# Graph algebras: Bridges between graph C*-algebras and Leavitt path algebras (13w5049)

Arriving in Banff, Alberta Sunday, April 21 and departing Friday April 26, 2013

## Organizers

Gene Abrams (University of Colorado at Colorado Springs)

Jason Bell (Simon Fraser University)

Soren Eilers (University of Copenhagen)

George Elliott (University of Toronto)

Marcelo Laca (University of Victoria)

Mark Tomforde (University of Houston)

## Objectives

A common theme throughout all branches of mathematics is to classify objects through the use of ``easily computable" invariants. Classification for $C^*$-algebras began with a seminal result of Elliott, in which he showed that $K$-theoretic data can be used to classify AF-algebras (i.e., $C^*$-algebras that are direct limits of finite-dimensional algebras [Ell2]. Similar results were established for the algebraic counterparts of these algebras (the ultramatricial algebras) in Goodearl's and Handelman's monograph [GH]. The use of $K$-theory to classify $C^*$-algebras has proven very beneficial, and Elliott initiated an ambitious program to classify all separable nuclear $C^*$-algebras by $K$-theoretic data. Through the work of Elliott and many other researchers, a great deal of progress has been made on this program in the past few decades. One of the main results was the Kirchberg-Phillips classification theorem, which classifies certain simple, purely infinite $C^*$-algebras by $K$-theory. (Unlike AF-algebras, which are close to being finite dimensional, purely infinite $C^*$-algebras contain many infinite-dimensional projections and are far from finite dimensional.) In addition, there have recently been many exciting results that have brought important cases in the program near to solution.

A majority of the results obtained in the classification of $C^*$-algebras have been for simple $C^*$-algebras. In recent years, more attention has been paid to nonsimple $C^*$-algebras, but frequently the nonsimple $C^*$-algebras studied fall into a class where all ideals and quotients of the $C^*$-algebra are that same class (e.g., AF-algebras, stably finite $C^*$-algebras, purely infinite $C^*$-algebras). Graph $C^*$-algebras (which are vast generalizations of the Cuntz-Krieger algebras [CK, Cun3] are one of the first examples of nonsimple $C^*$-algebras of mixed type that have been amenable to classification. Eilers and Tomforde have shown that graph $C^*$-algebras with a unique proper nonzero ideal are classified by $K$-theoretic data. Graph $C^*$-algebras form a remarkably suitable class to test the classification results because, for example, the simple graph $C^*$-algebras exhibit a dichotomy; the simple graph $C^*$-algebras are either AF or purely infinite, and thus are classified by either Elliott's theorem or the Kirchberg-Phillips classification theorem. We believe that graph $C^*$-algebras could provide techniques for studying more general mixed-type classes.

One project for this workshop is the further classification of nonsimple graph $C^*$-algebras. There are difficulties that occur when many ideals are allowed, and it is even unclear whether the ``obvious generalization" of the invariant is the correct one. Restorff has shown that the invariant correctly classifies nonsimple Cuntz-Krieger algebras with finitely many ideals (which are special cases of graph $C^*$-algebras). However, recent work of Meyer and Nest, and some conjectures of Katsura, cast into doubt whether this is true for general graph $C^*$-algebras.

A second project is to use these techniques to classify more general $C^*$-algebras of mixed type. It would be useful if one could identify the salient features of graph $C^*$-algebras necessary for the classification results, and then identify more general classes of $C^*$-algebras that have these properties. It is likely that the graph $C^*$-algebra techniques can be used to usher in more general methods for classification of mixed-type nonsimple $C^*$-algebras. A first place to look for these investigations are the classes that generalize graph $C^*$-algebras, such as the topological graph $C^*$-algebras of Katsura or the $k$-graph $C^*$-algebras of Kumjian and Pask.

A third project involves classification of Leavitt path algebras. Despite the results of Goodearl and Handelman for real ultramatricial algebras in [GH], classification of more general classes by $K$-theory has often been unsuccessful. Recent results, however, show that Leavitt path algebras provide a class of algebras where $K$-theory classification may be accomplished. It was shown in [ALPS] that there is a version of the Kirchberg-Phillips theorem for unital, purely infinite Leavitt path algebras, and that these algebras are classified by $K$-theoretic data together with the sign of the determinant of a matrix determined by the graph. It is unknown whether the sign of the determinant is a necessary part of the invariant. The proof of the result relies heavily on results from symbolic dynamics---specifically, the classification of shifts of finite type up to flow equivalence, contained largely in the recent work of Boyle [Boy] and Boyle and Huang [BoyHua], and the earlier work of Huang [Hua]. These symbolic dynamics results allow one to perform operations on graphs that turn out to be isomorphism or Morita invariant on the corresponding Leavitt path algebras. There are two important outstanding questions concerning these results. First: Is the sign of the determinant a necessary part of the invariant? Second: Can these results be extended to nonunital Leavitt path algebras? Our hope is that by bringing together researchers at BIRS who are experts in classification, algebra, $C^*$-algebras, and symbolic dynamics, we will be able to connect ideas from different areas and come up with a solution to this problem. Moreover, this solution would probably involve performing a lot of graph operations, connecting these to the associated algebras, and coming up with methods or transforming one graph into the other. This type of activity seems well suited for the interactive environment at BIRS.

