Classifying Leavitt path algebras up to involution preserving homotopy
aa r X i v : . [ m a t h . R A ] J a n CLASSIFYING LEAVITT PATH ALGEBRAS UP TOINVOLUTION PRESERVING HOMOTOPY
GUILLERMO CORTI ˜NAS
Abstract.
We prove that the Bowen-Franks group classifies the Leavitt pathalgebras of purely infinite simple finite graphs over a regular supercoherentcommutative ring with involution where 2 is invertible, equipped with theirstandard involutions, up to matricial stabilization and involution preservinghomotopy equivalence. We also consider a twisting of the standard involutionon Leavitt path algebras and obtain partial results in the same direction forpurely infinite simple graphs. Our tools are K -theoretic, and we prove severalresults about (Hermitian, bivariant) K -theory of Leavitt path algebras. Introduction
A directed graph E consists of a set E of vertices and a set E of edges togetherwith source and range functions r, s : E → E . This article is concerned with theLeavitt path algebra L ( E ) of a directed graph E over a commutative ring ℓ withinvolution ([2]). When ℓ = C , with complex conjugation as involution, L ( E ) is anormed ∗ -algebra; its completion is the graph C ∗ -algebra C ∗ ( E ). A graph E iscalled finite or countable if both E and E are finite or countable; a finite graph E is purely infinite simple if and only if C ∗ ( E ) is. A result of Cuntz and Rørdam([17, Theorem 6.5]) says that C ∗ -algebras of finite purely infinite simple graphs,i.e. the purely infinite simple Cuntz-Krieger algebras, are classified up to (stable)isomorphism by the Bowen-Franks group BF of the corresponding graph, definedas BF ( E ) = Coker( I − A tE ) . Here A E is the incidence matrix of E . It is an open question whether a similarresult holds for Leavitt path algebras [4].In the current article we prove that if E and F are finite, purely infinite graphssuch that BF ( E ) ∼ = BF ( F ), and ℓ is regular supercoherent where 2 is invertible,then their Leavitt path ℓ -algebras L ( E ) and L ( F ) are homotopy equivalent up tostabilization, by means of involution preserving homomorphisms and homotopies;see Theorems 1.4 and 1.8 for precise statements. For ℓ regular supercoherent (where2 may not be invertible) we obtain a not necessarily involution preserving homotopyequivalence (Theorem 1.9).In Theorems 1.4, 1.8, 1.9 and elsewhere, we use the following notations andvocabulary. We fix a commutative ring ℓ with involution; an algebra is an algebraover ℓ . A ∗ -algebra is an algebra together with an ℓ -semilinear involution ∗ : R → R op ; a ∗ -homomorphism is an involution preserving algebra homomorphismbetween ∗ -algebras. If R is a ∗ -algebra, we write M R for the matrix algebraequipped with conjugate transpose involution and ι : R → M R for upper lefthand corner inclusion. Two ∗ -homomorphisms φ, ψ : A → B are ∗ -homotopic ,and we write φ ∼ ∗ ψ , if they are connected by a finite sequence of involutionpreserving, polynomial homotopies A → B [ t ]; we write φ ∼ ∗ M ψ to mean that ι φ ∼ ∗ ι ψ . A projection in a ∗ -algebra R is a self adjoint idempotent. If R isunital, we say that a projection p is very full if there is an element x such that x ∗ x = 1 and pxx ∗ = xx ∗ p = p . If E is a graph and R a unital ∗ -algebra, then analgebra homomorphism φ : L ( E ) → R has property (P) if φ ( v ) and φ ( ee ∗ ) are veryfull projections for every v ∈ E and e ∈ E . If, for example, E is finite purelyinfinite simple, then every vertex of E is a very full projection (Lemma 8.7) and soa ∗ -homomorphism φ : L ( E ) → R has property (P) if and only if φ (1) is very full(Example 9.2).If R is a unital ∗ -algebra, we write K ( R ) ∗ for the group completion of the monoidof Murray-von Neumann equivalence classes of projections in M ∞ R . There is acanonical forgetful map K ( R ) ∗ → K ( R ) which is an isomorphism, for example,when R is a C ∗ -algebra ([8, Section 5.1]). If L ( E ) is unital there is also a canonicalmap(1.1) can ′ : BF ( E ) → K ( L ( E )) ∗ , induced by mapping each vertex to its Murray-von Neumann class. Observe thatcan ′ sends [1] E := [ X v ∈ E v ] [1 L ( E ) ] . The map (1.1) need not be an isomorphism for general E or ℓ , and for an arbitrary ∗ -homomorphism φ : L ( E ) → L ( F ), K ( φ ) ∗ ◦ can ′ need not factor through can : BF ( F ) → K ( L ( F )) ∗ . We say that a ∗ -homomorphism φ : L ( E ) → L ( F ) and agroup homomorphism(1.2) ξ : BF ( E ) → BF ( F )are compatible if the following diagram commutes(1.3) K ( L ( E )) ∗ K ( φ ) ∗ / / K ( L ( F )) ∗ BF ( L ( E )) can ′ O O ξ / / BF ( F ) . can ′ O O The following classification theorem follows from Theorem 14.2 (see Example14.3). A nonzero element a in a ∗ -algebra R is positive if for some n ≥ x , . . . , x n ∈ R such that a = P ni =1 x i x ∗ i . A unital ring A is regular if every A -module has a finite projective resolution and supercoherent if for every n ≥
0, the category of finitely presented A [ x , . . . , x n ]-modules is abelian. Theorem 1.4.
Let E and F be finite, purely infinite simple graphs and let ℓ beregular supercoherent such that is invertible and − is positive in ℓ . Let ξ : BF ( E ) ∼ −→ BF ( F ) be a group isomorphism. Then there exist ∗ -homomorphisms φ : LE → LF and ψ : LF → LE , compatible with ξ and ξ − , both with property(P), and such that ψ ◦ φ ∼ ∗ M id LE and φ ◦ ψ ∼ ∗ M id LF . If furthermore ξ ([1] E ) =[1] F , then φ and ψ can be chosen to be unital. The hypothesis that − ℓ = C withtrivial involution, but fails for C with complex conjugation as involution. Ourclassification result without assuming that − ℓ is Theorem 1.8; weexplain the notation and vocabulary involved in its formulation.If R is a ∗ -algebra, we write M ± R for the ∗ -algebra whose underlying algebra is M R , and whose involution is (cid:20) a bc d (cid:21) ∗ = (cid:20) a ∗ − c ∗ − b ∗ d ∗ (cid:21) . If R is unital, then − M ± R (Example 2.3.2). Theorem 1.8 guaranteesthat if (1.2) is an isomorphism then it can be lifted to a ∗ -homomorphism φ : LASSIFYING LEAVITT PATH ALGEBRAS 3 L ( E ) → M ± L ( F ) that is compatible with ξ in a sense that we shall presentlyexplain. The Hermitian K or Witt-Grothendieck group of R is K h ( R ) = K ( M ± R ) ∗ . The upper left hand corner inclusion ι + : R → M ± R induces a natural homomor-phism(1.5) K ( R ) ∗ → K h ( R ) . The map (1.5) is an isomorphism whenever − R (see Section 7).In particular, K h ( ι + ) is an isomorphism. Composing (1.5) with can ′ we obtainanother canonical map(1.6) can : BF ( E ) → K h ( L ( E )) . A group homomorphism (1.2) and a ∗ -homomorphism φ : L ( E ) → M ± L ( F ) are compatible if the following diagram commutes K h ( L ( E )) ∗ K h ( φ ) / / K h ( M ± L ( F )) BF ( L ( E )) can O O ξ / / BF ( F ) . K h ( ι + ) ◦ can O O A unital ∗ -algebra R is strictly properly infinite if there are elements s , s ∈ R such that s ∗ i s j = δ i,j . For example, L ( F ) is strictly properly infinite whenever F is finite and purely infinite simple (Corollary 8.5). If R is strictly properly infinite,then ⊞ : R × R → R , x ⊞ y = s xs ∗ + s ys ∗ is a ∗ -homomorphism. For any ∗ -algebra A , ⊞ makes the set [ A, R ] ∗ M of M - ∗ -homotopy classes of ∗ -homomorphisms intoan abelian monoid (Lemma 8.12). Two ∗ -homomorphisms φ, ψ : A → R are stably M -homotopic , and we write φ ∼ sM ψ , if their M - ∗ -homotopy classes go to thesame element under the group completion map(1.7) [ A, R ] ∗ M → ([ A, R ] ∗ M ) + . Theorem 14.1 implies the following (see Example 14.3).
Theorem 1.8.
Let E and F be purely infinite simple, finite graphs. Assume that is invertible in ℓ . Let ξ : BF ( E ) ∼ −→ BF ( F ) be an isomorphism. Then thereare ∗ -homomorphisms φ : L ( E ) → M ± L ( F ) and ψ : L ( F ) → M ± L ( E ) compatiblewith ξ and ξ − , both with property (P), and such that M ± ( ψ ) ◦ φ ∼ sM ι : LE → M ± M ± L ( E ) and M ± ( φ ) ◦ ψ ∼ sM ι : L ( F ) → M ± M ± L ( F ) . We remark that the maps M ± ( ψ ) ◦ φ and M ± ( φ ) ◦ ψ in Theorem 1.8 have property(P), whereas, in general, those labelled ι do not, unless of course − ℓ . Corollary 13.9 implies that if ℓ is regular supercoherent, E is finite andpurely infinite simple and R is K -regular and strictly properly infinite, and − R , then any ∗ -homomorphism φ : L ( E ) → R is stably M - ∗ -homotopicto a morphism φ P with property (P), and that moreover φ ∼ sM ψ if and only if φ P ∼ ∗ M ψ P . When F is finite and purely infinite simple, R = M ± L ( F ) satisfiesthose hypothesis; Theorem 1.8 identifies the M - ∗ -homotopy class of M ± ( ψ ) ◦ φ and of M ± ( φ ) ◦ ψ with that of the corresponding map ( ι ) P .Recall (e.g. from [5, Theorem 7.6]) that for regular supercoherent ℓ , K ( L ( E )) = BF ( E ) ⊗ K ( ℓ ) . Hence we have another canonical mapcan ” : BF ( E ) → K ( L ( E )) , x x ⊗ [1] . GUILLERMO CORTI˜NAS
Compatibility between a group homomorphism (1.2) and an algebra homomorphism φ : L ( E ) → L ( F ) is defined as in (1.3), using can ” instead of can ′ .The following theorem is a particular case of Theorem 14.4 and classifies Leavittpath algebras up to not necessarily involution preserving, M -homotopy equiv-alences. An idempotent p in a unital algebra R is very full if there are elements x, y ∈ R such that yx = 1 and xyp = pxy = xy . We write ∼ M for the M -homotopyrelation between not necessarily involution preserving algebra homomorphisms. Theorem 1.9.
