Amplified graph C*-algebras II: reconstruction
aa r X i v : . [ m a t h . OA ] J u l AMPLIFIED GRAPH C*-ALGEBRAS II:RECONSTRUCTION
SØREN EILERS, EFREN RUIZ, AND AIDAN SIMS
Abstract.
Let E be a countable directed graph that is amplifiedin the sense that whenever there is an edge from v to w , thereare infinitely many edges from v to w . We show that E can berecovered from C ∗ ( E ) together with its canonical gauge-action,and also from L K ( E ) together with its canonical grading. Introduction
The purpose of this paper is to investigate the gauge-equivariant iso-morphism question for C ∗ -algebras of countable amplified graphs, andthe graded isomorphism question for Leavitt path algebras of countableamplified graphs. A directed graph E is called an amplified graph iffor any two vertices v, w , the set of edges from v to w is either emptyor infinite.The geometric classification (that is, classification by the underlyinggraph modulo the equivalence relation generated by a list of allow-able graph moves) of the C ∗ -algebras of finite-vertex amplified graph C ∗ -algebras was completed in [12], and was an important precursorto the eventual geometric classification of all finite graph C ∗ -algebras[13]. But there has been increasing recent interest in understandingisomorphisms of graph C ∗ -algebras that preserve additional structure:for example the canonical gauge action of the circle; or the canonicaldiagonal subalgebra isomorphic to the algebra of continuous functionsvanishing at infinity on the infinite path space of the graph; or thesmaller coefficient algebra generated by the vertex projections; or somecombination of these (see, for example, [5, 6, 7, 8, 9, 10, 19]).A program of geometric classification for these various notions ofisomorphism was initiated by the first two authors in [11]. They discuss xyz -isomorphism of graph C ∗ -algebras, where x is 1 if we require exactisomorphism, and 0 if we require only stable isomorphism; y is 1 if theisomorphism is required to be gauge-equivariant, and 0 otherwise; and Mathematics Subject Classification.
Primary: 46L35.
Key words and phrases. amplified graph, graph C ∗ -algebra.This research was supported by Australian Research Council Discovery ProjectDP200100155, by DFF-Research Project 2 ‘Automorphisms and Invariants of Op-erator Algebras’, no. 7014-00145B, and by a Simons Foundation CollaborationGrant, z is 1 if the isomorphism is required to preserve the diagonal subalgebraand 0 otherwise. They also identified a set of moves on graphs thatpreserve various kinds of xyz -isomorphism, and conjectured that forall xyz other than x10 , the equivalence relation on graphs with finitelymany vertices induced by xyz -isomorphism of C ∗ -algebras is generatedby precisely those of their moves that induce xyz -isomorphisms.This was an important motivation for the present paper. None ofthe moves in [11] takes an amplified graph to an amplified graph.And although we know of one important instance where one amplifiedgraph can be transformed into another via a sequence of -preservingmoves passing through non-amplified graphs (see Diagram (3.1) in Re-mark 3.5), we had given up on envisioning such a sequence consistingonly of x1z -preserving moves. Based on the main conjecture of [11],this led us to expect that an amplified graph C ∗ -algebra together withits gauge action should remember the graph itself.Our main theorem shows that, indeed, any countable amplified graph E can be reconstructed from either the circle-equivariant K -group ofits C ∗ -algbra, or the graded K -group of its Leavitt path algebra overany field. That is: Theorem A.
Let E and F be countable amplified graphs and let K bea field. Then the following are equivalent: (1) E ∼ = F ; (2) there is a Z [ x, x − ] -module order-isomorphism K gr0 ( L K ( E )) ∼ = K gr0 ( L K ( F )) ; and (3) there is a Z [ x, x − ] -module order-isomorphism K T ( C ∗ ( E ) , γ )) ∼ = K T ( C ∗ ( F ) , γ )) of T -equivariant K -groups. We spell out a number of consequences of this theorem in Remark 3.9,Theorem 3.4, and Theorem 3.8. The headline is that for amplifiedgraphs, and for any x , z , the graph C ∗ -algebras C ∗ ( E ) and C ∗ ( F ) are x1z -isomorphic if and only if E and F are isomorphic. Combined withresults of [4, 13], this confirms [11, Conjecture 5.1] for amplified graphs(see Remark 3.5).Another immediate consequence is that, since ordered graded K isan isomorphism invariant of graded rings, and ordered T -equivariant K is an isomorphism invariant of C ∗ -algebras carrying circle actions, ourtheorem confirms a special case of Hazrat’s conjecture: ordered graded K is a complete graded-isomorphism invariant for amplified Leavittpath algebras; and we also obtain that ordered T -equivariant K is acomplete gauge-isomorphism invariant of amplified graph C ∗ -algebras.A third consequence is related to different graded stabilisations ofLeavitt path algebras (and different equivariant stabilisations of graph C ∗ -algebras). Each Leavitt path algebra has a canonical grading, and,as alluded to above, significant work led by Hazrat has been done on MPLIFIED GRAPH C*-ALGEBRAS II: RECONSTRUCTION 3 determining when graded K-theory completely classifies graded Leav-itt path algebras. Historically, in the classification program for C ∗ -algebras, significant progress has been made by first considering classi-fication up to stable isomorphism; so it is natural to consider the sameapproach to Hazrat’s graded classification question. But almost imme-diately, there is a difficulty: which grading on L K ( E ) ⊗ M ∞ ( K ) shouldwe consider? It seems natural enough to use the grading arising fromthe graded tensor product of the graded algebras L K ( E ) and M ∞ ( K ).But there are many natural gradings on M ∞ ( K ): given any δ ∈ Q i Z ,we obtain a grading of M ∞ ( K ) in which the m, n matrix unit is homo-geneous of degree δ m − δ n . Different