In addition, these classification questions for Leavitt path algebras are particularly timely. A well-known result, useful in much of the classification of $C^*$-algebras, is that $\mathcal{O}_2 \cong \mathcal{O}_2 \otimes \mathcal{O}_2$, where $\mathcal{O}_2$ is the Cuntz algebra, which is isomorphic to the $C^*$-algebra of the graph with one vertex and two edges. The algebraic analogue of $\mathcal{O}_2$ is the Leavitt algebra $L_2$, which is the Leavitt path algebra over $\mathbb{C}$ of the same graph. Earlier this year, it was established by Ara and Cortiñas that $L_2$ is not isomorphic (and, in fact, not even Morita equivalent) to $L_2 \cong L_2$ [AC]. This an important result in terms of classification, and shows that different techniques, and perhaps even different results, are expected to arise in the algebraic situation. Other researchers have been interested in this result, and there are currently three different proofs. These ``algebraic surprises" are the kinds of results of which a BIRS workshop can make the general graph algebra community aware. The workshop will also provide a forum for algebraists and functional analysts to exchange ideas on their significance.

A fourth project is more ambitious in scope, and deals with examining the relationship between graph $C^*$-algebras and Leavitt path algebras. There are stunning similarities between the two classes of graph algebras. For instance, in both theories one attempts to determine those graphs $E$ for which the graph $C^*$-algebra $C^*(E)$ and the Leavitt path algebra $L_K(E)$ have a specified $C^*$-algebraic or algebraic property. As it turns out, for nearly every graph-theoretic property of $E$ that is known to be equivalent to a $C^*$-algebraic property of $C^*(E)$, the same graph-theoretic property of $E$ is equivalent to the corresponding property of $L_K(E)$. For example, the graph-theoretic conditions under which $C^*(E)$ is a simple algebra (respectively, an AF-algebra, a purely infinite simple algebra, an exchange ring, a finite-dimensional algebra) in the category of $C^*$-algebras are precisely the same graph-theoretic conditions under which $L_K(E)$ is a simple algebra (respectively, an ultramatricial algebra, a purely infinite simple algebra, an exchange ring, a finite-dimensional algebra) in the category of $K$-algebras. Moreover, the Leavitt path algebra results seem to hold independently of the field $K$, and in particular for the field $K=\mathbb{C}$ of complex numbers.

Despite these similarities, the relationship between various structural properties of $C^*(E)$ and $L_K(E)$ is as mysterious as it is startling. In fact, there is currently no known way to take results from one class and deduce results for the other class. Instead, the theorems from each class have had to be proven using techniques different from the other class. The large number of similarities in properties and statements of results might suggest that theorems once obtained on either the graph $C^*$-algebra side or on the Leavitt path algebra side might then immediately be translated via some sort of Rosetta Stone to a corresponding theorem on the other side. Nonetheless, a vehicle to transfer information in this way remains elusive, and in fact, researchers are uncertain how to even formulate conjectures that would lead to such a vehicle.

A current, active, cross-disciplinary project related to this investigation focuses on finding a complete answer to what would seem to be a fundamental question in this context (the ``Isomorphism Question"): If $E$ and $F$ are graphs for which $L_\mathbb{C}(E)\cong L_\mathbb{C}(F)$, is it true that $C^*(E) \cong C^*(F)$? Conversely, if $C^*(E)\cong C^*(F)$, do we get $L_\mathbb{C}(E)\cong L_\mathbb{C}(F)$? Partial answers (covering specific classes of graphs) have been provided in [AT]. Also, determining the necessity of the sign of the determinant (which we mentioned earlier) is a test case for this question. At the proposed workshop we hope to gain a better understanding of these issues, find more general answers to the Isomorphism (and Morita Equivalence) Questions, and gain more insight into how the graph determines the structure of the associated graph $C^*$-algebra and Leavitt path algebra.

