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List of Open Problems
 The Cuntz Splice for Leavitt path algebras. If E is a graph, the operation of the Cuntz splice attaches a
portion to the graph that changes the sign of det (I–A^{t}), where A is the vertex matrix of E. Here is an example: In the following graph, we attach the Cuntz
Splice at the starred vertex, and label the adjoined vertices and edges.
The Cuntz splice is important in the classification of C*algebras of finite graphs. As a special case of the construction, we can consider
the graph with a single vertex and two edges, which we denote E_{2}, and the graph formed by performing a Cuntz splice to it, denoted
E_{2}^{–}.
In [Rør95] Rørdam showed that C*(E_{2}) is isomorphic to C*(E_{2}^{–}). It automatically follows
from this that if E is any finite graph whose associated algebra is simple and purely infinite, and if F is the graph formed by performing the
Cuntz splice at any vertex of F, then C*(E) is Morita equivalent to C*(F). (More generally, Eilers, Ruiz, and Sørensen have shown in a forthcoming preprint, that if
E is any graph with a finite number of vertices and if F is the graph formed by performing the
Cuntz splice at any vertex of E that is the base point of two distinct cycles, then C*(E) is Morita equivalent to C*(F).)
The classification for Leavitt path algebras of finite graphs has been hindered by the lack of analogous results concerning the Cuntz splice. In particular, the following are important open problems.
Question: Let E be a graph whose associated algebra is simple and purely infinite, and let K be any field. If F
is the graph formed by performing the Cuntz splice to any vertex of E, then is L_{K} (E) Morita equivalent to L_{K} (F)?
The answer to the above question is the last remaining piece in the classification of unital Leavitt path algebras. See [RT13, Section 8] for a discussion of these issues.
Question: If K is any field, then are L_{K}(E_{2}) and L_{K}(E_{2}^{–}) isomorphic as rings?
(Note: Sometimes in the notation, the field K is suppressed and one writes L_{2} and L_{2–} in place of
L_{K}(E_{2}) and L_{K}(E_{2}^{–}), respectively. The question is then often stated as: Are L_{2} and L_{2–} isomorphic?)
Comment: Since the algebraic K_{0}groups of L_{K}(E_{2}) and L_{K}(E_{2}^{–}) are both zero, these algebras will be Morita equivalent if and only if they are isomorphic. In addition, we mention that
all algebraic Kgroups of these two algebras are zero, so the Kgroups cannot be used to distinguish between the two.
Comment: One can verify that the homogeneous zero components of L_{K}(E_{2}) and L_{K}(E_{2}^{–}) have different K_{0}groups and hence are not isomorphic. Thus there does not exist a graded isomorphism from
L_{K}(E_{2}) onto L_{K}(E_{2}^{–}).
Comment: Unlike in the C*algebra case, an affirmative answer to the second question does not immediately imply an affirmative answer to the first. For a discussion
of what extra properties are needed, see [ALPS11, Section 2], particularly "The Hypothesis" listed there.
References
[ALPS11] G. Abrams, A. Louly, E. Pardo, and C. Smith, Flow invariants in the classification of Leavitt path algebras, J. Algebra 333 (2011), 202231.
[Rør95] M. Rørdam, Classification of CuntzKrieger algebras, KTheory 9 (1995), no. 1, 3158.
[RT13] E. Ruiz and M. Tomforde, Classification of unital simple Leavitt path algebras of infinite graphs, J. Algebra 384 (2013), 4583.
 Kgroups of unital Leavitt path algebras. In [RT13, Example 11.2] an example was given of graphs E and F that each have associated algebras that are unital, simple, purely infinite, and have the property that
K_{0} (L_{ℚ}(E)) ≅ K_{0} (L_{ℚ}(F)) and K_{1} (L_{ℚ}(E)) ≅ K_{1} (L_{ℚ}(F)), but
K_{2} (L_{ℚ}(E)) is not isomorphic to K_{2} (L_{ℚ}(F)). (Here, ℚ denotes the field of rational numbers.)
Question: For a given field K and any natural number N, do there exist graphs E and F, each with a finite number of vertices, an infinite number of edges, and associated algebras that are simple and purely infinite, such
that K_{i} (L_{K}(E)) ≅ K_{i} (L_{K}(F)) for all 1 ≤ i ≤ N1,
but K_{N} (L_{K}(E)) is not isomorphic to K_{N} (L_{K}(F))?