Let E and F be finite, purely infinite simple graphs, let ℓ be regularsupercoherent and let ξ : BF ( E ) ∼ −→ BF ( F ) be an isomorphism. Then there exist ℓ -algebra homomorphisms φ : LE → LF and ψ : LF → LE compatible with ξ and ξ − , such that φ (1) and ψ (1) are very full idempotents, and such that ψ ◦ φ ∼ M id LE and φ ◦ ψ ∼ M id LF . If furthermore ξ ([1] E ) = [1] F , then φ and ψ can be chosento be unital. Note that if ℓ is a field, then L ( E ) and L ( F ) are purely infinite simple rings[2, Theorem 3.1.10 and Section 5.6]. Since by [7, Proposition 1.5] any nonzeroidempotent of a purely infinite simple unital ring is very full, when ℓ is a fieldthe condition that φ (1) and ψ (1) be very full in Theorem 1.9 is equivalent to thecondition that φ and ψ be nonzero. Hence in this case Theorem 1.9 recovers themain theorem of [11].We also show, under mild assumptions on ℓ , that general, not necessarily bijectivehomomorphisms between Bowen-Franks groups can be lifted to ∗ -homomorphismsof the corresponding Leavitt path algebras. The following is a particular case ofTheorem 9.8. Theorem 1.10.
Assume that ℓ is regular supercoherent and that the canonical map Z → K ( ℓ ) is an isomorphism. Let E and F be graphs with finitely many verticesand let ξ : BF ( E ) → BF ( F ) be a group homomorphism. Assume that F is purelyinfinite simple. Then there is a ∗ -homomorphism φ : L ( E ) → L ( F ) with property ( P ) such that K ( φ ) = ξ ; if ξ ([1] E ) = [1] F then φ can be chosen to be unital. One of the main tools we use to prove the classification theorems above isthe bivariant Hermitian K -theory introduced in [14]. It consists of a functor j h : Alg ∗ ℓ → kk h from the category of ∗ -algebras to a triangulated category kk h which is homotopy invariant, maps ∗ -algebra extensions to distinguished triangles,is matricially stable and sends the natural transformation ι + to an isomorphism,and is universal initial among functors Alg ∗ ℓ → T to triangulated categories thathave those properties. Remark that j h is the Hermitian analogue of the bivariantalgebraic K -theory j : Alg ℓ → kk introduced in [13]. Bivariant algebraic Hermitian K -theory is defined under the following assumption. λ -assumption . The ground ring ℓ contains an element λ such that λ + λ ∗ = 1.For A, B ∈ Alg ∗ ℓ and n ∈ Z , we write kk hn ( A, B ) = hom kk h ( j h ( A ) , j h ( B )[ n ]) , kk h ( A, B ) = kk h ( A, B ) . Here [ n ] is the n -fold inverse suspension. It was shown in [14, Proposition 8.1] that kk h recovers Weibel-style homotopy algebraic Hermitian K -theory ([20]); for all n ∈ Z , we have(1.12) kk hn ( ℓ, B ) = KH hn ( B ) . There is a natural comparison map from Karoubi’s Hermitian K -theory(1.13) K hn ( B ) → KH hn ( B )which is an isomorphism whenever B is K h -regular, as is the case, for example,of L ( E ) when ℓ is regular supercoherent and 2 is invertible in ℓ [14, Lemma 3.8]. LASSIFYING LEAVITT PATH ALGEBRAS 5
Recall that a vertex v of a graph E is regular if it emits a nonzero, finite number ofedges, and is singular otherwise. We write sing( E ) ⊂ E for the subset of singularvertices. In Theorem 6.11 we obtain the following classification result, where weabuse notation and write can for the composite of the maps (1.6) and (1.13). Theorem 1.14.
Let E and F be graphs with finitely many vertices and such that | sing( E ) | = | sing( F ) | . Let ξ : BF ( E ) ∼ −→ BF ( F ) be an isomorphism. As-sume that ℓ satisfies the λ -assumption 1.11. Then there exists an isomorphism ξ : j h ( L ( E )) ∼ −→ j h ( L ( F )) such that KH h ( ξ ) ◦ can = can ◦ ξ . Because purely infinite simple graphs have no singular vertices, Theorem 1.14tells us, in particular, that in the situation of Theorem 1.8, L ( E ) and L ( F ) areisomorphic in kk h . To prove Theorem 1.8, we need to show further that such kk h isomorphism can be lifted to the homotopy class of a ∗ -homomorphism from L ( E )to M ± L ( F ) with property (P). The necessary lifting result is provided by the nexttheorem. For any graph E , the set [ L ( E ) , R ] PM of M - ∗ -homotopy classes of ∗ -homomorphisms L ( E ) → R with property (P) is naturally a sub-semigroup of themonoid [ L ( E ) , R ] ∗ M (see Remark 9.3). The following theorem is proved in Theorem13.7. Theorem 1.15.
Let E be a finite, purely infinite simple graph and R ∈ Alg ∗ ℓ a K h -regular, strictly properly infinite ∗ -algebra over a ∗ -ring ℓ satisying the λ -assumption. Assume that − is positive in R . Then the map (1.16) j h : [ L ( E ) , R ] PM → kk h ( L ( E ) , R ) is a semigroup isomorphism. It follows from Theorem 1.15 (see Corollary 8.5) that [ L ( E ) , R ] PM is an abeliangroup, isomorphic to the group completion (1.7).We show in Lemma 2.3.5 that − L ( E ) (equipped with its standardinvolution) if and only if it is positive in ℓ . Thus unless − ℓ , wecannot apply Theorem 1.15 with a Leavitt path algebra substituted for R ; this iswhy M ± appears in Theorem 1.8. Another way to get rid of the positivity problemis to modify the involution of L ( E ), so that for the new involution, − Z / Z grading, definedby | v | ≡ | e | ≡ | e ∗ | ≡ τ : L ( E ) → L ( E ), τ ( a + a ) = a − a . Composing it with theinvolution ∗ we get a new involution a = τ ( a ∗ ). We write L ( E ) for L ( E ) equippedwith this involution, which we call the signed involution. For example, e = − e ∗ , soif E is a finite regular graph, then − X e ∈ E ee. In particular, − L ( E ). We show moreover that L ( E ) is still strictlyproperly infinite whenever E is purely infinite simple (Corollary 8.5), so in this caseTheorem 1.15 applies to R = L ( E ). Furthermore we show in Theorem 13.10 thatfor E and R as in Theorem 1.15 the canonical semigroup homomorphism(1.17) j h : [ L ( E ) , R ] PM → kk h ( L ( E ) , R )is a monomorphism. Here the superscript P indicates classes of ∗ -homomorphismsthat send each projection of the form ee ∗ ( e ∈ E ) to a very full projection; thisimplies that every vertex goes to a very full projection, but the converse need notbe true, since r ( e ) and ee ∗ need not be Murray-von Neumann equivalent in L ( E ).The main tool to establish injectivity of (1.16) and (1.17) is Poincar´e duality . Weprove in Theorem 11.2 that if E is a finite graph without sinks or sources and E t GUILLERMO CORTI˜NAS is the dual graph, then for every
R, S ∈ Alg ∗ ℓ there are natural isomorphisms of KH h ( ℓ )-modules kk h ( R ⊗ LE, S ) ∼ −→ kk h ( R, S ⊗ L ( E t ))(1.18) kk h ( R ⊗ LE, S ) ∼ −→ kk h ( R, S ⊗ L ( E t )) . Here ⊗ = ⊗ ℓ . Observe that this is the purely algebraic analogue of Poincar´eduality for C ∗ -graph algebras, proved by Jerome Kaminker and Ian Putnam in[15]. The main tool for establishing the surjectivity of the map (1.16) is Theorem9.4, which says that if E is countable and R is strictly properly infinite, then anymap BF ( E ) → K ( R ) ∗ lifts to a ∗ -homomorphism L ( E ) → R with property (P).The proof relies on the fact that the edges of E are partial isometries in L ( E ),which is no longer the case in L ( E ); for e ∈ E we have eee = − e. We also obtain a partial analogue of Theorem 1.14 for the signed involution. Let σ be the generator of Z / Z , written multiplicatively; write Z [ σ ] for the group ring.We consider the cokernel of the matrix I − σA tE ∈ Z [ σ ] E × reg( E ) , which we call the Bowen-Franks Z [ σ ] -module BF ( E ) = Coker( I − σA tE ) . We prove in Theorem 6.14 that if E and F have finitely many vertices and BF ( E )is finite, then any isomorphism BF ( E ) ∼ −→ BF ( F )of Z [ σ ]-modules lifts to an isomorphism j h ( L ( E )) ∼ −→ j h ( L ( F )) . Proposition 6.15 shows that if BF ( E ) is finite and isomorphic to BF ( E ) as a Z [ σ ]-module, and 2 = xx ∗ for some invertible x ∈ ℓ , then(1.19) j h ( L ( E )) ∼ −→ Ker(1 − σ : j h ( LE ) → j h ( LE )) . By its very definition, the involution of L ( E ) is related to the Z / Z -grading of L ( E ). This is reflected by Hermitian K -theory in the following way. Let R be aunital, Z / Z -graded ∗ -algebra. The shift functor makes K gr0 ( R ) = (Gr Z / Z − Mod( R ))into a Z [ σ ]-module. It follows from results in [6] (see Section 5) that if ℓ is regularsupercoherent, then(1.20) K gr0 ( L ( E )) = BF ( E ) ⊗ K ( ℓ ) . Here ⊗ = ⊗ Z . In particular if ℓ is a field, K gr0 ( L ( E )) = BF ( E ). Let ι − : R → M ± R be the lower right corner inclusion; let σ act on K h ( R ) by tensoring with ι − (1).We show in Corollary 4.4 that if ℓ is a field with char( ℓ ) = 2, then(1.21) K h ( L ( E )) = BF ( E ) ⊗ Z [ σ ] K h ( ℓ ) . It follows from Theorem 1.8 that the Leavitt path algebras L of the graph R withone vertex and two loops and L − of its Cuntz’ splice R − are (stably) homotopyequivalent by means of involution preserving homomorphisms and homotopies. Bycontrast, using (1.20) and (1.21), we obtain the following (see Proposition 5.6). Proposition 1.22.
