path). For p ∈ E ∞ ,[ p ] denotes the ∼ equivalence class of p . An element p of E ∞ is rational in case p ∼ c ∞ for someirreducible closed path c ; otherwise p is irrational . For instance, in R = • ve f h h ,q = ef ef f ef f f ef f f f e · · · is an irrational infinite path. In any graph E for which there exists avertex having two distinct irreducible closed paths based at that vertex, it is not hard to show thatthere are uncountably many irrational infinite paths in E ∞ . Additionally, there are infinitely manyirreducible paths in such a situation (and thus infinitely many tail-inequivalent infinite rationalpaths); for instance, any path of the form ef i for i ∈ Z + is irreducible in R . Definition 4.9.
Let p be an infinite path in the graph E , and let K be any field. Let V [ p ] denote the K -vector space having basis [ p ], consisting of the distinct elements of E ∞ which are tail-equivalentto p . For v ∈ E , e ∈ E , and q = f f f · · · ∈ [ p ], define v · q = ( q if v = s ( f )0 otherwise, e · q = ( eq if r ( e ) = s ( f )0 otherwise,and e ∗ · q = ( τ > ( q ) if e = f Then the K -linear extension of this action to all of V [ p ] gives a left L K ( E )-module structure on V [ p ] . Theorem 4.10. ( [54, Theorem 3.3] ). Let E be any graph and K any field. Let p ∈ E ∞ . Then theleft L K ( E ) -module V [ p ] described in Definition 4.9 is simple. Moreover, if p, q ∈ E ∞ , then V [ p ] ∼ = V [ q ] as left L K ( E ) -modules if and only if p ∼ q , which happens precisely when V [ p ] = V [ q ] . A module of the form V [ p ] as in Theorem 4.10 is called a Chen simple L K ( E ) -module . In [39], Araand Rangaswamy describe those Leavitt path algebras L K ( E ) which admit at most countably manysimple left modules (Chen simples or otherwise) up to isomorphism. Building on an observationmade prior to Definition 4.9, one sees that the structure of K plays a role in this result, in thatwhen L K ( E ) has this property, and E contains cycles, then necessarily K must be countable.It is possible to explicitly describe projective resolutions for the Chen simple modules. Let β = e e · · · e n ∈ Path( E ) or β = e e · · · ∈ E ∞ . For each i ≥ i ≤ n − β = e · · · e n ∈ Path( E )), let X i ( β ) = { f ∈ E | s ( f ) = s ( e i +1 ) , and f = e i +1 } , and let J i ( β ) bethe left ideal P f ∈ X i ( β ) L K ( E ) f ∗ β ∗ i of L K ( E ). The following explicit description of projective resolu-tions of Chen simple modules follows from an elementwise analysis of the kernel of the appropriateright-multiplication map. (For an element m in a left L K ( E )-module M , and any left ideal I of L K ( E ), ρ m : I → M denotes right multiplication by m .) Theorem 4.11. [14]
Let E be any graph and K any field.(1) Let c be an irreducible closed path in E , with v = s ( c ) . Then V [ c ∞ ] is finitely presented (infact, singly presented); a projective resolution of V [ c ∞ ] is given by / / L K ( E ) v ρ c − v / / L K ( E ) v ρ c ∞ / / V [ c ∞ ] / / . (2) Let p ∈ E ∞ be an irrational infinite path in E for which no element of Vert( p ) is an infiniteemitter. Then / / ⊕ ∞ i =0 J i ( p ) / / L K ( E ) v ρ p / / V [ p ] / / is a projective resolution of V [ p ] . In particular, V [ p ] is finitely presented if and only if X i ( p ) isnonempty for at most finitely many i ∈ Z + . Theorem 4.11 sharpens and clarifies some of the results of [38]. The explicit description of pro-jective resolutions given in Theorem 4.11 can be used to (easily) show that V [ c ∞ ] is never projective,and that V [ p ] (for p irrational) is not projective when V [ p ] is not finitely presented (e.g., whenever E is a finite graph). Consequently, these two types of modules admit nontrivial extensions, some ofwhich are captured in the following result. Theorem 4.12. [14]
Let E be a finite graph and K any field. Let T be a Chen simple module.Denote by U ( T ) the set { v ∈ E | vT = { }} . For p ∈ E ∞ , denote by r ( X i ( p )) the set { r ( e i ) | e i ∈ X i ( p ) } .(1) Let d be an irreducible closed path in E with v = s ( d ) . Then Ext L K ( E ) ( V [ d ∞ ] , T ) = { } if and only if vT = { } . (2) Let p be an irrational infinite path in E . Then Ext L K ( E ) ( V [ p ] , T ) = { } if and only if r ( X i ( p )) ∩ U ( T ) = ∅ for infinitely many i ≥ . As a consequence of Theorem 4.12, whenever E is a graph containing at least one cycle, then(non-projective) indecomposable L K ( E )-modules of any desired finite length can be constructed.We close this subsection on simple L K ( E )-modules by noting that Rangaswamy [87] has given aconstruction of such modules arising from the infinite emitters v of E . EAVITT PATH ALGEBRAS: THE FIRST DECADE 31
Additional module-theoretic properties of L K ( E ) . The previous discussion in this sectionfirst focused on projective modules, then on non-projective simple modules, over Leavitt path alge-bras. We conclude the section by mentioning some monoid-theoretic properties of M = V ( L K ( E )).As the V -monoid of a ring, M is of course conical, and contains a distinguished element (as de-scribed prior to Theorem 1.6). But there are two important additional properties of V ( L K ( E )),both of which yield information about the decomposition of projective L K ( E )-modules.Suppose that M is a left R -module which admits two direct sum decompositions M = A ⊕ A = B ⊕ B . We ask whether there is necessarily some relationship between the two decompositions,indeed, whether there is some compatible “refinement” of these which allows for the systematicformation of each of the summands. More formally, suppose A ⊕ A = B ⊕ B as left R -modules.Then a refinement of this pair of direct sums consists of left R -modules M , M , M , and M ,for which: A = M ⊕ M , A = M ⊕ M ,B = M ⊕ M , B = M ⊕ M . A second type of decomposition of modules relates to cancellation of direct summands. Clearlyin general an isomorphism A ⊕ C ∼ = B ⊕ C of left R -modules need not imply A ∼ = B . A germaneexample here is this: if R = L K (1 , n ), and A = { } , B = R R n − , and C = R R , then we have A ⊕ C ∼ = B ⊕ C (since R R ∼ = R R n ), but obviously A = B . In various situations it is natural torequire a stronger relationship between such isomorphic direct sums, prior to trying to cancel C .One possible approach is as follows. A ring R is called separative in case it satisfies the followingproperty: If A, B, C ∈ V ( R ) satisfy A ⊕ C ∼ = B ⊕ C , and C is isomorphic to direct summandsof both A n and B n for some n ∈ N , then A ∼ = B . (Note that this additional condition obviouslyrenders moot the previous example.) Theorem 4.13.
Let E be a row-finite graph and K any field.1) [36, Proposition 4.4] The monoid M E is a refinement monoid. Consequently, V ( L K ( E )) is arefinement monoid.2) [36, Theorem 6.3] The monoid M E is separative. Consequently, the monoid V ( L K ( E )) , andthus the ring L K ( E ) , is separative.Sketch of Proof. (1) is established by a careful analysis of the generators and relations whichproduce the graph monoid M E . On the other hand, (2) follows in part from results of Brookfield[52] on primely generated refinement monoids. ✷ In fact, the class of primely generated refinement monoids satisfies many other nice cancellationproperties, e.g. unperforation . We will revisit refinement monoids at the end of Section 6.5.
Classes of algebras related to, or motivated by, Leavitt path algebras ofrow-finite graphs
Historically, Leavitt path algebras were first defined only in the context of row-finite graphs.Subsequently, the more general definition of Leavitt path algebras for countable graphs ([9]), andthen truly arbitrary graphs ([66]), appeared in the literature. The original notion of a Leavitt pathalgebra for row-finite graphs has been generalized in other ways as well, including: the constructionof Leavitt path algebras for separated graphs; Cohn path algebras; Kumjian-Pask algebras of higherranks graphs; Leavitt path rings; and more. In this section we give an overview of some of theseLeavitt-path-algebra-inspired structures.
Leavitt path algebras for arbitrary graphs.