A majority of the results obtained in the classification of $C^*$-algebras have been for simple $C^*$-algebras. In recent years, more attention has been paid to nonsimple $C^*$-algebras, but frequently the nonsimple $C^*$-algebras studied fall into a class where all ideals and quotients of the $C^*$-algebra are that same class (e.g., AF-algebras, stably finite $C^*$-algebras, purely infinite $C^*$-algebras). Graph $C^*$-algebras (which are vast generalizations of the Cuntz-Krieger algebras [CK, Cun3] are one of the first examples of nonsimple $C^*$-algebras of mixed type that have been amenable to classification. Eilers and Tomforde have shown that graph $C^*$-algebras with a unique proper nonzero ideal are classified by $K$-theoretic data. Graph $C^*$-algebras form a remarkably suitable class to test the classification results because, for example, the simple graph $C^*$-algebras exhibit a dichotomy; the simple graph $C^*$-algebras are either AF or purely infinite, and thus are classified by either Elliott's theorem or the Kirchberg-Phillips classification theorem. We believe that graph $C^*$-algebras could provide techniques for studying more general mixed-type classes.

One project for this workshop is the further classification of nonsimple graph $C^*$-algebras. There are difficulties that occur when many ideals are allowed, and it is even unclear whether the ``obvious generalization" of the invariant is the correct one. Restorff has shown that the invariant correctly classifies nonsimple Cuntz-Krieger algebras with finitely many ideals (which are special cases of graph $C^*$-algebras). However, recent work of Meyer and Nest, and some conjectures of Katsura, cast into doubt whether this is true for general graph $C^*$-algebras.

A second project is to use these techniques to classify more general $C^*$-algebras of mixed type. It would be useful if one could identify the salient features of graph $C^*$-algebras necessary for the classification results, and then identify more general classes of $C^*$-algebras that have these properties. It is likely that the graph $C^*$-algebra techniques can be used to usher in more general methods for classification of mixed-type nonsimple $C^*$-algebras. A first place to look for these investigations are the classes that generalize graph $C^*$-algebras, such as the topological graph $C^*$-algebras of Katsura or the $k$-graph $C^*$-algebras of Kumjian and Pask.

A third project involves classification of Leavitt path algebras. Despite the results of Goodearl and Handelman for real ultramatricial algebras in [GH], classification of more general classes by $K$-theory has often been unsuccessful. Recent results, however, show that Leavitt path algebras provide a class of algebras where $K$-theory classification may be accomplished. It was shown in [ALPS] that there is a version of the Kirchberg-Phillips theorem for unital, purely infinite Leavitt path algebras, and that these algebras are classified by $K$-theoretic data together with the sign of the determinant of a matrix determined by the graph. It is unknown whether the sign of the determinant is a necessary part of the invariant. The proof of the result relies heavily on results from symbolic dynamics---specifically, the classification of shifts of finite type up to flow equivalence, contained largely in the recent work of Boyle [Boy] and Boyle and Huang [BoyHua], and the earlier work of Huang [Hua]. These symbolic dynamics results allow one to perform operations on graphs that turn out to be isomorphism or Morita invariant on the corresponding Leavitt path algebras. There are two important outstanding questions concerning these results. First: Is the sign of the determinant a necessary part of the invariant? Second: Can these results be extended to nonunital Leavitt path algebras? Our hope is that by bringing together researchers at BIRS who are experts in classification, algebra, $C^*$-algebras, and symbolic dynamics, we will be able to connect ideas from different areas and come up with a solution to this problem. Moreover, this solution would probably involve performing a lot of graph operations, connecting these to the associated algebras, and coming up with methods or transforming one graph into the other. This type of activity seems well suited for the interactive environment at BIRS.

In addition, these classification questions for Leavitt path algebras are particularly timely. A well-known result, useful in much of the classification of $C^*$-algebras, is that $\mathcal{O}_2 \cong \mathcal{O}_2 \otimes \mathcal{O}_2$, where $\mathcal{O}_2$ is the Cuntz algebra, which is isomorphic to the $C^*$-algebra of the graph with one vertex and two edges. The algebraic analogue of $\mathcal{O}_2$ is the Leavitt algebra $L_2$, which is the Leavitt path algebra over $\mathbb{C}$ of the same graph. Earlier this year, it was established by Ara and Cortiñas that $L_2$ is not isomorphic (and, in fact, not even Morita equivalent) to $L_2 \cong L_2$ [AC]. This an important result in terms of classification, and shows that different techniques, and perhaps even different results, are expected to arise in the algebraic situation. Other researchers have been interested in this result, and there are currently three different proofs. These ``algebraic surprises" are the kinds of results of which a BIRS workshop can make the general graph algebra community aware. The workshop will also provide a forum for algebraists and functional analysts to exchange ideas on their significance.