It follows from [RT13, Theorem 7.1] that such a field K must necessarily be a field with free quotients, as defined in [RT13, Section 6].
References
[RT13] E. Ruiz and M. Tomforde, Classification of unital simple Leavitt path algebras of infinite graphs, J. Algebra 384 (2013), 4583.
 Twisted kgraph algebras. In [KPS12] and [KPS11] it was described how a
Τvalued 2cocycle c on a kgraph Λ can be incorporated into the relations defining the associated C*algebra to
obtain a twisted kgraph C*algebra C*(Λ, c). (Here Τ denotes the unit circle consisting of complex numbers of modulus one.) It was shown in [KPS13] that if C*(Λ) is a Kirchberg algebra,
then C*(Λ, c) ≅ C*(Λ) for any 2cocycle c. In [ACaHR13] the authors showed that for a kgraph Λ
and a ring R one can define a "higherrank Leavitt path algebra" KP_{R}(Λ), which they call a KumjianPask algebra over R.
For a 2cycle c it is possible to mimic the definition of a twisted kgraph C*algebra to define a "twisted KumjianPask algebra" KP_{R}(Λ,c).
This raises the following question.
Question: Let C denote the field of complex numbers. Does there exist a 2graph Λ and a Τvalued 2cocycle c on Λ such that C*(Λ) is a Kirchberg algebra,
and KP_{C}(Λ) is not isomorphic to KP_{C}(Λ, c)?
References
[ACaHR13] G. Aranda Pino, J. Clark, A. an Huef, and I. Raeburn, KumjianPask algebras of higherrank graphs, Trans. Amer. Math. Soc. 365 (2013), 36133641.
[KPS11] A. Kumjian, D. Pask and A. Sims, On twisted higherrank graph C*algebras, preprint 2011.
[KPS12] A. Kumjian, D. Pask, and A. Sims, Homology for higherrank graphs and twisted C*algebras, J. Funct. Anal. 263 (2012), 15391574.
[KPS13] A. Kumjian, D. Pask, and A. Sims, On the Ktheory of twisted higherrankgraph C*algebras, J. Math. Anal. Appl. 401 (2013), 104113.
 Moves on Graphs. If E and F are graphs with a finite number of vertices, it follows from results of
[CK80], [Rør95], and [Sør12] that C*(E) is Morita equivalent to C*(F) if and only if the graph E may be transformed into the graph F using
the following five moves and their inverses: (S) Source Removal, (O) Outsplitting, (I) Insplitting, (R) Reduction, and (C) Cuntz Splice.
Question: Is there a set of graph moves that generates Morita equivalence for all graph C*algebras? In other words, is there a set of graph moves such that if E and F are (possibly infinite) graphs, then C*(E) is Morita equivalent to C*(F) if and only if
E may be transformed into F using these moves?
Question: Is there a set of graph moves that generates isomorphism for graph C*algebras? In other words, is there a set of moves such that if E and F are (possibly infinite) graphs, then C*(E) is isomorphic to C*(F) if and only if
E may be transformed into F using these moves? (An answer to this question would be interesting even under the hypotheses that E and F are finite graphs whose associated C*algebras are simple.)
With regards to the first question, a starting point is to consider the graph
The C*algebra of this graph is nonunital, simple, and purely infinite. By computing the Ktheory of this C*algebra, we see that it has the same Kgroups as the Cuntz algebra
O_{∞}, and thus by the KirchbergPhillips classification theorem, this C*algebra is Morita equivalent to O_{∞}. Since O_{∞} is the C*algebra of the graph
an answer to the first question would require one to find a move that would change the first graph into the second. To date, no one has been able to show that the C*algebras of these two
graphs are Morita equivalent without using the KirchbergPhillips classification theorem. In addition, it is unknown if the Leavitt path algebras of these two graphs are Morita equivalent.
References
[CK80] J. Cuntz and W. Krieger, A class of C*algebras and topological Markov chains, Invent. Math. 56 (1980), 251268.
[Rør95] M. Rørdam, Classification of CuntzKrieger algebras, KTheory 9 (1995), no. 1, 3158.
[Sør12] A.P.W. Sørensen, Geometric classification of simple graph algebras, preprint 2012.
 Dependence of Isomorphism Class on the Field.
