Graded Steinberg algebras and their representations
aa r X i v : . [ m a t h . K T ] A p r GRADED STEINBERG ALGEBRAS AND THEIR REPRESENTATIONS
PERE ARA, ROOZBEH HAZRAT, HUANHUAN LI, AND AIDAN SIMS
Abstract.
We study the category of left unital graded modules over the Steinberg algebra of a graded ample Hausdorffgroupoid. In the first part of the paper, we show that this category is isomorphic to the category of unital left modulesover the Steinberg algebra of the skew-product groupoid arising from the grading. To do this, we show that theSteinberg algebra of the skew product is graded isomorphic to a natural generalisation of the the Cohen-Montgomerysmash product of the Steinberg algebra of the underlying groupoid with the grading group. In the second part of thepaper, we study the minimal (that is, irreducible) representations in the category of graded modules of a Steinbergalgebra, and establish a connection between the annihilator ideals of these minimal representations, and effectivenessof the groupoid.Specialising our results, we produce a representation of the monoid of graded finitely generated projective modulesover a Leavitt path algebra. We deduce that the lattice of order-ideals in the K -group of the Leavitt path algebrais isomorphic to the lattice of graded ideals of the algebra. We also investigate the graded monoid for Kumjian–Paskalgebras of row-finite k -graphs with no sources. We prove that these algebras are graded von Neumann regular rings,and record some structural consequences of this. Introduction
There has long been a trend of “algebraisation” of concepts from operator theory into algebra. This trend seemsto have started with von Neumann and Kaplansky and their students Berberian and Rickart to see what propertiesin operator algebra theory arise naturally from discrete underlying structures [33]. As Berberian puts it [13], “ifall the functional analysis is stripped away. . . what remains should stand firmly as a substantial piece of algebra,completely accessible through algebraic avenues”.In the last decade, Leavitt path algebras [2, 5] were introduced as an algebraisation of graph C ∗ -algebras [36, 41]and in particular Cuntz–Krieger algebras. Later, Kumjian–Pask algebras [11] arose as an algebraisation of higher-rank graph C ∗ -algebras [35]. Quite recently Steinberg algebras were introduced in [48, 21] as an algebraisation ofthe groupoid C ∗ -algebras first studied by Renault [44]. Groupoid C ∗ -algebras include all graph C ∗ -algebras andhigher-rank graph C ∗ -algebras, and Steinberg algebras include Leavitt and Kumjian–Pask algebras as well as inversesemigroup algebras. More generally, groupoid C ∗ -algebras provide a model for inverse-semigroup C ∗ -algebras, andthe corresponding inverse-semigroup algebras are the Steinberg algebras of the corresponding groupoids. All of theseclasses of algebras have been attracting significant attention, with particular interest in whether K -theoretic data canbe used to classify various classes of Leavitt path algebras, inspired by the Kirchberg–Phillips classification theoremfor C ∗ -algebras [40].In this note we study graded representations of Steinberg algebras. For a Γ-graded groupoid G , (i.e., a groupoid G with a cocycle map c : G →
Γ) Renault proved [44, Theorem 5.7] that if Γ is a discrete abelian group with Pontryagindual b Γ, then the C ∗ -algebra C ∗ ( G × c Γ) of the skew-product groupoid is isomorphic to a crossed-product C ∗ -algebra C ∗ ( G ) × b Γ. Kumjian and Pask [34] used Renault’s results to show that if there is a free action of a group Γ on a graph E , then the crossed product of graph C ∗ -algebra by the induced action is strongly Morita equivalent to C ∗ ( E/ Γ),where E/ Γ is the quotient graph.Parallelling Renault’s work, we first consider the Steinberg algebras of skew-product groupoids (for arbitrarydiscrete groups Γ). We extend Cohen and Montgomery’s definition of the smash product of a graded ring by thegrading group (introduced and studied in their seminal paper [24]) to the setting of non-unital rings. We then provethat the Steinberg algebra of the skew-product groupoid is isomorphic to the corresponding smash product. Thisallows us to relate the category of graded modules of the algebra to the category of modules of its smash product.Specialising to Leavitt path algebras, the smash product by the integers arising from the canonical grading yields an
Date : April 3, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Steinberg algebra, Leavitt path algebra, skew-product, smash product, graded irreducible representation,annihilator ideal, effective groupoid. ultramatricial algebra. This allows us to give a presentation of the monoid of graded finitely generated projectivemodules for Leavitt path algebras of arbitrary graphs. In particular, we prove that this monoid is cancellative. Thegroup completion of this monoid is called the graded Grothendieck group, K gr0 , which is a crucial invariant in studyof Leavitt path algebras. It is conjectured [31, § K gr0 and establish a lattice isomorphism between orderideals of K gr0 and graded ideals of Leavitt path algebras.We then apply the smash product to Kumjian–Pask algebras KP K (Λ). Unlike Leavitt path algebras, Kumjian–Pask algebras of arbitrary higher rank graphs are poorly understood, so we restrict our attention to row finite k -graphs with no sources. We show that the smash product of KP K (Λ) by Z k is also an ultramatricial algebra. Thisallows us to show that KP K (Λ) is a graded von Neumann regular ring and, as in the case of Leavitt path algebras, itsgraded monoid is cancellative. Several very interesting properties of Kumjian–Pask algebras follow as a consequenceof general results for graded von Neumann regular rings.We then proceed with a systematic study of the irreducible representations of Steinberg algebras. In [16], Chenused infinite paths in a graph E to construct an irreducible representation of the Leavitt path algebra E . Theserepresentations were further explored in a series of papers [4, 9, 10, 32, 43]. The infinite path representations ofKumjian–Pask algebras were also defined in [11]. In the setting of a groupoid G , the infinite path space becomes theunit space of the groupoid. For any invariant subset W of the unit space, the free module RW with basis W is arepresentation of the Steinberg algebra A R ( G ) [15]. These representations were used to construct nontrivial ideals ofthe Steinberg algebra, and ultimately to characterise simplicity.For the Γ-graded groupoid G , we introduce what we call Γ-aperiodic invariant subsets of the unit space of thegroupoid G . We obtain graded (irreducible) representations of the Steinberg algebra via these Γ-aperiodic invariantsubsets. We then describe the annihilator ideals of these graded representations and establish a connection betweenthese annihilator ideals and effectiveness of the groupoid. Specialising to the case of Leavitt and Kumjian–Paskalgebras we obtain new results about representations of these algebras.The paper is organised as follows. In Section 2, we recall the background we need on graded ring theory, andthen introduce the smash product A A , possibly without unit. We establish anisomorphism of categories between the category of unital left A A -modules. This theory is used in Section 3, where we consider the Steinberg algebra associated to a Γ-graded amplegroupoid G . We prove that the Steinberg algebra of the skew-product of G × c Γ is graded isomorphic to the smashproduct of A R ( G ) with the group Γ.In Section 4 we collect the facts we need to study the monoid of graded rings with graded local units. In Section 5and Section 6, we apply the isomorphism of categories in Section 2 and the graded isomorphism of Steinberg algebras(Theorem 3.4) on the setting of Leavitt path algebras and Kumjian–Pask algebras. Although Kumjian–Pask algebrasare a generalisation of Leavitt path algebras, we treat these classes separately as we are able to study Leavitt pathalgebras associated to any arbitrary graph, whereas for Kumjian–Pask algebras we consider only row-finite k -graphswith no sources, as the general case is much more complicated [42, 46]. We describe the monoids of graded finitelygenerated projective modules over Leavitt path algebras and Kumjian–Pask algebras, and obtain a new descriptionof their lattices of graded ideals. In Section 7, we turn our attention to the irreducible representations of Steinbergalgebras. We consider what we call Γ-aperiodic invariant subset of the groupoid G and construct graded simple A R ( G )-modules. This covers, as a special case, previous work done in the setting of Leavitt path algebras, and givesnew results in the setting of Kumjian–Pask algebras. We describe the annihilator ideals of the graded modules overa Steinberg algebra and prove that these ideals reflect the effectiveness of the groupoid.2. Graded rings and smash products
Graded rings.
Let Γ be a group with identity ε . A ring A (possibly without unit) is called a Γ -graded ring if A = L γ ∈ Γ A γ such that each A γ is an additive subgroup of A and A γ A δ ⊆ A γδ for all γ, δ ∈ Γ. The group A γ is called the γ - homogeneous component of A. When it is clear from context that a ring A is graded by group Γ , wesimply say that A is a graded ring . If A is an algebra over a ring R , then A is called a graded algebra if A is a gradedring and A γ is a R -submodule for any γ ∈ Γ. A Γ-graded ring A = L γ ∈ Γ A γ is called strongly graded if A γ A δ = A γδ for all γ, δ in Γ.The elements of S γ ∈ Γ A γ in a graded ring A are called homogeneous elements of A. The nonzero elements of A γ are called homogeneous of degree γ and we write deg( a ) = γ for a ∈ A γ \{ } . The set Γ A = { γ ∈ Γ | A γ = 0 } is calledthe support of A . We say that a Γ-graded ring A is trivially graded if the support of A is the trivial group { ε } —that RADED STEINBERG ALGEBRAS AND THEIR REPRESENTATIONS 3 is, A ε = A , so A γ = 0 for γ ∈ Γ \{ ε } . Any ring admits a trivial grading by any group. If A is a Γ-graded ring and s ∈ A , then we write s α , α ∈ Γ for the unique elements s α ∈ A α such that s = P α ∈ Γ s α . Note that { α ∈ Γ : s α = 0 } is finite for every s ∈ A .We say a Γ-graded ring A has graded local units if for any finite set of homogeneous elements { x , · · · , x n } ⊆ A ,there exists a homogeneous idempotent e ∈ A such that { x , · · · , x n } ⊆ eAe . Equivalently, A has graded local units,if A ε has local units and A ε A γ = A γ A ε = A γ for every γ ∈ Γ.Let M be a left A -module. We say M is unital if AM = M and it is Γ -graded if there is a decomposition M = L γ ∈ Γ M γ such that A α M γ ⊆ M αγ for all α, γ ∈ Γ. We denote by A -Mod the category of unital left A -modulesand by A -Gr the category of Γ-graded unital left A -modules with morphisms the A -module homomorphisms thatpreserve grading.For a graded left A -module M , we define the α - shifted graded left A -module M ( α ) as M ( α ) = M γ ∈ Γ M ( α ) γ , (2.1)where M ( α ) γ = M γα . That is, as an ungraded module, M ( α ) is a copy of M , but the grading is shifted by α . For α ∈ Γ, the shift functor T α : A -Gr −→ A -Gr , M M ( α )is an isomorphism with the property T α T β = T αβ for α, β ∈ Γ.2.2.
Smash products.
Let A be a Γ-graded unital R -algebra where Γ is a finite group. In the influential paper [24],Cohen and Montgomery introduced the smash product associated to A , denoted by A R [Γ] ∗ . They proved twomain theorems, duality for actions and coactions, which related the smash product to the ring A . In turn, thesetheorems relate the graded structure of A to non-graded properties of A . The construction has been extended to thecase of infinite groups (see for example [12, 45] and [38, § Definition 2.1.
For a Γ-graded ring A (possibly without unit), the smash product ring A P γ ∈ Γ r ( γ ) p γ , where r ( γ ) ∈ A and p γ are symbols. Addition is defined component-wise andmultiplication is defined by linear extension of the rule ( rp α )( sp β ) = rs αβ − p β , where r, s ∈ A and α, β ∈ Γ.It is routine to check that A p γ do not belong to A A has unit, then we regard the p γ as elements of A A p γ with p γ . Each p γ is then anidempotent element of A A A ∗ of [12]. If Γ is finite, then A A k [Γ] ∗ of [24]. Note that A A γ = X α ∈ Γ A γ p α . (2.2)Next we define a shift functor on A A -Gr (seeProposition 2.5). This does not seem to be exploited in the literature and will be crucial in our study of K -theoryof Leavitt path algebras ( § α ∈ Γ, there is an algebra automorphism S α : A −→ A S α ( sp β ) = sp βα for sp β ∈ A s ∈ A and β ∈ Γ. We sometimes call S α the shift map associatedto α . For M ∈ A α ∈ Γ, we obtain a shifted A S α ∗ M obtained by setting S α ∗ M := M as agroup, and defining the left action by a · S α ∗ M m := S α ( a ) · M m . For α ∈ Γ, the shift functor e S α : A −→ A , M
7→ S α ∗ M is an isomorphism satisfying e S α e S β = e S αβ for α, β ∈ Γ.If A is a unital ring then A Lemma 2.2.
Let A be a Γ -graded ring with graded local units. Then the ring A has graded local units. PERE ARA, ROOZBEH HAZRAT, HUANHUAN LI, AND AIDAN SIMS
Proof.
Take a finite subset X = { x , x , · · · x n } ⊆ A x i are homogeneous elements. Since homoge-neous elements of A rp α for r ∈ A a homogeneous element and α ∈ Γ, we mayassume that x i = r i p α i , 1 ≤ i ≤ n , where r i ∈ A are homogeneous of degree γ i and α i ∈ Γ. Since A has graded localunits, there exists a homogenous idempotent e ∈ A such that er i = r i e = r i for all i . Consider the finite set Y = { γ ∈ Γ | γ = α i or γ = γ i α i for 1 ≤ i ≤ n } , and let w = P γ ∈ Y ep γ . Since the idempotent e ∈ A is homogeneous, w is a homogeneous element of A w = w and wx i = x i = x i w for all i . (cid:3) As we will see in Sections 5 and 6, smash products of Leavitt path algebras or of Kumjian–Pask algebras areultramatricial algebras, which are very well-behaved. This allows us to obtain results about the path algebras viatheir smash product. For example, ultramatricial algebras are von Neumann regular rings. The following lemmaallows us to exploit this property (see Theorems 6.4, 6.5). Recall that a graded ring is called graded von Neumannregular if for any homogeneous element a , there is an element b such that aba = a . Lemma 2.3.
Let A be a Γ -graded ring (possibly without unit). Then A is graded von Neumann regular if andonly if A is graded von Neumann regular.Proof. Suppose A a ∈ A γ , for some γ ∈ G . Since ap e ∈ ( A γ (see (2.2)), there is anelement P α ∈ Γ b γ α p α ∈ ( A γ − with deg( b γ α ) = γ − , α ∈ Γ, such that ap e (cid:16) X α ∈ Γ b γ α p α (cid:17) ap e = ap e . This identity reduces to ab γ γ ap e = ap e . Thus ab γ γ a = a . This shows that A is graded regular.Conversely, suppose A is graded regular and x := P α ∈ Γ a γ α p α ∈ ( A γ . By (2.2) we have deg( a γ α ) = γ , α ∈ Γ.Then there are b γ − α ∈ A γ − such that a γ α b γ − α a γ α = a γ α , for α ∈ Γ. Consider the element y := P α ∈ Γ b γ − α p γα ∈ ( A γ − . One can then check that xyx = x . Thus A (cid:3) An isomorphism of module categories.
In this section we first prove that, for a Γ-graded ring A withgraded local units, there is an isomorphism between the categories A A -Gr (Proposition 2.5). This isa generalisation of [18, Theorem 2.2] and [12, Theorem 2.6]. We check that the isomorphism respects the shiftingin these categories. This in turn translates the shifting of modules in the category of graded modules to an actionof the group on the category of modules for the smash-product. Since graded Steinberg algebras have graded localunits, using this result and Theorem 3.4, we obtain a shift preserving isomorphism A R ( G × c Γ)-Mod ∼ = A R ( G )-Gr . In Section 5 we will use this in the setting of Leavitt path algebras to establish an isomorphism between thecategory of graded modules of L R ( E ) and the category of modules of L R ( E ), where E is the covering graph of E ( § Lemma 2.4.
Let A be a Γ -graded ring with a set of graded local units E . A left A -module M is unital if andonly if for every finite subset F of M , there exists w = P ni =1 up γ i with γ i ∈ Γ , and u ∈ E such that wx = x for all x ∈ F .Proof. Suppose that M is unital. Then each m ∈ F may be written as m = P n ∈ G m y n n for some finite G m ⊆ M and choice of scalars { y n : n ∈ G m } ⊆ A T := S m ∈ F G m . By Lemma 2.2, there exists a finite set Y of Γsuch that w = P γ ∈ Y up γ satisfies wy = y for all y ∈ T . So wm = m for all m ∈ F .Conversely, for m ∈ M , take F = { m } . Then there exists w such that m = wm ∈ ( A M ; that is, ( A M = M . (cid:3) RADED STEINBERG ALGEBRAS AND THEIR REPRESENTATIONS 5
Proposition 2.5.