Assume that ℓ is regular supercoherent and such that the canon-ical map Z → K ( ℓ ) is an isomorphism. Then there is no Z / Z -homogeneous unital ℓ -algebra homomorphism from L to L − nor in the opposite direction. If further-more ℓ is a field with char( ℓ ) = 2 such that the canonical map Z [ σ ] → K h ( ℓ ) is LASSIFYING LEAVITT PATH ALGEBRAS 7 an isomorphism, then there is no unital ∗ -homomorphism L → L − nor in theopposite direction. The hypothesis of Proposition 1.22 apply, for example, when ℓ = C equippedwith complex conjugation as involution.The rest of this paper is organized as follows. In the following summary, wefix a commutative unital ring ℓ with involution and work in the category Alg ∗ ℓ of ∗ -algebras over ℓ . Since kk h is defined only under the λ -assumption 1.11, whenever kk h or j h appear, it is implicitly assumed that ℓ satisfies it; when they do not, andno other assumptions are made explicit, ℓ can be arbitrary. Section 2 is devotedto some preliminaries on ∗ -algebras, Z / Z -gradings, positivity, bivariant Hermitian K -theory and Cohn and Leavitt path algebras. For a Z / Z -graded ∗ -algebra A wewrite A for A equipped with the signed involution a + a = a ∗ − a ∗ . There arenatural maps ∆ : A → M ± A and ∆ ′ : A → M (2.2.10) and we show in Corollary2.4.6 that upon inverting 2 in kk h , ∆ and ∆ ′ give rise to splittings j h [1 / A ) = j h [1 / A ) + ⊕ j h [1 / A ) − j h [1 / A ) = j h [1 / A ) + ⊕ j h [1 / A ) − such that j h [1 / A ) + ∼ = j h [1 / A ) + . Concerning positivity, we show that − L ( E ) if and only it is positivein ℓ (Lemma 2.5.6). In Section 3 we prove Theorem 3.2, which says that for anarbitrary graph E , both C ( E ) and C ( E ) are kk h -isomorphic to the ∗ -algebra ℓ ( E ) of finitely supported functions E → ℓ ; j h ( C ( E )) ∼ = j h ( C ( E )) ∼ = j h ( ℓ ( E ) ) . We use Theorem 3.2 together with the algebra extensions (2.5.4) and (2.5.5)0 → K ( E ) → C ( E ) → L ( E ) → → K ( E ) → C ( E ) → L ( E ) → L ( E ). We show in Theorem 3.3 that if T is a triangulatedcategory and X : Alg ∗ ℓ → T is any homotopy invariant, matricially stable, ι + -stableand excisive functor that commutes with direct sums of at most | E | summands,and R ∈ Alg ∗ ℓ , then applying X ( − ⊗ R ) to the algebra extensions above yieldsdistinguished triangles X ( R ) (reg( E )) I − A tE / / X ( R ) ( E ) / / X ( L ( E ) ⊗ R ) X ( R ) (reg( E )) I − σA tE / / X ( R ) ( E ) / / X ( L ( E ) ⊗ R ) . Here A tE ∈ Z ( E × reg( E )) is the transpose of the reduced incidence matrix and I isthe matrix obtained from the identity upon removing the columns correspondingto singular vertices. The triangles above apply in particular to X = j h when E isfinite (Theorem 3.5). It follows from this that the gauge automorphism τ becomesthe identity in kk h [1 /
2] (Corollary 3.6). The Bowen-Franks group and Z [ σ ]-moduleand their relation to the Hermitian K -theory of L ( E ) and L ( E ) are the subject ofSection 4. Theorem 4.2 establishes short exact sequences0 → BF ( E ) ⊗ KH hn ( R ) → KH hn ( L ( E ) ⊗ R ) → Ker(( I − A tE ) ⊗ KH hn − ( R )) → → BF ( E ) ⊗ Z [ σ ] KH hn ( R ) → KH hn ( L ( E ) ⊗ R ) → Ker(( I − σA tE ) ⊗ KH hn − ( R )) → ℓ is a field of char( ℓ ) = 2, then K h ( L ( E )) = BF ( E ) ⊗ K h ( ℓ ) , and K h ( L ( E )) = BF ( E ) ⊗ Z [ σ ] K h ( ℓ ) . GUILLERMO CORTI˜NAS
Also in this section we characterize when L ( E ) and L ( E ) are zero in kk h . We provein Lemma 4.6 that j h ( L ( E )) = 0 ⇐⇒ BF ( E ) ⊗ Z [ σ ] KH h ( ℓ ) = 0 j h ( L ( E )) = 0 ⇐⇒ BF ( E ) ⊗ KH h ( ℓ ) = 0 . For example, we have j h ( L ) = 0, but j h ( L ) = 0 in general (see (5.8)). We use thecharacterization above to give an example of a finite, purely infinite simple graphΥ such that j h ( L (Υ)) = j h ( L (Υ)) = 0 (see Examples 4.7 and 8.2). In Section 5 weprove Proposition 1.22 (Proposition 5.6). To do this, we compute BF ( R ) = Z / Z , BF ( R − ) = Z / Z then we use use (1.20) and (1.21). Section 6 is devoted to characterizing L ( E ) and L ( E ) up to kk h -isomorphism. We prove that if | E | < ∞ is such that(1.23) BF ( E ) = Z r ⊕ n M i =1 Z /d i Z , then for s = | sing( E ) | we have j h ( L ( E )) ∼ = j h ( L s ⊕ L r − s ⊕ n M i =1 L d i +1 ) . We remark that a similar structure theorem was proved in [10, Theorem 6.10]for usual, nonhermitian bivariant K -theory j : Alg ℓ → kk . Theorem 1.14 is alsoproved in this section 6.11. Under mild additional hypothesis, that are satisfied,for example, when ℓ is a field or a P ID , we show that for graphs E and F withfinitely many vertices, we have j h ( L ( E )) ∼ = j h ( L ( F )) ⇐⇒ j ( L ( E )) ∼ = j ( L ( F )) . Next we turn our attention to L ( E ). In Theorem 6.14 we prove the partial analogueof Theorem 1.14 mentioned before in this introduction; in particular, when E , F and BF ( E ) are finite, BF ( E ) ∼ = BF ( F ) ⇒ j h ( L ( E )) ∼ = j h ( L ( F )) . Proposition 6.15 proves that if 2 = xx ∗ for some invertible x ∈ ℓ , E is finiteand BF ( E ) is finite and Z [ σ ]-linearly isomorphic to BF ( E ), then (1.19) holds and j h ( L ( E )) is a split direct summand of j h ( L ( E )). Remark 6.17 shows that in general,1 − σ is nonzero on j h ( L ( E )) and so the summand j h ( L ( E )) is proper. Remark6.18 explains how to obtain graphs E satisfying the hypothesis of Proposition 6.15and gives a concrete example. The short Section 7 is concerned with the group K ( R ) ∗ some of its elementary properties are proved and its relation with K h ( R )and with the fixed points of the action of the involution on K ( R ) are discussed.Section 8 introduces strictly properly infinite unital ∗ -algebras, which are the purelyalgebraic counterpart of properly infinite unital C ∗ -algebras. We show that if E isfinite and purely infinite simple, then L ( E ) and L ( E ) are strictly properly infinite(Corollary 8.5). For a strictly properly infinite ∗ -algebra R , K ( R ) ∗ agrees with themonoid V f ( R ) of Murray-von Neummann equivalence classes of very full projections(Proposition 8.9), by the same argument as in the C ∗ -algebra case ([8, Theorem6.11.7]), and the set [ A, R ] ∗ M of M - ∗ -homotopy classes of ∗ -homomorphisms isnaturally an abelian monoid (Lemma 8.12). Theorem 1.10 is proved in Section 9(Theorem 9.8). We first prove, in Theorem 9.4, that if R is any strictly properlyinfinite ∗ -algebra and E is countable, then any homomorphism ξ : BF ( E ) → K ( R ) ∗ lifts to a ∗ -homorphism φ : L ( E ) → R with property (P) and that if | E | isfinite and ξ ([1] E ) = [1] then φ can be chosen to be unital. Section 10 introduces, LASSIFYING LEAVITT PATH ALGEBRAS 9 for a unital ∗ -ring R , the group K ( R ) ∗ = U ∞ ( R ) Ab , defined as the abelianizationof the group U ∞ ( R ) = S ∞ n =1 U n ( R ) of the invertible unitary matrices of finite size,where the elements of U n ( R ) are unitary for the standard form h x, y i = P ni =1 x ∗ i y i .We also introduce a homotopy invariant, Karoubi-Villamayor style variant which wecall KV ( R ) ∗ . For R strictly properly infinite we characterize K ( R ) ∗ and KV ( R ) ∗ as quotients of U ( R ) = U ( R ) modulo some explicit relations. Section 11 is devotedto establishing Poincar´e duality (1.18) which we prove in Theorem 11.2. In Section12 we prove a version of the universal coefficient theorem. We introduce the dualBowen-Franks group and the dual Bowen-Franks Z [ σ ]-module BF ∨ ( E ) = Coker( I t − A E ) , BF ∨ ( E ) = Coker( I t − σA E ) . We show in Theorem 12.2 that if E is a graph such that | E | < ∞ then for any ∗ -algebra R there are exact sequences0 → KH h ( R ) ⊗ BF ∨ ( E ) → kk ( L ( E ) , R ) ev −→ hom( BF ( E ) , KH h ( R )) → → KH h ( R ) ⊗ Z [ σ ] BF ∨ ( E ) → kk ( L ( E ) , R ) ev −→ hom Z [ σ ] ( BF ( E ) , KH h ( R )) → KH h and composing withthe canonical maps BF ( E ) → BF ( E ) ⊗ KH h ( ℓ ) → KH h ( L ( E )) and BF ( E ) → KH h ( L ( E )). Theorem 1.15 and the injectivity of the map (1.17) are proved inSection 13 as Theorems 13.7 and 13.10. The last section, Section 14, contains theproof of classification theorems 1.8, 1.4 and 1.9; they are Theorems 14.1, 14.2 and14.4. Acknowledgement.
I wish to express my gratitude towards Santiago Vega, whosecontributions go beyond our collaboration in [14]. Thanks to Guido Arnone, whoread several earlier drafts of this article and gave useful feedback. Thanks also toMax Karoubi, Jonathan Rosenberg, Marco Schlichting and Chuck Weibel, whom Ibothered (especially insistingly in the case of Marco) with several questions aboutHermitian K -theory. Dedication.
I dedicate this article to the memory of my parents, Julio Corti˜nas(1933–2019) and Lina Villar (1934–2020).2.
Preliminaries
Algebras and involutions.
A commutative unital ring ℓ with involution ∗ is fixed throughout the article. A ∗ -algebra is an ℓ -algebra R equipped with aninvolution ∗ : R → R op that is semilinear with respect to the ℓ -module action;( λa ) ∗ = λ ∗ a ∗ for all λ ∈ ℓ and a ∈ R . We use the term ∗ -ring for a ∗ - Z -algebra. A ∗ -homomorphism between ∗ -algebras is an algebra homomorphism which commuteswith involutions. We write Alg ℓ for the category of algebras and algebra homomor-phisms and Alg ∗ ℓ for the subcategory of ∗ -algebras and ∗ -homomorphisms.Tensor products of algebras are taken over ℓ ; we write ⊗ for ⊗ ℓ . We also use ⊗ for tensor products of abelian groups, e.g. K ( R ) ⊗ K ( S ) = K ( R ) ⊗ Z K ( S ). If R and S are ∗ -algebras, we regard R ⊗ S as a ∗ -algebra with the tensor productinvolution ( a ⊗ b ) ∗ = a ∗ ⊗ b ∗ . If L is a ∗ -algebra and A ∈ Alg ∗ ℓ , we shall often write LA for L ⊗ A . Example 2.1.1.