Suppose E is a graph which contains aninfinite emitter v ; that is, the set s − ( v ) = { e ∈ E | s ( e ) = v } is infinite. Then in a purely ring-theoretic context, the symbol P e ∈ s − ( v ) ee ∗ , which would be the natural generalization of the (CK2)relation imposed at v , is not defined. Even in the analytic context of graph C ∗ -algebras, whereconvergence properties might allow for some sort of appropriate interpretation of an infinite sum,an expression of the form P e ∈ s − ( v ) s e s ∗ e proves to be problematic, in part owing to the fact that { s e s ∗ e | e ∈ s − ( v ) } is an infinite set of orthogonal projections.So, somewhat cavalierly, we simply choose not to invoke any (CK2)-like relation at infinite emit-ters. We recall that a vertex v ∈ E is regular in case 0 < | s − ( v ) | < ∞ . Definition 5.1.
Let E = ( E , E , s, r ) be any graph, and K any field. Let b E denote the extendedgraph of E . The Leavitt path K -algebra L K ( E ) is defined as the path K -algebra K b E , modulo therelations:(CK1) e ∗ e ′ = δ e,e ′ r ( e ) for all e, e ′ ∈ E .(CK2) v = P { e ∈ E | s ( e )= v } ee ∗ for every regular vertex v ∈ E .Equivalently, we may define L K ( E ) as the free associative K -algebra on generators E ⊔ E ⊔ ( E ) ∗ ,modulo the relations(1) vv ′ = δ v,v ′ v for all v, v ′ ∈ E . (2) s ( e ) e = er ( e ) = e for all e ∈ E . (3) r ( e ) e ∗ = e ∗ s ( e ) = e ∗ for all e ∈ E . (4) e ∗ e ′ = δ e,e ′ r ( e ) for all e, e ′ ∈ E .(5) v = P { e ∈ E | s ( e )= v } ee ∗ for every regular v ∈ E . So the definition of a Leavitt path algebra for arbitrary graphs is essentially word-for-word iden-tical to that for row-finite graphs (since “regular” and “non-sink” are identical properties in therow-finite case); there is simply no (CK2) relation imposed at any vertex which is the source vertexof infinitely many edges.The generalization from Leavitt path algebras of row-finite graphs to those of arbitrary graphswas achieved in two stages. Owing to the hypotheses typically placed on the corresponding graphC ∗ -algebras (in order to ensure separability), the initial extension for Leavitt path algebras wasto graphs having countably many vertices and edges. It is shown in [9] that the Leavitt pathalgebra of any such countable graph is Morita equivalent to the Leavitt path algebra of a suitablydefined row-finite graph, using the desingularization process. Subsequently, the foundational resultsregarding Leavitt path algebras for arbitrary graphs were presented in [66]. Among other things,Goodearl established a suitable definition and context for morphisms between graphs (so-called CK-morphisms ). He was then able to show that direct limits exist in the appropriately defined graphcategory (denoted
CKGr ), and that the functor L K from CKGr to the category of K -algebraspreserves direct limits.The generalization to Leavitt path algebras of arbitrary graphs (from those of row-finite graphs)indeed expands the Leavitt path algebra universe. For instance, it was shown in [17] that L K ( E ) isMorita equivalent to L K ( F ) for some row-finite graph F if and only if E contains no uncountableemitters (i.e., in case the set s − ( v ) is at most countable for each v ∈ E ). So, for instance, let I be an uncountable set, and let D I denote the graph consisting of two vertices v, w , and edges { e i | i ∈ I } , where s ( e i ) = v and r ( e i ) = w . Then L K ( D I ) is isomorphic to the (unital) K -algebragenerated by M I ( K ) ⊔{ Id } , where Id is the I × I identity matrix. So L K ( D I ) is not Morita equivalent(let alone, isomorphic) to the Leavitt path algebra of any row-finite graph. Similarly, if R c denotes EAVITT PATH ALGEBRAS: THE FIRST DECADE 33 the “rose with uncountably infinitely many petals” graph, then L K ( R c ) is not Morita equivalent to L K ( F ) for any row-finite graph F .In this expanded universe of Leavitt path algebras for arbitrary graphs, many of the resultsestablished in the row-finite case generalize verbatim, but many do not. One of the main differencesis that in the general case, we may pick up many new idempotents inside L K ( E ) for which thereare no counterparts in the row-finite case. For instance, let v ∈ E , and let e ∈ s − ( v ). Then theelement x = v − ee ∗ of L K ( E ) is easily shown to be an idempotent. If v is a regular vertex, then x = P f ∈ s − ( v ) ,f = e f f ∗ by the (CK2) relation. On the other hand, if v is an infinite emitter, then x has no such analogous representation.We recall the graph-theoretic ideas given in Notation 3.2: a subset X of E is hereditary in case,whenever v ∈ X and w ∈ E and v ≥ w , then w ∈ X ; X is saturated in case, whenever v ∈ E isregular and r ( s − ( v )) ⊆ X , then v ∈ X . Definition 5.2.
Let E be any graph, and let H be a hereditary subset of E . A vertex v ∈ E isa breaking vertex of H in case v is in the set B H = { v ∈ E \ H | | s − ( v ) | = ∞ and 0 < | s − ( v ) ∩ r − ( E \ H ) | < ∞} . In words, B H consists of those vertices which are infinite emitters, which do not belong to H , andfor which the ranges of the edges they emit are all, except for a finite (but nonzero) number, inside H . For v ∈ B H , define v H = v − X e ∈ s − ( v ) ∩ r − ( E \ H ) ee ∗ , and, for any subset S ⊆ B H , define S H = { v H | v ∈ S } .Of course a row-finite graph contains no breaking vertices, so that this concept does not play arole in the study of Leavitt path algebras arising from such graphs. Also, we note that both B E and B ∅ are empty. To help clarify the concept of breaking vertex, we offer the following example. Example 5.3.
Let C N be the infinite clock graph pictured here • u • u • ve O O e < < ③③③③③③③③ e / / e " " ❉❉❉❉❉❉❉❉ (cid:15) (cid:15) (cid:127) (cid:127) • u • u Let U denote the set { u i | i ∈ N } = C N \ { v } . Any subset of U is a hereditary subset of C N . Wenote also that, since saturation applies only to regular vertices, any subset of U is saturated as well.If H ⊆ U has U \ H infinite, or if H = U , then B H = ∅ . On the other hand, if U \ H is finite,then B H = { v } , and in this situation, v H = v − P { i | r ( e i ) ∈ U \ H } e i e ∗ i . It is clear that for any hereditary saturated subset H of a graph E , and for any S ⊆ B H , the ideal I ( H ∪ S H ) is a graded ideal, as it is generated by elements of L K ( E ) of degree zero. It turns out thatthis process generates all the graded ideals of L K ( E ). We denote by L gr ( L K ( E )) the collection oftwo-sided graded ideals of L K ( E ), and by T E the collection of pairs ( H, S ) where H is a hereditarysaturated subset of E , and S ⊆ B H . Theorem 5.4. [97, Theorem 5.7]
Let E be an arbitrary graph and K any field. Then there is abijection ϕ : L gr ( L K ( E )) → T E , given by I ( I ∩ E , S ) where S = { v ∈ B H | v H ∈ I } for H = I ∩ E . The inverse is given by ϕ − : T E → L gr ( L K ( E )) , via ( H, S ) I ( H ∪ S H ) . There is an appropriate lattice structure which can be defined in T E so that the map ϕ is a latticeisomorphism. In addition, there is a generalization of Theorem 5.4 to the lattice of all ideals of L K ( E ), see [6, Theorem 2.8.10].We close the subsection by presenting a result which is of interest in its own right (it provided asystematic approach to answering a decades-old question of Kaplansky), and which will reappearlater in the context of the Rosetta Stone. An algebra A is called left primitive in case A admitsa faithful simple left module. It was shown in [44] that for row-finite graphs, L K ( E ) is primitiveif and only if E is downward directed and satisfies Condition (L). However, the extension of thisresult to arbitrary graphs requires an extra condition. The graph E has the Countable SeparationProperty in case there exists a countable set S ⊆ E with the property that for every v ∈ E thereexists s ∈ S for which v ≥ s . Theorem 5.5. [10, Theorem 5.7]
Let E be an arbitrary graph and K any field. Then L K ( E ) isprimitive if and only if E is downward directed, E satisfies Condition (L), and E has the CountableSeparation Property. Leavitt path algebras of separated graphs.