Let A be a Γ -graded ring with graded local units. Then there is an isomorphism of categories A - Gr ∼ −→ A - Mod such that the following diagram commutes for every α ∈ Γ . A - Gr ∼ / / T α (cid:15) (cid:15) A - Mod e S α (cid:15) (cid:15) A - Gr ∼ / / A - Mod (2.3)
Proof.
We first define a functor φ : A −→ A - Gr as follows. Fix a set E of graded local units for A . Let M be a unital left A M as a Γ-graded left A -module M ′ as follows. For each γ ∈ Γ, define M ′ γ := X u ∈ E up γ M. We first show that for α ∈ Γ, we have M ′ α ∩ P γ ∈ Γ ,γ = α M ′ γ = { } . Suppose this is not the case, so there exist finiteindex sets F and { F ′ γ : γ ∈ Γ } (only finitely many nonempty), elements { u i : i ∈ F } and { v γ,j : γ ∈ Γ and j ∈ F ′ γ } in E , and elements { m i : i ∈ F } and { n γ,j : γ ∈ Γ and j ∈ F ′ γ } such that x = X i ∈ F u i p α m i = X γ ∈ Γ ,γ = α X j ∈ F ′ γ v γ,j p γ n γ,j , Fix e ∈ E such that eu i = u i = u i e for all i ∈ F . Using that the u i are homogeneous elements of trivial degree atthe second equality, we have ep α x = X i ∈ F ( ep α u i p α ) m i = X i ∈ F eu i p α m i = x. We also have ep α x = X γ ∈ Γ \{ α } X j ∈ F ′ γ ep α v γ,j p γ n γ,j = 0 . Hence x = 0.For r ∈ A γ and m ∈ M ′ α , define rm := rp α m . This determines a left A -action on M ′ α . For u ∈ E satisfying ur = r = ru , we have up γα rm = ( up γα rp α ) m = urp α m = rm. Hence rm ∈ M ′ γα . One can easily check the associativity of the A -action. Using Lemma 2.4 we see that M = M ′ assets. We claim that M ′ is a unital A -module. For m ∈ M ′ γ , we write m = P u ∈ E ′ up γ m u , where E ′ ⊆ E is a finiteset and m u ∈ M . Since u is a homogeneous idempotent, u ( up γ m u ) = up γ ( up γ m u ) = up γ m u . Thus u ( up γ m u ) = up γ m u ∈ AM ′ implies that m ∈ AM ′ showing that M ′ = AM ′ . We can therefore define φ : Obj( A → Obj( A - Gr) by φ ( M ) = M ′ .To define φ on morphisms, fix a morphism f in A m = P γ ∈ Γ m γ ∈ M ′ such that m γ = P u ∈ F γ up γ m u with F γ a finite subset of E , we define f ′ : M ′ → N ′ by f ′ ( m γ ) = f (cid:16) X u ∈ F γ up γ m u (cid:17) = X u ∈ F γ up γ f ( m u ) = f ( m ) γ . (2.4)To see that f ′ is an A -module homomorphism, fix m ∈ M ′ γ and r ∈ A . Since f ( m ) ∈ M ′ γ , we have f ′ ( rm ) = f ( rp γ m ) = rp γ f ( m ) = rf ′ ( m ) . The definition (2.4) shows that it preserves the gradings. That is, f ′ is a Γ-graded A -module homomorphism. So wecan define φ on morphisms by φ ( f ) = f ′ . It is routine to check that φ is a functor.Next we define a functor ψ : A - Gr −→ A N = ⊕ γ ∈ Γ N γ be a Γ-graded unital left A -module.Let N ′′ be a copy of N as a group. Fix n ∈ N , and write n = P γ ∈ Γ n γ . Fix r ∈ A and α ∈ Γ, and define( rp α ) n = rn α . It is straightforward to check that this determines an associative left A N ′′ . We claim that N ′′ is a unital A n ∈ N ′′ . Since AN = N , we can express n = P li =1 r i n i , with the n i homogeneous in PERE ARA, ROOZBEH HAZRAT, HUANHUAN LI, AND AIDAN SIMS N and the r i ∈ A , and we can then write each r i as r i = P β ∈ Γ r i,β as a sum of homogeneous elements r i,β ∈ A β .For any γ ∈ Γ, n γ = l X i =1 r i,β ( n i ) β − γ = l X i =1 ( r i,β p β − γ ) n i ∈ ( A N ′′ . So we can define ψ : Obj( A - Gr) → Obj( A ψ ( N ) = N ′′ . Since ψ ( N ) = N ′′ is just a copy of N as amodule, we can define ψ on morphisms simply as the identity map; that is, if f : M → N is a homomorphism ofgraded A -modules, then for m ∈ M we write m ′′ for the same element regarded as an element of M ′′ , and we have ψ ( f )( m ′′ ) = f ( m ) ′′ . Again, it is straightforward to check that ψ is a functor.To prove that ψ ◦ φ = Id A and φ ◦ ψ = Id A - Gr , it suffices to show that ( M ′ ) ′′ = M for M ∈ A N ′′ ) ′ = N for N ∈ A -Gr; but this is straightforward from the definitions.To prove the commutativity of the diagram in (2.3), it suffices to show that the A ψ ◦ T α )( N ) = N ( α ) ′′ and ( e S α ◦ ψ )( N ) = N ′′ ( α ) coincide for any N ∈ A - Gr. Take any n ∈ N and sp β ∈ A s ∈ A and β ∈ Γ. For n ∈ N ′′ ( α ) and a typical spanning element sp β of A sp β ) n = ( sp βα ) n = sn βα . On the otherhand, for the same n regarded as an element of N ′′ , and the same sp β ∈ A sp β ) n = sn ′ β = sn βα . Since N ( α ) β = N βα by definition, this completes the proof. (cid:3) The Steinberg algebra of the skew-product
In this section, we consider the skew-product of an ample groupoid G carrying a grading by a discrete group Γ.We prove that the Steinberg algebra of the skew-product is graded isomorphic to the smash product by Γ of theSteinberg algebra associated to G . This result will be used in Section 5 to study the category of graded modules overLeavitt path algebras and give a representation of the graded finitely generated projective modules.3.1. Graded groupoids.
A groupoid is a small category in which every morphism is invertible. It can also beviewed as a generalization of a group which has partial binary operation. Let G be a groupoid. If x ∈ G , d ( x ) = x − x is the domain of x and r ( x ) = xx − is its range . The pair ( x, y ) is composable if and only if r ( y ) = d ( x ). Theset G (0) := d ( G ) = r ( G ) is called the unit space of G . Elements of G (0) are units in the sense that xd ( x ) = x and r ( x ) x = x for all x ∈ G . For U, V ∈ G , we define
U V = (cid:8) αβ | α ∈ U, β ∈ V and r ( β ) = d ( α ) (cid:9) . A topological groupoid is a groupoid endowed with a topology under which the inverse map is continuous, and suchthat composition is continuous with respect to the relative topology on G (2) := { ( β, γ ) ∈ G×G : d ( β ) = r ( γ ) } inheritedfrom G × G . An ´etale groupoid is a topological groupoid G such that the domain map d is a local homeomorphism.In this case, the range map r is also a local homeomorphism. An open bisection of G is an open subset U ⊆ G suchthat d | U and r | U are homeomorphisms onto an open subset of G (0) . We say that an ´etale groupoid G is ample ifthere is a basis consisting of compact open bisections for its topology.Let Γ be a discrete group and G a topological groupoid. A Γ-grading of G is a continuous function c : G →
Γsuch that c ( α ) c ( β ) = c ( αβ ) for all ( α, β ) ∈ G (2) ; such a function c is called a cocycle on G . In this paper, we shallalso refer to c as the degree map on G . Observe that G decomposes as a topological disjoint union F γ ∈G c − ( γ ) ofsubsets satisfying c − ( β ) c − ( γ ) ⊆ c − ( βγ ). We say that G is strongly graded if c − ( β ) c − ( γ ) = c − ( βγ ) for all β, γ .For γ ∈ Γ, we say that X ⊆ G is γ -graded if X ⊆ c − ( γ ). We always have G (0) ⊆ c − ( ε ), so G (0) is ε -graded. Wewrite B co γ ( G ) for the collection of all γ -graded compact open bisections of G and B co ∗ ( G ) = [ γ ∈ Γ B co γ ( G ) . Throughout this note we only consider Γ-graded ample Hausdorff groupoids.3.2.
Steinberg algebras.
Steinberg algebras were introduced in [48] in the context of discrete inverse semigroupalgebras and independently in [21] as a model for Leavitt path algebras. We recall the notion of the Steinberg algebraas a universal algebra generated by certain compact open subsets of an ample Hausdorff groupoid.
Definition 3.1.
Let G be a Γ-graded ample Hausdorff groupoid and B co ∗ ( G ) = S γ ∈ Γ B co γ ( G ) the collection of allgraded compact open bisections. Given a commutative ring R with identity, the Steinberg R -algebra associated to G , denoted A R ( G ), is the algebra generated by the set { t B | B ∈ B co ∗ ( G ) } with coefficients in R , subject to RADED STEINBERG ALGEBRAS AND THEIR REPRESENTATIONS 7 (R1) t ∅ = 0;(R2) t B t D = t BD for all B, D ∈ B co ∗ ( G ); and(R3) t B + t D = t B ∪ D whenever B and D are disjoint elements of B co γ ( G ) for some γ ∈ Γ such that B ∪ D is abisection.Every element f ∈ A R ( G ) can be expressed as f = P U ∈ F a U t U , where F is a finite subset of elements of B co ∗ ( G ).It was proved in [18, Proposition 2.3] (see also [21, Theorem 3.10]) that the Steinberg algebra defined above isisomorphic to the following construction: A R ( G ) = span { U : U is a compact open bisection of G} , where 1 U : G → R denotes the characteristic function on U . Equivalently, if we give R the discrete topology, then A R ( G ) = C c ( G , R ), the space of compactly supported continuous functions from G to R . Addition is point-wise andmultiplication is given by convolution ( f ∗ g )( γ ) = X { αβ = γ } f ( α ) g ( β ) . It is useful to note that 1 U ∗ V = 1 UV for compact open bisections U and V (see [48, Proposition 4.5(3)]) and the isomorphism between the two constructionsis given by t U U on the generators. By [18, Lemma 2.2] and [21, Lemma 3.5], every element f ∈ A R ( G ) can beexpressed as f = X U ∈ F a U U , (3.1)where F is a finite subset of mutually disjoint elements of B co ∗ ( G ).Recall from [23, Lemma 3.1] that if c : G →
Γ is a continuous 1-cocycle into a discrete group Γ, then the Steinbergalgebra A R ( G ) is a Γ-graded algebra with homogeneous components A R ( G ) γ = { f ∈ A R ( G ) | supp( f ) ⊆ c − ( γ ) } . The family of all idempotent elements of A R ( G (0) ) is a set of local units for A R ( G ) ([20, Lemma 2.6]). Here, A R ( G (0) ) ⊆ A R ( G ) is a subalgebra. Since G (0) ⊆ c − ( ε ) is trivially graded, A R ( G ) is a graded algebra with gradedlocal units. Note that any ample Hausdorff groupoid admits the trivial cocycle from G to the trivial group { ε } , whichgives rise to a trivial grading on A R ( G ).3.3. Skew-products.
Let G be an ample Hausdorff groupoid, Γ a discrete group, and c : G →
Γ a continuouscocycle. Then G admits a basis B of compact open bisections. Replacing B with B ′ = { U ∩ c − ( γ ) | U ∈ B , γ ∈ Γ } ,we obtain a basis of compact open homogeneous bisections.To a Γ-graded groupoid G one can associate a groupoid called the skew-product of G by Γ. The aim of this sectionis to relate the Steinberg algebra of the skew-product groupoid to the Steinberg algebra of G . We recall the notionof skew-product of a groupoid (see [44, Definition 1.6]). Definition 3.2.
Let G be an ample Hausdorff groupoid, Γ a discrete group and c : G →
Γ a continuous cocycle. The skew-product of G by Γ is the groupoid G × c Γ such that ( x, α ) and ( y, β ) are composable if x and y are composableand α = c ( y ) β . The composition is then given by (cid:0) x, c ( y ) β (cid:1)(cid:0) y, β (cid:1) = ( xy, β ) with the inverse ( x, α ) − = ( x − , c ( x ) α ).Note that our convention for the composition of the skew-product here is slightly different from that in [44,Definition 1.6]. The two determine isomorphic groupoids, but when we establish the isomorphism of Theorem 3.4,the composition formula given here will be more obviously compatible with the multiplication in the smash product. Lemma 3.3.
Let G be a Γ -graded ample groupoid. Then the skew-product G × c Γ is a Γ -graded ample groupoid underthe product topology on G × Γ and with degree map ˜ c ( x, γ ) := c ( x ) .Proof. We can directly check that under the product topology on
G ×
Γ, the inverse and composition of the skew-product
G × c Γ are continuous making it a topological groupoid. Since the domain map d : G −→ G (0) is a localhomeomorphism, the domain map (also denoted d ) from G × c Γ to G (0) × Γ is d × id Γ so restricts to a homeomorphismon U × Γ for any set U on which d is a homeomorphism. So d : G × c Γ → ( G × c Γ) (0) is a local homeomorphism.Since the inverse map is clearly a homeomorphism, it follows that the range map is also a local homeomorphism. PERE ARA, ROOZBEH HAZRAT, HUANHUAN LI, AND AIDAN SIMS If B is a basis of compact open bisections for G , then { B × { γ } | B ∈ B and γ ∈ Γ } is a basis of compactopen bisections for the topology on G × c Γ. Since composition on
G × c Γ agrees with composition in G in the firstcoordinate, it is clear that ˜ c is a continuous cocycle. (cid:3) The Steinberg algebra A R ( G × c Γ) associated to
G × c Γ is a Γ-graded algebra, with homogeneous components A R ( G × c Γ) γ = (cid:8) f ∈ A R ( G × c Γ) | supp( f ) ⊆ c − ( γ ) × Γ (cid:9) for γ ∈ Γ.We are in a position to state the main result of this section.
Theorem 3.4.