nonzero choices for δ correspond todifferent ways of stabilising L K ( E ) by modifying the graph E (for ex-ample by adding heads [23]), while taking δ = (0 , , , . . . ) correspondsto stabilising the associated groupoid by taking its cartesian productwith the (trivially graded) full equivalence relation N × N .In Section 3.2, we show that for amplified graphs it doesn’t mat-ter what value of δ we pick. Specifically, using results of Hazrat,we prove that K gr0 ( L K ( E ) ⊗ M ∞ ( K )( δ )) ∼ = K gr0 ( L K ( E )) regardless of δ . Consequently, for any choice of δ we have L K ( E ) ⊗ M ∞ ( K )( δ ) ∼ = L K ( F ) ⊗ M ∞ ( K )( δ ) if and only if there exists a Z [ x, x − ]-module order-isomorphism K gr0 ( L K ( E )) ∼ = K gr0 ( L K ( F )). A similar result holds for C ∗ -algebras with the gradings on Leavitt path algebras replaced bygauge actions on graph C ∗ -algebras, and the gradings of M ∞ ( K ) corre-sponding to different elements δ replaced by the circle actions on K ( ℓ )implemented by different strongly continuous unitary representationsof the circle on ℓ .We prove our main theorem in Section 2. We use general results tosee that the graded K -group of L K ( E ) and the equivariant K -group of C ∗ ( E ) are isomorphic as ordered Z [ x, x − ]-modules to the K -groups ofthe Leavitt path algebra and the graph C ∗ -algebra (respectively) of theskew-product graph E × Z . These are known to coincide, and theirlattice of order ideals (with canonical Z -action) is isomorphic to thelattice of hereditary subsets of ( E × Z ) with the Z -action of translationin the second variable. So the bulk of the work in Section 2 goes intoshowing how to recover E from this lattice. We then go on in Section 3.2to establish the consequences of our main theorem for stabilizations.Here the hard work goes into showing that K gr0 ( L K ( E ) ⊗ M ∞ ( K )( δ )) ∼ = K gr0 ( L K ( E )) for any δ ∈ Q i Z and that K T ( C ∗ ( E ) ⊗ K , γ E ⊗ Ad u ) ∼ = K gr0 ( L K ( E )) for any strongly continuous unitary representation u of T .2. Gauge-invariant classification of amplified graph C ∗ -algebras Throughout the paper, a countable directed graph E is a quadruple E = ( E , E , r, s ) where E is a countable set whose elements are called EILERS, RUIZ, AND SIMS vertices , E is a countable set whose elements are called edges , and r, s : E → E are functions. We think of the elements of E as pointsor dots, and each element e of E as an arrow pointing from the vertex s ( e ) to the vertex r ( e ). We follow the conventions of, for example [14],where a path is a sequence e . . . e n of edges in which s ( e n +1 ) = r ( e n ).This is not the convention used in Raeburn’s monograph [21], but is theconvention consistent with all of the Leavitt path algebra literature aswell as much of the graph C ∗ -algebra literature. In keeping with this,for v, w ∈ E and n ≥
0, we define vE = s − ( v ) , E w = r − ( w ) , and vE w = s − ( v ) ∩ r − ( w ) . We will also write vE n for the sets of paths of length n that are emittedby v , E n w for the set of paths of length n received by w , and vE n w for the set of paths of length n pointing from v to w .A vertex v is singular if vE is either empty or infinite, so v is either asink or an infinite emitter; and for any edge e , we have s ∗ e s e = p r ( e ) and p s ( e ) ≥ s e s ∗ e in the graph C ∗ -algebra C ∗ ( E ). We will also consider theLeavitt path algebras, L K ( E ) for any field K , the so-called algebraiccousin of graph C ∗ -algebras. Leavitt path algebras are defined viagenerators and relations similar to those for graph C ∗ -algebras (see[1]).Countable directed graphs E and F are isomorphic , denoted E ∼ = F ,if there is a bijection φ : E ⊔ E → F ⊔ F that restricts to bijections φ : E → F and φ : E → F such that φ ( r ( e )) = r ( φ ( e )) and φ ( s ( e )) = s ( φ ( e )) . In this paper, we consider amplified graphs. The classification ofamplified graph C ∗ -algebras was the starting point in the classificationof unital graph C ∗ -algebras via moves (see [12] and [13]). Definition 2.1 (Amplified Graph and Amplified graph algebra) . Adirected graph E is an amplified graph if for all v, w ∈ E , the set vE w = s − ( v ) ∩ r − ( w ) is either empty or infinite. An amplified graph C ∗ -algebra is a graph C ∗ -algebra of an amplified graph and an amplifiedLeavitt path algebra is a Leavitt path algebra of an amplified graph.Observe that in an amplified graph, every vertex is singular.Recall that a set H ⊆ E is hereditary if s ( e ) ∈ H implies r ( e ) ∈ H for every e ∈ E , and is saturated if whenever v is a regular vertexsuch that r ( vE ) ⊆ H , we have v ∈ H . Again since every vertex in anamplified graph is singular, every set of vertices is saturated.Recall from [18] that if E is a directed graph, then the skew-productgraph E × Z is the graph with vertices E × Z and edges E × Z with s ( e, n ) = ( s ( e ) , n ) and r ( e, n ) = ( r ( e ) , n + 1). If E is an amplifiedgraph, then so is E × Z .For a countable amplified graph, E , we write H ( E × Z ) for thelattice (under set inclusion) of hereditary subsets of the vertex-set of MPLIFIED GRAPH C*-ALGEBRAS II: RECONSTRUCTION 5 the skew-product graph E × Z . The action of Z on E × Z by given by n · ( e, m ) = ( e, n + m ) induces an action lt of Z on H ( E × Z ). There isalso a distinguished element H ∈ H ( E × Z ) given by H := { ( v, n ) : v ∈ E , n ≥ } ⊆ ( E × Z ) .Throughout this section, given v ∈ E and n ∈ Z , we write H ( v, n )for the smallest hereditary subset of ( E × Z ) containing ( v, n ). So H ( v, n ) = { ( r ( µ ) , n + | µ | ) : µ ∈ vE ∗ } is the set of vertices that can bereached from ( v, n ) in E × Z .If ( L , (cid:22) ) is a lattice, we say that L ∈ L has a unique predecessorif there exists K ∈ L such that K ≺ L , and every K ′ with K ′ ≺ L satisfies K ′ (cid:22) K . The next proposition is the engine-room of our mainresult. Proposition 2.2.