The (CK2) condition imposed at any regularvertex in the definition of a Leavitt path algebra may be modified in various ways. Such is themotivation for the discussion in both this and the following subsection. All of these ideas appearin [31].In the (CK2) condition which appears in the definition of the Leavitt path algebra L K ( E ), theedges emanating from a given regular vertex v are treated as a single entity, and the relation v = P e ∈ s − ( v ) ee ∗ is imposed. More generally, one may partition the set s − ( v ) into disjoint nonemptysubsets, and then impose a (CK2)-type relation corresponding exactly to those subsets. Moreformally, a separated graph is a pair ( E, C ), where E is a graph, C = ⊔ v ∈ E C v , and, for each v ∈ E , C v is a partition of s − ( v ) (into pairwise disjoint nonempty subsets). (In case v is a sink, C v istaken to be the empty family of subsets of s − ( v ).) Definition 5.6.
Let E be any graph and K any field. Let b E denote the extended graph of E , and K b E the path K -algebra of b E . The Leavitt path algebra of the separated graph ( E, C ) with coefficientsin the field K is the quotient of K b E by the ideal generated by these two types of relations:(SCK1) for each X ∈ C , e ∗ f = δ e,f r ( e ) for all e, f ∈ X , and(SCK2) for each non-sink v ∈ E , v = P e ∈ X ee ∗ for every finite X ∈ C v . So the usual Leavitt path algebra L K ( E ) is exactly L K ( E, C ), where each C v is defined to bethe subset { s − ( v ) } if v is not a sink, and ∅ otherwise. Leavitt path algebras of separated graphsinclude a much wider class of algebras than those which arise as Leavitt path algebras in thestandard construction. For instance, the algebras of the form L K ( m, n ) for m ≥ L K ( E ) for any graph E . On the other hand, as shown in [31,Proposition 2.12], L K ( m, n ) ( m ≥
2) appears as a full corner of the Leavitt path algebra of anexplicitly described separated graph (having two vertices and m + n edges). In particular, L K ( m, n )is Morita equivalent to the Leavitt path algebra of a separated graph.Of significantly more importance is the following Bergman-like realization result, which showsthat the collection of Leavitt path algebras of separated graphs is extremely broad. EAVITT PATH ALGEBRAS: THE FIRST DECADE 35
Theorem 5.7. [31, Section 4]
Let M be any conical abelian monoid. Then there exists a graph E ,and partition C = ⊔ v ∈ E C v , for which V ( L K ( E, C )) ∼ = M . Consequently, V ( L K ( E, C )) need not share the separativity nor the refinement properties of thestandard Leavitt path algebras L K ( E ). Furthermore, the ideal structure of L K ( E, C ) is in generalsignificantly more complex than that of L K ( E ), but a description of the idempotent-generated idealscan be achieved (solely in terms of graph-theoretic information).5.3. Cohn path algebras.
In the previous subsection we saw one way to modify the (CK2)relation, namely, by imposing it on subsets of s − ( v ) for v ∈ E .A second way to modify the (CK2) relation is to simply eliminate it. Definition 5.8.
Let E be any graph and K any field. The Cohn path algebra C K ( E ) is the path K -algebra K b E of the extended graph of E , modulo the relation(CK1) e ∗ f = δ e,f r ( e ) for each e, f ∈ E . The terminology “Cohn path algebra” postdates the Leavitt path algebra terminology, and owesto the fact that for each n ≥
1, the algebra C K ( R n ) (for R n the rose with n petals graph) is preciselythe algebra U ,n described and investigated by Cohn in [55].Indeed, even the case n = 1 is of interest here: C K ( R ) is the unital K -algebra A generated by anelement e for which e ∗ e = 1 (and no other relation involving e ). Thus we get that C K ( R ) is exactlythe Jacobson algebra described in Example 2.14, so that (using the computation presented in thatExample), we have C K ( R ) ∼ = L K ( T ), the Leavitt path algebra of the Toeplitz graph. Pictorially, C K ( • ) ∼ = L K ( • / / • ) . This isomorphism between a Cohn path algebra and a Leavitt path algebra is not a coincidence.
Theorem 5.9. [6, Section 1.5]
Let E be any graph. Then there exists a graph F (which is explicitlyconstructed from E ) for which C K ( E ) ∼ = L K ( F ) . That is, every Cohn path algebra is isomorphic toa Leavitt path algebra. In particular, the explicit construction mentioned in Theorem 5.9 of the graph F from the graph E in case E = R yields that F = T . So although at first glance the Cohn path algebra constructionseems less restrictive than the Leavitt path algebra construction, the collection of algebras whicharise as C K ( E ) is (properly) contained in the collection of algebras which arise as L K ( E ). (One wayto see that the containment is proper is to note that the Cohn path algebra C K ( E ) has InvariantBasis Number for any finite graph E ; see [12].)One may view the Leavitt path algebras and Cohn path algebras as occupying the opposite endsof a spectrum: in the former, we impose the (CK2) relation at all (regular) vertices, while, in thelatter, we do not impose it at any of the vertices. The expected middle-ground construction may beformalized: if X is any subset of the regular vertices Reg( E ) of E , then the Cohn path K -algebrarelative to X , denoted C XK ( E ), is the algebra C K ( E ), modulo the (CK2) relation imposed only atthe vertices v ∈ X . So C K ( E ) = C ∅ K ( E ), while L K ( E ) = C Reg( E ) K ( E ). Theorem 5.9 generalizesappropriately from Cohn path algebras to relative Cohn path algebras.5.4. Additional constructions.
We close this section with a description of four additional Leavitt-path-algebra-inspired constructions.
Cohn-Leavitt algebras.
The following (not unexpected) mixing-and-matching of the Leavitt pathalgebras of separated graphs with the relative Cohn path algebras has been defined and studied in[31].
Definition 5.10.
Let ( E, C ) be a separated graph. Let C fin denote the subset of C consisting ofthose X for which | X | is finite. Let S be any subset of C fin . Denote by CL K ( E, C, S ) the quotient ofthe path K -algebra K b E , modulo the relations (SCK1) of Definition 5.6, together with the relations(SCK2) for the sets X ∈ S . CL K ( E, C, S ) is called the Cohn-Leavitt algebra of the triple ( E, C, S ) .Kumjian-Pask algebras. Any directed graph E = ( E , E , s, r ) may be viewed as a category Γ E ;the objects of Γ E are the vertices E , and, for each pair v, w ∈ E , the morphism set Hom Γ E ( v, w )consists of those elements of Path( E ) having source v and range w . Composition is concatenation.As well, the set Z + is a category with one object, and morphisms given by the elements of Z + ,where composition is addition. In this level of abstraction, the length map ℓ : Path( E ) → Z + is afunctor, which satisfies the following factorization property: if λ ∈ Path( E ) and ℓ ( λ ) = m + n , thenthere are unique µ, ν ∈ Path( E ) such that ℓ ( µ ) = m, ℓ ( ν ) = n , and λ = µν . Conversely, we mayview a category as the morphisms of the category, where the objects are identified with the identitymorphisms. Then any category Λ which admits a functor d : Λ → Z + having the factorizationproperty can be viewed as a directed graph E Λ in the expected way.With these observations as motivation, one defines a higher rank graph, as follows. Definition 5.11.
Let k be a positive integer. View the additive semigroup ( Z + ) k as a category withone object, and view a category as the morphisms of the category, where the objects are identifiedwith the identity morphisms. A graph of rank k (or simply a k -graph) is a countable category Λ , together with a functor d : Λ → ( Z + ) k , which satisfies the factorization property: if λ ∈ Λ and d ( λ ) = m + n for some m, n ∈ ( Z + ) k , then there exist unique µ, ν ∈ Λ such that d ( µ ) = m, d ( ν ) = n ,and λ = µν . (So the usual notion of a graph is a -graph in this more general context.)Given any k -graph (Λ , d ) and field K , one may define the Kumjian-Pask K -algebra KP K (Λ , d ) .(We omit the somewhat lengthy details of the construction; see [40] for the complete description.)In case k = 1 , KP K (Λ , d ) is the Leavitt path algebra L K ( E Λ ) .The regular algebra of a graph. The following construction should be viewed not as a methodto generalize the notion of Leavitt path algebra, but rather to use the properties of Leavitt pathalgebras as a tool to answer what at first glance seems to be an unrelated question. The “RealizationProblem for von Neumann Regular Rings” asks whether every countable conical refinement monoidcan be realized as the monoid V ( R ) for some von Neumann regular ring R . It was shown in [16]that the only von Neumann regular Leavitt path algebras are those associated to acyclic graphs,so it would initially seem that Leavitt path algebras would not be fertile ground in the context ofthe Realization Problem. Nonetheless, Ara and Brustenga developed an elegant construction whichprovides the key connection. Using the algebra of rational power series on E , and appropriatelocalization techniques ( inversion ), they showed how to construct a K -algebra Q K ( E ) with thefollowing properties. Theorem 5.12. [26, Theorem 4.2]
Let E be a finite graph and K any field. Then there exists a K -algebra Q K ( E ) for which:(1) there is an inclusion of algebras L K ( E ) ֒ → Q K ( E ) ,(2) Q K ( E ) is unital von Neumann regular, and(3) V ( L K ( E )) ∼ = V ( Q K ( E )) . Consequently, using the Realization Theorem (Theorem 1.15 ′ ), Theorem 5.12 yields that anymonoid which arises as the graph monoid M E for a finite graph E has a positive solution to theRealization Problem. This result represented (at the time) a significant broadening of the class of EAVITT PATH ALGEBRAS: THE FIRST DECADE 37 monoids for which the Realization Problem had a positive solution. The result extends relativelyeasily to row-finite graphs (see [26, Theorem 4.3]), with the proviso that Q K ( E ) need not be unitalin that generality. Non-field coefficients.