Let G be a Γ -graded ample, Hausdorff groupoid and R a unital commutative ring. Then there isan isomorphism of Γ -graded algebras A R ( G × c Γ) ∼ = A R ( G ) , assigning U ×{ α } to U p α for each compact openbisection U of G and α ∈ Γ .Proof. We first define a representation { t U | U ∈ B co ∗ ( G × c Γ) } in the algebra A R ( G ) U isa graded compact open bisection of G × c Γ, say U ⊆ ˜ c − ( α ), then for each γ ∈ Γ, the set U ∩ G × { γ } is a compactopen bisection. Since these are mutually disjoint and U is compact, there are finitely many (distinct) γ , . . . , γ l ∈ Γsuch that U = F li =1 U ∩ G × { g i } . Each U ∩ G × { g i } has the form U i × { γ i } where U i ⊆ U is compact open. The U i have mutually disjoint sources because the domain map on G × c Γ is just d × id, and U is a bisection. So each U i ∈ B co α ( G ), and U = F li =1 U i × { γ i } . Using this decomposition, we define t U = l X i =1 U i p γ i . We show that these elements t U satisfy (R1)–(R3). Certainly if U = ∅ , then t U = 0, giving (R1). For (R2), take V ∈ B co β ( G × c Γ), and decompose V = S mj =1 V j × { γ ′ j } as above. Then t U t V = l X i =1 U i p γ i m X j =1 V j p γ ′ j = l X i =1 m X j =1 U i p γ i V j p γ ′ j = m X j =1 l X i =1 U i p γ i V j p γ ′ j = m X j =1 X { ≤ i ≤ l | γ i ( γ ′ j ) − = β } U i V j p γ ′ j . (3.2)On the other hand, by the composition of the skew-product G × c Γ, we have
U V = l [ i =1 m [ j =1 U i × { γ i } · V j × { γ ′ j } = m [ j =1 l [ i =1 U i × { γ i } · V j × { γ ′ j } = m [ j =1 [ { ≤ i ≤ l | γ i = βγ ′ j } U i V j × { γ ′ j } . For each 1 ≤ j ≤ m , there exists at most one 1 ≤ i ≤ l such that γ i = βγ ′ j and U i V j ∈ B co αβ ( G ). It follows that t UV = P sj =1 P { ≤ i ≤ l | γ i ( γ ′ j ) − = β } U i V j p γ ′ j . Comparing this with (3.2), we obtain t U t V = t UV .To check (R3), suppose that U and V are disjoint elements of B co ω ( G × c Γ) for some ω ∈ Γ such that U ∪ V isa bisection of G × c Γ. Write them as U = S li =1 U i × { γ i } and V = S mj =1 V i × { γ ′ j } as above. We have t U + t V = P li =1 U i p γ i + P mj =1 V j p γ ′ j . On the other hand U ∪ V = ( S li =1 U i × { γ i } ) S ( S mj =1 V i × { γ ′ j } ). If γ i = γ ′ j , then U i × { γ i } ∪ V j × { γ ′ j } = ( U i ∪ V j ) × { γ i } . Since U and V are disjoint and U ∪ V is a bisection, we deduce that r ( U i ) ∩ r ( V j ) = ∅ = d ( U i ) ∩ d ( V j ) so that U i ∪ V j is a bisection. So t U i ×{ γ i }∪ V j ×{ γ ′ j } = t U i ∪ V j ×{ γ i } = 1 U i ∪ V j p γ i = 1 U i p γ i + 1 V j p γ i = 1 U i p γ i + 1 V j p γ ′ j . This shows that after combining pairs where γ i = γ ′ j as above, we obtain t U + t V = t U ∪ V .By the universality of Steinberg algebras, we have an R -homomorphism, φ : A R ( G × c Γ) −→ A R ( G ) φ (1 U ×{ α } ) = 1 U p α for each compact open bisection U of G and α ∈ Γ. From the definition of φ , it isevident that φ preserves the grading. Hence, φ is a homomorphism of Γ-graded algebras. RADED STEINBERG ALGEBRAS AND THEIR REPRESENTATIONS 9
Next we prove that φ is an isomorphism. For any element ap γ ∈ A R ( G ) a ∈ A R ( G ) and γ ∈ Γ, there is afinite index set T , elements { r i | i ∈ T } of R , and compact open bisections K i ∈ B co ∗ ( G ) such that ap γ = X i ∈ T r i K i p γ = X i ∈ T r i φ (1 K i ×{ γ } ) ∈ Im φ. So φ is surjective. It remains to prove that φ is injective. Take an element x ∈ A R ( G × c Γ) such that φ ( x ) = 0. Since φ is graded, we can assume that x is homogeneous, say x ∈ A R ( G × c Γ) γ . By (3.1), there is a finite set F , mutuallydisjoint B i ∈ B co γ ( G × c Γ) indexed by i ∈ F and coefficients r i ∈ R indexed by i ∈ F such that x = X i ∈ F r i B i . For each B i , we write B i = S k ∈ F i B ik × { δ ik } such that F i is a finite set and the δ ik indexed by k ∈ F i are distinct.Set ∆ = { δ ik | i ∈ F, k ∈ F i } . For each δ ∈ ∆, let F δ ⊆ F be the collection F δ = (cid:8) i ∈ F : δ ∈ { δ ik : k ∈ F i } (cid:9) . Then φ ( x ) = X i ∈ F r i φ (1 B i ) = X i ∈ F X k ∈ F i r i B ik p δ ik = X δ ∈ ∆ X i ∈ F δ r i B i,k ( δ ) p δ = 0 . For any δ ∈ ∆, we obtain P i ∈ F δ r i B i,k ( δ ) = 0. Since the B i are mutually disjoint, for any element g ∈ G , we have (cid:16) X i ∈ F δ r i B i,k ( δ ) (cid:17) ( g ) = ( r i , if g ∈ B i,k ( δ ) for some i ∈ F δ ;0 , otherwise . Then r i = 0 for any i ∈ F δ , giving x = 0. (cid:3) C ∗ -algebras and crossed-products. In the groupoid- C ∗ -algebra literature, it is well-known that if G is aΓ-graded groupoid, and Γ is abelian, then the C ∗ -algebra C ∗ ( G ×
Γ) of the skew-product groupoid is isomorphic tothe crossed product C ∗ -algebra C ∗ ( G ) × α c b Γ, where α c is the action of the Pontryagin dual b Γ such that α cχ ( f )( g ) = χ ( c ( g )) f ( g ) for f ∈ C c ( G ), χ ∈ b Γ, and g ∈ G . This extends to nonabelian Γ via the theory of C ∗ -algebraic coactions.In this subsection, we reconcile this result with Theorem 3.4 by showing that there is a natural embedding of A C ( G ) C ∗ ( G ) × α c b Γ when Γ is abelian.
Lemma 3.5.
Suppose that Γ is a discrete abelian group and that G is a Γ -graded groupoid with grading cocycle c : G → Γ . For a ∈ A C ( G ) and γ ∈ Γ , define a · ˆ γ ∈ C ( b Γ , C ∗ ( G )) ⊆ C ∗ ( G ) × α c Γ by (cid:0) a · ˆ γ (cid:1) ( χ ) = χ ( γ ) a. Then there is a homomorphism A C ( G ) ֒ → C ∗ ( G ) × α c b Γ that carries ap γ to a · ˆ γ .Proof. The multiplication in the crossed-product C ∗ -algebra is given on elements of C ( b Γ , C ∗ ( G )) by ( F ∗ G )( χ ) = R b Γ F ( ρ ) α cρ ( G ( ρ − χ )) dµ ( ρ ), where µ is Haar measure on b Γ.The action of b Γ induces a Γ-grading of C ∗ ( G ) × α c b Γ such that for a ∈ C ∗ ( G ) × α c b Γ and γ ∈ Γ, the correspondinghomogeneous component a γ of a is given by a γ = Z b Γ χ ( γ ) α cχ ( a ) dµ ( χ ) . There is certainly a linear map i : A C ( G ) → C ∗ ( G ) × α c b Γ satisfying i ( ap γ ) = a · ˆ γ ; we just have to check thatit is multiplicative. For this, fix a, b ∈ A C ( G ) and γ, β ∈ Γ and χ ∈ b Γ, and calculate (cid:0) i ( ap γ ) i ( bp β ) (cid:1) ( χ ) = Z b Γ i ( ap γ )( ρ ) α cρ ( i ( bp β )( ρ − χ )) dµ ( ρ ) = Z b Γ a · ˆ γ ( ρ ) α cρ ( b ˆ β ( ρ − χ )) dµ ( ρ )= Z b Γ ρ ( γ ) a ( ρ − χ )( β ) α cρ ( b ) dµ ( ρ ) = χ ( β ) a Z b Γ ρ ( γ − β ) α cρ ( b ) dµ ( ρ )= χ ( β ) ab γ − β = ( ab γ − β ) · ˆ β = i ( ab γ − β p β ) = i ( ap γ bp β ) . So i is multiplicative as required. (cid:3) Non-stable graded K -theory For a unital ring A , we denote by V ( A ) the abelian monoid of isomorphism classes of finitely generated projectiveleft A -modules under direct sum. In general for an abelian monoid M and elements x, y ∈ M , we write x ≤ y if y = x + z for some z ∈ M . An element d ∈ M is called distinguished (or an order unit ) if for any x ∈ M , we have x ≤ nd for some n ∈ N . A monoid is called conical , if x + y = 0 implies x = y = 0. Clearly V ( A ) is conical with adistinguished element [ A ]. For a finitely generated conical abelian monoid M containing a distinguished element d ,Bergman constructed a “universal” K -algebra B (here K is a field) for which there is an isomorphism φ : V ( B ) → M ,such that φ ([ B ]) → d ([14, Theorem 6.2]).For a (finite) directed graph E , one defines an abelian monoid M E generated by the vertices, identifying a vertexwith the sum of vertices connected to it by edges (see § L K ( E ) associated to thegraph E , i.e., V ( L K ( E )) ∼ = M E . Leavitt path algebras of directed graphs have been studied intensively since theirintroduction [2, 5]. The classification of such algebras is still a major open topic and one would like to find a completeinvariant for such algebras. Due to the success of K -theory in the classification of graph C ∗ -algebras [40], one wouldhope that the Grothendieck group K with relevant ingredients might act as a complete invariant for Leavitt pathalgebras; particularly since K ( L K ( E )) is the group completion of V ( L K ( E )). However, unless the graph consists ofonly cycles with no exit, V ( L K ( E )) is not a cancellative monoid (Lemma 5.5) and thus V ( L K ( E )) → K ( L K ( E )) isnot injective, reflecting that K might not capture all the properties of L K ( E ).For a graded ring A one can consider the abelian monoid of isomorphism classes of graded finitely generatedprojective modules denoted by V gr ( A ). Since a Leavitt path algebra has a canonical Z -graded structure, one canconsider V gr ( L K ( E )). One of the main aims of this paper is to show that the graded monoid carries substantialinformation about the algebra.In Sections 5 and 6 we will use the results on smash products obtained in Section 3 to study the graded monoid ofLeavitt path algebras and Kumjian–Pask algebras. In this section we collect the facts we need on the graded monoidof a graded ring with graded local units.4.1. The monoid of a graded ring with graded local units.
For a ring A with unit, the monoid V ( A ) isdefined as the set of isomorphism classes [ P ] of finitely generated projective A -modules P , with addition given by[ P ] + [ Q ] = [ P ⊕ Q ].For a non-unital ring A , we consider a unital ring e A containing A as a two-sided ideal and define V ( A ) = { [ P ] | P is a finitely generated projective e A -module and P = AP } . (4.1)This construction does not depend on the choice of e A , as can be seen from the following alternative description: V ( A ) is the set of equivalence classes of idempotents in M ∞ ( A ), where e ∼ f in M ∞ ( A ) if and only if there are x, y ∈ M ∞ ( A ) such that e = xy and f = yx ([37, pp. 296]).When A has local units, V ( A ) = { [ P ] | P is a finitely generated projective A -module in A - Mod } . (4.2)To see this, recall that the unitisation ring e A of a ring A is a copy of Z × A with componentwise addition, andwith multiplication given by( n, a )( m, b ) = ( nm, ma + nb + ab ) for all n, m ∈ Z and a, b ∈ A .The forgetful functor provides a category isomorphism from e A - Mod to the category of arbitrary left A -modules [26,Proposition 8.29B]. Any A -module N can be viewed as a e A -module via ( m, b ) x = mx + bx for ( m, b ) ∈ e A and x ∈ N .By [6, Lemma 10.2], the projective objects in A - Mod are precisely those which are projective as e A -modules; thatis, the projective e A -modules P such that AP = P . Any finitely generated e A -module M with AM = M is a finitelygenerated A -module. In fact, suppose that M is generated as an e A -module by x , · · · , x n . Since AM = M , each x i can be written as x i = P t i j =1 b j x ij for some b j ∈ A and x ij ∈ M . Now any m ∈ M can be written m = n X i =1 a i x i = n X i =1 t i X j =1 a i b j x ij RADED STEINBERG ALGEBRAS AND THEIR REPRESENTATIONS 11 So { x ij | ≤ i ≤ n and 1 ≤ j ≤ t i } generates M as an A -module. Clearly any finitely generated A -module is afinitely generated e A -module. So the definitions of V ( A ) in (4.1) and (4.2) coincide.We need a graded version of (4.2) as this presentation will be used to study the monoid associated to the Leavittpath algebras of arbitrary graphs.Recall that for a group Γ and a Γ-graded ring A with unit, the monoid V gr ( A ) consists of isomorphism classes [ P ]of graded finitely generated projective A -modules with the direct sum [ P ] + [ Q ] = [ P ⊕ Q ] as the addition operation.For a non-unital graded ring A , a similar construction as in (4.1) can be carried over to the graded setting (see [31, § e A be a Γ-graded ring with identity such that A is a graded two-sided ideal of A . For example, consider e A = Z × A . Then e A is Γ-graded with e A = Z × A , and e A γ = 0 × A γ for γ = 0 . Define V gr ( A ) = { [ P ] | P is a graded finitely generated projective e A -module and AP = P } , (4.3)where [ P ] is the class of graded e A -modules, graded isomorphic to P , and addition is defined via direct sum. Then V gr ( A ) is isomorphic to the monoid of equivalence classes of graded idempotent matrices over A [31, pp. 146].Let A be a Γ-graded ring with graded local units. We will show that V gr ( A ) = { [ P ] | P is a graded finitely generated projective A -module in A - Gr } . (4.4)For this we need to relate the graded projective modules to modules which are projective. A graded A -module P in A - Gr is called a graded projective A -module if for any epimorphism π : M −→ N of graded A -modules in A - Gr andany morphism f : P −→ N of graded A -modules in A - Gr, there exists a morphism h : P −→ M of graded A -modulessuch that π ◦ h = f .In the case of unital rings, a module is graded projective if and only if it is graded and projective [31, Prop. 1.2.15].We need a similar statement in the setting of rings with local units. Lemma 4.1.
Let A be a Γ -graded ring with graded local units and P a graded unital left A -module. Then P is agraded projective left A -module in A - Gr if and only if P is a graded left A -module which is projective in A - Mod .Proof.
First suppose that P is a graded projective A -module in A - Gr. It suffices to prove that P is projective in A - Mod. For any homogeneous element p ∈ P of degree δ p , there exists a homogeneous idempotent e p ∈ A suchthat e p p = p . Let L p ∈ P h Ae p ( − δ p ) be the direct sum of graded A -modules where deg( e p ) = δ p and P h is the set ofhomogeneous elements of P . Then there exists a surjective graded A -module homomorphism f : M p ∈ P h Ae p ( − δ p ) −→ P such that f ( ae p ) = ae p p = ap for a ∈ Ae p . Since P is graded projective, there exists a graded A -module homomor-phism g : P −→ L p ∈ P h Ae p ( − δ p ) such that f g = Id P . Forgetting the grading, P is a direct summand of L p ∈ P h Ae p as an A -module. By [51, 49.2(3)], L p ∈ P h Ae p is projective in A - Mod. So P is projective in A - Mod.Conversely, suppose that P is a graded and projective A -module. Let π : M −→ N be an epimorphism of graded A -modules in A - Gr and f : P −→ N a morphism of graded A -modules in A - Gr. We first claim that any epimorphism π : M −→ N of graded A -modules in A - Gr is surjective. To prove the claim, write A h for the set of all homogeneouselements of A . Let X = { x ∈ N | A h x ⊆ π ( M ) } ⊆ N (cf. [27, § X is a graded submodule of N . Wedenote by q : N −→ N/X the quotient map. Then q ◦ π = 0. Hence, q = 0, giving N = X . It follows that N = π ( M ).So the epimorphism π : M −→ N of graded A -modules in A - Gr is surjective. Forgetting the grading, π : M −→ N isa surjective morphism of A -modules in A - Mod. Since P is projective in A - Mod, there exists h : P −→ M such that π ◦ h = f . By [31, Lemma 1.2.14], there exists a morphism h ′ : P −→ M of graded A -modules such that π ◦ h ′ = f .Thus, P is a graded projective left A -module in A - Gr. (cid:3) Thus for a Γ-graded ring A with graded local units, combining Lemma 4.1 with [6, Lemma 10.2] (i.e., projectiveobjects in A - Mod are precisely those that are projective as e A -modules), P is a graded finitely generated projective e A -module with AP = P if and only if P is a finitely generated A -module which is graded projective in A - Gr. Thisshows that the definitions of V gr ( A ) by (4.3) and (4.4) coincide. Application: Leavitt path algebras
In this section we study the monoid V gr ( L K ( E )) of the Leavitt path algebra of a graph E (4.4). Using the resultson smash products of Steinberg algebras obtained in Section 3, we give a presentation for this monoid in line with M E (see § V gr ( L K ( E )) is a cancellative monoid and thus the natural map V gr ( L K ( E )) → K gr0 ( L K ( E )) is injective (Corollary 5.8). It follows that there is a lattice correspondence between thegraded ideals of L K ( E ) and the graded ordered ideals of K gr0 ( L K ( E )) (Theorem 5.11). This is further evidence forthe conjecture that the graded Grothendieck group K gr0 may be a complete invariant for Leavitt path algebras [29].5.1. Leavitt path algebras modelled as Steinberg algebras.
We briefly recall the definition of Leavitt pathalgebras and establish notation. We follow the conventions used in the literature of this topic (in particular the pathsare written from left to right).A directed graph E is a tuple ( E , E , r, s ), where E and E are sets and r, s are maps from E to E . Wethink of each e ∈ E as an arrow pointing from s ( e ) to r ( e ). We use the convention that a (finite) path p in E is asequence p = α α · · · α n of edges α i in E such that r ( α i ) = s ( α i +1 ) for 1 ≤ i ≤ n −
1. We define s ( p ) = s ( α ), and r ( p ) = r ( α n ). If s ( p ) = r ( p ), then p is said to be closed. If p is closed and s ( α i ) = s ( α j ) for i = j , then p is called acycle. An edge α is an exit of a path p = α · · · α n if there exists i such that s ( α ) = s ( α i ) and α = α i . A graph E iscalled acyclic if there is no closed path in E .A directed graph E is said to be row-finite if for each vertex u ∈ E , there are at most finitely many edges in s − ( u ). A vertex u for which s − ( u ) is empty is called a sink , whereas u ∈ E is called an infinite emitter if s − ( u )is infinite. If u ∈ E is neither a sink nor an infinite emitter, then it is called a regular vertex . Definition 5.1.