A fourth project is more ambitious in scope, and deals with examining the relationship between graph $C^*$-algebras and Leavitt path algebras. There are stunning similarities between the two classes of graph algebras. For instance, in both theories one attempts to determine those graphs $E$ for which the graph $C^*$-algebra $C^*(E)$ and the Leavitt path algebra $L_K(E)$ have a specified $C^*$-algebraic or algebraic property. As it turns out, for nearly every graph-theoretic property of $E$ that is known to be equivalent to a $C^*$-algebraic property of $C^*(E)$, the same graph-theoretic property of $E$ is equivalent to the corresponding property of $L_K(E)$. For example, the graph-theoretic conditions under which $C^*(E)$ is a simple algebra (respectively, an AF-algebra, a purely infinite simple algebra, an exchange ring, a finite-dimensional algebra) in the category of $C^*$-algebras are precisely the same graph-theoretic conditions under which $L_K(E)$ is a simple algebra (respectively, an ultramatricial algebra, a purely infinite simple algebra, an exchange ring, a finite-dimensional algebra) in the category of $K$-algebras. Moreover, the Leavitt path algebra results seem to hold independently of the field $K$, and in particular for the field $K=\mathbb{C}$ of complex numbers.

Despite these similarities, the relationship between various structural properties of $C^*(E)$ and $L_K(E)$ is as mysterious as it is startling. In fact, there is currently no known way to take results from one class and deduce results for the other class. Instead, the theorems from each class have had to be proven using techniques different from the other class. The large number of similarities in properties and statements of results might suggest that theorems once obtained on either the graph $C^*$-algebra side or on the Leavitt path algebra side might then immediately be translated via some sort of Rosetta Stone to a corresponding theorem on the other side. Nonetheless, a vehicle to transfer information in this way remains elusive, and in fact, researchers are uncertain how to even formulate conjectures that would lead to such a vehicle.

A current, active, cross-disciplinary project related to this investigation focuses on finding a complete answer to what would seem to be a fundamental question in this context (the ``Isomorphism Question"): If $E$ and $F$ are graphs for which $L_\mathbb{C}(E)\cong L_\mathbb{C}(F)$, is it true that $C^*(E) \cong C^*(F)$? Conversely, if $C^*(E)\cong C^*(F)$, do we get $L_\mathbb{C}(E)\cong L_\mathbb{C}(F)$? Partial answers (covering specific classes of graphs) have been provided in [AT]. Also, determining the necessity of the sign of the determinant (which we mentioned earlier) is a test case for this question. At the proposed workshop we hope to gain a better understanding of these issues, find more general answers to the Isomorphism (and Morita Equivalence) Questions, and gain more insight into how the graph determines the structure of the associated graph $C^*$-algebra and Leavitt path algebra.

#### Bibliography

- [ALPS] G. Abrams, A. Louly, E. Pardo, and C. Smith,
*Flow invariants in the classification of Leavitt path algebras*, J. Algebra**333**(2011), 202--231. - [AT] G. Abrams and M. Tomforde,
*Isomorphism and Morita equivalence of graph algebras*, Trans. Amer. Math. Soc.**363**(2011), 3733--3767. - [AC] P. Ara and G. Cortiñas,
*Tensor products of Leavitt path algebras*, preprint. arXiv:1108.0352 - [Boy] M. Boyle,
*Flow equivalence of shifts of finite type via positive factorizations*, Pacific J. Math.**204**(2002), no. 2, 273--317. - [BoyHua] M. Boyle and D. Huang,
*Poset block equivalence of integral matrices*, Trans. Amer. Math. Soc.**355**(2003), no. 10, 3861--3886 - [Cun3] J. Cuntz,
*A class of $C^*$-algebras and topological Markov chains II: reducible chains and the Ext-functor for $C^*$-algebras*, Invent. Math.**63**(1981), 25--40. - [CK] J. Cuntz and W. Krieger,
*A class of $C^*$-algebras and topological Markov chains*, Invent. Math.**56**(1980), 251--268. - [Ell2] G. A. Elliott,
*On the classification of inductive limits of sequences of semisimple finite-dimensional algebras*, J. Algebra**38**(1976), 29--44. - [GH] K. R. Goodearl and D. E. Handelman,
*Classification of ring and $C^*$-algebra direct limits of finite-dimensional semisimple real algebras*. Mem. Amer. Math. Soc.**69**(1987). - [Hua] D. Huang,
*Flow equivalence of reducible shifts of finite type and Cuntz-Krieger algebras*, J. Reine Angew. Math.**462**(1995), 185--217.