While nearly all of the energy expended on understand L K ( E ) has focusedon the graph E , one may also relax the requirement that the coefficients be taken from a field K .For a commutative unital ring R and graph E one may form the path ring RE of E with coefficientsin R in the expected way; it is then easy to see how to subsequently define the Leavitt path ring L R ( E ) of E with coefficients in R . While some of the results given when R is a field do not holdverbatim in the more general setting (e.g., the Simplicity Theorem), one can still understand muchof the structure of L R ( E ) in terms of the properties of E and R ; see e.g. [98].With these many generalizations of Leavitt path algebras having now been noted, a comment onthe extremely robust interplay between algebras and C ∗ -algebras is in order. In some situations,the C ∗ -ideas preceded the algebra ideas; in other situations, the opposite; and in still others, theideas were introduced simultaneously. • Leavitt [76] built the Leavitt algebras L C (1 , n ) (1962); subsequently, Cuntz [56] built their C ∗ -counterparts, the Cuntz algebras O n (1977). (Cuntz’s results were achieved independently from thework of Leavitt.) • Graph C ∗ -algebras of row-finite graphs were then introduced in [47] (2000); these in turnmotivated the definition of Leavitt path algebras of row-finite graphs in [7] and [36] (2005). • Graph C ∗ -algebras of countable graphs which contain infinite emitters were introduced in [63](2000); these motivated the definition of Leavitt path algebras of such graphs in [9] (2006). • Leavitt path algebras for arbitrary graphs were first given complete consideration in [66] (2009).The initial study of C ∗ -algebras corresponding to arbitrary graphs appears in [18] (2013), wherethis notion was utilized to give the first systematic construction of C ∗ -algebras which are prime butnot primitive. • C ∗ -algebras of higher rank graphs were formalized in [73] (2000); the corresponding Kumjian-Pask algebras were introduced in [40] (2014). • In the context of separated graphs, both Leavitt path algebras and graph C ∗ -algebras of theseobjects were introduced essentially simultaneously in the articles [30] (2011), and [31] (2012).6. Current lines of research in Leavitt path algebras
The Classification Question for purely infinite simple Leavitt path algebras, a.k.a.
The Algebraic Kirchberg Phillips Question . We start with what is generally agreed to be the mostcompelling unresolved question in the subject of Leavitt path algebras, stated concisely as:The Algebraic Kirchberg Phillips Question:
Can we drop the hypothesis on the determinants in Theorem 4.8?
More formally, the Algebraic KP Question is the following “Classification Question”. Let E and F be finite graphs, and K any field. Suppose L K ( E ) and L K ( F ) are purely infinite simple. If K ( L K ( E )) ∼ = K ( L K ( F )) via an isomorphism for which [ L K ( E )] [ L K ( F )] , is it necessarily thecase that L K ( E ) ∼ = L K ( F )?The name given to the Question derives from the previously mentioned Kirchberg Phillips Theo-rem for C ∗ -algebras (see the discussion prior to Theorem 1.12), which yields as a special case that if E and F are finite graphs, and if C ∗ ( E ) and C ∗ ( F ) are purely infinite simple graph C ∗ -algebras with K ( C ∗ ( E )) ∼ = K ( C ∗ ( F )) via an isomorphism for which [ C ∗ ( E )] [ C ∗ ( F )] , then C ∗ ( E ) ∼ = C ∗ ( F )(homeomorphically). In particular, the determinants of the appropriate matrices play no role.Intuitively, the Question asks whether or not the integer det( I − A E ) can be “seen” or “recovered”inside L K ( E ) as an isomorphism invariant. There is indeed a way to interpret det( I − A E ) in termsof the cycle structure of E , see e.g. [92]; but this interpretation has not (yet?) been useful in thiscontext.With the Restricted Algebraic Kirchberg Phillips Theorem having been established, there arethree possible answers to the Algebraic Kirchberg Phillips Question: No.
That is, if the two graphs E and F have det( I − A E ) = det( I − A F ), then L K ( E ) = L K ( F )for any field K . Yes.
That is, the existence of an isomorphism of the indicated type between the K groups issufficient to yield an isomorphism of the associated Leavitt path algebras, for any field K . Sometimes.
That is, for some pairs of graphs E and F , and/or for some fields K , the answer isNo, and for other pairs the answer is Yes.One of the elegant aspects of the Algebraic KP Question is that its answer will be interesting,regardless of which of the three possibilities turns out to be correct. If the answer is No , thenisomorphism classes of purely infinite simple Leavitt path algebras will match exactly the flowequivalences classes of the germane set of graphs, which would suggest that there is some deeper,as-of-yet-not-understood connection between the two subjects. If the answer is Yes , this wouldyield further compelling evidence for the existence of a Rosetta Stone, since then the Leavitt pathalgebra and graph C ∗ -algebra results would be exactly analogous. If the answer is Sometimes , then(in addition to providing quite a surprise to those of us working in the field) this would likelyrequire the development and utilization of a completely new set of tools in the subject. (Indeed,the
Sometimes answer might be the most interesting of the three.)Using a standard tool (the Smith Normal Form of an integer-valued matrix), it is not hard toshow that the cardinality of the group K ( L K ( E )) is | det( I − A E ) | in case K ( L K ( E )) is finite, andthe cardinality is infinite precisely when det( I − A E ) = 0 . So the Algebraic KP Question admits asomewhat more concise version: If the signs of det( I − A E ) and det( I − A F ) are different, is it thecase that L K ( E ) = L K ( F )?The analogous question about Morita equivalence asks whether or not we can drop the determi-nant hypothesis from Theorem 4.7. But the two questions will have the same answer: if isomorphic K groups yields Morita equivalence of the Leavitt path algebras, then the Morita equivalencetogether with Huang’s Theorem will yield isomorphism of the algebras. EAVITT PATH ALGEBRAS: THE FIRST DECADE 39
Suppose E is a finite graph for which L K ( E ) is purely infinite simple. There is a way to as-sociate with E a graph E − , for which L K ( E − ) is purely infinite simple, for which K ( L K ( E )) ∼ = K ( L K ( E − )), and for which det( I − A E ) = − det( I − A E − ). This is called the “Cuntz splice” process,which appends to a vertex v ∈ E two additional vertices and six additional edges, as shown herepictorially: E − = ((( E ))) • v • • $ $ d d $ $ d d (cid:1) (cid:1) b b . Although the isomorphism between K ( L K ( E )) and K ( L K ( E − )) need not in general send [1 L K ( E ) ]to [1 L K ( E − ) ], the Cuntz splice process allows us an easy way to produce many specific examples ofpairs of Leavitt path algebras to analyze in the context of the Algebraic KP Question. The most“basic” pair of such algebras arises from the following two graphs: E = • u ) ) * * • v (cid:8) (cid:8) j j and E = • u ) ) * * • v (cid:8) (cid:8) ) ) j j • (cid:5) (cid:5) ) ) j j • f f i i We note that E = ( E ) − . It is not hard to establish that( K ( L ( E )) , [1 L ( E ) ]) = ( { } ,
0) = ( K ( L ( E )) , [1 L ( E ) ]);det( I − A E ) = −
1; and det( I − A E ) = 1 . Is L K ( E ) ∼ = L K ( E )?Here is an alternate approach to establishing the (analytic) Kirchberg Phillips Theorem (Theorem1.12) in the limited context of graph C ∗ -algebras. Using the same symbolic-dynamics techniques asthose used to establish Theorem 4.8, one can establish the C ∗ -version of the Restricted AlgebraicKirchberg Phillips Theorem (i.e., one which involves the determinants). One then “crosses thedeterminant gap” for a single pair of algebras, by showing that C ∗ ( E ) ∼ = C ∗ ( E ); this is done usinga powerful analytic tool (KK-theory). Finally, again using analytic tools, one shows that this oneparticular crossing of the determinant gap allows for the crossing of the gap for all germane pairsof graph C ∗ -algebras. But neither KK-theory, nor the tools which yield the extension from onecrossing to all crossings, seem to accommodate analogous algebraic techniques.The pair { E , E } can appropriately be viewed as the “smallest” pair of graphs of interest in thiscontext, as follows. We say a graph has Condition (Sing) in case there are no parallel edges in thegraph (i.e., that the incidence matrix A E consists only of 0’s and 1’s). It can be shown that, upto graph isomorphism, there are 2 (resp., 34) graphs having two (resp., three) vertices, and havingCondition (Sing), and for which the corresponding Leavitt path algebras are purely infinite simple.(See [4].) For each of these graphs E , det( I − A E ) ≤
0. So finding an appropriate pair of graphswith (Sing) and with unequal (sign of the) determinant requires at least one of the two graphs tocontain at least four vertices.To the author’s knowledge, no Conjecture regarding what the answer to the Algebraic KP Ques-tion should be has appeared in the literature.6.2.