Let E be a directed graph and R a commutative ring with unit. The Leavitt path algebra L R ( E ) of E is the R -algebra generated by the set { v | v ∈ E } ∪ { e | e ∈ E } ∪ { e ∗ | e ∈ E } subject to the following relations:(0) uv = δ u,v v for every u, v ∈ E ;(1) s ( e ) e = er ( e ) = e for all e ∈ E ;(2) r ( e ) e ∗ = e ∗ = e ∗ s ( e ) for all e ∈ E ;(3) e ∗ f = δ e,f r ( e ) for all e, f ∈ E ; and(4) v = P e ∈ s − ( v ) ee ∗ for every regular vertex v ∈ E .Let Γ be a group with identity ε , and let w : E −→ Γ be a function. Extend w to vertices and ghost edges bydefining w ( v ) = ε for v ∈ E and w ( e ∗ ) = w ( e ) − for e ∈ E . The relations in Definition 5.1 are compatible with w ,so there is a grading of L R ( E ) such that e ∈ L R ( E ) w ( e ) and e ∗ ∈ L R ( E ) w ( e ) − for all e ∈ E , and v ∈ L R ( E ) ε for all v ∈ E . The set of all finite sums of distinct elements of E is a set of graded local units for L R ( E ) [2, Lemma 1.6].Furthermore, L R ( E ) is unital if and only if E is finite.Leavitt path algebras associated to arbitrary graphs can be realised as Steinberg algebras. We recall from [23,Example 2.1] the construction of the groupoid G E from an arbitrary graph E , which was introduced in [36] forrow-finite graphs and generalised to arbitrary graphs in [39]. We then realise the Leavitt path algebra L R ( E ) as theSteinberg algebra A R ( G ). This allows us to apply Theorem 3.4 to the setting of Leavitt path algebras.Let E = ( E , E , r, s ) be a directed graph. We denote by E ∞ the set of infinite paths in E and by E ∗ the set offinite paths in E . Set X := E ∞ ∪ { µ ∈ E ∗ | r ( µ ) is not a regular vertex } . Let G E := { ( αx, | α | − | β | , βx ) | α, β ∈ E ∗ , x ∈ X, r ( α ) = r ( β ) = s ( x ) } . We view each ( x, k, y ) ∈ G E as a morphism with range x and source y . The formulas ( x, k, y )( y, l, z ) = ( x, k + l, z )and ( x, k, y ) − = ( y, − k, x ) define composition and inverse maps on G E making it a groupoid with G (0) E = { ( x, , x ) | x ∈ X } which we identify with the set X .Next, we describe a topology on G E . For µ ∈ E ∗ define Z ( µ ) = { µx | x ∈ X, r ( µ ) = s ( x ) } ⊆ X. For µ ∈ E ∗ and a finite F ⊆ s − ( r ( µ )), define Z ( µ \ F ) = Z ( µ ) \ [ α ∈ F Z ( µα ) . RADED STEINBERG ALGEBRAS AND THEIR REPRESENTATIONS 13
The sets Z ( µ \ F ) constitute a basis of compact open sets for a locally compact Hausdorff topology on X = G (0) E (see[50, Theorem 2.1]).For µ, ν ∈ E ∗ with r ( µ ) = r ( ν ), and for a finite F ⊆ E ∗ such that r ( µ ) = s ( α ) for α ∈ F , we define Z ( µ, ν ) = { ( µx, | µ | − | ν | , νx ) | x ∈ X, r ( µ ) = s ( x ) } , and then Z (( µ, ν ) \ F ) = Z ( µ, ν ) \ [ α ∈ F Z ( µα, να ) . The sets Z (( µ, ν ) \ F ) constitute a basis of compact open bisections for a topology under which G E is a Hausdorffample groupoid. By [23, Example 3.2], the map π E : L R ( E ) −→ A R ( G E ) (5.1)defined by π E ( µν ∗ − P α ∈ F µαα ∗ ν ∗ ) = 1 Z (( µ,ν ) \ F ) extends to an algebra isomorphism. We observe that the isomor-phism of algebras in (5.1) satisfies π E ( v ) = 1 Z ( v ) , π E ( e ) = 1 Z ( e,r ( e )) , π E ( e ∗ ) = 1 Z ( r ( e ) ,e ) , (5.2)for each v ∈ E and e ∈ E .5.2. Covering of a graph.
In this section we show that the smash product of a Leavitt path algebra is isomorphicto the Leavitt path algebra of its covering graph. We briefly recall the concept of skew product or covering of agraph (see [28, §
2] and [34, Def. 2.1]).Let Γ be a group and w : E −→ Γ a function. As in [28, § covering graph E of E with respect to w is givenby E = { v α | v ∈ E and α ∈ Γ } , E = { e α | e ∈ E and α ∈ Γ } ,s ( e α ) = s ( e ) α , and r ( e α ) = r ( e ) w ( e ) − α . Example 5.2.
Let E be a graph and define w : E → Z by w ( e ) = 1 for all e ∈ E . Then E (sometimes denoted E × Z ) is given by E = (cid:8) v n | v ∈ E and n ∈ Z (cid:9) , E = (cid:8) e n | e ∈ E and n ∈ Z (cid:9) ,s ( e n ) = s ( e ) n , and r ( e n ) = r ( e ) n − . As examples, consider the following graphs E : ue f ! ! vg b b F : u e e e f q q Then E : . . . u e / / f ' ' PPPPPPPPP u e / / f ' ' ❖❖❖❖❖❖❖❖❖ u − e − / / f − ' ' ❖❖❖❖❖❖❖❖❖ · · · . . . v g ♥♥♥♥♥♥♥♥♥ v g ♦♦♦♦♦♦♦♦♦ v − g − ♦♦♦♦♦♦♦♦♦ · · · and F : . . . u f ! ! e : : u f e u − f − " " e − ; ; · · · If E is any graph, and w : E → Γ any function, we extend w to E ∗ by defining w ( v ) = 0 for v ∈ E , and w ( α · · · α n ) = w ( α ) · · · w ( α n ). We obtain from [34, Lemma 2.3] a continuous cocycle e w : G E −→ Γ satisfying e w ( αx, | α | − | β | , βx ) = w ( α ) w ( β ) − . By Lemma 3.3 the skew-product groupoid G E × Γ is a Γ-graded ample groupoid. For each (possibly infinite) path x = e e e · · · ∈ E and each γ ∈ Γ there is a path x γ of E given by x γ = e γ e w ( e ) − γ e w ( e e ) − γ . . . . (5.3) There is an isomorphism f : G E × Γ −→ G E of groupoids such that f (( x, k, y ) , γ ) = ( x e w ( x,k,y ) γ , k, y γ ) (see [34, Theorem 2.4]). The grading of the skew-product G E × Γ induces a grading of G E , and the isomorphism f respects the gradings of the two groupoids, and so inducesa graded isomorphism of Steinberg algebras e f : A R ( G E × Γ) −→ A R ( G E ) . Set g = e f − : A R ( G E ) −→ A R ( G E × Γ). Then g (1 Z ( v γ ) ) = 1 Z ( v ) ×{ γ } for v ∈ E and γ ∈ Γ ,g (1 Z ( e α ,r ( e ) w ( e ) − α ) ) = 1 Z ( e,r ( e )) ×{ w ( e ) − α } for e ∈ E and α ∈ Γ , (5.4) g (1 Z ( r ( e ) w ( e ) − α ,e α ) ) = 1 Z ( r ( e ) ,e ) ×{ α } for e ∈ E and α ∈ Γ . Let φ : A R ( G E × Γ) → A R ( G E ) g : A R ( G E ) → A R ( G E × Γ) be theisomorphism (5.4), let π E : L R ( E ) → A R ( G E ) and π E : L R ( E ) → A R ( G E ) be as in (5.1), and let e π E : L R ( E ) → A R ( G E ) e π E ( xp γ ) = π E ( x ) p γ for x ∈ L R ( E ) and γ ∈ Γ. Define φ ′ := e π − E ◦ φ ◦ g ◦ π E . Then we havethe following commuting diagram: L R ( E ) φ ′ / / π E (cid:15) (cid:15) L R ( E ) e π E (cid:15) (cid:15) A R ( G E ) φ ◦ g / / A R ( G E ) . (5.5) Corollary 5.3.
The map φ ′ : L R ( E ) −→ L R ( E ) is an isomorphism of Γ -graded algebras such that φ ′ ( v β ) = vp β , φ ′ ( e α ) = ep w ( e ) − α and φ ′ ( e ∗ α ) = e ∗ p α for v ∈ E , e ∈ E , and α, β ∈ Γ .Proof. Since all the homomorphisms in the diagram (5.5) preserve gradings of algebras, the map φ ′ : L R ( E ) −→ L R ( E ) v γ ∈ E and each edge e α ∈ E , we have φ ′ ( v γ ) = ( e π − E ◦ φ ◦ g )(1 Z ( v γ ) ) = ( e π − E ◦ φ )(1 Z ( v ) ×{ γ } ) = e π − E (1 Z ( v ) p γ ) = vp γ ,φ ′ ( e α ) = ( e π − E ◦ φ ◦ g )(1 Z ( e α ,r ( e ) w ( e ) − α ) ) = ( e π − E ◦ φ )(1 Z ( e,r ( e )) ×{ w ( e ) − α } ) = e π − E (1 Z ( e,r ( e )) p w ( e ) − α ) = ep w ( e ) − α , and φ ′ ( e ∗ α ) = ( e π − E ◦ φ ◦ g )(1 Z ( r ( e ) w ( e ) − α ,e α ) ) = ( e π − E ◦ φ )(1 Z ( r ( e ) ,e ) ×{ α } ) = e π − E (1 Z ( r ( e ) ,e ) p α ) = e ∗ p α . (cid:3) In [34], Kumjian and Pask show that given a free action of a group Γ on a graph E , the crossed product C ∗ ( E ) × Γby the induced action is strongly Morita equivalent to C ∗ ( E/ Γ), where E/ Γ is the quotient graph and obtained anisomorphism similar to Corollary 5.3 for graph C ∗ -algebras. Corollary 5.3 shows that this isomorphism alreadyoccurs on the algebraic level (see § L C ( E ) (cid:15) (cid:15) / / L C ( E ) (cid:15) (cid:15) C ∗ ( E ) / / C ∗ ( E ) × Γ . Remark 5.4.
In [28], Green showed that the theory of coverings of graphs with relations and the theory of gradedalgebras are essentially the same. For a Γ-graded algebra A , Green constructed a covering of the quiver of A andshowed that the category of representations of the covering satisfying a certain set of relations is equivalent to thecategory of finite dimensional graded A -modules.For any graph E and a function w : E −→ Γ, we consider the smash product of a quotient algebra of the pathalgebra of E with the group Γ. Let K be a field, E a graph and w : E −→ Γ a weight map. Denote by KE thepath algebra of E . A relation in E is a K -linear combination P i k i q i with q i paths in E having the same source andrange. Let r be a set of relations in E and h r i the two sided ideal of KE generated by r . Set A r ( E ) = KE/ h r i . RADED STEINBERG ALGEBRAS AND THEIR REPRESENTATIONS 15
We denote by r the lifting of r in E . For each finite path p = e e · · · e n in E and γ ∈ Γ, there is a path p γ of E given by p γ = e Q ni =1 w ( e i ) γ · · · e n − Q ni = n − w ( e i ) γ e nw ( e n ) γ , similar as (5.3). More precisely, for each relation P i k i q i ∈ r and each γ ∈ Γ, we have P i k i q γi ∈ r. Set A r ( E ) = KE/ h r i . We prove that A r ( E ) ∼ = A r ( E ) h : KE −→ A r ( E ) h ( v γ ) = vp γ and h ( e α ) = ep w ( e ) − α for v ∈ E , e ∈ E and α, γ ∈ Γ. Since h annihilates the relations r , it induces a homomorphism h : A r ( E ) −→ A r ( E ) . We show that h is an isomorphism. For injectivity, suppose that x = P mi =1 λ i ξ i ∈ A r ( E ) with λ i ∈ K and ξ i pairwise distinct paths in E . Each ξ i has the form of ( ξ ′ i ) α i for some ξ ′ i ∈ E ∗ and α i ∈ Γ. If h ( x ) = 0, then h ( x ) = P mi =1 λ i ξ ′ i p α i = 0. Suppose that the α i are not distinct; so by rearranging, we can assume that α = · · · = α k for some k ≤ m . Then P ki =1 λ i ξ ′ i = 0 in A r ( E ). Observe that P ki =1 λ i ξ ′ i = 0 in A r ( E ) implies P ki =1 λ i ξ i = 0 in A r ( E ). Hence x = 0, implying h is injective. For surjectivity, fix η in E ∗ and γ ∈ Γ. Then h ( η γ ) = ηp γ by definition.Since the elements { ηp γ | η ∈ E ∗ , γ ∈ Γ } span A r ( E ) h is surjective. Thus h is an isomorphismas claimed.5.3. The monoid V gr ( L K ( E )) V gr ( L K ( E )) V gr ( L K ( E )) . In this subsection, we consider the Leavitt path algebra L K ( E ) over a field K . Ara,Moreno and Pardo [5] showed that for a Leavitt path algebra associated to the graph E , the monoid V ( L K ( E )) isentirely determined by elementary graph-theoretic data. Specifically, for a row-finite graph E , we define M E to bethe abelian monoid presented by E subject to v = X e ∈ s − ( v ) r ( e ) (5.6)for every v ∈ E that is not a sink. Theorem 3.5 of [5] says that V ( L K ( E )) ∼ = M E .There is an explicit description [5, §
4] of the congruence on the free abelian monoid given by the definingrelations of M E . Let F be the free abelian monoid on the set E . The nonzero elements of F can be writtenin a unique form up to permutation as P ni =1 v i , where v i ∈ E . Define a binary relation −→ on F \ { } by P ni =1 v i −→ P i = j v i + P e ∈ s − ( v j ) r ( e ) whenever j ∈ { , · · · , n } is such that v j is not a sink. Let −→ be the transitiveand reflexive closure of −→ on F \ { } and ∼ the congruence on F generated by the relation −→ . Then M E = F/ ∼ .Ara and Goodearl defined analogous monoids M ( E, C, S ) and constructed natural isomorphisms M ( E, C, S ) ∼ = V ( CL K ( E, C, S )) for arbitrary separated graphs (see [6, Theorem 4.3]). The non-separated case reduces to that ofordinary Leavitt path algebras, and extends the result of [5] to non-row-finite graphs.Following [6, 7], we recall the definition of M E when E is not necessarily row-finite. In [7, § v ∈ E of the abelian monoid M E for E are supplemented by generators q Z as Z runs through all nonempty finitesubsets of s − ( v ) for infinite emitters v . The relations are(1) v = P e ∈ s − ( v ) r ( e ) for all regular vertices v ∈ E ;(2) v = P e ∈ Z r ( e ) + q Z for all infinite emitters v ∈ E and all(3) q Z = P e ∈ Z \ Z r ( e ) + q Z for all nonempty finite sets Z ⊆ Z ⊆ s − ( v ), where v ∈ E is an infinite emitter.An abelian monoid M is cancellative if it satisfies full cancellation, namely, x + z = y + z implies x = y , forany x, y, z ∈ M . In order to prove that the graded monoid associated to any Leavitt path algebra is cancellative(Corollary 5.8), we will need to know that the monoid associated to Leavitt path algebras of acyclic graphs arecancellative. Lemma 5.5.