Let E be a countable amplified graph. Define H vert ⊆H ( E × Z ) to be the subset H vert = { H ∈ H ( E × Z ) : H has a unique predecessor } . Then H vert = { H ( v, n ) : v ∈ E and n ∈ Z } . Let E := { H ∈ H vert : H ⊆ H and H lt ( H ) } . Define E := { ( H, n, K ) :
H, K ∈ E , lt ( K ) ⊆ H, and n ∈ N } .Define ¯ s, ¯ r : E → E by ¯ s ( H, n, K ) = H and ¯ r ( H, n, K ) = K . Then E := ( E , E , ¯ r, ¯ s ) is a countable amplifed directed graph, and thereis an isomorphism E ∼ = E that carries each v ∈ E to the hereditarysubset of ( E × Z ) generated by ( v, .Proof. The argument of [12, Lemma 5.2] shows that H vert = { H ( v, n ) : v ∈ E , n ∈ Z } .We clearly have H ( v, n ) ⊆ H if and only if n ≥
0, and H ( v, n ) ⊆ lt ( H ) if and only if n ≥
1, so E = { H ( v,
0) : v ∈ E } . Since E × Z is acyclic, the H ( v,
0) are distinct, and we deduce that θ : v H ( v, E to E .Fix v, w ∈ E . We have lt ( H ( w, H ( w, w, ∈ H ( v,
0) if and only if vE w = ∅ , we have H ( w, ⊆ H ( v,
0) if and onlyif vE w = ∅ , in which case vE w is infinite because E is amplified.It follows that | H ( v, E H ( w, | = | vE w | for all v, w , so we canchoose a bijection θ : E → E that restricts to bijections vE w → θ ( v ) E θ ( w ) for all v, w ∈ E . The pair ( θ , θ ) is then the desiredisomorphism E ∼ = E . (cid:3) In order to use Proposition 2.2 to prove Theorem A, we need to knowthat if ( H ( E × Z ) , lt E ) is order isomorphic to ( H ( F × Z ) , lt F ) thenthere is an isomorphism from ( H ( E × Z ) , lt E ) to ( H ( F × Z ) , lt F ) thatcarries H E to H F . We do this by showing that if E is connected, thenwe can recognise the sets lt n ( H ) amongst all the hereditary subsets of( E × Z ) using just the order-structure and the action lt. EILERS, RUIZ, AND SIMS
Recalling that vE n w denotes the set of paths of length n from v to w , we have(2.1) H ( w, n ) ⊆ H ( v, m ) if and only if vE n − m w = ∅ . Recall that a graph E is said to be connected if the smallest equiva-lence relation on E containing { ( s ( e ) , r ( e )) : e ∈ E } is all of E × E .Let E be a connected, countable amplified graph. The set V := { H ( v,
0) : v ∈ E } is exactly the set of maximal elements of the col-lection { H ∈ H vert : H ⊆ H } . The sets H and V have the followingproperties: • for each H ∈ H vert there is a unique n ∈ Z such that lt n ( H ) ∈ V ; • the smallest equivalence relation on V containing { ( H, K ) :lt ( K ) ⊆ H } is all of V × V ; and • if H, K are distinct elements of V , and if n ≥
0, then H lt n ( K ).The next lemma shows that for connected graphs, these propertiescharacterise H up to translation. Lemma 2.3.
Suppose that E is a connected, countable amplified graph.Take H ∈ H ( E × Z ) , and let V H be the set of maximal elements of { K ∈ H vert : K ⊆ H } with respect to set inclusion. Suppose that (1) for each K ∈ H vert there is a unique n ∈ Z such that lt n ( K ) ∈ V H ; (2) the smallest equivalence relation on V H containing { ( H, K ) :lt ( K ) ⊆ H } is all of V H × V H ; and (3) if K, K ′ are distinct elements of V H , and if n ≥ , then K lt n ( K ′ ) .Then there exists n ∈ Z such that H = lt n ( H ) .Proof. For each v ∈ E , item (1) applied to K = H ( v,
0) shows thatthere exists a unique n v ∈ Z such that H ( v, n v ) = lt n v ( K ) ∈ V H .So V H = { H ( v, n v ) : v ∈ E } . We must show that n v = n w for all v, w ∈ E . To do this, it suffices to show that for any u ∈ E , we have n w ≥ n u for all w ∈ E .So fix u ∈ E . Define L u := { v ∈ E : n v < n u } and G u := { w ∈ E : n w ≥ n u } We prove that if v ∈ L u and w ∈ G u , then(2.2) lt ( H ( v, n v )) H ( w, n w ) and lt ( H ( w, n w )) H ( v, n v ) . For this, fix v ∈ L u and w ∈ G u ; note that in particular v = w .To see that lt ( H ( v, n v )) H ( w, n w ), suppose otherwise for con-tradiction. Then H ( v, n v + 1) ⊆ H ( w, n w ). Hence (2.1) shows that wE n v +1 − n w v = ∅ , which forces n v ≥ n w −
1. Since v ∈ L u and w ∈ G u , MPLIFIED GRAPH C*-ALGEBRAS II: RECONSTRUCTION 7 we also have n v ≤ n w −
1, and we conclude that n v + 1 − n w = 0. Thisforces wE v = ∅ , contradicting that v = w .To see that lt ( H ( w, n w )) H ( v, n v ), we first claim that there is no e ∈ E satisfying s ( e ) ∈ L u and r ( e ) ∈ G u . To see this, fix x ∈ L u and y ∈ G u . Then n y > n x , and in particular n y − − n x ≥
0. HenceItem (3) shows that H ( y, n y ) lt n y − − n x ( H ( x, n x )). Applying lt − n y on both sides shows that lt ( H ( y, H ( x, xE y = ∅ .This proves the claim.Since v ∈ L u , applying the claim n w +1 − n v times shows that for anypath µ ∈ vE n w +1 − n v , we have r ( µ ) ∈ L u . In particular, vE n w +1 − n v w = ∅ . Thus (2.1) implies that lt ( H ( w, n w )) H ( v, n v ).We have now established (2.2). Set L u = { H ( v, n v ) : v ∈ L u } and G u = { H ( w, n w ) : w ∈ G u } . Then (2.2) shows that ( L u × L u ) ⊔ ( G u × G u ) is an equivalence relationon V H containing { ( H, K ) : lt ( K ) ⊆ H } . Thus item (2) implies thateither L u or G u is empty. Since H ( u, n u ) ∈ G u , we deduce that L u = ∅ which implies that L u = ∅ . Hence G u = E , and so n w ≥ n u for all w ∈ E as required. (cid:3) Corollary 2.4.