The Classification Question for graphs with finitely many vertices and infinitelymany edges.
We consider now the collection S of those graphs E having finitely many vertices,but (countably) infinitely many edges, and for which L K ( E ) is (necessarily unital) purely infinitesimple. The Purely Infinite Simplicity Theorem (Theorem 4.3) extends to this generality, so we canfairly easily determine whether or not a given graph E is in S . Unlike the case for finite graphs, a description of K ( L K ( E )) for E ∈ S cannot be given in terms of the cokernel of an integer-valued matrix transformation from Z | E | to Z | E | . Nonetheless, there is still a relatively easy wayto determine K ( L K ( E )), so that this group remains a useful player in this context.For a graph E let Sing( E ) denote the set of singular vertices of E , i.e., the set of vertices whichare either sinks, or infinite emitters. Ruiz and Tomforde in [90] achieved the following. Theorem 6.1.
Let
E, F ∈ S . If K ( L K ( E )) ∼ = K ( L K ( F )) and | Sing( E ) | = | Sing( F ) | , then L K ( E ) is Morita equivalent to L K ( F ) . So, while “the determinant of I − A E ” is clearly not defined here in the usual sense (because atleast one of the entries would be the symbol ∞ ), the isomorphism class of K together with thenumber of singular vertices is enough information to determine Morita equivalence. Although thisis quite striking, it is not completely satisfying, in that it remains unclear whether or not | Sing( E ) | is an algebraic property of L K ( E ).Continuing the search for a Classification Theorem which is cast completely in terms of algebraicproperties of the underlying algebras, the authors were able to show that for a certain type of field(those with no free quotients ), there is such a result. In a manner similar to the computation of K ( L K ( E )) for E ∈ S , there is a way to easily compute K ( L K ( E )) as well. Theorem 6.2. [90, Theorem 7.1]
Suppose
E, F ∈ S , and suppose that K is a field with no freequotients. Then L K ( E ) is Morita equivalent to L K ( F ) if and only if K ( L K ( E )) ∼ = K ( L K ( F )) and K ( L K ( E )) ∼ = K ( L K ( F )) . The collection of fields having no free quotients includes algebraically closed fields, R , finitefields, perfect fields of positive characteristic, and others. However, the field Q is not included inthis list. Indeed, the authors in [90, Example 10.2] give an example of graphs E, F ∈ S for which K ( L Q ( E )) ∼ = K ( L Q ( F )) and K ( L Q ( E )) ∼ = K ( L Q ( F )), but L Q ( E ) is not Morita equivalent to L Q ( F ). There are many open questions here. For instance, might there be an integer N for which,if K i ( L K ( E )) ∼ = K i ( L K ( F )) for all 0 ≤ i ≤ N , then L K ( E ) and L K ( F ) are Morita equivalent for allfields K ? Of note in this context is that, unlike the situation for graph C ∗ -algebras (in which “Bottperiodicity” yields that K and K are the only distinct K -groups), there is no analogous result forthe K -groups of Leavitt path algebras. Further, although a long exact sequence for the K -groupsof L K ( E ) has been computed in [27, Theorem 7.6], this sequence does not yield easily recognizableinformation about K i ( L K ( E )) for i ≥ K is a finite extensionof Q , then the pair consisting of ( K ( L K ( E )) , K ( L K ( E ))) provides a complete invariant for theMorita equivalence classes of Leavitt path algebras arising from graphs in S , while none of the pairs( K ( L K ( E )) , K i ( L K ( E ))) for 1 ≤ i ≤ Graded Grothendieck groups, and the corresponding Graded Classification Ques-tion.
The Algebraic Kirchberg Phillips Question, motivated by the corresponding C ∗ -algebra result,is not the only natural classification-type question to ask in the context of Leavitt path algebras.Having in mind the importance that the Z -grading on L K ( E ) has been shown to play in the mul-tiplicative structure, Hazrat in [68] has built the machinery which allows for the casting of ananalogous question from the graded point of view.There is a very well developed theory of graded modules over group-graded rings, see, e.g., [80].(The theory is built for all groups, and is particularly robust in case the group is Z , the case ofinterest for Leavitt path algebras.) If A = ⊕ t ∈ Z A t is a Z -graded ring and M is a left A -module,then M is graded in case M = ⊕ i ∈ Z M i , and a t m i ∈ M t + i whenever a t ∈ A t and m i ∈ M i . If M is a EAVITT PATH ALGEBRAS: THE FIRST DECADE 41 Z -graded A -module, and j ∈ Z , then the suspension module M ( j ) is a graded A -module, for which M ( j ) = M as A -modules, with Z -grading given by setting M ( j ) i = M j + i for all i, j ∈ Z .In a standard way, one can define the notion of a graded finitely generated projective module, andsubsequently build the monoid V gr of isomorphism classes of such modules, with ⊕ as operation. If[ M ] ∈ V gr , then [ M ( j )] ∈ V gr for each j ∈ Z , which yields a Z -action on V gr , and thus by extensiongives V gr the structure of a Z [ x, x − ]-module. In a manner completely analogous to the non-gradedcase, one may define the graded Grothendieck groups K gr i for each i ≥
0; the suspension operationyields a Z [ x, x − ]-module structure on these as well.From this graded-module point of view, one can now ask about structural information of the Z -graded K -algebra L K ( E ) which might be gleaned from the K gr i groups. A reasonable initialquestion might be to see whether the graded version of the Kirchberg Phillips Theorem holds. Thatis, suppose that E and F are finite graphs for which L K ( E ) and L K ( F ) are purely infinite simple,and suppose K gr0 ( L K ( E )) ∼ = K gr0 ( L K ( F )) as Z [ x, x − ]-modules, via an isomorphism which takes[ L K ( E )] to [ L K ( F )]. Is it necessarily the case that L K ( E ) ∼ = L K ( F ) as Z -graded K -algebras?As it turns out, the purely infinite simple hypothesis is not the natural one to start with in thegraded context. In fact, Hazrat in [68] makes the following Conjecture, which at first glance mightseem somewhat audacious. Conjecture 6.3.
Let E and F be any pair of row-finite graphs. Then L K ( E ) ∼ = L K ( F ) as Z -graded K -algebras if and only if K gr0 ( L K ( E )) ∼ = K gr0 ( L K ( F )) as Z [ x, x − ] -modules, via an order-preservingisomorphism which takes [ L K ( E )] to [ L K ( F )] . So Hazrat’s conjecture, slightly rephrased, asserts that the graded K (viewed with the Z [ x, x − ]-module structure induced by the suspension operation), together with the natural order and positionof the regular module, is a complete graded isomorphism invariant for the collection of all Leavittpath algebras over row-finite graphs. (The order on K ( R ) is induced by viewing the nonzeroelements of V ( R ) as the positive elements. The order on K ( R ) plays no role in purely infinitesimple rings, because every nonzero element of V ( R ) is positive in that case.)In [68, Theorem 4.8], Hazrat verifies Conjecture 6.3 in case the graphs E and F are polycephalic (essentially, mixtures of acyclic graphs, or graphs which can be described as “multiheaded comets”or “multiheaded roses” in which the cycles and/or roses have no exits.)As mentioned in the Historical Plot Line A [ t + , t − , α ]. Recast in the language ofLeavitt path algebras, the discussion in [29, Example 2.5] yields that, when E is an essential graph(i.e., has no sinks or sources), then L K ( E ) = L K ( E ) [ t + , t − , α ] for suitable elements t + , t − ∈ L K ( E ),and a corner-isomorphism α of the zero component L K ( E ) .When E is a finite graph with no sinks, then L K ( E ) is strongly graded ([69, Theorem 2]), whichyields (by a classical theorem of Dade) that the category of graded modules over L K ( E ) is equivalentto the category of (all) modules over the zero component L K ( E ) . Thus, when E has no sinks, wehave reason to expect that the zero component might play a role in the graded theory. In a deepresult (which relies heavily on ideas from symbolic dynamics), Ara and Pardo [37, Theorem 4.1]prove the following modified version of Conjecture 6.3. Theorem 6.4.