Let E be an arbitrary graph. The monoid M E is cancellative if and only if no cycle in E has an exit.In particular, if E is acyclic, then M E is cancellative.Proof. We first claim that M E is cancellative for any row-finite acyclic graph E . By [5, Lemma 3.1], the row-finitegraph E is a direct limit of a directed system of its finite complete subgraphs { E i } i ∈ X . In turn, the monoid M E isthe direct limit of { M E i } i ∈ X ([5, Lemma 3.4]). We claim that M E is cancellative. Let x + u = y + u in M E , where x, y, u are sum of vertices in E . By [5, Lemma 4.3], there exists b ∈ F (sum of vertices in E ) such that x + u −→ b and y + u −→ b . Observe that vertices involved in this transformations are finite. Thus there is a finite graph E i suchthat all these vertices are in E i . It follows that in M E i we have x + u −→ b and y + u −→ b . Thus x + u = y + u in M E i . Since the subgraph E i of E is finite and acyclic, M E i is a direct sum of copies of N (as L K ( E i ) is a semi-simplering) and thus is cancellative. So x = y in M E i and so the same in M E . Hence, M E is cancellative.We now show that it suffices to consider the case where E is a row-finite graph in which no cycle has an exit.To see this, let E be any graph, and let E d be its Drinen–Tomforde desingularisation [25], which is row-finite. Then L K ( E ) and L K ( E d ) are Morita equivalent, and so M E ∼ = M E d [3, Theorem 5.6]. So M E has cancellation if and onlyif M E d has cancellation. Since no cycle in E has an exit if and only if E d has the same property, it therefore sufficesto prove the result for row-finite graph E in which no cycle has an exit.Finally, we show that for any row-finite graph E in which no cycle has an exit, the monoid M E is cancellative.For this, fix a set C ⊆ E such that C contains exactly one edge from every cycle in E [47]. Let F be the subgraphof E obtained by removing all the edges in C . We claim that M F ∼ = M E . To see this, observe that they have thesame generating set F = E , and the generating relation F −→ is contained in E −→ . So it suffices to show that E −→ ⊆ F −→ .For this, note that for v ∈ E , we have s − E ( v ) = s − F ( v ) unless v = s ( e ) for some e ∈ C , in which case s − E ( v ) = { e } and s − F ( v ) = ∅ . So it suffices to show that for e ∈ C , we have s ( e ) F −→ r ( e ). Let p = eα α . . . α n be the cycle in E containing e . Then r ( e ) F −→ s ( α ) F −→ r ( α ) F −→ s ( α ) F −→ r ( α ) F −→ · · · F −→ s ( α n ) F −→ r ( α n ) F −→ s ( e ) . So M F ∼ = M E as claimed. So the preceding paragraphs show that M E is cancellative.Now suppose that E has a cycle with an exit; say p = α . . . α n has an exit α . Without loss of generality, s ( α ) = s ( α n ) and α = α n . Write s ( p ) E ≤ n = { q ∈ E ∗ : s ( q ) = s ( p ) , and | q | = n or | q | < n and r ( q ) is not regular } . Let p ′ := α . . . α n − α and X := s ( p ) E ≤ n \ { p, p ′ } . A simple induction shows that s ( p ) −→ X q ∈ s ( α ) E ≤ n r ( q ) = r ( p ) + r ( p ′ ) + X q ∈ X r ( q ) = s ( p ) + r ( p ′ ) + X q ∈ X r ( q ) . Since r ( p ′ ) = 0 in M E , it follows that M E does not have cancellation. (cid:3) In order to compute the monoid V gr ( L K ( E )) for an arbitrary graph E , we define an abelian monoid M gr E suchthat the generators { a v ( γ ) | v ∈ E , γ ∈ Γ } are supplemented by generators b Z ( γ ) as γ ∈ Γ and Z runs through allnonempty finite subsets of s − ( u ) for infinite emitters u ∈ E . The relations are(1) a v ( γ ) = P e ∈ s − ( v ) a r ( e ) ( w ( e ) − γ ) for all regular vertices v ∈ E and γ ∈ Γ;(2) a u ( γ ) = P e ∈ Z a r ( e ) ( w ( e ) − γ ) + b Z ( γ ) for all γ ∈ Γ, infinite emitters u ∈ E and nonempty finite subsets Z ⊆ s − ( u );(3) b Z ( γ ) = P e ∈ Z \ Z a r ( e ) ( w ( e ) − γ ) + b Z ( γ ) for all γ ∈ Γ, infinite emitters u ∈ E and nonempty finite subsets Z ⊆ Z ⊆ s − ( u ).The group Γ acts on the monoid M gr E as follows. For any β ∈ Γ, β · a v ( γ ) = a v ( βγ ) and β · b Z ( γ ) = b Z ( βγ ) . (5.7)There is a surjective monoid homomorphism π : M gr E → M E such that π ( a v ( γ )) = v and π ( b Z ( γ )) = q Z for v ∈ E and nonempty finite subset Z ⊂ s − ( u ), where u is an infinite emitter. π is Γ-equivariant in the sense that π ( β · x ) = π ( x ) for all β ∈ Γ and x ∈ M gr E .Recall the covering graph E from § L K ( E )-Mod be the category of unital left L K ( E )-modules and L K ( E )- Gr the category of graded unital left L K ( E )-modules. The isomorphism φ ′ : L K ( E ) ∼ −→ L K ( E ) L K ( E )-Gr −→ L K ( E )-Mod . (5.8) Lemma 5.6.
Let E be an arbitrary graph, Γ a group and w : E −→ Γ a function. RADED STEINBERG ALGEBRAS AND THEIR REPRESENTATIONS 17 (1) Fix a path η in E , and β ∈ Γ , and let η = η β − be the path in E defined at (5.3) . Then Φ (( L K ( E ) ηη ∗ )( β )) ∼ = L K ( E ) ηη ∗ . In particular, Φ(( L K ( E ) v )( β )) ∼ = L K ( E ) v β − .(2) Let u ∈ E be an infinite emitter, and let Z ⊆ s − E ( u ) be a nonempty finite set. Fix β ∈ Γ , and let Z = { e β − | e ∈ Z } . Then u β − is an infinite emitter in E and Z is a nonempty finite subset of s − E ( u β − ) .Moreover, Φ( L K ( E )( u − P e ∈ Z ee ∗ )( β )) ∼ = L K ( E )( u β − − P f ∈ Z f f ∗ ) .Proof. We prove (1). By the isomorphism of algebras in Corollary 5.3, we have L K ( E ) ηη ∗ ∼ = ( L K ( E ) ηη ∗ p β − . We claim that f : Φ(( L K ( E ) ηη ∗ )( β )) −→ ( L K ( E ) ηη ∗ p β − given by f ( y ) = yp β − is an isomorphism of left L K ( E )-modules. It is clearly a group isomorphism. To see that it is an L K ( E )-module morphism, note that( rp γ ) y = ry γ for y ∈ ( L K ( E ) ηη ∗ )( β ) and y γ a homogeneous element of degree γ . We have y ∈ L K ( E ) γβ ηη ∗ , yielding f (( rp γ ) y ) = ry γ p β − = ( rp γ )( yp β − ) = rp γ f ( y ). The proof for (2) is similar. (cid:3) Recall from § e S α on L K ( E ) α ∈ Γ. So the isomorphism φ ′ : L K ( E ) ∼ −→ L K ( E ) T α on L K ( E )- Mod. This in turn induces ahomomorphism T α : V ( L K ( E )) −→ V ( L K ( E )), giving a Γ-action on the monoid V ( L K ( E )).Fix v γ ∈ E , an infinite emitter u β ∈ E , and a finite Z ⊆ s − E ( u β ). Write Z · α − = { e βα − | e β ∈ Z } . We claimthat T α ([ L K ( E ) v γ ]) = [ L K ( E ) v γα − ] and T α ([ L K ( E )( u β − X e ∈ Z ee ∗ )]) = [ L K ( E )( u βα − − X f ∈ Z · a − f f ∗ )] . (5.9)To see the first equality in (5.9), we use Lemma 5.6 to see thatΦ( L K ( E ) v ( γ − )) = L K ( E ) v γ and Φ( L K ( E ) v ( αγ − )) = L K ( E ) v γα − . Using the commutative diagram (2.3) at the second equality, we see that T α ( L K ( E ) v γ ) = ( T α ◦ Φ)( L K ( E ) v ( γ − )) = (Φ ◦ T α )( L K ( E ) v ( γ − )) = Φ( L K ( E ) v ( αγ − )) = L K ( E ) v γα − . The proof for the second equality in (5.9) is similar.The group Γ acts on the monoid M E as follows. Again fix v γ ∈ E , an infinite emitter u β ∈ E , and a finite Z ⊆ s − E ( u β ), and write Z · α − = { e βα − | e β ∈ Z } . Then α · v γ = v γα − and α · q Z = q Z · α − . (5.10) Proposition 5.7.
Let E be an arbitrary graph, K a field, Γ a group and w : E −→ Γ a function. Let A = L K ( E ) and A = L K ( E ) . Then the monoid V gr ( A ) is generated by [ A v ( α )] and [ A ( u − P e ∈ Z ee ∗ )( β )] , where v ∈ E , α, β ∈ Γ and Z runs through all nonempty finite subsets of s − ( u ) for infinite emitters u ∈ E . Given an infinite emitter u ∈ E , a finite nonempty set Z ⊆ s − ( u ) , and β ∈ Γ , write Z β − := { e β − : e ∈ Z } ⊆ s − E ( u β − ) . Then there are Γ -module isomorphisms V gr ( A ) ∼ = V ( A ) ∼ = M E ∼ = M gr E , (5.11) that satisfy [ A v ( α )] [ A v α − ] [ v α − ] [ a v ( α )] for all v ∈ E and α ∈ Γ , and [ A ( u − X e ∈ Z ee ∗ )( β )] [ A ( u β − − X e ∈ Z β − ee ∗ )] [ q Z β − ] [ b Z ( β )] for every infinite emitter u , finite nonempty Z ⊆ s − ( u ) , and β ∈ Γ .Proof. Let P be a graded finitely generated projective left A -module. We claim that the isomorphism Φ : A -Gr −→A -Mod in (5.8) preserves the finitely generated projective objects. Since Φ is an isomorphism of categories, Φ( P ) isprojective. Observe that P has finite number of homogeneous generators x , · · · , x n of degree γ i . By the A -actionof Φ( P ), we have the following equalities: (1) if v ∈ E , γ ∈ Γ, then v γ x i = vp γ x i = ( vx i , if γ i = γ ;0 , otherwise; (5.12)(2) if e : u −→ v ∈ E , w ( e ) = β and γ ∈ Γ, then e γ x i = ep β − γ x i = ( ex i , if γ i = β − γ ;0 , otherwise; and (5.13)(3) if e : u −→ v ∈ E , w ( e ) = β and γ ∈ Γ, then e ∗ γ x i = e ∗ p γ x i = ( e ∗ x i , if γ i = γ ;0 , otherwise. (5.14)So for y ∈ Φ( P ), we can express y = P ni =1 r i x i for some r i ∈ A . Fix i ≤ n and paths η, τ in E satisfying r ( η ) = r ( τ ).Then (5.12), (5.13), and (5.14) give τ η ∗ x i = τ w ( τ ) w ( η ) − γ i ( η γ i ) ∗ x i . (5.15)Since y = P ni =1 r i x i = P ni =1 P h ∈ Γ r i,h x i with r i,h a homogeneous element of degree h , equation (5.15) gives y ∈A (Φ( P )). Thus Φ( P ) is a finitely generated projective A -module.By (4.2) and (4.4), there exists a homomorphism V gr ( A ) −→ V ( A ) sending [ P ] to [Φ( P )] for a graded finitelygenerated projective left A -module P . Applying [6, Theorem 4.3] for the non-separated case, we obtain the secondmonoid isomorphism V ( A ) ∼ −→ M E in (5.11). Then for each graded finitely generated projective left A -module P , themodule Φ( P ) in A -Mod is generated by the elements A v α and A ( u β − P e ∈ Z ′ ee ∗ ) that it contains. Combining thiswith Lemma 5.6 gives the first isomorphism of monoids. The last monoid isomorphism M E ∼ = M gr E follows directlyby their definitions. By (5.7), (5.9) and (5.10), the monoid isomorphisms in (5.11) are Γ-module isomorphisms. (cid:3) Recall the following classification conjecture [1, 8, 29]. Let E and F be finite graphs. Then there is an orderpreserving Z [ x, x − ]-module isomorphism φ : K gr0 ( L K ( E )) → K gr0 ( L K ( F )) if and only if L K ( E ) is graded Moritaequivalent to L K ( F ). Furthermore, if φ ([ L K ( E )] = [ L K ( F )] then L K ( E ) ∼ = gr L K ( F ) . Note that K ( L K ( E )) and K gr0 ( L K ( E )) are the group completions of V ( L K ( E )) and V gr ( L K ( E )), respectively.Let Γ = Z and let w : E → Z be the function assigning 1 to each edge. Then Proposition 5.7 implies that there isan order preserving Z [ x, x − ]-module isomorphism K gr0 ( L K ( E )) ∼ = K ( L K ( E )), thus relating the study of a Leavittpath algebra over an arbitrary graph to the case of acyclic graphs (see Example 5.2).The following corollary is the first evidence that K gr0 ( L K ( E )) preserves all the information of the graded monoid. Corollary 5.8.
Let E be an arbitrary graph. Consider L K ( E ) as a graded ring with the grading determined by thefunction w : E → Z such that w ( e ) = 1 for all e . Then V gr ( L K ( E )) is cancellative.Proof. By Proposition 5.7, we have V gr ( L K ( E )) ∼ = M E . Since E = E × Z is an acyclic graph, the monoid M E iscancellative by Lemma 5.5. Hence V gr ( L K ( E )) is cancellative. (cid:3) For the next result we need to recall the notion of order-ideals of a monoid. An order-ideal of a monoid M is asubmonoid I of M such that x + y ∈ I implies x, y ∈ I . Equivalently, an order-ideal is a submonoid I of M that ishereditary in the sense that x ≤ y and y ∈ I implies x ∈ I . The set L ( M ) of order-ideals of M forms a (complete)lattice (see [5, § I of K gr0 ( A ), we write I + = I ∩ K gr0 ( A ) + . We say that I is a graded orderedideal if I is closed under the action of Z [ x, x − ], I = I + − I + , and I + is an order-ideal.Let E be a graph. Recall that a subset H ⊆ E is said to be hereditary if for any e ∈ E we have that s ( e ) ∈ H implies r ( e ) ∈ H . A hereditary subset H ⊆ E is called saturated if whenever 0 < | s − ( v ) | < ∞ , then { r ( e ) : e ∈ E and s ( e ) = v } ⊆ H implies v ∈ H . If H is a hereditary subset, a breaking vertex of H is a vertex v ∈ E \ H such that | s − ( v ) | = ∞ but 0 < | s − ( v ) \ r − ( H ) | < ∞ . We write B H := { v ∈ E \ H | v is a breaking vertex of H } .We call ( H, S ) an admissible pair in E if H is a saturated hereditary subset of E and S ⊆ B H .Let E be a row-finite graph. Isomorphisms between the lattice of saturated hereditary subsets of E , the lattice L ( M E ), and the lattice of graded ideals of L K ( E ) were established in [5, Theorem 5.3]. Tomforde used the admissiblepairs ( H, S ) of vertices to parameterise the graded ideals of L K ( E ) for a graph E which is not row-finite (see [49,Theorem 5.7]). In analogy, Ara and Goodearl [6] proved that the lattice of those ideals of Cohn-Leavitt algebras CL K ( E, C, S ) generated by idempotents is isomorphic to a certain lattice A C,S of admissible pairs (
H, G ), where
RADED STEINBERG ALGEBRAS AND THEIR REPRESENTATIONS 19 H ⊆ E and G ⊆ C (see [6, Definition 6.5] for the precise definition). There is also a lattice isomorphism between A C,S and the lattice L ( M ( E, C, S )) of order-ideals of M ( E, C, S ). Specialising to the non-separated graph E , thereis a lattice isomorphism H ∼ = L ( M E ) (5.16)between the lattice H of admissible pairs ( H, S ) of E and the lattice L ( M E ) of order-ideals of the monoid M E .Let E be a finite graph with no sinks. There is a one-to-one correspondence [30, Theorem 12] between the set ofhereditary and saturated subsets of E and the set of graded ordered ideals of K gr0 ( L K ( E )). The main theorem ofthis section describes a one-to-one correspondence between the set of admissible pairs ( H, S ) of vertices and the setof graded ordered ideals of K gr0 ( L K ( E )) for an arbitrary graph E . To prove it, we first need to extend [5, Lemma4.3] to arbitrary graphs. This may also be useful in other situations. Lemma 5.9.