Suppose that E and F are amplified graphs. If thereexists an isomorphism ρ : ( H ( E × Z ) , ⊆ , lt E ) ∼ = ( H ( F × Z ) , ⊆ , lt F ) ,then there exists an isomorphism ρ : ( H ( E × Z ) , ⊆ , lt E ) → ( H ( F × Z ) , ⊆ , lt F ) such that ρ ( H E ) = H F .Proof. First suppose that E and F are connected as in Lemma 2.3.Since H ∈ H E vert if and only if H has a unique predecessor in H ( E × Z )and similarly for F , the map ρ restricts to an inclusion-preserving bijec-tion ρ : H E vert → H F vert . Since H E and V E satisfy (1)–(3) of Lemma 2.3,so do ρ ( H E ) and { ρ ( H ) : H ∈ V E } . So Lemma 2.3 shows that ρ ( H E ) = lt n ( H F ) for some n ∈ Z , and therefore ρ := lt − n ◦ ρ is thedesired isomorphism.Now suppose that E and F are not connected. Let WC ( E ) denotethe set of equivalence classes for the equivalence relation on E gen-erated by { ( s ( e ) , r ( e )) : e ∈ E } ; so the elements of WC ( E ) are theweakly connected components of E . Similarly, let WC ( F ) be the setof weakly connected components of F .Using that vE ∗ w is nonemtpy if and only if lt n ( H ( w, ⊆ H ( v, n ∈ Z , we see that vE ∗ w = ∅ if and only if S n lt n ( H ( w, i )) ⊆ S n lt n ( H ( v, j )) for some (equivalently for all) i, j ∈ Z . Since the same istrue in F , we see that for v, w ∈ E , writing x, y ∈ F for the elementssuch that ρ ( H ( v, ∈ lt Z ( H ( x, ρ ( H ( w, ∈ lt Z ( H ( y, vE ∗ w = ∅ if and only if xF ∗ y = ∅ . Now an induction showsthat there is a bijection ˜ ρ : WC ( E ) → WC ( F ) such that for each C ∈ WC ( E ), we have ρ ( { H ( v, n ) : v ∈ C, n ∈ Z } ) = { H ( w, m ) : w ∈ ˜ ρ ( C ) , m ∈ Z } . For each C ∈ WC ( E ), write E C for the subgraph EILERS, RUIZ, AND SIMS ( C, CE C, r, s ) of E and similarly for F . Then the inclusions E C ֒ → E induce inclusions ( H ( E C × Z ) , lt) ֒ → ( H ( E × Z ) , lt) whose rangesare lt-invariant and mutually incomparable with respect to ⊆ . Hence ρ induces isomorphisms ρ C : ( H ( E C × Z ) , lt) ∼ = ( H ( F ˜ ρ ( C ) × Z ) , lt).The first paragraph then shows that for each C ∈ WC ( E ) there isan isomorphism ρ C : ( H ( E C × Z ) , lt) → ( H ( F ˜ ρ ( C ) × Z ) , lt) that car-ries H E C to H F ˜ ρ ( C ) , and these then assemble into an isomorphsm ρ :( H ( E × Z ) , ⊆ , lt E ) → ( H ( F × Z ) , ⊆ , lt F ) such that ρ ( H E ) = H F . (cid:3) We are now ready to prove Theorem A.
Proof of Theorem A.
That (1) implies (2) and that (1) implies (3) areclear.By [3, Proposition 5.7] the graded V -monoid V gr ( L K ( E )) is isomor-phic to the V -monoid V ( L K ( E × Z )), and that this isomorphism isequivariant for the canonical Z [ x, x − ] actions arising from the grad-ing on V gr ( L K ( E )) and from the action on V ( L K ( E × Z )) inducedby translation in the Z -coordinate in E × Z . Hence K gr0 ( L K ( E )) isorder isomorphic to K ( L K ( E × Z )) as Z [ x, x − ]-modules. Hence con-dition (2) holds if and only if K ( L K ( E × Z )) ∼ = K ( L K ( F × Z )) asordered Z [ x, x − ]-modules.Likewise [20, Theorem 2.7.9] shows that the equivariant K -theorygroup K T ( C ∗ ( E )) is order isomorphic, as a Z [ x, x − ]-module, to the K -group K ( C ∗ ( E ) × γ T ). The canonical isomorphism C ∗ ( E ) × γ T ∼ = C ∗ ( E × Z ) is equivariant for the dual action ˆ γ of Z on the formerand the action of Z on the latter induced by translation in E × Z . Ittherefore induces an isomorphism K ( C ∗ ( E ) × γ T ) ∼ = K ( C ∗ ( E × Z ))of ordered Z [ x, x − ]-modules. So condition (3) holds if and only if K ( C ∗ ( E × Z )) ∼ = K ( C ∗ ( F × Z )) as ordered Z [ x, x − ]-modules.By [16, Theorem 3.4 and Corollary 3.5] (see also [2]), for any di-rected graph E there is an isomorphism K ( L K ( E )) ∼ = K ( C ∗ ( E )) thatcarries the class of the module L K ( E ) v to the class of the projection p v in C ∗ ( E ) for each v ∈ E . It follows that K ( L K ( E × Z )) ∼ = K ( C ∗ ( E × Z )) as ordered Z [ x, x − ]-modules. This shows that con-ditions (2) and (3) are equivalent. So it now suffices to show that(2) implies (1).So suppose that (2) holds. Since E , and therefore E × Z , is anamplified graph, it admits no breaking vertices with respect to anysaturated hereditary set, and every hereditary subset of E × Z is asaturated hereditary subset. So the lattice H ( E × Z ) of hereditary setsis identical to the lattice of admissible pairs in the sense of [22] via themap H ( H, ∅ ). By [3, Theorem 5.11], there is a lattice isomorphismfrom H ( E × Z ) to the lattice of order ideals of K ( L K ( E × Z )) thatcarries a hereditary set H to the class of the module L K ( E × Z ) H . thisisomorphism clearly intertwines the action of Z induced by the modulestructure on K ( L K ( E × Z )) and the action lt E of Z on H ( E × Z ) MPLIFIED GRAPH C*-ALGEBRAS II: RECONSTRUCTION 9 induced by translation. By the same argument applied to F , we seethat (cid:0) H ( E × Z ) , ⊆ , lt E ) ∼ = (cid:0) H ( F × Z ) , ⊆ , lt F (cid:1) .Now Corollary 2.4 implies that (cid:0) H ( E × Z ) , lt E , H E ) ∼ = (cid:0) H ( F × Z ) , lt F , H F (cid:1) . This isomorphism induces an isomorphism E ∼ = F ofthe graphs constructed from these data in Proposition 2.2. Thus twoapplications of Proposition 2.2 give E ∼ = E ∼ = F ∼ = F , which is (1). (cid:3) Equivariant K -theory and graded K -theory are stableinvariants In this section, we prove that equivariant K -theory and graded K -theory are stable invariants. We suspect that these are well-knownresults but we have been unable to find a reference in the literature.For the convenience of the reader, we include their proofs here. Weuse these results