Let A be a ring. Put inv( A ) = A ⊕ A op for the ∗ -ring withthe coordinatewise operations and involution ( a, b ) ( b, a ). If B is a ∗ -ring, theninv( A ) ⊗ Z B → inv( A ⊗ Z B ) ( a ⊗ b , a ⊗ b ) ( a ⊗ b , a ⊗ b ∗ ) is a ∗ -isomorphism.If A is an algebra over a ground ring ℓ , then inv( A ) is a ∗ -inv( ℓ ) algebra; this givesrise to a category equivalence inv : Alg ℓ ∼ −→ Alg inv( ℓ ) . The polynomial ring B [ t ] = B ⊗ Z Z [ t ] with coefficients in a ∗ -algebra is equippedwith the tensor product involution, where Z [ t ] has the trivial involution.An elementary ∗ -homotopy between two ∗ -homomorphisms f , f : A → B isa ∗ -homomorphism H : A → B [ t ] such that ev i ◦ H = f i ( i = 0 , ∗ -homomorphisms f, g : A → B are ∗ -homotopic , and write f ∼ ∗ g , if thereis a finite sequence f = f , . . . , f n = g such that for each i there is an elementary ∗ -homotopy between f i and f i +1 . We write [ A, B ] ∗ for the set of ∗ -homotopy classesof ∗ -homomorphisms A → B .A ∗ -ideal in a ∗ -algebra R is a two-sided ideal I ⊳ R such that I ∗ = I . Anelement a in a ∗ -algebra R is self-adjoint if a ∗ = a .Let X be a set; consider Karoubi’s cone algebra Γ X of square matrices indexedby X whose set of coefficients is finite and which have a global bound on the size ofthe support of its rows and columns. Viewing X × X -matrices with coefficients in ℓ as functions X × X → ℓ , the elements of Γ X are all the functions a : X × X → ℓ that satisfy the following conditions: | Im( a ) | < ∞ and ( ∃ N ) max {| supp a ( x, − ) | , | supp( a ( − , x )) | : x ∈ X } ≤ N. We equip Γ X with the product of matrices, or what is the same, with the convolutionproduct of functions. Observe that Γ X contains the algebra of finitely supportedmatrices as an ideal M X = { a : X × X → ℓ : | supp( a ) | < ∞} ⊳ Γ X . The standard involution on Γ X is ∗ : Γ X → Γ X , a ∗ ( x, y ) = a ( y, x ) ∗ ( x, y ∈ X ) . Observe that M X is a ∗ -ideal with respect to the standard involution. Karoubi’ssuspension algebra is the quotient ∗ -algebraΣ X = Γ X /M X . If ( x, y ) ∈ X × X , write ǫ x,y for the matrix unit and ι x : R → M X R for the cornerembedding ι x ( a ) = ǫ x,x a . Observe that ι x is a ∗ -homomorphism for the standardinvolution on M X as well as for any other involution which makes ǫ x,x self-adjoint.When X = { , . . . , n } , we write M n for M X . If R is a ∗ -algebra, we write Γ X R and M X R for Γ X ⊗ R and M X ⊗ R .2.2. Z / Z -gradings. If G is a group, by a G -graded ∗ -algebra we understand a G -graded algebra equipped with a involution that is homogeneous of degree 0 withrespect to the grading. If A = A ⊕ A is a Z / Z -graded ∗ -algebra, then(2.2.1) τ : A → A, τ ( a + a ) = a − a is a ∗ -automorphism. Composing it with the involution, we obtain a new involution(2.2.2) : A → A op , a + a = a ∗ − a ∗ . Write A for A equipped with the involution . If B = B ⊕ B is another Z / Z -graded ∗ -algebra and f : A → B is a homogeneous ∗ -homomorphism of degree 0,then f ( a ) = f ( a ) for all a ∈ A . Hence the function f defines a ∗ -homomorphism(2.2.3) f : A → B, f ( a ) = f ( a ) . Example 2.2.4.
The algebra M admits a Z / Z -grading, where | ǫ i,j | ≡ i − j mod (2). We write M ± = M . Thus M ± is the algebra of 2 × ℓ , equipped with the following involution(2.2.5) (cid:20) a bc d (cid:21) ∗ = (cid:20) a ∗ − c ∗ − b ∗ d ∗ (cid:21) . LASSIFYING LEAVITT PATH ALGEBRAS 11
We write ι + and ι − for the upper left and lower right corner inclusions ℓ → M ± ,respectively. For a ∗ -algebra R , we put M ± R = M ± ⊗ R for the matrix algebra withthe tensor product involution; we also abuse notation and write ι ± : R → M ± R ,for ι ± ⊗ id R . Example 2.2.6. If X is a set then any function l : X → { , } induces a Z / Z -grading on Γ X that makes M X a homogeneous ideal, where | ǫ x,y | ≡ l ( x ) + l ( y ) mod (2) . The grading of M in Example 2.2.4 is particular case of this. Observe also that,by passage to the quotient, we also obtain a Z / Z -grading on Σ X . Example 2.2.7. If A is any Z / Z -graded ∗ -algebra, then so are M A and M ± A ,with the tensor product grading. The involutions of M A and M ± A agree on thecommon ∗ -subalgebra(2.2.8) ˆ A = (cid:20) A A A A (cid:21) We have a ∗ -homomorphism(2.2.9) A → ˆ A, a + a (cid:20) a a a a (cid:21) . Composing with the inclusions ˆ A ⊂ M ± A and ˆ A ⊂ M A we get ∗ -homomorphisms(2.2.10) ∆ : A → M ± A, ∆ ′ : A → M A. A calculation shows that the following diagrams commute M A M ∆ ′ (cid:15) (cid:15) A ∆ ′ o o ( ι + ι ) ◦ ∆ ′ { { ✇✇✇✇✇✇✇✇✇ M M A A ∆ / / ( ι + + ι − ) ◦ ∆ ′ (cid:15) (cid:15) M ± A M ± ∆ (cid:15) (cid:15) M ± M A M ± M ± A. ∼ o o (2.2.11) Remark . Let σ be the generator of Z / Z , written multiplicatively. Then Z [ σ ] = Z [ Z / Z ]is a Hopf ring, and a Z / Z -graded ring is the same thing as a comodule-algebraover Z [ σ ]. One checks that the algebra ˆ A of (2.2.8) is the crossed product of A with Z / Z under this coaction, as defined for example in [6, Definition 2.1]. In particular,ˆ A is equipped with a Z / Z action; the generator σ acts by the automorphism (cid:20) a bc d (cid:21) (cid:20) d cb a (cid:21) . Observe that (2.2.9) is an isomorphism from A onto the fixed ring under the auto-morphism above. Next assume that A is unital. Let M = M ⊕ M be a gradedleft A -module. Regarding an element m + m ∈ M ⊕ M as a column vector andusing the matricial product, one defines an ˆ A -module structure on M . This givesa functor(2.2.13) Gr Z / Z Mod A → Mod ˆ A If A has local units, then by [6, Proposition 2.5], (2.2.13) is an isomorphism ofcategories, and maps the shift functor M ∗ M ∗ +1 to the action of σ . Positivity.
Let a ∈ A ∈ Alg ∗ ℓ and n ≥
1; we call a n -positive if a = 0 and canbe written as a sum a = P ni =1 x i x ∗ i for some x , . . . , x n ∈ A , and positive if it is n -positive for some n . We call a negative if − a is positive. In a general ∗ -algebrait can happen that an element is positive and negative at the same time. Example 2.3.1.
Assume that ℓ is a field of char( k ) = 2. Then every self-adjointelement in ℓ can be written as a difference of two 1-positive elements. Hence − ℓ is positive. Example 2.3.2.
The element x = (cid:20) −
11 0 (cid:21) ∈ M ± is self-adjoint and satisfies x = −
1. Hence if R is any unital ∗ -algebra, then − M ± R . Example 2.3.3.
Let A = L n ∈ Z A n be a Z -graded ∗ -algebra; regard A as Z / Z -graded via its even-odd grading and let A be as in Subsection 2.2. Let R be a unital ∗ -algebra; assume that R contains a central element x such that xx ∗ = −
1. Then θ x : A ⊗ R → A ⊗ R, θ x ( a ⊗ b ) = a ⊗ x | a | b ( a ∈ A ∗ , b ∈ R )is a ∗ -isomorphism. Similarly, the map θ x : Hom Alg ∗ ℓ ( A, R ) → Hom
Alg ∗ ℓ ( A, R ) , θ x ( f )( a ) = f ( a ) x | a | ( a ∈ A ∗ )is bijective. Remark . The hypothesis that x be central –which is not satisfied in Example2.3.2– is essential in Example 2.3.3. For example if A is unital, M ± A and M ± A havethe same Hermitian K -theory as A and A , respectively, but in general, K h ∗ ( A ) = K h ∗ ( A ); see Example 4.5. Lemma 2.3.5.
Let S be a set; equip Γ S with the standard involution. Then − ispositive in Γ S if and only if it is positive in ℓ .Proof. The if direction is clear. Assume conversely that − S . Thenthere are elements y (1) , . . . , y ( n ) ∈ Γ S such that − P ni =1 y ( i ) ∗ y ( i ). Hence for N = max {| supp( y ( i ) ∗ , ) | : 1 ≤ i ≤ n } , we have the following identity betweenelements of ℓ − n X i =1 y ( i ) ∗ y ( i )) , = n X i =1 N X j =1 ( y ( i ) j, ) ∗ y ( i ) j, . Thus − ℓ . (cid:3) Hermitian bivariant K -theory. In this subsection we assume that ℓ satis-fies the λ -assumption 1.11.An extension of ∗ -algebras is a sequence of ∗ -homomorphisms(2.4.1) ( E ) A / / i B p / / / / C with i injective and p surjective and which is exact as a sequence of ℓ -modules. Anextension is semi-split if it is split as a sequence of ℓ -modules. Under our standingAssumption 1.11, if (2.4.1) is semisplit, then there exists an involution preservinglinear map s : C → B such that p ◦ s = id C . Let T be a triangulated category;write [ − n ] for the n -fold suspension in T . Let E be the class of all semi-splitextensions (2.4.1). An excisive homology theory on Alg ∗ ℓ with values in T is a functor LASSIFYING LEAVITT PATH ALGEBRAS 13 H : Alg ∗ ℓ → T together with a family of maps { ∂ E : H ( C )[1] → H ( A ) | E ∈ E} suchthat for every E ∈ E , H ( C )[1] ∂ E / / H ( A ) H ( i ) / / H ( B ) H ( p ) / / H ( C )is a triangle in T , and such that ∂ is compatible with maps of extensions in thesense of [13, Section 6.6]. Let H : Alg ∗ ℓ → T be an excisive homology theory, X an infinite set, x ∈ X and A ∈ Alg ∗ ℓ . Consider the natural evaluation and cornerinclusion maps ev : A [ t ] → A and ι + : A → M ± A , ι x : A → M X A . We say thata H is homotopy invariant , ι + -stable and M X -stable if for every A ∈ Alg ∗ ℓ , themaps H (ev ), H ( ι + ) and H ( ι x ) are isomorphisms in T . By [14, Lemma 2.4.1], M X -stability is independent of the choice of the element x in the previous definition.It was proved in [14, Proposition 6.2.7] that for any infinite set X , there existsan excisive, homotopy invariant, ι + -stable and M X -stable homology theory j h :Alg ∗ ℓ → kk h , depending on X , such that if H : Alg ∗ ℓ → T is any other excisive,homotopy invariant, ι + -stable and M X -stable homology theory, then there exists aunique triangulated functor ¯ H : kk h → T such that ¯ H ◦ j h = H . We fix such an X and for A, B ∈ Alg ∗ ℓ and n ∈ Z , we write kk hn ( A, B ) = hom kk h ( j h ( A ) , j h ( B )[ n ]) , kk h ( A, B ) = kk h ( A, B ) . The suspension in kk h is represented by the Karoubi suspension; for any infiniteset Y with | Y | ≤ | X | and any A ∈ Alg ∗ ℓ , we have j h ( A )[ −
1] = j h (Σ Y A ) . The inverse suspension is obtained by tensoring with Ω = (1 − t ) tℓ [ t ]; for all A ∈ Alg ∗ ℓ we have j h ( A )[+1] = j h (Ω A ) . It was shown in [14, Proposition 8.1] that kk h recovers Weibel-style homotopyalgebraic Hermitian K -theory; we have(2.4.2) kk hn ( ℓ, B ) = KH hn ( B ) . For a definition of KH h and its relation to the more standard Hermitian K -theorydefined by Karoubi (sometimes called Grothendieck-Witt theory) see [14, Section 3]; K h is discussed in Section 7. There is a natural map(2.4.3) K hn ( B ) → KH hn ( B ) . Recall that if F is a functor defined on a subcategory C ⊂ Alg ℓ closed undertensoring with ℓ [ t ] and A ∈ C , then A is F -regular if for every n ≥ F maps theinclusion A ⊂ A [ t , . . . , t n ] to an isomorphism. A is K h or K -regular if it is K hm or K m -regular for every m . The map (2.4.3) is an isomorphism for all n whenever B is K h -regular. If B = inv( B ) for some ring B , then B is K h -regular if and onlyif it is K -regular. If multiplication by 2 is invertible in B and B is K -regular and K -excisive then it is K h -regular [14, Lemma 3.8].It follows from (2.4.2) that for all A, B ∈ Alg ∗ ℓ , the groups kk h ( A, B ) are modulesover the ring kk h ( ℓ, ℓ ) = KH h ( ℓ ), and thus over K h ( ℓ ) via the canonical map K h ( ℓ ) → KH h ( ℓ ). As will be recalled in Section 7 below, K h ( ℓ ) is the groupcompletion of the monoid of equivalence classes of self-adjoint idempotent finitematrices over M ± . There is a unital ring homomorphism Z [ σ ] → K h ( ℓ ) mapping 1to the class of ι + (1) and σ to the class of ι − (1). Thus kk h ( A, B ) is a Z [ σ ]-modulefor all A, B ∈ Alg ∗ ℓ . If − ℓ , then σ acts trivially in kk h ; σξ = ξ forall ξ ∈ kk h ( A, B ) and all
A, B ∈ Alg ∗ ℓ . Lemma 2.4.4.