Let E be an arbitrary graph and denote by F the free abelian group generated by E ∪ { q Z } , where Z ranges over all the nonempty finite subsets of s − ( v ) for infinite emitters v . Let ∼ be the congruence on F suchthat F/ ∼ = M E . Let → be the relation on F defined by v + α → P e ∈ s − ( v ) r ( e ) + α if v is a regular vertex in E , v + α → r ( z ) + q { z } + α if v ∈ E is an infinite emitter and z ∈ s − ( v ) , and also q Z + α → r ( z ) + q Z ∪{ z } + α , if Z is a non-empty finite subset of s − ( v ) for an infinite emitter v and z ∈ s − ( v ) \ Z . Let → be the transitive andreflexive closure of → . Then α ∼ β in F if and only if there is γ ∈ F such that α → γ and β → γ .Proof. As in [7, Alternative proof of Theorem 4.1], we write M E = lim M ( E ′ , C ′ , T ′ ), where E ′ ranges over all thefinite complete subgraphs of E and C ′ = { s − E ′ ( v ) | v ∈ ( E ′ ) , | s − E ′ ( v ) | > } , T ′ = { s − E ′ ( v ) ∈ C ′ | v ∈ ( E ′ ) , < | s − E ( v ) | < ∞} . Applying [6, Construction 5.3], we get that M ( E ′ , C ′ , T ′ ) = M e E for some finite graph e E . The vertices of e E are thevertices of E and the elements of the form q Z , where Z ∈ C ′ \ T ′ , and there is a new edge e Z : v → q Z if the source of Z is v . If α ∼ β in F , then [ α ] = [ β ] in M E , and so there is ( E ′ , C ′ , T ′ ) as above such that [ α ] = [ β ] in M ( E ′ , C ′ , T ′ ).But since M ( E ′ , C ′ , T ′ ) = M e E , and e E is finite, we conclude from [5, Lemma 4.3] that there is an element γ in thefree monoid on ( E ′ ) ∪ { q Z | Z ∈ C ′ \ T ′ } such that α → γ and β → γ . This implies that α → γ and β → γ in F . (cid:3) Lemma 5.10.
Let E be an arbitrary graph and K a field. Consider L K ( E ) as a graded ring with the gradingdetermined by the function w : E → Z such that w ( e ) = 1 for all e . Let L c ( M gr E ) be the set of order-ideals of M gr E which are closed under the Z -action. Let π : M gr E → M E be the canonical surjective homomorphism. Then the map φ : L ( M E ) → L c ( M gr E ) defined by φ ( I ) = π − ( I ) is a lattice isomorphism.Proof. It is easy to show that the map φ is well-defined. The key to show the result is to prove the equality π − ( π ( J )) = J for any J ∈ L c ( M gr E ). The inclusion J ⊆ π − ( π ( J )) is obvious. To show the reverse inclusion π − ( π ( J )) ⊆ J , denote by F the free abelian group on E ∪ { q Z } , where Z ranges over all the nonempty finitesubsets of s − ( v ) for infinite emitters v . Take z ∈ π − ( π ( J )). Then there is y ∈ J such that π ( z ) = π ( y ). Now write z = X i a v i ( γ i ) + X j b Z j ( λ j ) , y = X i a v ′ i ( γ ′ i ) + X j b Z ′ j ( λ ′ j ) . Then we have P i v i + P j q Z j = π ( z ) = π ( y ) = P i v ′ i + P j q Z ′ j . By Lemma 5.9, there is x = P i w i + P j q W j suchthat π ( z ) → x and π ( y ) → x in F . Now using the same changes than in the paths π ( y ) → x and π ( z ) → x , butlifted to M gr E , we obtain that y = P i a w i ( η i ) + P j b W j ( ν j ) in M gr E and z = P i a w i ( η ′ i ) + P j b W j ( ν ′ j ) in M gr E . Butnow y ∈ J and J is an order ideal of M gr E , so it follows that a w i ( η i ) ∈ J for all i and b W j ( ν j ) ∈ J for all j . Usingthat J is invariant, we obtain a w i ( η ′ i ) ∈ J for all i and b W j ( ν ′ j ) ∈ J for all j . Thus z = P i a w i ( η ′ i ) + P j b W j ( ν ′ j ) ∈ J and we conclude the proof.Now using that J = π − ( π ( J )), we can easily show that π ( J ) is an order-ideal of M E and that the map φ isbijective, with φ − ( J ) = π ( J ). (cid:3) We can now state the main theorem of this section, which indicates that the graded K -group captures the latticestructure of graded ideals of a Leavitt path algebra. Theorem 5.11.
Let E be an arbitrary graph and K a field. Consider L K ( E ) as a graded ring with the gradingdetermined by the function w : E → Z such that w ( e ) = 1 for all e . Then there is a one-to-one correspondencebetween the admissible pairs of E and the graded ordered ideals of K gr0 ( L K ( E )) . Proof.
Let H be the set of all admissible pairs of E and L ( K gr0 ( A )) the set of all graded ordered ideals of K gr0 ( A ),where A = L K ( E ). We first claim that there is a one-to-one correspondence between the order-ideals of M E andorder-ideals of M gr E which are closed under the Z -action. Let L c ( M gr E ) be the set of order-ideals of M gr E which areclosed under the Z -action.The map φ : L ( M E ) −→ L c ( M gr E ) has been defined in Lemma 5.10, where it is proved that it is a lattice isomor-phism.By Corollary 5.8, we have an injective homomorphism V gr ( A ) −→ K gr0 ( A ). By Proposition 5.7, there is a one-to-one corespondence between the order-ideals of M gr E which are closed under the Z -action and the graded orderedideals of K gr0 ( A ). Finally by (5.16), we have lattice isomorphisms H ∼ = L ( M E ) ∼ = L c ( M gr E ) ∼ = L ( K gr0 ( A )) . (cid:3) Application: Kumjian–Pask algebras
In this section we will use our result on smash products (Theorem 3.4) to study the structure of Kumjian–Paskalgebras [11] and their graded K -groups. We will see that the graded K -group remains a useful invariant for studyingKumjian–Pask algebras. We deal exclusively with row-finite k -graphs with no sources: our analysis for arbitrarygraphs relied on constructions like desingularisation that are not available in general for k -graphs. We briefly recallthe definition of Kumjian–Pask algebras and establish our notation. We follow the conventions used in the literatureof this topic (in particular the paths are written from right to left).Recall that a graph of rank k or k -graph is a countable category Λ = (Λ , Λ , r, s ) together with a functor d : Λ → N k , called the degree map , satisfying the following factorisation property: if λ ∈ Λ and d ( λ ) = m + n forsome m, n ∈ N k , then there are unique µ, ν ∈ Λ such that d ( µ ) = m, d ( ν ) = n , and λ = µν . We say that Λ is rowfinite if r − ( v ) ∩ d − ( n ), abbreviated v Λ n is finite for all v ∈ Λ and n ∈ N k ; we say that Λ has no sources if each v Λ n is nonempty.An important example is the k -graph Ω k defined as a set by Ω k = { ( m, n ) ∈ N k × N k : m ≤ n } with d ( m, n ) = n − m , Ω k = N k , r ( m, n ) = m , s ( m, n ) = n and ( m, n )( n, p ) = ( m, p ). Definition 6.1.
Let Λ be a row-finite k -graph without sources and K a field. The Kumjian–Pask K -algebra of Λis the K -algebra KP K (Λ) generated by Λ ∪ Λ ∗ subject to the relations(KP1) { v ∈ Λ } is a family of mutually orthogonal idempotents satisfying v = v ∗ ,(KP2) for all λ, µ ∈ Λ with r ( µ ) = s ( λ ), we have λµ = λ ◦ µ, µ ∗ λ ∗ = ( λ ◦ µ ) ∗ , r ( λ ) λ = λ = λs ( λ ) , s ( λ ) λ ∗ = λ ∗ = λ ∗ r ( λ ) , (KP3) for all λ, µ ∈ Λ with d ( λ ) = d ( µ ), we have λ ∗ µ = δ λ,µ s ( λ ) , (KP4) for all v ∈ Λ and all n ∈ N k \ { } , we have v = X λ ∈ v Λ n λλ ∗ . Let Λ be a a row-finite k -graph without sources and KP K (Λ) the Kumjian–Pask algebra of Λ. Following [35, § x : Ω k −→ Λ. Denote the set of all infinite paths by Λ ∞ . We definethe relation of tail equivalence on the space of infinite path Λ ∞ as follows: for x, y ∈ Λ ∞ , we say x is tail equivalentto y , denoted, x ∼ y , if x ( n, ∞ ) = y ( m, ∞ ), for some n, m ∈ N k . This is an equivalence relation. For x ∈ Λ ∞ , wedenote by [ x ] the equivalence class of x , i.e., the set of all infinite paths which are tail equivalent to x . An infinitepath x is called aperiodic if x ( n, ∞ ) = x ( m, ∞ ), n, m ∈ N k , implies n = m .We can form the skew-product k -graph, or covering graph, Λ = Λ × d Z k which is equal as a set to Λ × Z k , hasdegree map given by d ( λ, n ) = d ( λ ), range and source maps r ( λ, n ) = ( r ( λ ) , n ) and s ( λ, n ) = ( s ( λ ) , n + d ( λ )) andcomposition given by ( λ, n )( µ, n + d ( λ )) = ( λµ, n ).As in the theory of Leavitt path algebras, one can model Kumjian–Pask algebras as Steinberg algebras via theinfinite-path groupoid of the k -graph (see [22, Proposition 5.4]). For the k -graph Λ, G Λ = (cid:8) ( x, l − m, y ) ∈ Λ ∞ × Z k × Λ ∞ | x ( l, ∞ ) = y ( m, ∞ ) (cid:9) . Define range and source maps r, s : G Λ −→ Λ ∞ by r ( x, n, y ) = x and s ( x, n, y ) = y . For ( x, n, y ) , ( y, l, z ) ∈ G Λ ,the multiplication and inverse are given by ( x, n, y )( y, l, z ) = ( x, n + l, z ) and ( x, n, y ) − = ( y, − n, x ). G Λ is a RADED STEINBERG ALGEBRAS AND THEIR REPRESENTATIONS 21 groupoid with Λ ∞ = G (0)Λ under the identification x ( x, , x ). For µ, ν ∈ Λ with s ( µ ) = s ( ν ), let Z ( µ, ν ) := { ( µx, d ( µ ) − d ( ν ) , νx ) : x ∈ Λ ∞ , x (0) = s ( µ ) } . Then the sets Z ( µ, ν ) comprise a basis of compact open sets for anample Hausdorff topology on G Λ . There is a continuous 1-cocycle c : G Λ → Z k given by c ( x, m, y ) = m .For the skew-product k -graph Λ = Λ × d Z k , we have G Λ ∼ = G Λ × c Z k (see [35, Theorem 5.2]). Thus specialisingTheorem 3.4 to this setting, we have KP K (Λ) ∼ = KP K (Λ) Z k . (6.1)We will show that KP K (Λ) is an ultramatricial algebra. Lemma 6.2.
For n ∈ Z k define B n ⊆ KP K (Λ) by B n = span K (cid:8) ( λ, n − d ( λ ))( µ, n − d ( µ )) ∗ | λ, µ ∈ Λ , s ( λ ) = s ( µ ) (cid:9) . Then B n is a subalgebra of KP K (Λ) and there is an isomorphism B n ∼ = L v ∈ Λ M Λ v ( K ) that carries ( λ, n − d ( λ ))( µ, n − d ( µ )) ∗ to the matrix unit e λ,µ .Proof. For the first statement we just have to show that for any λ, µ, η, ζ ∈ Λ we have( λ, n − d ( λ ))( µ, n − d ( µ )) ∗ ( η, n − d ( η ))( ζ, n − d ( ζ )) ∗ ∈ B n . This follows from the argument of [35, Lemma 5.4]. To wit, we have ( µ, n − d ( µ )) ∗ ( η, n − d ( η )) = 0 unless r ( µ, n − d ( µ )) = r ( η, n − d ( η )), which in turn forces d ( µ ) = d ( η ). But then d ( µ, n − d ( µ )) = d ( η, n − d ( η )), and then theCuntz–Krieger relation forces ( µ, n − d ( µ )) ∗ ( η, n − d ( η )) = δ µ,η ( s ( µ ) , n ). Hence( λ, n − d ( λ ))( µ, n − d ( µ )) ∗ ( η, n − d ( η ))( ζ, n − d ( ζ )) ∗ = δ µ,η ( λ, n − d ( λ ))( ζ, n − d ( ζ )) ∗ ∈ B n . For each v ∈ Λ , M Λ( v,n ) ( K ) ∼ = M Λ v ( K ). So the elements ( λ, n − d ( λ ))( µ, n − d ( µ )) ∗ satisfy the same multiplicationformula as the matrix units e λ,µ in L v ∈ Λ M Λ v ( K ). Hence the uniqueness of the latter shows that there is anisomorphism as claimed. (cid:3) Lemma 6.3.
For m ≤ n ∈ Z k , we have B m ⊆ B n , and in particular for each v ∈ Λ , we have ( v, m ) = P α ∈ v Λ n − m ( α, m )( α, m ) ∗ .Proof. Again, this follows from the proof of [35, Lemma 5.4]. We just apply the Cuntz–Krieger relation, using at thefirst equality that Λ has no sources:( λ, m − d ( λ ))( µ, m − d ( µ )) ∗ = ( λ, m − d ( λ )) (cid:16) X α ∈ s ( λ )Λ n − m ( α, m )( α, m ) ∗ (cid:17) ( µ, m − d ( µ )) ∗ = X α ∈ s ( λ )Λ n − m ( λα, m − d ( λ ))( µα, m − d ( µ )) ∗ ∈ B n . This gives the first assertion, and the second follows by taking λ = µ = v . (cid:3) Theorem 6.4.
Let Λ be a row-finite k -graph with no sources and K a field. Then the Kumjian–Pask algebra KP K (Λ) is a graded von Neumann regular ring.Proof. Lemma 2.3 shows that KP K (Λ) is graded regular if and only if KP K (Λ) Z k is graded regular. By (6.1)KP K (Λ) Z k ∼ = KP K (Λ) and the latter is an ultramatricial algebra by Lemma 6.3. Since ultramatricial algebras areregular, the theorem follows. (cid:3) Since KP K (Λ) is graded von Neumann regular, we immediately obtain the following statements. Theorem 6.5.
Let Λ be a row-finite k -graph with no sources and K a field. Then the Kumjian–Pask algebra A = KP K (Λ) has the following properties: (1) any finitely generated right (left) graded ideal of A is generated by one homogeneous idempotent; (2) any graded right (left) ideal of A is idempotent; (3) any graded ideal is graded semi-prime; (4) J ( A ) = J gr ( A ) = 0 ; and (5) there is a one-to-one correspondence between the graded right (left) ideals of A and the right (left) ideals of A .Proof. All the assertions are the properties of a graded von Neumann regular ring [31, § (cid:3) For the next result, given a k -graph Λ, and given m ≤ n ∈ Z k , we define φ m,n : N Λ → N Λ by φ m,n ( v ) = P w ∈ Λ | v Λ n − m w | w . Corollary 6.6.
Let Λ be a row-finite k -graph with no sources and K a field. There is an isomorphism V (KP K (Λ)) ∼ = lim −→ Z k (cid:0) N Λ , φ m,n ) that carries [( v, n )] to the copy of v in the n th copy of N Λ . Fathermore, the monoid V (KP K (Λ)) is cancellative.Proof. It is standard that there is an isomorphism V (cid:0) L v ∈ Λ M Λ v ( K ) (cid:1) ∼ = N Λ that takes e λ,λ to s ( λ ) for all λ . SoLemma 6.2 implies that there is an isomorphism V ( B n ) → N Λ that carries [( λ, n − d ( λ ))( λ, n − d ( λ )) ∗ ] to s ( λ ) forall λ . Let S n be a copy N Λ × { n } of the monoid N Λ (so ( a, n ) + ( b, n ) = ( a + b, n ) in S n ). Lemma 6.3 shows thatthese isomorphisms of monoids carry the inclusions B m ֒ → B n to the maps ( v, m ) P λ ∈ v Λ n − m ( s ( λ ) , n ), which isprecisely given by the formula φ m,n for m ≤ n ∈ Z k . Since the monoid of a direct limit is the direct limit of themonoids of the approximating algebras, we have an isomorphism V (KP K (Λ)) ∼ = lim −→ Z k S n , which sends [( v, n )] to( v, n ) ∈ S n .Suppose that x + z = y + z in V ( KP K (Λ)). By the isomorphism V (KP K (Λ)) ∼ = lim −→ Z k S n , there exist images x ′ , y ′ , z ′ of x, y, z , respectively, in S n = N Λ × { n } for some n ∈ Z k such that x ′ + z ′ = y ′ + z ′ . The monoid N Λ is cancellative, so V ( KP K (Λ)) is too. (cid:3) Corollary 6.7.
Let Λ be a row-finite k -graph with no sources and K a field. Then V gr (KP K (Λ)) ∼ = lim −→ Z k (cid:0) N Λ , φ m,n ) .Proof. Recall from (6.1) that KP K (Λ) ∼ = KP K (Λ) Z k . Specialising Proposition 2.5 to Kumjian–Pask algebras, wehave the isomorphism of categories Ψ : KP K (Λ)- Gr ∼ −→ KP K (Λ)- Mod. We argue as in the directed-graph situationthat Ψ preserves finitely generated projective objects. By (4.2) and (4.4), we have V gr (KP K (Λ)) ∼ = V (KP K (Λ)). (cid:3) The graded representations of the Steinberg algebra
In this section, for a Γ-graded groupoid G and its associated Steinberg algebra A R ( G ), we construct graded simple A R ( G )-modules. Specialising our results to the trivial grading, we obtain irreducible representations of (ungraded)Steinberg algebras. We determine the ideals arising from these representations and prove that these ideals relate tothe effectiveness or otherwise of the groupoid.7.1. Representations of a Steinberg algebra.