Let E and F be finite essential graphs. Write L K ( E ) = L K ( E ) [ t + , t − , α ] as described above. Then the following are equivalent.(1) K ( L K ( E ) ) ∼ = K ( L K ( F ) ) via an order-preserving K [ x, x − ] -module isomorphism whichtakes [1 L K ( E ) ] to [1 L K ( F ) ] . (2) There exists a locally inner automorphism g of L K ( E ) for which L K ( F ) ∼ = L K ( E ) [ t + , t − , g ◦ α ] as Z -graded K -algebras. A complete resolution of Conjecture 6.3 currently remains elusive.6.4.
Connections to noncommutative algebraic geometry.
One of the basic ideas of (stan-dard) algebraic geometry is the correspondence between geometric spaces and commutative algebras.Over the past few decades, significant research energy has been focused on appropriately extendingthis correspondence to the noncommutative case; the resulting theory is called noncommutativealgebraic geometry. Suppose A is a Z + -graded algebra (i.e., a Z -graded algebra for which A n = { } for all n < A ) denote the category of Z -graded left A -modules (with graded homomorphisms), and letFdim( A ) denote the full subcategory of Gr( A ) consisting of the graded A -modules which are the sumof their finite dimensional submodules. Denote by QGr( A ) the quotient category Gr( A ) / Fdim( A ).The category QGr( A ) turns out to be one of the fundamental constructions in noncommutativealgebraic geometry. In particular, if E is a directed graph, then the path algebra KE is Z + -gradedin the usual way (by setting deg( v ) = 0 for each vertex v , and deg( e ) = 1 for each edge e ), and soone may construct the category QGr( KE ).Let E nss denote the graph gotten by repeatedly removing all sinks and sources (and their incidentedges) from E . Theorem 6.5. [94, Theorem 1.3]
Let E be a finite graph. Then there is an equivalence of categories QGr( KE ) ∼ Gr( L K ( E nss )) . Moreover, since L K ( E nss ) is strongly graded, then these categories are also equivalent to the fullcategory of modules over the zero-component ( L K ( E nss )) . So the Leavitt path algebra construction arises naturally in the context of noncommutative alge-braic geometry. (The appearance of Leavitt path algebras in this setting is clarified by the notionof a Universal Localization, see e.g. [91].)In general, when the Z + -graded K -algebra A arises as an appropriate graded deformation of thestandard polynomial ring K [ x , ..., x n ], then QGr( A ) shares many similarities with projective n -space P n ; parallels between them have been studied extensively (see e.g. [96]). However, in general,an algebra of the form KE does not arise in this way; and for these, as asserted in [95], “it is muchharder to see any geometry hiding in QGr( KE ).” In specific situations there are some geometricperspectives available (see e.g. [93]), but the general case is not well understood.6.5. Tensor products.
As described in Section 3.5, the algebras L K (1 , ⊗ L K (1 ,
2) and L K (1 , ϕ : L K (1 , ⊗ K L K (1 , → L K (1 , L K (1 , ⊗ K L K (1 ,
2) isomorphic to L K (1 , ⊗ K L K (1 , Thanks to S. Paul Smith for providing much of the information contained in this subsection.
EAVITT PATH ALGEBRAS: THE FIRST DECADE 43
The Realization Problem for von Neumann regular rings.
Although significant progresshas been made in resolving the Realization Problem for von Neumann regular rings (see the discus-sion prior to Theorem 5.12), there is as of yet not a complete answer. An excellent survey of themain ideas relevant to this endeavor can be found in [25].Using direct limit arguments, one can show that the graph monoid M E corresponding to acountable graph E can be realized as V ( R ) for a von Neumann regular algebra R . Indeed, M E is constructed as a direct limit of monoids of the form M F , where the graphs F are finite; inparticular, M E is a direct limit of finitely generated refinement monoids. Furthermore, R can beconstructed as a direct limit of (von Neumann regular) quotient algebras of the form Q K ( F ) for F finite.More generally, one can divide the (countable refinement) monoids arising in the RealizationProblem into two types: tame (those which can be constructed as direct limits of finitely generatedrefinement monoids), and the others (called wild ). Investigations (by Ara and Goodearl, see [32])continue into whether or not every finitely generated refinement monoid is realizable; whether ornot the realization passes to direct limits; and whether or not there are wild monoids which are notrealizable.7. Appendix 1: Some properties of L K ( E ) and C ∗ ( E ) which suggest the existenceof a Rosetta Stone It has become apparent that there is a strong, but mysterious, relationship between the structureof the Leavitt path algebra L C ( E ) and the corresponding graph C ∗ -algebra C ∗ ( E ). In this contextit is helpful to keep in mind that while L C ( E ) may always be viewed as a dense ∗ -subalgebra of C ∗ ( E ) (see Proposition 2.3), the two algebras are in general clearly different as rings: indeed, theycoincide only when E is finite and acyclic.We focus in this Appendix on finite graphs, so that the corresponding Leavitt path algebra L C ( E )or graph C ∗ -algebra C ∗ ( E ) is unital (and C ∗ ( E ) is separable as well). But many of the observationswe make here hold more generally.Any C ∗ -algebra A wears two hats: not only is A a ring, but A comes equipped with a topology aswell, so that one may view the ring-theoretic structure of A from a topological/analytic viewpoint.The standard example is this: one may define the (algebraic) simplicity of the C ∗ -algebra eitheras a ring (no nontrivial two-sided ideals), or the (topological) simplicity as a topological ring (nonontrivial closed two-sided ideals). In general, the algebraic and topological properties of a givenC ∗ -algebra A need not coincide.The graph E is called cofinal in case every vertex of E connects to every cycle and every sink of E . (This turns out to be equivalent to E having the property that the only hereditary saturatedsubsets of E are ∅ and E .)As a reminder: E has Condition (L) if every cycle in E has an exit; E has Condition (K) ifthere is no vertex v of E which has exactly one simple closed path based at v ; and E is downwarddirected if for each pair of vertices v, w of E there exists a vertex y for which v ≥ y and w ≥ y . Property 1: Simplicity
Algebraic:
No nontrivial two-sided ideals.