to deduce the consequences of Theorem A for gradedstable isomorphisms of amplified Leavitt path algebras, and gauge-equivariant stable isomorphisms of amplified graph C ∗ -algebras.3.1. Stability of equivariant K -theory.Theorem 3.1. Let G be a compact group and let α be an action of G ona C ∗ -algebra A . If A has an increasing approximate identity consistingof G -invariant projections, then the natural R ( G ) -module isomorphismfrom K G ( A, α ) to K ( C ∗ ( G, A, α )) is an order isomorphism.Proof. First suppose A has a unit. Then the theorem follows fromthe proof of Julg’s Theorem [17] (see also [20, Theorem 2.7.9]). Theisomorphism is given by the composition of two isomorphisms: K G ( A, α ) → K ( L ( G, A, α )) and K ( L ( G, A, α )) → K ( C ∗ ( G, A, α )) . The proof that these maps are isomorphisms shows that the mapsare order isomorphisms (see the proof of [20, Lemma 2.4.2 and Theo-rem 2.6.1]).Now suppose A has an increasing approximate identity S consistingof G -invariant projections. Fix p ∈ S . Let λ A : K G ( A, α ) → K ( C ∗ ( G, A, α )) , and λ p : K G ( pAp, α ) → K ( C ∗ ( G, pAp, α )) , p ∈ S be the natural R ( G )-isomorphisms given in Julg’s Theorem. Note that α does indeed induce an action on pAp since p is G -invariant. Let ι p be the G -equivariant inclusion of pAp into A and let e ι p be the induce ∗ -homomorphism from C ∗ ( G, pAp, α ) to C ∗ ( G, A, α ).Let x ∈ K G ( A, α ) + . By [20, Corollary 2.5.5], there exist p ∈ S and x ′ ∈ K G ( pAp, α ) + such that ( ι p ) ∗ ( x ′ ) = x . Naturality of themaps λ A and λ p gives λ A ( x ) = ( e ι p ) ∗ ◦ λ p ( x ′ ). Consequently, λ A ( x ) ∈ K ( C ∗ ( G, A, α )) + since ( e ι p ) ∗ ◦ λ p ( x ′ ) ∈ K ( C ∗ ( G, A, α )) + . Fix y ∈ K ( C ∗ ( G, A, α )) + . For f ∈ L ( G ) and a ∈ A we write f ⊗ a : G → A for the function ( f ⊗ a )( g ) = f ( g ) a . Since S is an approximate identityof A and since { f ⊗ a : f ∈ L ( G ) , a ∈ A } is dense in C ∗ ( G, A, α ), the set S p ∈ S e ι p ( C ∗ ( G, pAp, α )) is dense in C ∗ ( G, A, α ). Thus, there exists a projection p ∈ S and there exists y ′ ∈ K ( C ∗ ( G, pAp, α )) + such that ( e ι p ) ∗ ( y ′ ) = x . Since λ p is an order iso-morphism, λ − p ( y ′ ) ∈ K G ( pAp, α ) + . Then ( ι p ) ∗ ◦ λ − p ( y ′ ) ∈ K G ( A, α ) + .Naturality of the maps λ A and λ p implies that λ A ◦ ( ι p ) ∗ ◦ λ − p ( y ′ ) = y .We have shown that λ A ( K G ( A, α ) + ) = K ( C ∗ ( G, A, α )) + which impliesthat λ A is an order isomorphism. (cid:3) Lemma 3.2.
Let G be a compact group and let A be a separable C ∗ -algebra and let α be an action of G on A . If B is a hereditary subalgebraof A such that (1) B has an increasing approximate identity of G -invariant pro-jections, (2) A has an increasing approximate identity of G -invariant pro-jections, (3) ABA = A , and (4) α g ( B ) ⊆ B for all g ∈ G ,then the inclusion ι : B → A induces an isomorphism K G ( B ) ∼ = K G ( A ) of ordered R ( G ) -modules.Proof. Since B is G -invariant, α is also an action on B and the in-clusion ι is G -equivariant. Let λ B : K G ( B, α ) → K ( C ∗ ( G, B, α )) and λ A : K G ( A ) → K ( C ∗ ( G, A, α )) be the natural R ( G )-module order iso-morphisms given in Theorem 3.1. Naturality of λ B and λ A implies thatthe diagram K G ( B ) ι ∗ / / λ B (cid:15) (cid:15) K G ( A ) λ A (cid:15) (cid:15) K ( C ∗ ( G, B, α )) e ι ∗ / / K ( C ∗ ( G, A, α ))is commutative. As in the proof of [20, Proposition 2.9.1], C ∗ ( G, B, α )is a hereditary subalgebra of C ∗ ( G, A, α ) such that the closed two-sidedideal of C ∗ ( G, A, α ) generated by C ∗ ( G, B, α ) is C ∗ ( G, A, α ). This e ι ∗ is an order isomorphism, and so ι ∗ is also an order isomorphism. (cid:3) The corollary below implies that the equivariant K -group is a stableinvariant. Corollary 3.3.
Let G be a compact group, let α be an action of G ona separable C ∗ -algebra A , and let β be an action of G on K ( ℓ ) . If both MPLIFIED GRAPH C*-ALGEBRAS II: RECONSTRUCTION 11 A and K ( ℓ ) admit increasing approximate identities consisting of G -invariant projections, then there is a R ( G ) -module order isomorphismfrom K G ( A, α ) to K G ( A ⊗ K ( ℓ ) , α ⊗ β ) .In particular, if u : G → U ( ℓ ) is a continuous (in the strong operatortopology) unitary representation of G and β g = Ad( u g ) , then there is a R ( G ) -module order isomorphism from K G ( A, α ) and K G ( A ⊗K ( ℓ ) , α ⊗ β ) Proof.
Let { p n } n ∈ N be an increasing approximate identity consistingof G -invariant projections in K ( ℓ ). We may assume p = 0. Then A ⊗ p is a G -invariant hereditary subalgebra of A ⊗ K ( ℓ ) such that( A ⊗ K ( ℓ ))( A ⊗ p )( A ⊗ K ( ℓ )) = A ⊗K ( ℓ ). From the assumption on A and K ( ℓ ), both A ⊗ p and A ⊗ K ( ℓ ) have increasing approximateidentities consisting of G -invariant projections. Lemma 3.2 implies thatthere is an R ( G )-module order isomorphism from K G ( A ⊗ p , α ⊗ β ) to K G ( A ⊗ K ( ℓ ) , α ⊗ β ). The result now follows since the map a a ⊗ p is a G -equivariant ∗ -isomorphism from A to A ⊗ p .For the last part of the theorem, since G is compact, u is a direct sumof finite dimensional representations. Thus, K ( ℓ ) has an increasingapproximate identity consisting of G -invariant projections. (cid:3) To finish this subsection, we describe the consequences of Theorem Afor equivariant stable isomorphism of amplified graph C ∗ -algebras. Forthe following theorem, given a strong-operator continuous unitary rep-resentation u : T → U ( ℓ ) of T on a Hilbert space H , we will write β u for the action of T on B ( ℓ ) given by β uz = Ad( u z ). Theorem 3.4.