Question: Do there exist graphs E and F and fields K and K' such that L_{K} (E) ≅ L_{K} (F),
but L_{K'} (E) is not isomorphic to L_{K'} (F)? Likewise, do there exist graphs E and F and a field K such that L_{K} (E) ≅ L_{K} (F),
but L_{ℤ}(E) is not isomorphic to L_{ℤ} (F), where ℤ denotes the ring of integers?
See [Tom11] for the definition of and basic results for Leavitt path algebras over rings.
References
[Tom11] M. Tomforde, Leavitt path algebras with coefficients in a commutative ring, J. Pure Appl. Algebra 215 (2011), 471484.
 The Isomorphism and Morita Equivalence Conjectures. These conjectures
were first posed by Abrams and Tomforde in [AT11].
The Isomorphism Conjecture: Let ℂ denote the field of complex numbers. If E and F are graphs and if L_{ℂ}(E) and L_{ℂ}(F) are
isomorphic as rings, then are C*(E) and C*(F) isomorphic as C*algebras?
The Morita Equivalence Conjecture: Let ℂ denote the field of complex numbers. If E and F are graphs and if L_{ℂ}(E) and L_{ℂ}(F) are
Morita equivalent as rings, then are C*(E) and C*(F) strongly Morita equivalent as C*algebras?
See [AT11] for the first appearance of these conjectures. Also see [AT11] and [RT13] for partial progress, and lists of classes of graphs where the conjectures are known to hold.
References
[AT11] G. Abrams and M. Tomforde, Isomorphism and Morita equivalence of graph algebras, Trans. Amer. Math. Soc. 363 (2011), 37333767.
[RT13] E. Ruiz and M. Tomforde, Idealrelated Ktheory for Leavitt path algebras and graph C*algebras, Indiana Univ. Math J., to appear.
 Continuous Orbit Equivalence. If E is a graph,
the subalgebra D(E) is defined to be the closure of span { S_{α} S_{α}* : α is a finite path }.
One can show that if the graph E satisfies Condition (L), then D(E) is a MASA of C*(E).
Given a graph E, we let Σ_{E} denote the onesided shift space consisting of onesided infinite paths in E together with
the canonical shift map.
In [Mat10, Theorem 1], Matsumoto proves that if E and F finite graphs with no sinks and whose associated C*algebras are simple, then
Σ_{E} and Σ_{F} are continuously orbit equivalent if and only if there an isomorphism
φ : C*(E) > C*(F) with φ (D(E)) = D(F). (The definition of "continuously orbit equivalent" can be found in
[Mat10].)
This raises the following questions.
Question: If E and F are graphs and C*(E) ≅ C*(F), then is there an isomorphism
φ : C*(E) > C*(F) with φ (D(E)) = D(F)?
Question: If E and F are graphs and C*(E) is strongly Morita Equivalent to C*(F), then is there
a Morita equivalence between C*(E) and C*(F) that preserves D(E) and D(F)?
In [Mat13, Theorem 4.3] Matsumoto has proven that if E and F are finite graphs with no sinks and whose associated C*algebras are simple,
and if the sign of det (I–A^{t}) is equal to the sign of det (I–B^{t}) (where A and B are the vertex matrices of E and F,
respectively), then C*(E) is isomorphic to C*(F) if and only if Σ_{E} and Σ_{F} are continuously orbit equivalent.
In [Mat13, Section 6] Matsumoto states that there are no known examples of onesided shifts Σ_{E} and Σ_{F}
that are continuously orbit equivalent and with sign of det (I–A^{t}) not equal to the sign of det (I–B^{t}). This has led
Matsumoto to make the following conjecture.
Matsumoto's Conjecture: If E is a graph with onesided shift Σ_{E} and vertex matrix A, then det (I–A^{t}) is an invariant of the continuous orbit
equivalence class of Σ_{E}.
Matsumoto points out in [Mat13, Section 6] that if this conjecture is true, the triple (K_{0} (C*(E)), [1]_{0}, det (I–A^{t}))
would be a complete invariant for the continuous orbit equivalence class of the onesided shift Σ_{E}. This would imply
that two onesided topological Markov shifts Σ_{E} and Σ_{F} are
continuously orbit equivalent if and only if the graph C*algebras C*(E) and C*(F)
are isomorphic and det (I–A^{t}) = det (I–B^{t}).
References
[Mat10] K. Matsumoto, Orbit equivalence of topological Markov shifts and CuntzKrieger algebras, Pacific J. Math. 246 (2010), 199225.