Analytic:
No nontrivial closed two-sided ideals.By Theorem 1.20, L C ( E ) is simple if and only if E is cofinal and has Condition (L). By [47, Proposition 5.1] (for the case without sources), and [86] (for the general case), C ∗ ( E ) is(topologically) simple if and only if E is cofinal and has Condition (L).By [57, p. 215], for any unital C ∗ -algebra A , A is topologically simple if and only if A isalgebraically simple. Result : These are equivalent for any finite graph E :(i) L C ( E ) is simple.(ii) C ∗ ( E ) is (topologically) simple.(iii) C ∗ ( E ) is (algebraically) simple.(iv) E is cofinal, and satisfies Condition (L). Property 2: The V -monoid (Much of this discussion is taken directly from [36, Sections 2 and 7].) Algebraic : For a ring R , V ( R ) is the monoid of isomorphism classes of finitely generated projectiveleft R -modules, with operation ⊕ . By [50, Chapter 3], V ( R ) can be viewed as the set of equivalenceclasses V ( e ) of idempotents e in the (nonunital) infinite matrix ring M N ( R ), with operation V ( e ) + V ( f ) = V ( (cid:18) e f (cid:19) ) . Analytic : For an operator algebra A , V MvN ( A ) is the monoid of Murray - von Neumann equiva-lence classes of projections in M N ( A ).By [50, 4.6.2 and 4.6.4], whenever A is a C ∗ -algebra, then V ( A ) agrees with V MvN ( A ).By [36, Theorem 7.1], the natural inclusion ψ : L C ( E ) → C ∗ ( E ) induces a monoid isomorphism V ( ψ ) : V ( L C ( E )) → V ( C ∗ ( E )).By [36, Theorem 3.5], the monoid V ( L K ( E )) is independent of the field K ; specifically, V ( L K ( E )) ∼ = M E , the graph monoid of E . Result : For any finite graph E and any field K , the following semigroups are isomorphic.(i) the graph monoid M E (ii) V ( L K ( E ))(iii) V ( C ∗ ( E ))(iv) V MvN ( C ∗ ( E )) Property 3: Purely infinite simplicity
Algebraic : R is purely infinite simple in case R is simple and every nonzero right ideal of R contains an infinite idempotent. (Source: [35, Definitions 1.2].) Analytic : The simple C ∗ -algebra A is called purely infinite (simple) if for every positive x ∈ A ,the subalgebra xAx contains an infinite projection. (Source: [58, p. 186].)By [35, Theorem 1.6], (algebraic) purely infinite simplicity for unital rings is equivalent to: R isnot a division ring, and for all nonzero x ∈ R there exist α, β ∈ R for which αxβ = 1.By [50, Proposition 6.11.5], (topological) purely infinite simplicity for unital C ∗ -algebras is equiv-alent to: A = C and for every x = 0 in A there exist α, β ∈ A for which αxβ = 1. (Remark:Blackadar defines purely infinite simplicity this way, and then shows this definition is equivalent toCuntz’ definition given in [58].) Easily, for any graph E , C ∗ ( E ) is a division ring if and only if E is a single vertex, in which case C ∗ ( E ) = C .Thus we have, for graph C ∗ -algebras, C ∗ ( E ) is (algebraically) purely infinite simple if and onlyif C ∗ ( E ) is (topologically) purely infinite simple. EAVITT PATH ALGEBRAS: THE FIRST DECADE 45
By [8, Theorem 11], L C ( E ) is purely infinite simple if and only if L C ( E ) is simple, and E has theproperty that every vertex connects to a cycle.By [47, Proposition 5.3], C ∗ ( E ) is (topologically) purely infinite simple if and only if C ∗ ( E ) issimple, and E has the property that every vertex connects to a cycle. Result : These are equivalent for any finite graph E :(i) L C ( E ) is purely infinite simple.(ii) C ∗ ( E ) is (topologically) purely infinite simple.(iii) C ∗ ( E ) is (algebraically) purely infinite simple.(iv) E is cofinal, every cycle in E has an exit, and every vertex in E connects to a cycle. Property 4: Exchange
Algebraic : R is an exchange ring if for any a ∈ R there exists an idempotent e ∈ R for which e ∈ Ra and 1 − e ∈ R (1 − a ). (Note: The original definition of exchange ring was given by Warfield,in terms of a property on direct sum decomposition of modules; this property clarifies the genesisof the name exchange . The definition given here is equivalent to Warfield’s; this equivalence wasshown independently by Goodearl and Warfield in [67, discussion on p. 167], and by Nicholson in[81, Theorem 2.1].) Analytic : For every x > p such that p ∈ Ax and 1 − p ∈ A (1 − x ). (Wecall this condition “topological exchange”. Note: There does not seem to be an explicit definitionof “topological exchange ring” in the literature.)By [43, Theorem 4.5]. L C ( E ) is an exchange ring if and only if E satisfies Condition (K).By [71, Theorem 4.1] C ∗ ( E ) has real rank zero if and only if E satisfies Condition (K).By [33, Theorem 7.2], for a unital C ∗ -algebra A , A has real rank zero if and only if A is atopological exchange ring if and only if A is an exchange ring. Result : These are equivalent for a finite graph E :(i) L C ( E ) is an exchange ring.(ii) C ∗ ( E ) is a (topological) exchange ring.(iii) C ∗ ( E ) is an (algebraic) exchange ring.(iv) E satisfies Condition (K). Property 5: Primitivity
Algebraic : R is (left) primitive if there exists a simple faithful left R -module. Analytic : A is (topologically) primitive if there exists an irreducible faithful ∗ -representation of A . (That is, there is a faithful irreducible representation π : A → B ( H ) for a Hilbert space H .)It is shown in [44, Theorem 4.6] that L C ( E ) is left (and / or right) primitive if and only if E isdownward directed and satisfies Condition (L).It is shown in [46, Proposition 4.2] that C ∗ ( E ) is (topologically) primitive if and only if E isdownward directed and satisfies Condition (L).It is shown in [62, Corollary to Theorem 2.9.5] that a C ∗ -algebra is algebraically primitive if andonly if it is topologically primitive. Result : These are equivalent for finite graphs:(i) L C ( E ) is primitive.(ii) C ∗ ( E ) is (topologically) primitive.(iii) C ∗ ( E ) is (algebraically) primitive.(iv) E is downward directed and satisfies Condition (L). (We note that the first three properties have been shown to be equivalent for arbitrary graphs aswell, with the fourth condition being replaced by: E satisfies Condition (L), is downward directed,and has the Countable Separation Property. See Theorem 5.5 and [19].)It is interesting to note that in the situations in which we have a result which suggests the existenceof a Rosetta Stone, the algebraic and topological conditions on C ∗ ( E ) are identical. Perhaps thereis something in this observation which will lead to a deeper understanding of why there seems tobe such a strong relationship between the properties of L C ( E ) and C ∗ ( E ).There are indeed situations where the analogies between the Leavitt path algebras and graph C ∗ algebras are not as tight as those presented above. A property for which the algebraic and analytic results are not identical: Primeness
Algebraic : R is a prime ring in case { } is a prime ideal of R ; that is, in case for any two-sidedideals I, J of R , I · J = { } if and only if I = { } or J = { } . Analytic : A is a prime C ∗ -algebra in case { } is a prime ideal of A ; that is, in case for any closedtwo-sided ideals I, J of R , I · J = { } if and only if I = { } or J = { } .In [44, Corollary 3.10] it is shown that L C ( E ) is prime if and only if E is downward directed.But by [61, Corollaire 1], any separable C ∗ -algebra is (topologically) prime if and only if it is(topologically) primitive. So (for finite E ) C ∗ ( E ) is prime if and only if C ∗ ( E ) is primitive, whichby the previous discussion is if and only if E is downward directed and satisfies Condition (L). (Wenote that since I · J = { } implies I · J = { } , it is straightforward to show that A is algebraicallyprime if and only if A is analytically prime.)So for example if E is the graph with one vertex and one loop, then L C ( E ) is prime (it’s anintegral domain, in fact), but C ∗ ( E ) is not prime. (It’s not hard to write down nonzero continuousfunctions on the circle which are orthogonal.)There are a few other situations where the properties of L C ( E ) and C ∗ ( E ) do not match upexactly. For instance, the only possible values of the (algebraic) stable rank of L C ( E ) are 1 ,
2, and ∞ ; as well, the only possible values of the (topological) stable rank of C ∗ ( E ) are 1 ,
2, and ∞ . Butamong individual graphs, the values may be different: if E = R , then the stable rank of L C ( R ) is2, while the stable rank of C ∗ ( R ) is 1.In addition, we have seen in Section 3.5 that O ⊗ O ∼ = O , but L C (1 , ⊗ C L C (1 , = L C (1 , Summary of Appendix 1 : A “Rosetta Stone for graph algebras” refers to an overarchingprinciple which would allow an understanding as to why there is such an extremely tight (but notperfect) relationship between various properties of Leavitt path algebras and graph C ∗ -algebras, assuggested by the examples given in this section. Does such a Rosetta Stone exist?8. Appendix 2: A number-theoretic observation
Let L K (1 , n ) denote the Leavitt algebra of order n ; so R = L K (1 , n ) has the property that R R ∼ = R R n as left R -modules. By Theorem 3.10 we have L K (1 , n ) ∼ = M d ( L K (1 , n )) ⇔ g . c . d . ( d, n −
1) = 1 . Indeed, when the appropriate number-theoretic condition is satisfied then the isomorphism may beexplicitly constructed.The key to constructing such an isomorphism lies in considering a partition of { , , ..., d } intotwo nonempty disjoint subsets S ⊔ S , described as follows. EAVITT PATH ALGEBRAS: THE FIRST DECADE 47
Suppose n is an integer having g . c . d . ( d, n −
1) = 1. Write n = qd + r with 1 ≤ r ≤ d . Asg . c . d . ( d, n −
1) = 1 we get g . c . d . ( d, r −
1) = 1.For the current discussion we focus only on r . Note we have r ≥
1. Let s = d − ( r − . c . d . ( d, r −
1) = 1 we easily see g . c . d . ( d, s ) = 1. Now consider the sequence Σ d,r , given byΣ d,r = 1 , s, s, ..., d − s of d integers, interpreted mod d . (Here we interpret 0 mod d as d mod d .) Since g . c . d . ( d, s ) = 1,elementary number theory gives that, as a set, the elements of Σ d,r form a complete set of residuesmod d .In particular, for some i r (1 ≤ i r ≤ d ) we have 1 + ( i r − s ≡ r − d .Now consider these two sequences:Σ d,r = 1 , s, ..., i r − s Σ d,r = 1 + i r s, i r + 1) s, ..., d − s. So Σ d,r is just the first i r elements of Σ d,r , and Σ d,r is the remaining d − i r elements.We can also consider the partition S d,r = S d,r ⊔ S d,r of { , , ..., d } which corresponds to theelements of these two sequences: S d,r = { , s, ..., i r − s } S d,r = { i r s, i r + 1) s, ..., d − s } So in particular | S d,r | = i r and | S d,r | = d − i r . Clearly S d,r = ∅ . But S d,r = ∅ as well, since d ∈ S d,r . This is because the first element 1 + i r s of Σ d,r is always d , as 1 + i r s = (1 + ( i r − s ) + s =( r −
1) + ( d − ( r − d .For notational convenience, if d, r are fixed then we drop the superscript d, r in the sequencesand subsets. Example 8.1.