Let E and F be countable amplified graphs. Then thefollowing are equivalent: (1) E ∼ = F ; (2) ( C ∗ ( E ) , γ E ) ∼ = ( C ∗ ( F ) , γ F ) ; (3) ( C ∗ ( E ) ⊗K , γ E ⊗ β u ) ∼ = ( C ∗ ( F ) ⊗K , γ F ⊗ β u ) , for every stronglycontinuous representation u : T → U ( ℓ ) ; (4) there exists a strongly continuous unitary representation u : T → U ( ℓ ) such that ( C ∗ ( E ) ⊗ K , γ E ⊗ β u ) ∼ = ( C ∗ ( F ) ⊗ K , γ F ⊗ β u ) ; and (5) there exist strongly continuous unitary representations u, v : T → U ( ℓ ) such that ( C ∗ ( E ) ⊗ K , γ E ⊗ β u ) ∼ = ( C ∗ ( F ) ⊗ K , γ F ⊗ β v ) .Proof. If φ : E → F is an isomorphism, it induces an isomorphism C ∗ ( E ) ∼ = C ∗ ( F ), which is gauge invariant because it carries generatorsto generators. This gives (1) = ⇒ (2).If (2) holds, say φ : C ∗ ( E ) → C ∗ ( F ) is a gauge-equivariant iso-morphism, then for any u the map φ ⊗ id K is an equivariant isomor-phism from ( C ∗ ( E ) ⊗ K , γ E ⊗ β u ) to ( C ∗ ( F ) ⊗ K , γ F ⊗ β u ), giving (3).Clearly (3) implies (4). And if (4) holds for a given u : T → B ( ℓ ), then (5) holds with u = v . Finally, if (5) holds, then two applicationsof Corollary 3.3 show that K T ( C ∗ ( E ) , γ E ) ∼ = K T ( C ∗ ( E ) ⊗ K ( ℓ ) , γ E ⊗ β u ) ∼ = K T ( C ∗ ( F ) ⊗ K ( ℓ ) , γ F ⊗ β v ) ∼ = K T ( C ∗ ( F ) , γ F )as ordered Z [ x, x − ]-modules, and so Theorem A gives (1). (cid:3) Remark . In this remark, we use the notation, moves, and drawingconventions of [11]; we refer the reader there for details.Combined with the results of others, Theorem 3.4 confirms, for theclass of amplified graphs, [11, Conjecture 5.1]. The conjecture statesthat for all xyz other than and , the equivalence relation xyz isgenerated by the moves from { (O) , (I-) , (I+) , (R+) , (S) , (C+) , (P+) } that preserve it.Theorem 3.4 shows that for amplified graphs,( E, F ) ∈ = ⇒ E ∼ = F = ⇒ ( E, F ) ∈ Since we trivially have ⊆ x1z ⊆ for all x , z , we deduce thatthe four equivalence relations x1z are identical and coincide with graphisomorphism. In particular, for amplified graphs, each x1z is triviallycontained in the relation generated by the moves that preserve it. Forthe reverse containment, note that the only moves in the list abovethat preserve any x1z -equivalences are (O), (I+) and (I-) . Of these,neither (I+) nor (I-) can be applied to an amplified graph, and [11,Theorem 3.2] shows that h (O) i ⊆ x1z for all x, z . So we confirm [11,Conjecture 5.1] for amplified graphs for the relations x1z .We now show that a similar result holds for the relations x0z . Recallfrom [12] that if E is an amplified graph then its amplified transitiveclosure tE is the amplified graph with tE = E and v ( tE ) w = ∅ if and only if vE ∗ w \ { v } 6 = ∅ . Theorem 1.1 of [12] shows that foramplified graphs, if ( E, F ) ∈ , then tE ∼ = tF . We claim that thisforces ( E, F ) ∈ . To see this, first note that by [11, Theorems3.2 and 3.10], moves (0) and (R+) preserve . So it suffices to showthat the graph move t that, given vertices u, v, w such that uE v and vE w are infinite, adds infinitely many new edges to uE w , can beobtained using (0) and (R+) . This is achieved as follows:(3.1) (O) (R+) (R+) (O) So as above, for amplified graphs, we see that the four equivalence re-lations x0z are identical, coincide with isomorphism of amplified tran-sitive closures of the underlying graphs, and are generated by (O) and (R+) , and in particular by the moves from [11] that are x0z -invariant.
MPLIFIED GRAPH C*-ALGEBRAS II: RECONSTRUCTION 13
The results of [11] give the reverse containment, so we have confirmed[11, Conjecture 5.1] for amplified graphs for the relations x0z .3.2.
Stability of graded algebraic K . Next we establish the stableinvariance of graded K -theory. Let Γ be an additive abelian group andlet A be a Γ-graded ring. For δ ∈ Γ n , we write M n ( A )( δ ) for the Γ-graded ring M n ( A ) with grading given by ( a i,j ) ∈ M n ( A ) λ if and onlyif a i,j ∈ A λ + δ j − δ i . Similarly, for δ ∈ Q n Γ, we write M ∞ ( A )( δ ) for theΓ-graded ring M ∞ ( A ) with grading given by ( a i,j ) ∈ M ∞ ( A )( δ ) λ if andonly if a i,j ∈ A λ + δ j − δ i .Since the tensor product of two graded modules will be key in theproof, we recall the construct given in [15, Section 1.2.6]. Let Γ be anadditive abelian group, let A be a Γ-graded ring, let M be a gradedright A -module, and let N be a graded left A -module. Then M ⊗ A N is defined to be M ⊗ A N modulo the subgroup generated by { ma ⊗ n − m ⊗ an : m ∈ M, n ∈ N, and a ∈ A are homogeneous } with grading induced by the grading on M ⊗ A N given by( M ⊗ A N ) λ = (X i m i ⊗ n i : m i ∈ M α i , n i ∈ N β i with α i + β i = λ ) . Theorem 3.6.
Let Γ be an additive abelian group, let A be a unital Γ -graded ring, and let δ = ( δ , δ , . . . , δ n ) ∈ Γ n . Then the inclusion ι : A → M n ( A )( δ ) into the e , corner induces a Z [Γ] -module orderisomorphism K gr0 ( ι ) : K gr0 ( A ) → K gr0 ( M n ( A )( δ )) given by K gr0 ( ι )([ P ]) =[ P ⊗ A M n ( A )( δ )] (the left A -module structure on M n ( A )( δ ) is given bythe inclusion ι ).Proof. Let α = (0 , δ − δ , . . . , δ n − δ ). By [15, Corollary 2.1.2], thereis an equivalence of categories φ : Pgr- A → Pgr- M n ( A )( α ) given by φ ( P ) = P ⊗ A A n ( α ). Moreover, φ commutes with the suspension map.Since M n ( A )( α ) λ = A λ A λ + α − α · · · A λ + α n − α A λ + α − α A λ · · · A λ + α n − α ... ... . . . ... A λ + α − α n A λ + α − α n · · · A λ = A λ A λ + δ − δ · · · A λ + δ n − δ A λ + δ − δ A λ · · · A λ + δ n − δ ... ... . . . ... A λ + δ − δ n A λ + δ − δ n · · · A λ = M n ( A )( δ ) λ , we have M n ( A )( α ) = M n ( A )( δ ). Therefore, φ ( P ) = P ⊗ A A n ( α ) isan equivalence of categories from Pgr- A to Pgr- M n ( A )( δ ) and φ com-mutes with the suspension map. Hence, φ induces a Z [Γ]-module orderisomorphism from K gr0 ( A ) to K gr0 ( M n ( A )( δ )). We claim that φ = K gr0 ( ι ). Let M be a graded right A -module. Wewill show that M ⊗ A A n ( α ) and M ⊗ A M n ( A )( δ ) are isomorphic asgraded modules. Since 1 A ∈ A and M A = M , M ⊗ A M n ( A )( δ ) ∼ = gr M ⊗ A ι (1 A ) M n ( A )( δ ) = M ⊗ A e , M n ( A )( δ ) . By the definitions of the gradings on e , M n ( A )( δ ) and A n ( α ), the right M n ( A )-module isomorphism e , X ( x , , x , , . . . , x ,n )is a graded isomorphism. Hence, M ⊗ A M n ( A )( δ ) ∼ = gr M ⊗ A e , M n ( A )( δ ) ∼ = gr M ⊗ A A n ( α ) . Thus, φ = K gr0 ( ι ). Consequently, K gr0 ( ι ) is a Z [Γ]-module order iso-morphism. (cid:3) Corollary 3.7.