Let G be an ample Hausdorff groupoid, let Γ be a discrete groupwith identity ε , and let c : G →
Γ be a continuous 1-cocycle. A subset U of the unit space G (0) of G is invariant if d ( γ ) ∈ U implies r ( γ ) ∈ U ; equivalently, r ( d − ( U )) = U = d ( r − ( U )) . Given an element u ∈ G (0) , we denote by [ u ] the smallest invariant subset of G (0) which contains u . Then r ( d − ( u )) = [ u ] = d ( r − ( u )) . That is, for any v ∈ [ u ], there exists x ∈ G such that d ( x ) = u and r ( x ) = v ; equivalently, for any w ∈ [ u ], thereexists y ∈ G such that d ( y ) = w and r ( y ) = u . Thus for any v, w ∈ [ u ], there exists x ∈ G such that d ( x ) = v and r ( x ) = w . We call [ u ] an orbit . Observe that an invariant subset U ⊆ G (0) is an orbit if and only if for any v, w ∈ U ,there exists x ∈ G such that d ( x ) = v and r ( x ) = w . Lemma 7.1.
Let u , u , · · · , u n be pairwise distinct elements of G (0) with n ≥ . Then there exist disjoint compactopen bisections B i ⊆ G (0) such that u i ∈ B i for each i = 1 , · · · , n .Proof. Since G (0) is a Hausdorff space, there exist disjoint open subsets X i of G (0) such that u i ∈ X i for all i . Since G is ample, we can choose compact open bisections B i ⊆ X i such that u i ∈ B i for all i . (cid:3) The isotropy group at a unit u of G is the group Iso( u ) = { γ ∈ G | d ( γ ) = r ( γ ) = u } . A unit u ∈ G (0) is calledΓ -aperiodic if Iso( u ) ⊆ c − ( ε ), otherwise u is called Γ -periodic . For an invariant subset W ⊆ G (0) , we denote by W ap the collection of Γ-aperiodic elements of W and by W p the collection of Γ-periodic elements of W . Then W = W ap G W p . If W = W ap , we say that W is Γ -aperiodic ; If W = W p , we say that W is Γ -periodic . RADED STEINBERG ALGEBRAS AND THEIR REPRESENTATIONS 23
Remark 7.2.
Let E be a directed graph. Let G E be the associated graph groupoid and c : G E → Z the canonicalcocycle c ( x, m, y ) = m . It was shown in [36] that c − (0) is a principal groupoid, in the sense that Iso( c − (0)) = G (0) E .Hence x ∈ G (0) E = E ∞ is Z -aperiodic if and only if Iso( x ) = { x } . It is standard that Iso( x ) = { x } if and only if x = µλ ∞ for any cycle λ in E . So x is Z -aperiodic if and only if x = µλ ∞ for any cycle λ . Lemma 7.3.
Let W ⊆ G (0) be an invariant subset. Then W ap and W p are both invariant subsets of G (0) .Proof. For x ∈ G , let u = d ( x ) and v = r ( x ). Suppose that u ∈ W ap . If c ( y ) = ε for some y ∈ Iso( v ), then x − yx ∈ Iso( u ) and ε = c ( y ) = c ( x ) c ( x − yx ) c ( x ) − , forcing c ( x − yx ) = ε , a contradiction. Hence, v = r ( x ) isΓ-aperiodic. Since W is invariant, we have v ∈ W ap . So W ap is invariant. Since W = W ap ⊔ W p , it follows that W p is also invariant. (cid:3) By the proof of Lemma 7.3, u ∈ G (0) is Γ-aperiodic if and only if its orbit [ u ] is Γ-aperiodic. Example 7.4.
In this example we construct a Z -aperiodic invariant subset which is neither open nor closed in G (0) .Let E be the following directed graph. 1 · β , , α λ / / · γ r r δ l l Let u be the infinite path αβα βα β · · · . Then u is an element in G (0) E . The orbit [ u ] consists of all infinite pathstail equivalent to u . So α n u ∈ [ u ] for all n ∈ N . The sequence α n u converges to α ∞ , which does not belong to [ u ].So [ u ] is not closed. Similarly, the points u n := αβα β · · · α n βα ∞ all belong to G (0) \ [ u ], but u n → u , so [ u ] is notopen. In particular, neither [ u ] nor its complement is the invariant subset of G (0) corresponding to any saturatedhereditary subset of E .We will employ Γ-aperiodic invariant subsets of G (0) to obtain graded representations for the Steinberg algebra A R ( G ). For any invariant subset U ⊆ G (0) and a unital commutative ring R , we denote by RU the free R -module withbasis U . For every compact open bisection B ⊆ G , there is a function f B : G (0) −→ RU which has support containedin d ( B ) ∩ U and f B ( d ( γ )) = r ( γ ) for all γ ∈ B ∩ d − ( U ). There is a unique representation π U : A R ( G ) −→ End R ( RU )such that π U (1 B )( u ) = f B ( u ) (7.1)for every compact open bisection B and u ∈ U . This representation makes RU an A R ( G )-module (see [15, Proposition4.3]). An A R ( G )-submodule V ⊆ RU is called a basic submodule of RU if whenever r ∈ R \ { } and ru ∈ V , we have u ∈ V . We say an A R ( G )-module is basic simple if it has no non-trivial basic submodules.We can state one of the main results of this section. Theorem 7.5.
Let U be an invariant subset of G (0) . Then U is a Γ -aperiodic orbit if and only if RU is a gradedbasic simple A R ( G ) -module. Furthermore, RU is a graded basic simple A R ( G ) -module if and only if it is graded andbasic simple.Proof. Suppose that u ∈ G (0) satisfies U = [ u ], and that [ u ] is a Γ-aperiodic orbit. We first show that R [ u ] is aΓ-graded A R ( G )-module. For any γ ∈ Γ, set[ u ] γ = { v ∈ [ u ] | there exists x ∈ G such that c ( x ) = γ, d ( x ) = u and r ( x ) = v } . We claim that [ u ] γ ∩ [ u ] γ ′ = ∅ implies γ = γ ′ . Indeed, if v ∈ [ u ] γ ∩ [ u ] γ ′ , then there exist x ∈ c − ( γ ) and y ∈ c − ( γ ′ ) such that d ( x ) = d ( y ) = u and r ( x ) = r ( y ) = v . Now x − y ∈ Iso( u ). Since u is Γ-aperiodic this forces γ − γ ′ = c ( x − y ) = ε , and so γ = γ ′ . This gives a partition [ u ] = ⊔ γ ∈ Γ [ u ] γ . Therefore A R ( G )-module R [ u ] has adecomposition of R -modules R [ u ] = M γ ∈ Γ ( R [ u ]) γ , where ( R [ u ]) γ is a free R -module with basis [ u ] γ .We show that A R ( G ) α · ( R [ u ]) γ ⊆ ( R [ u ]) αγ , for α, γ ∈ Γ. Fix v ∈ [ u ] γ and B ∈ B co α ( G ). We use · to denote theaction of A R ( G ) on RU . We have 1 B · v = ( r ( b ) , if b ∈ B satisfies d ( b ) = v ;0 , if v d ( B ). Clearly 0 ∈ ( R [ u ]) αγ , so suppose that b ∈ B satisfies d ( b ) = v . Since v ∈ [ u ] γ , there exists x ∈ G such that c ( x ) = γ , d ( x ) = u , and r ( x ) = v . Now d ( bx ) = u , r ( bx ) = r ( b ), and c ( bx ) = c ( b ) c ( x ) = αγ . So r ( b ) ∈ [ u ] αγ . Since elements ofthe form 1 B where B ∈ B co α ( G ) span A R ( G ) α , we deduce that A R ( G ) α · ( R [ u ]) γ ⊆ ( R [ u ]) αγ as claimed.Next we show that R [ u ] is a basic simple A R ( G )-module. Suppose that V = 0 is a basic A R ( G )-submodule of R [ u ].Take a nonzero element x ∈ V . Fix nonzero elements r i ∈ R and pairwise distinct u i ∈ [ u ] such that x = P mi =1 r i u i .By Lemma 7.1, there exist disjoint compact open bisections B i ⊆ G (0) such that u i ∈ B i for all i = 1 , · · · , m . Now1 B · x = 1 B · m X i =1 r i u i = m X i =1 r i (1 B · u i ) = r f B ( u ) . Thus u = f B ( u ) ∈ V , because V is a basic submodule. Fix v ∈ [ u ] and choose x ∈ G such that d ( x ) = u and r ( x ) = v . Fix a compact open bisection D containing x . Then 1 D · u = f D ( u ) = r ( x ) = v ∈ V , giving V = R [ u ].Thus R [ u ] is basic simple, and consequently graded basic simple.For the converse suppose that RU is a graded basic simple A R ( G )-module. We first show that U is Γ-aperiodic.Let u ∈ U . We claim that there exists r ∈ R \ { } such that ru is a homogeneous element of RU . To see this, express u = P li =1 h i , where h i = u are homogeneous elements. For each i , express h i = P s i j =1 λ ij u ij with λ ij ∈ R \ { } andthe u ij ∈ U pairwise distinct. We first show that u ∈ { u ij | i = 1 , · · · , l ; j = 1 , · · · , s i } ; for if not, then Lemma 7.1gives compact open bisections B, B ij such that u ∈ B and u / ∈ B ij for all i, j . So 1 B · u = 0, whereas1 B · u = 1 B · l X i =1 h i = 1 B · l X i =1 s i X j =1 λ ij u ij = l X i =1 s i X j =1 λ ij B · u ij = 0 . This is a contradiction. So u = u ij for some i, j as claimed; without loss of generality, u = u . Hence h = λ u + P s j =2 λ j u j . There exist compact open bisections B ′ , B ′ j ⊆ G (0) ⊆ c − ( ε ) such that u ∈ B ′ but u / ∈ B ′ j for j = 1. Hence r := λ belongs to R \ { } , and ru = λ B ′ · u = 1 B ′ · h is homogeneous as claimed. Now suppose that u is not Γ-aperiodic. Then there exists x ∈ Iso( u ) with c ( x ) = ε . Fix D ∈ B co c ( x ) ( G ) containing x . Then 1 D · ru = r D · u = ru is homogeneous. Thus 1 D ∈ A R ( G ) ε , forcing c ( x ) = ε . Thisis a contradiction. Thus U is Γ-aperiodic.For the last part of the theorem we prove that U is an orbit. If not then there exist u, v ∈ G (0) with [ u ] ∩ [ v ] = ∅ and [ u ] ⊔ [ v ] ⊆ U . Hence R [ u ] ⊆ RU \ R [ v ] is a nontrivial proper graded basic submodule of RU by the first part ofthe theorem. This is a contradiction. So U is an orbit. The last statement of the theorem follows from the first partof the proof. (cid:3) Corollary 7.6.
Let G be an ample Hausdorff groupoid. U be an invariant subset of G (0) . Then U is an orbit of G (0) if and only if RU is a basic simple A R ( G ) -module.Proof. Apply Theorem 7.5 with c : G → { ε } the trivial grading. (cid:3) Specialising Theorem 7.5 to the case of Leavitt path algebras we obtain irreducible representations for thesealgebras.Let K be a field. For an infinite path p in a graph E , Chen constructed the left L K ( E )-module F [ p ] of the spaceof infinite paths tail-equivalent to p and proved that it is an irreducible representation of the Leavitt path algebra(see [16, Theorem 3.3]). These were subsequently called Chen simple modules and further studied in [4, 9, 10, 32, 43].In the groupoid setting, the infinite path p is an element in G (0) E . Thus q belongs to the orbit [ p ] if and only if q istail-equivalent to p . Applying Corollary 7.6, we immediately obtain that K [ p ] = F [ p ] is an irreducible representationof the Leavitt path algebra. Furthermore, by Theorem 7.5, p is an aperiodic infinite path (irrational path) if andonly if F [ p ] is a graded module (see [32, Proposition 3.6]).Recall from [16, Theorem 3.3] that End L K ( E ) ( F [ p ] ) ∼ = K . We claim that End A R ( G ) ( R [ u ]) ∼ = R for u ∈ G (0) E .Indeed, let f : R [ u ] −→ R [ u ] be a nonzero homomorphism of A R ( G )-modules. Then Ker f is a basic submodule of R [ u ]. Since R [ u ] is basic simple, we deduce that f is injective. For v ∈ [ u ], we write f ( v ) = P ni =1 r i v i with 0 = r i ∈ R and v i are distinct. We prove that n = 1 and v = v . For if not, then we may assume that v = v . By Lemma 7.1,there exist disjoint compact open bisections B, B ⊆ G (0) such that v ∈ B , v ∈ B and v i / ∈ B for i = 1. Then1 B · f ( v ) = f (1 B · v ) = 0. But, 1 B · f ( v ) = 1 B · P ni =1 r i v i = r v which is a contradiction. RADED STEINBERG ALGEBRAS AND THEIR REPRESENTATIONS 25
Likewise, Theorem 7.5 specialises to k -graph groupoids, giving new information about Kumjian–Pask algebras. Corollary 7.7.
Let Λ be a row-finite k -graph without sources and KP K (Λ) the Kumjian–Pask algebra of Λ . Then (1) for an infinite path x ∈ Λ ∞ , K [ x ] is a simple left KP K (Λ) -module; (2) for x, y ∈ Λ ∞ , we have K [ x ] ∼ = K [ y ] if and only if x ∼ y ; and (3) for x ∈ Λ ∞ , K [ x ] is a graded module if and only if x is an aperiodic path.Proof. For (1), the equivalence class of x is the orbit of G (0)Λ which contains x . By (7.1) and Corollary 7.6, thestatement follows directly. For (2), let φ : F ([ x ]) → F ([ y ]) be an isomorphism. Write φ ( x ) = P li =1 r i y i , where y i ∼ y are all distinct. If x = y i , for some i , then by transitivity of ∼ , x ∼ y and we are done. Otherwise one can choose n ∈ N k such that all y i (0 , n ) and x (0 , n ) are distinct. Setting a = y (0 , n ), we have 0 = φ ( a ∗ x ) = a ∗ φ ( x ) = y ( n, ∞ ),which is not possible unless x = y and l = 1. This gives that x ∼ y . The converse is clear. The statement (3)follows immediately by Theorem 7.5. (cid:3) The annihilator ideals and effectiveness of groupoids.
In this section, we describe the annihilator idealsof the graded modules over a Steinberg algebra and prove that these ideals reflect the effectiveness of the groupoid.As in previous sections, we assume that G is a Γ-graded ample Hausdorff groupoid which has a basis of gradedcompact open bisections. Let R be a commutative ring with identity and A R ( G ) the Γ-graded Steinberg algebraassociated to G .Let W ⊆ G (0) be an invariant subset. We write G W := d − ( W ) which coincides with the restriction G| W = { x ∈G | d ( x ) ∈ W, r ( x ) ∈ W } . Notice that G W is a groupoid with unit space W .Observe that the interior W ◦ of an invariant subset W is invariant. Indeed, r ( d − ( W ◦ )) is an open subset of G (0) ,since W ◦ is an open subset of G (0) . Since W is invariant, r ( d − ( W ◦ )) ⊆ W . Thus r ( d − ( W ◦ )) ⊆ W ◦ . It follows thatthe closure W − of W is also an invariant subset of G (0) , since W − = G (0) \ ( G (0) \ W ) ◦ .Recall from (7.1) that π W : A R ( G ) −→ End R ( RW )makes RW an A R ( G )-module. Lemma 7.8.
Let W ⊆ G (0) be an invariant subset of the unit space of G , and let U = ( G (0) \ W ) ◦ . Then A R ( G U ) ⊆ Ann A R ( G ) ( RW ) . Proof.