Assume that ℓ satisfies the λ -assumption 1.11. Let X be an infiniteset, and let j h : Alg ∗ ℓ → kk h be the universal excisive, homotopy invariant, ι + -stable and M X -stable homology theory. Let l : X → { , } be a function and let x ∈ X .Equip M X with the Z / Z -grading induced by l , as in Example 2.2.6. Consider thecorner embedding ι x : ℓ → M X , ι x ( a ) = ǫ x,x a . Then j h ( ι x ) is an isomorphism forevery x ∈ X , and if l ( x ) = l ( y ) , then j h ( ι x ) = j h ( ι y ) .Proof. Let i ∈ { , } , X i = l − { i } and let inc : M X i → M X be the map induced bythe inclusion X i ⊂ X . We have a commutative diagram M X ℓ ι x > > ⑤⑤⑤⑤⑤⑤⑤⑤⑤ ι x / / M X i . inc O O The map j h ( ι x ) is an isomorphism by M X -stability; by [14, Lemma 2.4.1] it isindependent of x ∈ X i . The map j h (inc) is an isomorphism by [14, Lemma 2.4.3]. (cid:3) Proposition 2.4.5.
Let A be a Z / Z -graded ∗ -algebra and let ∆ = ∆ A : A → M ± A and ∆ ′ : A → M A be as in (2.2.10) . Set z}|{ ∆ = j h ( ι + ) − ◦ j h (∆) ∈ kk h ( A, A ) and z}|{ ∆ ′ = j h ( ι ) − ◦ j h (∆ ′ ) ∈ kk h ( A, A ) . Then z}|{ ∆ A ◦ z}|{ ∆ A = (1 + σ ) z}|{ ∆ ′ A , z}|{ ∆ ′ = 2 z}|{ ∆ ′ . If furthermore is invertible and -positive in ℓ , then for τ as in (2.2.1) , we have z}|{ ∆ ′ = 1 + j h ( τ ) .Proof. The displayed identities follow from the commutative diagrams (2.2.11) us-ing Lemma 2.4.4. Next assume that 2 = xx ∗ for some invertible x ∈ ℓ , and consider u = (cid:20) /x /x /x ∗ − /x ∗ (cid:21) . A calculation shows that ad( u ) ◦ ∆ ′ = ι + ι τ , whence z}|{ ∆ ′ = 1 + j h ( τ ). (cid:3) In the next corollary and elsewhere we write kk h [1 /
2] for the idempotent comple-tion of the Verdier quotient of kk h by the full subcategory of those objects C suchthat j h (id C ) is 2-torsion, j h [1 /
2] : Alg ∗ ℓ → kk h → kk h [1 /
2] for the composite ofthe canonical functors, and kk h [1 / A, B ) = hom kk h [1 / ( j h [1 / A ) , j h [1 / B ))for A, B ∈ Alg ∗ ℓ . Corollary 2.4.6.
Let p + = (1 + σ ) / ∈ Z [1 / , σ ] . If A is Z / Z -graded then ( z}|{ ∆ A / and ( z}|{ ∆ A / induce inverse isomorphisms Im( p + ◦ ( z}|{ ∆ ′ A / ↔ Im( p + ◦ ( z}|{ ∆ ′ A / . Cohn and Leavitt path algebras.
A (directed) graph is a quadruple E =( s, r : E ⇒ E ) consisting of sets E and E of vertices and edges and source and range maps s and r . A vertex v ∈ E is a sink if s − ( { v } ) = ∅ , a source if r − ( { v } ) = ∅ , an infinite emitter if s − ( { v } ) is infinite and a singular vertex itis either a sink or an infinite emitter. Vertices which are not singular are called regular . We write sink( E ) , sour( E ) , inf( E ) for the sets of sinks, sources, and infiniteemitters, and sing( E ) and reg( E ) for those of singular and regular vertices. We saythat E is regular if E = reg( E ). A graph E is countable or finite if both E and E are. The reduced incidence matrix of a graph E is the matrix A E with nonnegative LASSIFYING LEAVITT PATH ALGEBRAS 15 integer coefficients, indexed by reg( E ) × E , whose ( v, w ) entry is the number ofedges with source v and range w :( A E ) v,w = | s − ( v ) ∩ r − ( w ) | . Our conventions are such that we will mainly deal with the transpose A tE . Weabuse notation and write I for the E × reg( E )-matrix obtained from the identitymatrix of M E Z upon removing the columns corresponding to the singular vertices.Thus I − A tE is a well-defined integral matrix indexed by E × reg( E ).We write C ( E ) and L ( E ) for the Cohn and Leavitt path algebras over ℓ [2,Definitions 1.5.1 and 1.2.3]. Each of these carries a standard ℓ -semilinear involution a a ∗ which fixes the vertices and maps each edge e to the corresponding phantomedge e ∗ . Let P = P ( E ) be the set of all finite paths in E ([2, Definitions 1.2.2]);we write | α | for the length of a path α ∈ P ( E ). For v ∈ E , set(2.5.1) P v = { µ ∈ P | r ( µ ) = v } , P v = { µ ∈ P | s ( µ ) = v } . Let ρ : C ( E ) → Γ P , (2.5.2) ρ ( v ) = X α ∈P v ǫ α,α , ρ ( e ) = X α ∈P r ( e ) ǫ eα,α ,ρ ( e ∗ ) = X α ∈P r ( e ) ǫ α,eα , ( v ∈ E , e ∈ E ) . Observe that ρ is a ∗ -homomorphism for the standard involutions on C ( E ) andΓ P ( E ) . Recall that C ( E ) carries a natural Z -grading, where C ( E ) n is generated byall αβ ∗ with | α | − | β | = n . Hence we may regard C ( E ) as Z / Z -graded, via theeven/odd grading. The twisted involution ¯ of (2.2.2) is the algebra homomorphism¯ : C ( E ) → C ( E ) op , v = v, e = − e ∗ , e ∗ = − e, ( v ∈ E , e ∈ E ) . (2.5.3)Similarly, the length modulo 2 induces a Z / Z -grading on Γ P ( E ) , and ρ is homoge-neous for this grading. Because ρ is homogeneous and a ∗ -homomorphism for thestandard involution, it defines a ∗ -homomorphism ρ : C ( E ) → Γ P ( E ) as in (2.2.3).Let K ( E ) = Ker( C ( E ) → L ( E )) be the kernel of the canonical surjection. Wehave semi-split extensions of ∗ -algebras0 → K ( E ) → C ( E ) → L ( E ) → → K ( E ) → C ( E ) → L ( E ) → . (2.5.5)An important feature of the involution a a is that if E is finite and regular,then the following identity holds in L ( E ) − X e ∈ E ee. This says that −
1, which is clearly negative, is also positive in LE . By contrast, − L ( E ) if and only if this happens already in ℓ , as shown by the nextlemma. Lemma 2.5.6.
Let E be a graph with finitely many vertices. Then − is positivein L ( E ) if and only if it is positive in ℓ . If − is -positive in ℓ , then L ( E ) ∼ = L ( E ) in Alg ∗ ℓ .Proof. The if direction is clear. To prove the converse, it suffices, in view of Lemma2.3.5, to find a set X and a unital ∗ -homomorphism f : L ( E ) → Γ X . Let S ( E ) = { αβ ∗ : α, β ∈ P ( E ) } be the inverse semigroup associated to E . Let X be an infiniteset of cardinality | X | ≥ | E | and I ( X ) the inverse semigroup of all partially defined injections X ⊃ Dom( f ) f −→ X . Proceed as in the proof of [12, Proposition 4.11]to find a semigroup homomorphism µ : S ( E ) → I ( X ) such that the associatedaction of S ( E ) on X is tight in the sense of Exel [12, Section 3]. By [12, Lemma3.1] and [1, Lemma 6.1], µ induces an algebra homomorphism L ( E ) → Γ X . Onechecks further that µ is a ∗ -homomorphism. This completes the proof of the firstassertion. The second assertion is immediate from Example 2.3.3. (cid:3) Remark . Let E be a regular, finite graph. It follows from Lemma 2.5.6 that if − ℓ then there can be no unital ∗ -homomorphism L ( E ) → L ( E ).However there are injective ∗ -homomorphisms L ( E ) → M ± L ( E ) and L ( E ) → M ± L ( E ) as shown in Example 2.2.7.3. Leavitt path algebras in kk h Throughout this section, we assume that ℓ satisfies the λ -assumption 1.11.For a set X and a ∗ -algebra R , we write R X for the ∗ -algebra of all functions X → R with pointwise operations and pointwise involution, and R ( X ) ⊂ R X forthe ∗ -ideal of finitely supported functions. If x ∈ X and a ∈ R , we write aχ x forthe function supported in { x } which maps x a .Let E be a graph and C ( E ) the Cohn algebra of E . The assignment(3.1) ℓ ( E ) → C ( E ) , χ v v defines ∗ -homomorphisms φ : ℓ ( E ) → C ( E ) and φ : ℓ ( E ) → C ( E ).We shall say that a homology theory is E -stable if it is stable with respect to aset X of cardinality | E ⊔ E ⊔ N | . Theorem 3.2.