[Mat13] K. Matsumoto, Classification of CuntzKrieger algebras by orbit equivalence of topological Markov shifts, Proc. Amer. Math. Soc. 141 (2013), 23292342.
 Analogues of O_{2} ⊗ O_{2} ≅ O_{2}.
A famous and important theorem of Elliott (which was exposited by Rørdam in [Rør95]) states that O_{2} ⊗ O_{2} ≅ O_{2}.
It is natural to ask if an analogue of this result holds for the Leavitt algebra with two generators. If K is a field, and L_{2} denotes the Leavitt algebra with two generators over K (i.e.,
the Leavitt path algebra of the graph with one vertex and two edges), then Ara and Cortiñas showed in [AC12] that L_{2} ⊗ L_{2} and L_{2}
are not isomorphic, and indeed not even Morita equivalent. However, one can still ask the following question.
Question: Is L_{2} ⊗ L_{2} isomorphic to a subalgebra of L_{2}?
Equivalently, one may ask if there exists an injective homomorphism φ : L_{2} ⊗ L_{2} > L_{2}, and since L_{2} ⊗ L_{2}
is simple it suffices to produce a nonzero homomorphism φ : L_{2} ⊗ L_{2} > L_{2}. Moreover, since eL_{2}e is
isomorphic to L_{2} for any nonzero idempotent e in L_{2}, the existence of a nonzero homomorphism from L_{2} ⊗ L_{2} into L_{2}
implies the existence of a unital injective homomorphism from L_{2} ⊗ L_{2} into L_{2}.
References
[AC12] P. Ara and G. Cortiñas, Tensor products of Leavitt path algebras, Centre de Recerca Matematica, Preprint 1042, 2011.
[Rør95] M. Rørdam, A short proof of Elliott's theorem: O_{2} ⊗ O_{2} ≅ O_{2}. C. R. Math. Rep. Acad. Sci. Canada 16 (1994), 3136.
 L^{p}versions of the Cuntz algebras and UHF algebras.
Chris Phillips has defined L^{p}versions of the Cuntz algebras for p ∈ [1,∞], which are denoted O_{d}^{p}. Definitions and basic facts for these objects can be found in
his paper [Phi12] as well as the slides from his talk at the BIRS Workshop on Graph Algebras.
Phillips has created a long list of open problems, which we make available in PDF form here:
Chris Phillips' list of problems on L^{p}versions of the Cuntz algebras and UHF algebras
We highlight one problem from this list (see Problem 5.3 in the PDF), which is particularly intriguing.
Question: For which values of p ∈ [1,∞] is it true that O_{2}^{p} ⊗_{p} O_{2}^{p}
is isomorphic to O_{2}^{p}?
References
[Phi12] N.C. Phillips, Analogs of Cuntz algebras on L^{p}spaces, preprint (2013).
 Onesided and twosided shift spaces.
If E is a finite graph with no sinks or sources, we let Σ_{E} denote the onesided shift space consisting of onesided infinite
paths in E, and we let X_{E} denote the twosided shift space consisting of twosided biinfinite
paths in E. It is well known that for irreducible graphs, the Morita equivalence class of C*(E) is closely related to flow equivalence class of
X_{E} (see [Rør95]). In particular, if E and F are finite graphs with no sinks whose associated algebras are simple, then
X_{E} is flow equivalent to X_{F} if and only if C*(E) is Morita equivalent to C*(F) and the sign of
det (I–A^{t}) is equal to the sign of det (I–B^{t}) (where A and B are the vertex
matrices of E and F, respectively). One may ask how other dynamical properties of Σ_{E} and X_{E}
are related to the isomorphism class and Morita equivalence class of C*(E).
Question: Let E and F be finite graphs with no sinks or sources. It is known that if the twosided shifts X_{E} and X_{F}
are conjugate, then there is a gaugeinvariant Morita equivalence between C*(E) and C*(F). Can a converse, or partial converse, to this result be obtained? In other words,
if there is a gaugeinvariant Morita equivalence between C*(E) and C*(F), what additional hypotheses are needed to ensure that X_{E} and X_{F}
are conjugate?
Question: Let E and F be finite graphs with no sinks. It is known that if the onesided shifts Σ_{E} and Σ_{F}
are conjugate, then there is a gaugeinvariant isomorphism between C*(E) and C*(F). Can a converse, or partial converse, to this result be obtained? In other words,
if there is a gaugeinvariant isomorphism between C*(E) and C*(F), what additional hypotheses are needed to ensure that Σ_{E} and Σ_{F}
are conjugate?