For any f ∈ A R ( G U ), we write f = P mk =1 r k B k with B k ⊆ G U compact open bisections of G and r k ∈ R nonzero scalars. Since d ( B k ) ⊆ U , we have d ( B k ) ∩ W = ∅ . Thus f · w = 0 for any w ∈ W , and hence f ∈ Ann A R ( G ) ( RW ). (cid:3) From now on, W ⊆ G (0) is a Γ-aperiodic invariant subset. We have W = [ u ∈ W [ u ] . Of course, two elements of W may belong to the same orbit.Recall from Theorem 7.5 that if u ∈ G (0) is Γ-aperiodic, then R [ u ] is a Γ-graded A R ( G )-module. Therefore RW is a Γ-graded A R ( G )-module. In order to construct graded representations for A R ( G ), we need to con-sider the “closed” subgroups of End R ( F W ) defined in (7.1). Namely, we consider the subgroup END R ( RW ) = L γ ∈ Γ Hom R ( RW, RW ) γ , where each component Hom R ( RW, RW ) γ consists of R -maps of degree γ .Then the map π W : A R ( G ) −→ END R ( RW ) (7.2)given by the A R ( G )-module action is a homomorphism of Γ-graded algebras. To prove that π W preserves the grading,fix α ∈ Γ and B ∈ B co α ( G ). Take u ∈ W and v ∈ [ u ]. Fix x ∈ G with d ( x ) = u and r ( x ) = v , and put β = c ( x ) sothat v ∈ [ u ] β . Then π W (1 B )( v ) = ( r ( γ ) if v = d ( γ ) for some γ ∈ B ;0 otherwise . Since c ( γx ) = αβ , we obtain π W (1 B ) ∈ Hom R ( RW, RW ) α . Recall that an ample Hausdorff groupoid G is effective if Iso( G ) ◦ = G (0) , where Iso( G ) = F u ∈G (0) Iso( u ). It followsthat G is effective if and only if for any nonempty B ∈ B co ∗ ( G ) with B ∩ G (0) = ∅ , we have B Iso( G ) (see [15,Lemma 3.1] for other equivalent conditions).We need the following graded uniqueness theorem for Steinberg algebras established in [18, Theorem 3.4]. Lemma 7.9.
Let G be a Γ -graded ample Hausdorff groupoid such that c − ( ε ) is effective. If π : A R ( G ) −→ A is agraded R -algebra homomorphism with Ker( π ) = 0 then there is a compact open subset B ⊆ G (0) and r ∈ R \ { } suchthat π ( r B ) = 0 . The following key lemma will be used to determine the annihilator ideal of the A R ( G )-module RW . This is ageneralisation of [15, Proposition 4.4] adapted to the graded setting. Recall that if G is a graded groupoid withgrading given by the continuous 1-cocycle c : G →
Γ, then c − ( ε ) is a (trivially graded) clopen subgroupoid of G . Lemma 7.10.
Let W ⊆ G (0) be a Γ -aperiodic invariant subset and π W : A R ( G ) −→ END R ( RW ) the homomorphismof Γ -graded algebras given in (7.2) . Then π W is injective if and only if W is dense in G (0) and c − ( ε ) is effective.Proof. Suppose π W is injective and there exists an open subset K of G (0) such that K ∩ W = ∅ . We have K = S i B i ,where B i are compact open bisections of G . So B i ∩ W = ∅ for each i , giving π W (1 B i ) = 0, a contradiction. Thusfor any open subset K of G (0) , K ∩ W = ∅ . Therefore W is dense in G (0) .Suppose now that c − ( ε ) is not effective. Then there exists a nonempty compact open bisection B ⊆ c − ( ε ) \ G (0) such that d ( b ) = r ( b ) for all b ∈ B . We have that d ( B ) = B and that B is a compact open bisection of G . Thus1 B − d ( B ) ∈ Ker( π W ). This is a contradiction. Hence, c − ( ε ) is effective.For the converse, Lemma 7.9 implies that it suffices to prove that for any compact open subset B ⊆ G (0) and r ∈ R \ { } , π W ( r B ) = 0. Since W is dense in G (0) , we have B ∩ W = ∅ . There exists w ∈ B ∩ W such that π W ( r B )( w ) = 0, proving π W ( r B ) = 0. (cid:3) If the group Γ is trivial, then by Lemma 7.10, for an invariant subset W ⊆ G (0) , the homomorphism π W : A R ( G ) −→ End R ( RW ) is injective if and only if W is dense in G (0) and the groupoid G is effective.The following is the main result of this section. Theorem 7.11.
Let G be a Γ -graded ample Hausdorff groupoid, R a commutative ring with identity and A R ( G ) theSteinberg algebra associated to G . The following statements are equivalent:(i) Let W ⊆ G (0) be a Γ -aperiodic invariant subset and W − the closure of W . Then the groupoid (cid:0) c | G W − (cid:1) − ( ε ) is effective;(ii) For any Γ -aperiodic invariant subset W ⊆ G (0) , Ann A R ( G ) ( RW ) = A R ( G U ) , where U = ( G (0) \ W ) ◦ is the interior of the invariant subset G (0) \ W .Proof. ( i ) ⇒ ( ii ) Let W ⊆ G (0) be a Γ-aperiodic invariant subset. By Theorem 7.5, RW is a graded A R ( G )-module.By Lemma 7.8, we have A R ( G U ) ⊆ Ann A R ( G ) ( RW ) with U = ( G (0) \ W ) ◦ . It follows that RW is an A R ( G ) /A R ( G U )-module. By [19, Lemma 3.6], we have an exact sequence of canonical ring homomorphisms0 −→ A R ( G U ) −→ A R ( G ) −→ A R ( G D ) −→ . The homomorphisms are induced by extensions from G U to G and restrictions from G to G D , respectively. One caneasily check that the homomorphisms are graded. It therefore follows that the quotient algebra A R ( G ) /A R ( G U ) isgraded isomorphic to A R ( G D ), where D = G (0) \ U . It follows that RW is a Γ-graded A R ( G D )-module (this alsofollows from Theorem 7.5). We denote by b π W : A R ( G D ) −→ END R ( RW ) the induced graded homomorphism. Observethat ( G D ) (0) = D is the closure of W . Thus by Lemma 7.10, the homomorphism b π W is injective. This implies that RW is a faithful A R ( G D )-module. Hence, the annihilator ideal of RW as an A R ( G )-module is A R ( G U ).( ii ) ⇐ ( i ) Let D denote the closure of W in G (0) . Then RW is a faithful A R ( G D )-module. So the result followsfrom Lemma 7.10. (cid:3) Recall that a groupoid G is strongly effective if for every nonempty closed invariant subset D of G (0) , the groupoid G D is effective. RADED STEINBERG ALGEBRAS AND THEIR REPRESENTATIONS 27
Remark 7.12. (1) If c − ( ε ) is strongly effective, then Theorem 7.11(i) holds. In fact, a closed invariant subset D of the unit space of G is in particular a closed c − ( ε )-invariant subset of G (0) . We have c − ( ε ) D = c − ( ε ) ∩ G D = (cid:0) c | G D (cid:1) − ( ε ). Hence, Theorem 7.11(i) follows directly. Example 7.13 below, on the other hand,shows that Theorem 7.11(i) does not imply that c − ( ε ) is strongly effective.(2) Resume the notation of Example 7.4, so u = αβα β · · · ∈ E ∞ . Let D be the closure of the Z -aperiodicinvariant subset [ u ] ⊆ G (0) E . As we saw in that example, D is not itself Z -aperiodic, because it contains α ∞ . Example 7.13.
It is easy to construct examples of Γ-graded groupoids with no Γ-aperiodic points. For example,let X be the Cantor set. Regard G = X × Z as a groupoid with unit space X × { } identified with X by setting r ( x, m ) = x = d ( x, m ) and defining composition and inverses by ( x, n )( x, m ) = ( x, m + n ) and ( x, m ) − = ( x, − m ).The map c : G −→ Z given by c ( x, ( m , m )) = m is a continuous 1-cocycle. We have c − (0) = X × ( { } × Z ),which is not effective (for example X × { (0 , } is a compact open bisection contained in the isotropy subgroupoidof c − (0)). Moreover, G (0) has no Z -aperiodic points because { u } × ( Z × { } ) ⊆ Iso( u ) \ c − (0) for all u ∈ G (0) ; soevery u ∈ G (0) is Z -periodic.Applying Theorem 7.11 to the trivial grading, we obtain a new characterisation of strong effectiveness. Corollary 7.14.
Let G be an ample Hausdorff groupoid, and R be a commutative ring with identity. Then G isstrongly effective if and only if for any invariant subset W of G (0) , the annihilator of the A R ( G ) -module RW is A R ( G U ) , where U = ( G (0) \ W ) ◦ . Acknowledgements
The authors would like to acknowledge Australian Research Council grants DP150101598 and DP160101481. Thefirst-named author was partially supported by DGI-MINECO (Spain) through the grant MTM2014-53644-P.
References [1] G. Abrams,
Leavitt path algebras: the first decade , Bull. Math. Sci. (2015), no. 1, 59–120. 18[2] G. Abrams, G. Aranda Pino, The Leavitt path algebra of a graph , J. Algebra (2) (2005), 319–334. 1, 10, 12[3] G. Abrams, G. Aranda Pino,
The Leavitt path algebra of arbitrary graphs , Houston J. Math. (2008), 423–442. 16[4] G. Abrams, F. Mantese, A. Tonolo, Extensions of simple modules over Leavitt path algebras , J. Algebra (2015), 78–106. 2, 24[5] P. Ara, M.A. Moreno, E. Pardo,
Nonstable K -theory for graph algebras , Algebr. Represent. Theory (2) (2007), 157–178. 1, 10,15, 16, 18, 19[6] P. Ara, K.R. Goodearl, Leavitt path algebras of separated graphs,
J. reine angew. Math. (2012), 165-224. 10, 11, 15, 18, 19[7] P. Ara, K.R. Goodearl,
Tame and wild refinement monoids,
Semigroup Forum (2015), 1–27. 15, 19[8] P. Ara, E. Pardo, Towards a K-theoretic characterization of graded isomorphisms between Leavitt path algebras , J. K-Theory (2014), no. 2, 203–245. 18[9] P. Ara, K. M. Rangaswamy, Finitely presented simple modules over Leavitt path algebras,
J. Algebra (2014), 333–352. 2, 24[10] P. Ara, K.M. Rangaswamy,
Leavitt path algebras with at most countably many irreducible representations , Rev. Mat. Iberoam. (2015), no. 4, 1263–1276. 2, 24[11] G. Aranda Pino, J. Clark, A. an Huef, I. Raeburn, Kumjian–Pask algebras of higher-rank graphs , Trans. Amer. Math. Soc. (2013), no. 7, 3613–3641. 1, 2, 20[12] M. Beattie,
A generalization of the smash product of a graded ring , J. Pure Appl. Algebra (1988), 219–226. 3, 4[13] S. Berberian, Baer ∗ -rings. Die Grundlehren der mathematischen Wissenschaften, Band 195. Springer-Verlag, New York-Berlin, 1972.1[14] G. Bergman, Modules over coproducts of rings , Trans. Amer. Math. Soc. (1974), 1–32. 10[15] J. Brown, L.O. Clark, C. Farthing, A. Sims,
Simplicity of algebras associated to ´etale groupoids , Semigroup Forum (2014), no. 2,433–452. 2, 23, 26[16] X.W. Chen, Irreducible representations of Leavitt path algebras , Forum Math. (1) (2015), 549–574. 2, 24[17] L.O. Clark, D.M. Barquero, C.M. Gonzalez, M.S. Molina, Using Steinberg algebras to study decomposability of Leavitt path algebras ,arXiv:1603.01033v1.[18] L.O. Clark, C. Edie-Michell,
Uniqueness theorems for Steinberg algebras , Algebr. Represent. Theory (2015), no. 4, 907–916. 4, 7,26[19] L.O. Clark, C. Edie-Michell, A. an Huef, A. Sims, Ideals of Steinberg algebras of strongly effective groupoids, with applications toLeavitt path algebras , arXiv:1601.07238v1. 26[20] L.O. Clark, R. Exel, E. Pardo,
A generalised uniqueness theorem and the graded ideal structure of Steinberg algebras ,arXiv:1609.02873v1. 7[21] L.O. Clark, C. Farthing, A. Sims, M. Tomforde,
A groupoid generalisation of Leavitt path algebras , Semigroup Forum (2014),501–517. 1, 6, 7[22] L.O. Clark, Y.E.P. Pangalela, Kumjian–Pask algebras of finitely-alighed higher-rank graphs , arXiv:1512.06547v1. 20[23] L.O. Clark, A. Sims,
Equivalent groupoids have Morita equivalent Steinberg algebras , J. Pure Appl. Algebra (2015), 2062–2075.7, 12, 13 [24] M. Cohen, S. Montgomery,
Group-graded rings, smash products, and group actions , Trans. Amer. Math. Soc. (1984), 237–258.1, 3[25] D. Drinen, M. Tomforde,
The C ∗ -algebras of arbitrary graphs , Rocky Mountain J. Math. (2005), 105–135. 16[26] C. Faith, Algebra: rings, modules and categories I, New York (1973), Springer–Verlag. 10[27] K.R. Goodearl, Leavitt path algebras and direct limits , Contemp. Math. (2009), 165–187. 11[28] E.L. Green,
Graphs with relations, coverings and group-graded algebras , Trans. Amer. Math. Soc. (1983), no. 1, 297–310. 13, 14[29] R. Hazrat,
A note on the isomorphism conjectures for Leavitt path algebras , J. Algebra (2013), 33–40. 12, 18[30] R. Hazrat,
The graded Grothendieck group and the classication of Leavitt path algebras , Math. Annalen (2013), 273–325. 19[31] R. Hazrat, Graded rings and graded Grothendieck groups, London Math. Society Lecture Note Series, Cambridge University Press,2016. 2, 11, 21[32] R. Hazrat, K.M. Rangaswamy,
On graded irreducible representations of Leavitt path algebras , J. Algebra (2016), 458–486. 2, 24[33] I. Kaplansky, Rings of operators. W. A. Benjamin, Inc., New York-Amsterdam 1968. 1[34] A. Kumjian, D. Pask, C ∗ -algebras of directed graphs and group actions , Ergod. Th. Dynam. Systems (1999), 1503–1519. 1, 13,14[35] A. Kumjian, D. Pask, Higher rank graph C ∗ -algebras , New York J. Math. (2000), 1–20. 1, 20, 21[36] A. Kumjian, D. Pask, I. Raeburn, J. Renault, Graphs, groupoids, and Cuntz–Krieger algebras , J. Funct. Anal. (1997), 505–541.1, 12, 23[37] P. Menal, J. Moncasi,
Lifting units in self-injective rings and an index theory for Rickart C ∗ -algebras , Pacific J. Math. (1987),295–329. 10[38] C. N˘ast˘asescu, F. Van Oystaeyen, Methods of graded rings, Lecture Notes in Mathematics, 1836, Springer-Verlag, Berlin, 2004. 3[39] A.L.T. Paterson, Graph inverse semigroups, groupoids and their C ∗ -algebras. J. Operator Theory (2002), 645–662. 12[40] N.C. Phillips, A classification theorem for nuclear purely infinite simple C ∗ -algebras , Doc. Math. (2000), 49–114. 1, 10[41] I. Raeburn, Graph algebras. CBMS Regional Conference Series in Mathematics, 103. The American Mathematical Society, Provi-dence, RI, 2005. 1[42] I. Raeburn, A. Sims, T. Yeend, The C ∗ -algebras of finitely aligned higher-rank graphs , J. Funct. Anal. (2004), 206–240. 2[43] K.M. Rangaswamy, Leavitt path algebras with finitely presented irreducible representations , J. Algebra (2016), 624–648. 2, 24[44] J. Renault, A groupoid approach to C ∗ -algebras, Lecture Notes in Mathematics, 793. Springer, Berlin, 1980. 1, 7[45] L. Shaoxue, F. van Oystaeyen, Group graded rings, smash products and additive categories , In van Oystaeyen, Le Bruyn (ed.),Perspectives in Ring Theory, 311–320, 1988. 3[46] A. Sims,
Gauge-invariant ideals in the C ∗ -algebras of finitely aligned higher-rank graphs , Canad. J. Math. (2006), 1268–1290. 2[47] A. Sims, The co-universal C ∗ -algebra of a row-finite graph , New York J. Math. (2010), 507–524. 16[48] B. Steinberg, A groupoid approach to discrete inverse semigroup algebras , Adv. Math. (2010), 689–727. 1, 6, 7[49] M. Tomforde,
Uniqueness theorems and ideal structure for Leavitt path algebras , J. Algebra (2007), 270–299. 18[50] S.B.G. Webster,
The path space of a directed graph , Proc. Amer. Math. Soc. (2014), 213–225. 13[51] R. Wisbauer,
Foundations of module and ring theory , Revised and translated from the 1988 German edition, Algebra, Logic andApplications, vol. 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991. Department of Mathematics, Universitat Auto´noma de Barcelona, 08193 Bellaterra (Barcelona), Spain
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