Assume that ℓ satisfies the λ -assumption 1.11. Let j h : Alg ∗ ℓ → kk h be the universal homotopy invariant, excisive, Hermitian stable and E -stablehomology theory. Let E be a graph and let φ be as in (3.1) . Then j h ( φ ) and j h ( φ ) are isomorphisms. In particular, j h ( C ( E )) ∼ = j h ( C ( E )) .Proof. The analogue statement for the universal homotopy invariant, excisive and E -stable homology theory j : Alg ℓ → kk was proved in [10, Theorem 4.2]. Thesame proof goes through here with minor adjustments and works for both choicesof involution on C ( E ). The adjustments are in the homotopies occurring in [10,Lemma 4.17] and [10, Lemma 4.21], which are not ∗ -homomorphisms. In both casesthe problem is fixed by applying the trick of [14, Lemma 5.4], as we shall explainpresently. Lemma 4.17 of [10] says that a certain map ˆ ι τ : C ( E ) → M P ( E ) C ( E ) issent by j h to the same isomorphism as ι α for any α ∈ P ( E ). Observe that Lemma2.4.4 guarantees that j h ( ι α ) = j h ( ι β ) only when | α | ≡ | β | mod (2). This is howevernot a problem, as the lemma in [10] is needed only to establish that j h (ˆ ι τ ) is anisomorphism, so it suffices to prove it when | α | = 0. In the proof of [10, Lemma4.17], a vertex w ∈ E is fixed and elements A v , B v ∈ M P ( E ) C ( E ) are definedfor each v ∈ E . With notation as in [14, Lemma 5.4], put C v = c ( A v , B v ) andlet H : C ( E ) → M ± M P ( E ) C ( E )[ t ] be the homomorphism determined by H ( v ) = C v ι + ( ǫ v,v ⊗ v ) C ∗ v , H ( e ) = C s ( e ) ι + ( ǫ s ( e ) ,r ( e ) ⊗ e ) C ∗ r ( e ) , H ( e ∗ ) = H ( e ) ∗ . One checksthat H is a ∗ -algebra homomorphism for both choices of involution. It follows that H is an elementary ∗ -homotopy between the composite of ι + with the maps ˆ ι τ and ι w of [10, Lemma 4.17], again for both choices of involution. Next we pass tothe analogue of [10, Lemma 4.21]. With notations as in loc.cit. , for each e ∈ E consider the following elements of A [ t ] U e = ǫ s ( e ) ,s ( e ) (1 − t ) ee ∗ + ǫ e,s ( e ) te ∗ ,V e = ǫ s ( e ) ,s ( e ) (1 − t ) ee ∗ + ǫ s ( e ) ,e (2 t − t ) e. LASSIFYING LEAVITT PATH ALGEBRAS 17
One checks that the homotopy H + : C ( E ) → D [ t ] defined in the proof of [10,Lemma 4.21] satisfies the following identity for each e ∈ E . H + ( e ) = ( em r ( e ) , U e ǫ s ( e ) ,r ( e ) e ) , H + ( e ∗ ) = ( m r ( e ) e ∗ , ǫ r ( e ) ,s ( e ) e ∗ V e ) . Put W e = c ( U e , V e ). Let H : C ( E ) → M ± D [ t ] ,H ( e ) = ι + ( em r ( e ) ,
0) + (0 , W e )(0 , ι + ( ǫ s ( e ) ,r ( e ) e )) ,H ( e ∗ ) = H ( e ) ∗ , H ( v ) = ( m v ,
0) ( v ∈ E , e ∈ E ) . One checks that H is a ∗ -algebra homomorphism for both choices of involution, sothat for both choices of involution, H is a ∗ -homotopy between the maps ψ and ψ / of [10, Lemma 4.21]. This finishes the proof. (cid:3) Let S be a set, T a triangulated category, and H : Alg ∗ ℓ → T an excisive homologytheory. We say that H is S -additive if direct sums of at most | S | factors existin T , and for any set T with cardinality | T | ≤ | S | and any family of ∗ -algebras { A t : t ∈ T } , the canonical map L t ∈ T H ( A t ) → H ( L t ∈ T A t ) is an isomorphism. Theorem 3.3.
Assume that ℓ satisfies the λ -assumption 1.11. Let X : Alg ∗ ℓ → T be an excisive, homotopy invariant, Hermitian stable, E -stable and E -additivehomology theory and let R ∈ Alg ∗ ℓ . Then (2.5.4) and (2.5.5) induce the followingdistinguished triangles in T X ( R ) (reg( E )) I − A tE / / X ( R ) ( E ) / / X ( L ( E ) ⊗ R ) X ( R ) (reg( E )) I − σA tE / / X ( R ) ( E ) / / X ( L ( E ) ⊗ R ) . Proof.
It follows from the universal property of j h that ⊗ R induces a triangulatedfunctor in kk h . Hence upon replacing X by X ( − ⊗ R ) if necessary, we may assumethat R = ℓ . Let inc : K ( E ) → C ( E ) and inc : K ( E ) → C ( E ) be the inclusions. Foreach vertex w ∈ E , let p w : ℓ ( E ) ⇆ ℓ : χ w be the projection onto the w -coordinateand the inclusion into the w -summand. If v ∈ reg( E ), write m v = P s ( e )= v ee ∗ ∈ C ( E ) and let q : ℓ (reg( E )) → K ( E ), v q v = v − m v . We shall abuse notationand write q v also for the map ℓ → K ( E ), v q v . In view of Theorem 3.2 and theadditivity hypothesis on X , it suffices to identify, for each pair ( v, w ) ∈ reg( E ) × E ,the composite j h ( p w ) j h ( φ ) − j h (inc ◦ q v ) ∈ KH h ( ℓ ), and similarly for φ , inc and q v subsituted for φ , inc and q v . In the case of the standard involution, for each e ∈ E the projection ee ∗ is M-vN equivalent to e ∗ e = r ( e ) and the same calculation as in[10, Proposition 5.2] goes through. However in the case of the twisted involution,this is no longer true. In fact, taking into account (2.2.5) and (2.5.3) and writing ∗ for the involution of M ± C ( E ), we obtain the following identities ι + ( ee ∗ ) = (cid:20) e (cid:21) · (cid:20) e ∗ (cid:21) = (cid:20) e (cid:21) · (cid:20) e (cid:21) ∗ (3.4) ι − ( r ( e )) = (cid:20) e (cid:21) ∗ · (cid:20) e (cid:21) It follows that [ ee ∗ ] = σ [ r ( e )] in KH h ( C ( E )), so from the orthogonal sum v = q v + m v we get that j h (inc q v ) = j h ( φχ v ) − P s ( e )= v σj h ( φχ r ( e ) ). Hence j h ( φ ) − ◦ j h (inc) ◦ j h ( q ) = I − σA tE . (cid:3) Theorem 3.5.
Let E be a graph. Assume that | E | < ∞ and that ℓ satisfies the λ -assumption 1.11. Then there are distinguished triangles in kk h j h ( ℓ reg( E ) ) I − A tE / / j h ( ℓ E ) / / j h ( LE ) j h ( ℓ reg( E ) ) I − σA tE / / j h ( ℓ E ) / / j h ( LE ) . Proof.
Apply Theorem 3.3 with X = j h . (cid:3) Corollary 3.6.
Let E be as in Theorem 3.5 τ : L ( E ) → L ( E ) as in (2.2.1) and τ as in (2.2.3) . Then j h ( τ ) − j h (id L ( E ) )) = 2( j h ( τ ) − j h (id L ( E ) )) = 0 . Proof.
The restrictions of τ : C ( E ) → C ( E ) to the images of φ : ℓ E → C ( E ) and q : ℓ reg( E ) → K ( E ) ⊂ C ( E ) are the identity maps. Hence writing 1 for all identitymaps, we have a map of triangles j h ( ℓ reg( E ) ) (cid:15) (cid:15) I − A tE / / j h ( ℓ E ) (cid:15) (cid:15) p / / j h ( LE ) j h ( τ ) (cid:15) (cid:15) ∂ / / j h ( ℓ reg( E ) )[ − (cid:15) (cid:15) j h ( ℓ reg( E ) ) I − A tE / / j h ( ℓ E ) p / / j h ( LE ) ∂ / / j h ( ℓ reg( E ) )[ − kk h ( LE, − ) and kk h ( − , LE ) to thetriangles above we obtain factorizations 1 − j h ( τ ) = p ◦ ξ = η ◦ ∂ . It follows that0 = η ◦ ∂ ◦ p ◦ ξ = (1 − j h ( τ )) = 2(1 − j h ( τ )) . The same argument shows that 2(1 − j h ( τ )) = 0. (cid:3) Let E be a finite graph; let B E ∈ { , } ( E ) × ( E ⊔ sink( E )) ,(3.7) ( B E ) e,x = (cid:26) δ r ( e ) ,s ( x ) x ∈ E δ r ( e ) ,x x ∈ sink( E )Also let J ∈ Z ( E ⊔ sink( E )) × E ,(3.8) J x,e = (cid:26) δ s ( x ) ,r ( e ) x ∈ E δ x,r ( e ) x ∈ sink( E ) Corollary 3.9.
Let E be a finite graph and B E and J as in (3.7) and (3.8) . Thenthere are distinguished triangles in kk h ℓ E J − B tE / / ℓ E ⊔ sink( E ) / / LEℓ E J − σB tE / / ℓ E ⊔ sink( E ) / / LE.
Proof.
As observed in [10, Remark 5.7], B E = A E s for the out-split graph E s of[2, Definition 6.3.23]. An explicit ∗ -algebra isomorphism f : LE ∼ −→ L ( E s ) isconstructed in the proof of [3, Theorem 2.8]; one checks that f ( a ) = f ( a ) for all a ∈ L ( E ). Given all this, the corollary is immediate from Theorem 3.5. (cid:3) LASSIFYING LEAVITT PATH ALGEBRAS 19 Hermitian K -theory and Bowen-Franks groups The
Bowen-Franks group of a graph E is BF ( E ) = Coker( I − A tE ) . We shall also consider the following Z [ σ ]-module BF ( E ) = Coker( I − σA tE ) . Remark . Let E be a regular graph and let E be the graph with the samevertices and where an edge is a path of length 2 in E . Then we have a groupisomorphism BF ( E ) ∼ = Coker( I − ( A tE ) ) = BF ( E ) . Under the isomorphism above, the action of σ becomes multiplication by A tE . Theorem 4.2.
Let E be a graph and R ∈ Alg ∗ ℓ . Then there are exact sequences → BF ( E ) ⊗ KH hn ( R ) → KH hn ( L ( E ) ⊗ R ) → Ker(( I − A tE ) ⊗ KH hn − ( R )) → → BF ( E ) ⊗ Z [ σ ] KH hn ( R ) → KH hn ( L ( E ) ⊗ R ) → Ker(( I − σA tE ) ⊗ KH hn − ( R )) → Proof.