Question: Let E and F be finite graphs with no sinks. Is there a notion of flow equivalence for the onesided shift
Σ_{E}? If not, can one be developed? Is the flow equivalence class of Σ_{E}
related to the isomorphism class or the Morita equivalence class of C*(E)?
Note: This problem and the questions it asks are related to Problem 13 below.
References
[CK80] J. Cuntz and W. Krieger, A class of C*algebras and topological Markov chains, Invent. Math. 56 (1980), 251268.
[Rør95] M. Rørdam, Classification of CuntzKrieger algebras, KTheory 9 (1995), no. 1, 3158.
 Separated Graph Algebras.
In [AG11] and [AG12] Ara and Goodearl introduced separated Leavitt path algebras (generalizing the Leavitt algebras L(m,n)), and they also introduced their
C*algebra counterparts. If (E,C) is a separated graph, we let M(E,C) denote the abelian monoid with generators
{a_{v} : v ∈ E^{0}} satisfying the relations a_{v} = Σ_{e ∈ X} a_{r(e)}
for all v ∈ E^{0} and all X ∈ C_{v}. It was shown in [AG12, Theorem 4.3] that there is a natural map
M(E,C) > V(L(E,C)) sending a_{v} to [v] ∈ V(L(E,C)), where V(L(E,C)) is the abelian monoid
of Murrayvon Neumann equivalence classes of projections in matrices over L(E,C). If we let V(C*(E,C)) denote the abelian monoid
of Murrayvon Neumann equivalence classes of projections in matrices over C*(E,C), there is a similar natural map M(E,C) > V(C*(E,C)) sending a_{v} to [v] ∈ V(C*(E,C)).
Question: Is the natural map M(E,C) > V(C*(E,C)) an isomorphism? (This is equivalent to asking if the natural map
V(L(E,C)) > V(C*(E,C)) is an isomorphism.)
The result is certainly true for nonseparated graphs [AMP07, Theorem 7.1]. In addition, if the answer to this question is positive,
it would follow, as in [AG12, Corollary 4.5], that every conical abelian monoid is isomorphic to V(C*(E,C)) for some finitely separated graph (E,C). If the answer is negative,
one would still like to know whether this map is always injective.
References
[AG11] P. Ara and K.R. Goodearl, C*algebras of separated graphs, J. Funct. Anal. 261 (2011), 25402568.
[AG12] P. Ara and K.R. Goodearl, Leavitt path algebras of separated graphs, J. Reine Angew. Math. 669 (2012), 165224.
[AMP07] P. Ara, M.A. Moreno, and E. Pardo, Nonstable Ktheory for graph algebras, Algebr. Represent. Theory 10 (2007), 157178.
 Phantom CuntzKrieger Algebras.
If A is a C*algebra, we say that A "looks like a CuntzKrieger algebra" if all of the following conditions are satisfied:
 A is unital, purely infinite, nuclear, separable, and has real rank zero.
 A has finitely many ideals.
 Every subquotient of A has finitely generated K_{0}group, finitely generated and free K_{1}group,
and rank of the K_{0}group equal to rank of the K_{1}group.
 The simple subquotients of A are in the bootstrap class of Rosenberg and Schochet.
A "phantom CuntzKrieger algebra" is then defined to be a C*algebra that looks like a CuntzKrieger algebra but is not isomorphic to a CuntzKrieger algebra.
Question: Do phantom CuntzKrieger algebras exist?
Arklint has obtained a number of results concerning phantom CuntzKrieger algebras in [Ark13]. She has also shown
that simple phantom CuntzKrieger algebras do not exist [Ark13, Corollary 3.3], and that phantom
CuntzKrieger algebras with exactly one ideal do not exist [Ark13, Corollary 3.7].
References
[Ark13] S. Arklint, Do phantom CuntzKrieger algebras exist?, preprint.
 The Graded Grothendieck Group as a Classification Tool
Roozbeh Hazrat has outlined problems related to the
graded Grothendieck group and formally conjectured that this group
serves as a complete invariant for the graded isomorphism of Leavitt path
algebras. He has also formulated an analytic version of the conjecture
for graph C*algebras. More details can be found in this writeup:
Conjectures regarding the Graded Grothendieck Group (and LaTeX source), a
writeup by Roozbeh Hazrat
Note: This problem is related to Problem 10 above.