Let Γ be an additive abelian group and let A be a Γ -graded ring with a sequence of idempotents { e n } ∞ n =1 ⊆ A such that e n e n +1 = e n for all n , and S n e n Ae n = A . For δ ∈ Q i Γ , the embedding ι : A → M ∞ ( A )( δ ) into the e , corner of M ∞ ( A )( δ ) induces a Z [Γ] -module order isomorphism K gr0 ( ι ) : K gr ( A ) → K gr0 ( M ∞ ( A )( δ )) .In particular, if E is a countable directed graph and δ ∈ Q i Z , thenthe inclusion of ι : L K ( E ) → M ∞ ( L K ( E ))( δ ) of L K ( E ) into the e , cor-ner of M ∞ ( L K ( E ))( δ ) induces a Z [ x, x − ] -module order isomorphismfrom K gr0 ( L K ( E )) to K gr0 ( M ∞ ( L K ( E ))( δ )) for any field K .Proof. Let ι n : e n Ae n → M ∞ ( e n Ae n )( δ ) be the inclusion of e n Ae n intothe e , corner of M ∞ ( e n Ae n )( δ ). Observe that A = lim −→ e n Ae n , that M ∞ ( A ) = lim −→ M ∞ ( e n Ae n ), and that the diagram e n Ae n ⊆ / / ι n (cid:15) (cid:15) A ι (cid:15) (cid:15) M ∞ ( e n Ae n )( δ ) ⊆ / / M ∞ ( A )( δ )commutes. Therefore, if each K gr0 ( ι n ) is a Z [Γ]-module order isomor-phism, then K gr0 ( ι ) is a Z [Γ]-module order isomorphism since the graded K -group respects direct limits ([15, Theorem 3.2.4]). Hence, withoutloss of generality, we may assume that A is a unital Γ-graded ring.Let δ n = ( δ , δ , . . . , δ n ). Let j n : A → M n ( A )( δ n ) be the inclusionof A into the e , corner of M n ( A )( δ n ), and let ι n : M n ( A )( δ n ) → M ∞ ( A )( δ ) be the inclusion map. Then lim −→ M n ( A )( δ n ) = M ∞ ( A )( δ )and the diagram A j n (cid:15) (cid:15) ι ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ M n ( A )( δ n ) ι n / / M ∞ ( A )( δ ) MPLIFIED GRAPH C*-ALGEBRAS II: RECONSTRUCTION 15 commutes. By Theorem 3.6, K gr0 ( j n ) is a Z [Γ]-module order isomor-phism. Since the graded- K functor respects direct limits, K gr0 ( ι ) is Z [Γ]-module order isomorphism.For the last part of the corollary, let { X n } be a sequence of finitesubsets of E such that X n ⊆ X n +1 and S n X n = E . Then e n := P v ∈ X n v defines idempotents of degree zero such that S n e n L K ( E ) e n = L K ( E ). (cid:3) As in the preceding subsection, we finish by describing the con-sequences of Theorem A for graded stable isomorphism of amplifiedLeavitt path algebras.
Theorem 3.8.
Let E and F be countable amplified graphs and let K be a field. Then the following are equivalent: (1) E ∼ = F (2) L K ( E ) ∼ = gr L K ( F ) ; (3) L K ( E ) ⊗ M ∞ ( K )( δ ) ∼ = gr L K ( F ) ⊗ M ∞ ( K )( δ ) for every δ ∈ Q i Z ; (4) L K ( E ) ⊗ M ∞ ( K )( δ ) ∼ = gr L K ( F ) ⊗ M ∞ ( K )( δ ) for some δ ∈ Q i Z ;and (5) L K ( E ) ⊗ M ∞ ( K )( δ ) ∼ = gr L K ( F ) ⊗ M ∞ ( K )( ε ) for some δ, ε ∈ Q i Z .Proof. The argument is very similar to that of Theorem 3.4, so wesummarise. Any isomorphism of graphs induces a graded isomor-phism of their Leavitt path algebras, and any graded isomorphism φ : L K ( E ) ∼ = L K ( F ) amplifies to a graded isomorphism φ ⊗ id : L K ( E ) ⊗ M ∞ ( K )( δ ) ∼ = L K ( F ) ⊗ M ∞ ( K )( δ ), giving (1) = ⇒ (2) = ⇒ (3). Theimplications (3) = ⇒ (4) = ⇒ (5) are trivial. The second statement ofCorollary 3.7 shows that if (5) holds then K gr ( L K ( E )) ∼ = K gr ( L K ( F ))as ordered Z [ x, x − ]-modules, and then Theorem A gives (1). (cid:3) Remark . Since statement (1) of Theorem 3.8 does not depend onthe field K , we deduce that each of the other four statements holdsfor some field K if and only if holds for every field K . In particularthe graded-isomorphism problem for amplified Leavitt path algebras isfield independent, so it suffices, for example, to consider the field F . Remark . Let E and F be amplified graphs. Theorem 3.4 showsthat the existence of an isomorphism ( C ∗ ( E ) ⊗ K , γ E ⊗ β u ) ∼ = ( C ∗ ( F ) ⊗K , γ F ⊗ β u ) for every u is equivalent to the existence of such an iso-morphism for some u , and indeed to the existence of an isomorphism( C ∗ ( E ) ⊗ K , γ E ⊗ β u ) ∼ = ( C ∗ ( F ) ⊗ K , γ F ⊗ β v ) for some u, v . All ofthese conditions are formally weaker than the existence of isomoprhisms( C ∗ ( E ) ⊗ K , γ E ⊗ β u ) ∼ = ( C ∗ ( F ) ⊗ K , γ F ⊗ β v ) for every pair of stronglycontinuous representations u, v : T → U ( ℓ ), and this in turn is clearlyequivalent to the existence of an isomorphism ( C ∗ ( E ) ⊗ K , γ E ⊗ β u ) ∼ =( C ∗ ( F ) ⊗ K , γ F ⊗ id) for every u . So it is natural to ask for which amplified graphs E, F and which strongly continuous representations u : T → U ( ℓ ) we have ( C ∗ ( E ) ⊗ K , γ E ⊗ β u ) ∼ = ( C ∗ ( F ) ⊗ K , γ F ⊗ id).This is an intriguing question to which we do not know a completeanswer, but we can certainly show that the condition that ( C ∗ ( E ) ⊗K , γ E ⊗ β u ) ∼ = ( C ∗ ( F ) ⊗ K , γ F ⊗ id) for every u is in general strictlystronger than the equivalent conditions of Theorem 3.4. Specifically,let E = F be the directed graph with E = { v, w } and E = { e n : n ∈ N } with s ( e n ) = v and r ( e n ) = w for all N . Then the onlynonzero spectral subspaces for the gauge action on C ∗ ( E ) are thosecorresponding to − , , −