Apply Theorem 3.3 to the functor KH h from Alg ∗ ℓ to the homotopy categoryof spectra that sends A ∈ Alg ∗ ℓ to the homotopy Hermitian K -theory spectrum KH h ( A ); then take homotopy groups. (cid:3) Recall that a unital ring R is said to be coherent if its finitely presented modulesform an abelian subcategory of the category of all modules, and supercoherent if R [ t , . . . , t n ] is coherent for every n ≥
0. We say that R is regular supercoherent if itis supercoherent and in addition any finitely presented module has finite projectivedimension. This implies that R [ t , . . . , t n ] is again regular supercoherent, by theargument of [5, beginning of Section 7]. If R is regular supercoherent, then L Z ( E ) ⊗ Z R is K -regular, again by the argument of [5, beginning of Section 7]. Corollary 4.3.
Let R ∈ Alg ∗ ℓ be unital. Assume that is invertible in R and that R is regular supercoherent. Then the maps K h ∗ ( L ( E ) ⊗ R ) → KH h ∗ ( L ( E ) ⊗ R ) and K h ∗ ( L ( E ) ⊗ R ) → KH h ∗ ( L ( E ) ⊗ R ) of (2.4.3) are isomorphisms and the exactsequences of Theorem 4.2 also hold for L ( E ) ⊗ R and L ( E ) ⊗ R with KH h replacedby K h .Proof. Because R is unital, L ( E ) ⊗ R is s -unital and therefore K -excisive. Because R is regular supercoherent, by [10, Example 5.5], L ( E ) ⊗ R is K -regular. Now apply[14, Lemma 3.8]. (cid:3) Corollary 4.4.
Assume that ℓ is a field of characteristic char( ℓ ) = 2 . Then K h ( L ( E )) = BF ( E ) ⊗ Z K h ( ℓ ) , K h ( L ( E )) = BF ( E ) ⊗ Z [ σ ] K h ( ℓ ) . Proof.
In view of Corollary 4.3, it suffices to show that K h − ( ℓ ) = 0. By [19,Theorems 7.1 and 8.1] and [18, Proposition 6.3] K h − ( ℓ ) agrees with Ranicki’s U − ( ℓ )which vanishes by [16, Proposition]. (cid:3) Example 4.5.
Let ℓ be a field of char( ℓ ) = 2. Assume that the canonical map Z [ σ ] → KH h ( ℓ ) is an isomorphism. It follows from Corollary 4.4 that K h ( L ( E )) = BF ( E ) and K h ( L ( E )) = BF ( E ) ⊗ Z [ σ ]. This is the case, for example, when ℓ = C equipped with complex conjugation as involution. Lemma 4.6.
Let E and ℓ be as in Theorem 3.5. Then i) j h ( L ( E )) = 0 ⇐⇒ BF ( E ) ⊗ Z [ σ ] KH h ( ℓ ) = 0 . ii) j h ( L ( E )) = 0 ⇐⇒ BF ( E ) ⊗ KH h ( ℓ ) = 0 . iii) j h ( L ( E ) = 0 ⇒ j h ( L ( E )) = 0 . iv) If j h ( L ( E )) = 0 then E is regular.Proof. By Theorem 3.5, j h ( L ( E )) = 0 if and only if the image of I − A tE in KH h ( ℓ ) E × reg( E ) under the map induced by Z ⊂ Z [ σ ] → KH h ( ℓ ) is an invert-ible matrix. Because KH h ( ℓ ) is a commutative ring, the latter condition impliesthat reg( E ) = E and is equivalent to having BF ( L ( E )) ⊗ KH h ( ℓ ) = 0. Simi-larly, j h ( L ( E )) = 0 is equivalent to BF ( L ( E )) ⊗ Z [ σ ] KH h ( ℓ ) = 0 and implies that j h ( L ( E )) = 0, since by definition BF ( E ) is a quotient of BF ( E ). (cid:3) Example 4.7.
Let Υ be the following graph • $ $ ( ( • h h Then det( I − σA t Υ ) = − σ , so j h ( L (Υ)) = j h ( L (Υ)) = 0.5. Hermitian K -theory of L ( E ) vs. Z / Z -graded K -theory of L ( E )Let E be a graph and let ˆ E be the graph with ˆ E = E i × Z / Z , ( i = 0 ,
1) withsource and range functions s ( e, i ) = ( s ( e ) , i ), r ( e, i ) = ( r ( e ) , i + 1). Observe that Z / Z acts on ˆ E by translation on the second component. We have an isomorphismof Z [ σ ]-modules Z ( ˆ E ) ∼ = Z [ σ ] ⊗ Z ( E ) which restricts to an isomorphism Z reg( ˆ E ) ∼ = Z [ σ ] ⊗ Z (reg( E )) . Under these identifications, A ˆ E becomes σA E . Hence we have anisomorphism of Z [ σ ]-modules(5.1) BF ( ˆ E ) ∼ = BF ( E ) . Hence by [6, Corollary 5.3] and Remark 2.2.12, for [ L ( E ) as in Example 2.2.7, wehave a ∗ -isomorphism(5.2) L ( ˆ E ) ∼ −→ [ L ( E ) . Write K gr for the graded K -theory of Z / Z -rings. It follows from (5.2), Remark2.2.12 and Theorem 4.2 applied to inv( ℓ ), that if E has finitely many vertices and ℓ is regular supercoherent, then(5.3) K gr0 ( L ( E )) = BF ( E ) ⊗ K ( ℓ ) . In particular, if ℓ is as in Example 4.5, then we have(5.4) K h ( L ( E )) ∼ = K gr0 ( L ( E )) . Remark . It follows from Example 2.2.7 that L ( ˆ E ) is a ∗ -subalgebra of M ± L ( E ).Write inc for inclusion map; we have a commutative diagram BF ( E ) ⊗ KH h ( ℓ ) (cid:15) (cid:15) (cid:15) (cid:15) / / KH h ( L ( ˆ E )) KH h ( ι + ) − ◦ KH h (inc) (cid:15) (cid:15) BF ( E ) ⊗ Z [ σ ] KH h ( ℓ ) / / KH h ( L ( E )) . Here the rows are the monomorphisms of Theorem 4.2 and the left column is thecanonical surjection. In particular, KH h (inc) is not injective in general.`In the next proposition and elsewhere, we write R n for the graph consisting of asingle vertex and n loops, R n − for its Cuntz splice ([4, Definition 2.11]), L n = L ( R n )and L n − = L ( R n − ). LASSIFYING LEAVITT PATH ALGEBRAS 21
Proposition 5.6.
Assume that ℓ is regular supercoherent and such that the canoni-cal map Z → K ( ℓ ) is an isomorphism. Then there is no Z / Z -homogeneous unital ℓ -algebra homomorphism from L to L − nor in the opposite direction. If further-more, ℓ is a field with char( ℓ ) = 2 and Z [ σ ] → K h ( ℓ ) is an isomorphism, then thereis no unital ∗ -algebra homomorphism L → L − nor in the opposite direction.Proof. By (5.3) and the hypothesis on ℓ , we have K ( L ( E )) = BF ( E ) for any graph E . Next use Remark 4.1 to compute(5.7) BF ( R ) = Z / Z , BF ( R − ) = Z / Z . By (5.1) and Remark 2.2.12, any unital, Z / Z -homogeneous algebra homomorphismbetween L and L − would induce a nonzero group homomorphism between Z / Z and Z / Z , but there is no such homomorphism. This proves the first assertion. If ℓ is as in the the second assertion, then (5.4) and (5.7) together imply(5.8) K h ( L ) = Z / Z , K h ( L − ) = Z / Z . The proof is now immediate. (cid:3) Structure theorems for Leavitt path algebras in kk h Let n , n ≥ M ∈ Z n × n , and(6.1) j h ( ℓ ) n M / / j h ( ℓ ) n / / j h ( R )a distinguished triangle in kk h . Applying kk h ( ℓ, − ) we obtain a monomorphismCoker( M ) ⊗ KH h ( ℓ ) → KH h ( R ); composing it with − ⊗ [1] : Coker( M ) → Coker( M ) ⊗ KH h ( ℓ ) we obtain a canonical map(6.2) can : Coker( M ) → KH h ( R ) . Hence for every S ∈ Alg ∗ ℓ , there is an evaluation map(6.3) ev : kk h ( R, S ) → hom(Coker( M ) , KH h ( S )) , ev( ξ ) = KH h ( ξ ) ◦ can . Lemma 6.4.
Let E be a graph with | E | < ∞ , and let n , n ≥ , M ∈ Z n × n ,and R be as in (6.1) . Assume that rk(Ker M ) = rk(Ker( I − A tE )) . Let ξ : BF ( E ) ∼ −→ Coker( M ) be a group isomorphism. Then there exists an isomorphism ξ : j h ( L ( E )) ∼ −→ j h ( R ) such that ev( ξ ) = can ◦ ξ .Proof. Put m = | E | , m = | reg( E ) | . Because the free abelian groups Ker( M )and Ker( I − A tE ) have the same rank, there is an isomorphism ξ : Ker( I − A tE ) ∼ −→ Ker( M ). Because Im( I − A tE ) and Im( M ) are free, the surjections Z m ։ Im( I − A tE ) and Z n ։ Im( M ) admit sections s and t . Let f : Z m → Z n be any lift of ξ ; put(6.5) f ( x ) = ξ ( x − s ( I − A tE ) x ) + t ( f (( I − A tE ) x )) . Then Z m f (cid:15) (cid:15) I − A tE / / Z m f (cid:15) (cid:15) Z n M / / Z n is a quasi-isomorphism f with H i ( f ) = ξ i ( i = 0 , Z → KH h ( ℓ ) we can associate to f a commutative solid arrow diagram j h ( ℓ ) m f (cid:15) (cid:15) I − A tE / / j h ( ℓ ) m p / / f (cid:15) (cid:15) j h ( L ( E )) ξ (cid:15) (cid:15) ∂ / / j h ( ℓ ) m [ − f [ − (cid:15) (cid:15) j h ( ℓ ) n M / / j h ( ℓ ) n / / j h ( R ) / / j h ( ℓ ) n [ − kk h is triangulated there exists a filler ξ as above. By construction, thefollowing diagram commutes BF ( E ) ⊗ KH h ( ℓ ) ∪ / / ξ ⊗ (cid:15) (cid:15) KH h ( L ( E )) KH h ( ξ ) (cid:15) (cid:15) Coker( M ) ⊗ KH h ( ℓ ) ∪ / / KH h ( R )It follows that ev( ξ ) = can ξ . It remains to show that ξ is an isomorphism. Pro-ceeding as above, we obtain a quasi-isomorphism g : Z n ∗ → Z m ∗ with H ∗ ( g ) = ξ − ∗ ,such that for all y ∈ Z n ,(6.6) g ( y ) = ξ − ( y − tM y ) + s ( g ( M y )) . As before, the chain map g induces a map of triangles j h ( ℓ ) n g (cid:15) (cid:15) M / / j h ( ℓ ) n / / g (cid:15) (cid:15) j h ( R ) η (cid:15) (cid:15) / / j h ( ℓ ) n [ − g [ − (cid:15) (cid:15) j h ( ℓ ) m I − A tE / / j h ( ℓ ) m p / / j h ( L ( E )) ∂ / / j h ( ℓ ) m [ − . It follows from (6.5) and (6.6) that g f restricts to the identity on Ker( I − A tE )and that there is a homomorphism h : Z m → Z m with h ◦ ( I − A tE ) = id − g f and M ◦ h = id − g f . Hence ∂ (1 − ηξ ) = (1 − g f [ − ∂ = h (1 − A tE ) p = 0(1 − ηξ ) p = p (1 − f g ) = p (1 − A tE ) h = 0 . Therefore there exist ζ ∈ kk h ( L ( E ) , ℓ m ) and ζ ∈ kk h ( ℓ m , L ( E )) such that 1 − ηξ = pζ = ζ ∂ . In particular, (1 − ηξ ) = ζ ∂pζ = 0, and therefore ηξ isan isomorphism. Similarly, ξη is an isomorphism, and so ξ is an isomorphism,concluding the proof. (cid:3) Remark . One may also ask whether an isomorphism ξ : BF ( E ) ⊗ KH h ( ℓ ) ∼ −→ BF ( F ) ⊗ KH h ( ℓ ) lifts to a kk h -isomorphism ξ : j h ( L ( E )) ∼ −→ j h ( L ( F )). The argu-ment of Lemma 6.4 proves that this is indeed the case if we additionally assume thatTor Z ( BF ( E ) , KH h ( ℓ )) = 0. The proof of Lemma 6.4 does not work for the analogueof the lemma with a matrix M with coefficients in Z [ σ ] and BF ( E ) substituted for BF ( E ). This is because a submodule of a free Z [ σ ]-module need not be free or evenprojective. However the problem disappears if Ker( M ) = Ker( I − σA tE ) = 0, andwe have the following. Lemma 6.8.