The case d = 3 , r = 2. g . c . d . (3 , −
1) = 1. r − , s = d − ( r −
1) = 3 − s = 2 each step, and we interpret mod 3 (1 ≤ i ≤ , , . Since r − = 1 Σ = 3 , S = { } , S = { , } . Example 8.2.
The case d = 3 , r = 3. g . c . d . (3 , −
1) = 1. r − , s = d − ( r −
1) = 3 − s = 1 each step, and we interpret mod 3 (1 ≤ i ≤ , , . Since r − = 1 , = 3and so S = { , } S = { } . Example 8.3.
The case d = 13 , r = 9. g . c . d . (13 , −
1) = 1. r − , s = d − ( r −
1) = 13 − s = 5 each step, and we interpret mod 13 (1 ≤ i ≤ , , , , , , , , , , , , . Since r − = 1 , , , , = 13 , , , , , , , S = { , , , , } S = { , , , , , , , } . By solving the congruence 1 + ( i r − s ≡ r − d , we easily get Lemma 8.4. i r ≡ ( r − − mod d . In particular, if we have 1 ≤ r = r ′ ≤ d for which g . c . d . ( d, r −
1) = 1 = g . c . d . ( d, r ′ − S d,r ⊔ S d,r and S d,r ′ ⊔ S d,r ′ of { , , ..., d } are necessarily different (since 1 is in S ,and the sizes of S d,r and S d,r ′ are unequal).Given d , there exist ϕ ( d ) (Euler ϕ -function) remainders which are relatively prime to d . So thereexist ϕ ( d ) distinct partitions of { , , ..., d } which arise as S d,r ⊔ S d,r for some r having g . c . d . ( d, r −
1) = 1.We note that for any d we always get these two partitions arising in the form S d,r : { } ⊔ { , , ..., d } = S d, , and { , , ..., d − } ⊔ { d } = S d,d . It is easy to see that there are (2 d − / d − − { , , ..., d } intotwo nonempty subsets S ⊔ S for which 1 ∈ S . Since ϕ ( d ) < d − − d ≥
3, we see thatnot all such two-nonempty-set partitions of { , , ..., d } can arise as S d,r ⊔ S d,r for some r havingg . c . d . ( d, r −
1) = 1. For example, when d = 3, the partition { , } ⊔ { } of { , , } does not arisein this way. We are interested in two related questions regarding the sequences described in thisAppendix. (1) For fixed d, r having g . c . d . ( d, r −
1) = 1, do the sequences Σ d,r and Σ d,r arise in contexts otherthan that of isomorphisms between matrix rings over Leavitt algebras?(2) Do the ϕ ( d ) partitions of { , , ..., d } of the form { , , ..., d } = S d,r ⊔ S d,r (for some r havingg . c . d . ( d, r −
1) = 1) play a special role in any sorts of number-theoretic investigations?
Remark 8.5.
Referring back to how these sequences and partitions arose in the context of Theorem3.10, in that setting we start with d, n having g . c . d . ( d, n −
1) = 1, write n = qd + r with 1 ≤ r ≤ d ,and then consider the partition S d,r ⊔ S d,r of { , , ..., d } . We then use this partition of { , , ..., d } to build a partition of { , , ..., n } by simply extending the partition S d,r ⊔ S d,r , mod d . So, forinstance, if n = 5, d = 3 then we get 5 = 1 · { , , } = S , ⊔ S , = { }⊔{ , } , as described in Example 8.1. This then yields the partition { , }⊔{ , , } of { , , , , } by simply extending mod 3.Specifically, in the proof of Theorem 3.10, the ordering properties of the sequences Σ and Σ arenot utilized, rather, only the partition S d,r ⊔ S d,r of { , , ..., d } as sets is used. EAVITT PATH ALGEBRAS: THE FIRST DECADE 49 Appendix 3: The graph moves
We give in this Appendix the formal definitions of each of the six “graph moves” which arise inthe symbolic dynamics analysis associated to the Restricted Algebraic Kirchberg Phillips Theorem.We conclude by presenting the “source elimination” process as well.
Definition 9.1.
Let E = ( E , E , r, s ) be a directed graph. For each v ∈ E with s − ( v ) = ∅ ,partition the set s − ( v ) into disjoint nonempty subsets E v , . . . , E m ( v ) v where m ( v ) ≥
1. (If v is asink, then we put m ( v ) = 0.) Let P denote the resulting partition of E . We form the out-splitgraph E s ( P ) from E using the partition P as follows: E s ( P ) = { v i | v ∈ E , ≤ i ≤ m ( v ) } ∪ { v | m ( v ) = 0 } ,E s ( P ) = { e j | e ∈ E , ≤ j ≤ m ( r ( e )) } ∪ { e | m ( r ( e )) = 0 } , and define r E s ( P ) , s E s ( P ) : E s ( P ) → E s ( P ) for each e ∈ E is ( e ) by s E s ( P ) ( e j ) = s ( e ) i and s E s ( P ) ( e ) = s ( e ) i , r E s ( P ) ( e j ) = r ( e ) j and r E s ( P ) ( e ) = r ( e ) . Conversely, if E and G are graphs, and there exists a partition P of E for which E s ( P ) = G , then E is called an out-amalgamation of G . Definition 9.2.
Let E = ( E , E , r, s ) be a directed graph, and let v ∈ E . Let v ∗ and f besymbols not in E ∪ E . We form the expansion graph E v from E at v as follows: E v = E ∪ { v ∗ } E v = E ∪ { f } s E v ( e ) = v if e = fv ∗ if s E ( e ) = vs E ( e ) otherwise r E v ( e ) = (cid:26) v ∗ if e = fr E ( e ) otherwiseConversely, if E and G are graphs, and there exists a vertex v of E for which E v = G , then E iscalled a contraction of G . Definition 9.3.
Let E = ( E , E , r, s ) be a directed graph. For each v ∈ E with r − ( v ) = ∅ ,partition the set r − ( v ) into disjoint nonempty subsets E v , . . . , E vm ( v ) where m ( v ) ≥
1. (If v is asource then we put m ( v ) = 0.) Let P denote the resulting partition of E . We form the in-splitgraph E r ( P ) from E using the partition P as follows: E r ( P ) = { v i | v ∈ E , ≤ i ≤ m ( v ) } ∪ { v | m ( v ) = 0 } ,E r ( P ) = { e j | e ∈ E , ≤ j ≤ m ( s ( e )) } ∪ { e | m ( s ( e )) = 0 } , and define r E r ( P ) , s E r ( P ) : E r ( P ) → E r ( P ) by s E r ( P ) ( e j ) = s ( e ) j and s E r ( P ) ( e ) = s ( e ) r E r ( P ) ( e j ) = r ( e ) i and r E r ( P ) ( e ) = r ( e ) i where e ∈ E r ( e ) i . Conversely, if E and G are graphs, and there exists a partition P of E for which E r ( P ) = G , then E is called an in-amalgamation of G . Definition 9.4.
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