1, and so the same is true for the spectralsubspaces of C ∗ ( E ) ⊗ K with respect to γ E ⊗ id. On the other hand, if u : T → B ( ℓ ( Z )) is given by u z e n = z n e n , then each spectral subspaceof C ∗ ( E ) ⊗ K for γ E ⊗ β u is nonempty, so ( C ∗ ( E ) ⊗ K , γ E ⊗ β u ) =( C ∗ ( E ) ⊗ K , γ E ⊗ id). We do not, however, know of an example inwhich C ∗ ( E ) is simple.A similar question can be posed for amplified Leavitt path algebras:for which amplified graphs E, F and elements δ ∈ Q i Z do we have L K ( E ) ⊗ M ∞ ( K )( δ ) ∼ = gr L K ( F ) ⊗ M ∞ ( K )(0)? The same example showsthat the existence of such an isomorphism for every δ is in generalstrictly stronger than the equivalent conditions of Theorem 3.8. References [1] Gene Abrams and Gonzalo Aranda Pino,
The Leavitt path algebras of arbitrarygraphs , Houston J. Math., (2008), 423–442.[2] Pere Ara and Kenneth R. Goodearl, Leavitt path algebras of separated graphs ,J. reine angew. Math. (2012), 165–224.[3] Pere Ara, Roozbeh Hazrat, Huanhuan Li, and Aidan Sims,
Graded Steinbergalgebras and their representations , Algebra Number Th. (2018), 131–172.[4] Sara E. Arklint, Søren Eilers, and Efren Ruiz, Geometric classification of iso-morphism of unital graph C ∗ -algebras , arXiv:1910.11514.[5] Kevin Aguyar Brix, Balanced strong shift equivalence, balanced in-splits andeventual conjugacy , preprint 2019, arXiv:1912.05212.[6] Nathan Brownlowe, Toke Carlsen, and Michael F. Whittaker,
Graph algebrasand orbit equivalence , Ergodic Theory Dynam. Systems (2017), 389–417.[7] Nathan Brownlowe, Marcelo Laca, Dave Robertson, and Aidan Sims, Recon-structing directed graphs from generalised gauge actions on their Toeplitz alge-bras , Proc. Roy. Soc. Edinburgh Sect. A, to appear, arXiv:1812.08903.[8] Toke Meier Carlsen, Søren Eilers, Eduard Ortega, and Gunnar Restorff,
Flowequivalence and orbit equivalence for shifts of finite type and isomorphism oftheir groupoids , J. Math. Anal. Appl. (2019) 1088–1110.[9] Toke Meier Carlsen, Efren Ruiz, and Aidan Sims,
Equivalence and stable iso-morphism of groupoids, and diagonal-preserving stable isomorphisms of graph C ∗ -algebras and Leavitt path algebras , Proc. Amer. Math. Soc., (2017),1581–1592.[10] Adam Dor-On, Søren Eilers, and Shirly Geffen, Classification of irreversibleand reversible Pimsner operator algebras , Compositio Math., to appear,arXiv:1907.01366.
MPLIFIED GRAPH C*-ALGEBRAS II: RECONSTRUCTION 17 [11] Søren Eilers and Efren Ruiz,
Refined moves for structure-preserving isomor-phism of graph C ∗ -algebras , arXiv:1908.03714.[12] Søren Eilers, Efren Ruiz, and Adam P. W. Sørensen, Amplified graph C ∗ -algebras , M¨unster J. Math. (2012), 121–150.[13] Søren Eilers, Gunnar Restorff, Efren Ruiz, and Adam P. W. Sørensen, Thecomplete classification of unital graph C ∗ -algebras: Geometric and strong ,arXiv:1611.07120.[14] Neal J. Fowler, Marcelo Laca, and Iain Raeburn, The C ∗ -algebras of infinitegraphs , Proc. Amer. Math. Soc. (2000), 2319–2327.[15] Roozbeh Hazrat, Graded rings and graded Grothendieck groups , London Math.Soc. Lect. Note Series , Cambridge University Press 2016.[16] Damon Hay, Marissa Loving, Martin Montgomery, Efren Ruiz, and KatherineTodd,
Non-stable K -theory for Leavitt path algebras , Rocky Mountain J. Math. (2014), 1817–1850.[17] Pierre Julg, K-th´eorie ´equivariante et produits crois´es (French. English sum-mary) [Equivariant K-theory and crossed products] C. R. Acad. Sci. Paris Sr.I Math. (1981), 629–632.[18] Alex Kumjian and David Pask, C ∗ -algebras of directed graphs and group ac-tions , Ergod. Th. Dynam. Sys. (1999), 1503–1519.[19] Kengo Matsumoto and Hiroki Matui, Continuous orbit equivalence of topo-logical Markov shifts and Cuntz–Krieger algebras , Kyoto J. Math. 54 (2014),863–877.[20] N. Christopher Phillips,
Equivariant K -theory and Freeness of group actionson C ∗ -algebras . Lect. Notes in Math. Springer-Verlag Berlin Heidelberg1987.[21] Iain Raeburn,
Graph algebras , CBMS Regional Conference Series in Mathe-matics, vol. 103, Published for the Conference Board of the Mathematical Sci-ences, Washington, DC; by the American Mathematical Society, Providence,RI, 2005.[22] Mark Tomforde,
Uniqueness theorems and ideal structure for Leavitt path al-gebras , J. Algebra (2007), 270–299.[23] Mark Tomforde,
Stability of C ∗ -algebras associated to graphs , Proc. Amer.Math. Soc. (2004), 1787–1795. E-mail address , S. Eilers: [email protected] (S. Eilers)
Department of Mathematical Sciences, University of Co-penhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark
E-mail address , E. Ruiz: [email protected] (E. Ruiz)
Department of Mathematics, University of Hawaii, Hilo,200W. Kawili St., Hilo, Hawaii, 96720-4091 USA
E-mail address , A. Sims: [email protected] (A. Sims)(A. Sims)