Let E be a graph with | E | < ∞ , and let n , n ≥ , M ∈ Z [ σ ] n × n ,and R as in (6.1) . Assume that Ker M = Ker( I − σA tE ) = 0 . Let ξ : BF ( E ) ∼ −→ Coker( M ) be a group isomorphism. Then there exists an isomorphism ξ : j h ( L ( E )) ∼ −→ j h ( R ) such that ev( ξ ) = can ◦ ξ .Proof. The argument of the proof of Lemma 6.4 shows this. (cid:3)
LASSIFYING LEAVITT PATH ALGEBRAS 23
Let E be a graph with | E | < ∞ . Then BF ( E ) is finitely generated. Thus thereare r, n ≥ ≤ d , . . . , d n with d i \ d i +1 for all i such that(6.9) BF ( E ) = Z r ⊕ n M i =1 Z /d i Z . Theorem 6.10.
Let E be a graph such that E is finite. Assume that ℓ satisfiesthe λ -assumption 1.11. Let r , n and d , . . . , d n be as in (6.9) and let s = | sing( E ) | be the number of singular vertices. Then j h ( L ( E )) ∼ = j h ( L s ⊕ L r − s ⊕ n M i =1 L d i +1 ) Proof.
Apply Lemma 6.4 with M the Smith normal form of I − A tE and R = L s ⊕ L r − s ⊕ L ni =1 L d i +1 . (cid:3) As previously recalled from [10] and [14], for any graph E we have E -stablevariants of the universal homology theories j : Alg ℓ → kk and j h : Alg ∗ ℓ → kk h .In the following corollary, as well as in any other statement involving two graphs E and F with possibly infinitely many edges, j and j h are understood to be the E ⊔ F -stable ones. Theorem 6.11.
Let E and F be graphs with finitely many vertices and such that | sing( E ) | = | sing( F ) | . Let ξ : BF ( E ) ∼ −→ BF ( F ) be an isomorphism. As-sume that ℓ satisfies the λ -assumption 1.11. Then there exists an isomorphism ξ : j h ( L ( E )) ∼ −→ j h ( L ( F )) such that ev( ξ ) = can ξ .Proof. Apply Lemma 6.4 with M = I − A tF and R = L ( F ). (cid:3) Theorem 6.12.
Let E and F be graphs with finitely many vertices. Assume that ℓ satisfies the λ -assumption 1.11, that KH − ( ℓ ) = 0 and that the canonical map Z → KH ( ℓ ) is an isomorphism. Then the following are equivalent. i) j ( LE ) ∼ = j ( LF ) in kk . ii) j h ( LE ) ∼ = j h ( LF ) in kk h .Proof. If ii) holds, then the forgetul functor kk h → kk sends the isomorphism j h ( LE ) ∼ = j h ( LF ) to an isomorphism j ( LE ) ∼ = j ( LF ). If i) holds, then by [10,Corollary 6.11], BF ( E ) ∼ = BF ( F ) and | sing( E ) | = | sing( F ) | , so j h ( LE ) ∼ = j h ( LF )by Theorem 6.11. (cid:3) Remark . The analogue of Theorem 6.12 with LE and LF substituted for LE and LF does not hold. Indeed, for any ℓ , j ( L ) = j ( L − ) = 0 but for ℓ as inExample 4.5, j h ( L ) = j h ( L − ), by (5.8). Theorem 6.14.
Let E and F be graphs with finitely many vertices and such that Ker( I − σA tE ) = Ker( I − σA tF ) = 0 . Let ξ : BF ( E ) ∼ −→ BF ( F ) be an isomorphism.Assume that ℓ satisfies the λ -assumption 1.11. Then there exists an isomorphism ξ : j h ( L ( E )) ∼ −→ j h ( L ( F )) such that ev( ξ ) = can ξ .Proof. Apply Lemma 6.8 with M = I − σA tF and R = L ( F ). (cid:3) Proposition 6.15.
Assume that is invertible and -positive in ℓ . Let E be a finitegraph such that the Z [ σ ] -modules BF ( E ) and BF ( E ) are finite and isomorphic. Let z}|{ ∆ be as in Proposition 2.4.5. Then L ( E ) and L ( E ) are invertible elementsof kk h ( L ( E ) , L ( E )) and kk h ( L ( E ) , L ( E )) , and we have ( z}|{ ∆ L ( E ) ) / ◦ ( z}|{ ∆ L ( E ) /
2) = j h (id L ( E ) ) . In particular, j h ( L ( E )) is a retract of j h ( L ( E )) , and the following sequence is exact / / j h ( L ( E )) z}|{ ∆ L ( E ) / / j h ( L ( E )) − σ / / j h ( L ( E )) . Proof. If BF ( E ) is finite, then E is regular, so by Remark 4.1, BF ( E ) is isomorphicas a Z [ σ ]-module to M = Coker( I − ( A tE ) ) where σ acts as multiplication by A tE .Then M ∼ = BF ( E ) as Z [ σ ]-modules ⇐⇒ A tE acts trivially on M , in which case I + A tE descends to multiplication by 2, and we have(6.16) Im( I − A tE ) = Im( I − ( A tE ) ) = ( I + A tE )(Im( I − A tE )) . Hence I + A tE restricts to an injection on Im( I − A tE ), which has rank | E | since BF ( E ) is finite. Thus det( I + A tE ) = 0, which combined with (6.16) implies that( I + A tE ) x goes to zero in M if and only if x does. It follows that multiplicationby 2 on M is injective and therefore bijective since M is finite. Hence I + A tE is invertible and n = | M | is odd. Write n = 2 q −
1; then by the argument ofLemma 6.4, n j h (id L ( E ) ) = n j h (id L ( E ) ) = 0, so 2 j h (id L ( E ) ) and 2 j h (id L ( E ) ) areisomorphisms. The same argument but with Z [ σ ] substituted for Z , shows that2(1 − σ ) id L ( E ) = (1 − σ ) id L ( E ) = 0 , which by what we have already seen implies that (1 − σ ) id L ( E ) = 0. The propositionnow follows from Proposition 2.4.5 and Corollary 3.6. (cid:3) Remark . Let E and ℓ be as in Proposition 6.15. Observe that σ acts as 1 ⊗ σ on BF ( E ) ⊗ KH h ( ℓ ) ⊂ KH h ( L ( E )); this action is nontrivial in general, so in particular0 = 1 − σ ∈ kk h ( L ( E ) , L ( E )). For example ℓ = C satifies the hypothesis of theproposition, and σ acts on BF ( E ) ⊕ σ BF ( E ) = BF ( E ) ⊗ KH h ( C ) by interchangingthe summands; this action is nontrivial whenever BF ( E ) = 0. Remark . The proof of Proposition 6.15 shows that if a regular finite graph E with incidence matrix A satisfies the hypothesis of the proposition and BF ( E ) = 0,then(6.19) det( I + A ) ∈ {± } and det( I − A ) ∈ Z \ { , ± } . The converse is also true; if (6.19) holds, then I + A t : BF ( E ) → BF ( E ) ∼ = BF ( E )is a Z [ σ ]-module isomorphism. A concrete example is the graph with incidencematrix A E = (cid:20) (cid:21) . K invariants for ∗ -algebras Let A be a ∗ -algebra and x ∈ A . A projection in A is a self-adjoint idempotentelement; we write Proj ( A ) for the set of all projections in A . If p, q ∈ Proj ( A ), wewrite p ≥ q whenever the identities q = pq = qp hold. Two projections p, q ∈ A are(Murray-von Neumann) equivalent –and we write p ∼ q – if there is an element x ∈ A such that x ∗ x = p , xx ∗ = q . Such x can be chosen to be in pAq , in which case wecall it an ( M vN -) equivalence and write x : p ∼ −→ q . All the basic properties aboutequivalence of idempotents proved in [8, Section 4.2] hold verbatim for projections,with the same proofs. A partial isometry is an element x ∈ A such that xx ∗ x = x .Let R be a unital ∗ -algebra. An isometry in R is an element u such that u ∗ u = 1;a unitary is an invertible isometry. For 1 ≤ n ≤ ∞ , put Proj n ( R ) = Proj ( M n R )and consider the set of equivalence classes V n ( R ) ∗ = Proj n ( R ) / ∼ . If n < m , thenthe map V n ( R ) ∗ → V m ( R ) ∗ is injective, and V ∗∞ ( R ) = S n V ∗ n ( R ). The set V ∞ ( R ) ∗ forms an abelian monoid under orthogonal sum ; if x, y ∈ Proj ∞ ( R ) then there are LASSIFYING LEAVITT PATH ALGEBRAS 25 orthogonal projections p, q ∈ Proj ∞ ( R ) with x = [ p ] and y = [ q ] and x + y :=[ p + q ]. This is well-defined by the projection analogue of [8, Proposition 4.2.4]. Aprojection p ∈ Proj ( R ) is strictly full if there exist n ≥ x , . . . , x n ) ∈ R n such that P ni =1 x ∗ i px i = 1. Lemma 7.1.
Let R be a unital ∗ -algebra and p ∈ Proj ( R ) a strictly full projection.Then the inclusion pRp ⊂ R induces an isomorphism V ∞ ( pRp ) ∗ ∼ −→ V ∞ ( R ) ∗ .Proof. Consider the category P ( R ) whose set of objects is Proj ∞ ( R ) and wherea homomorphism q → q is an element x ∈ M ∞ R such that x ∗ x = q and xx ∗ ≤ q . We have a functor to abelian monoids F : P ( R ) → AbMon whichsends q
7→ V ( qM ∞ Rq ) ∗ . Note that any two homomorphisms q → q in P ( R )induce the same monoid homomorphism upon applying F . Hence if C ⊂ P ( R )is a subcategory such that for every q ∈ P ( R ) there is q → q ′ with q ′ ∈ C , thencolim C F = colim P ( R ) F . Apply this to the subcategory I ( R ) ⊂ P ( R ) whose ob-jects are the identity matrices 1 n and where a homomorphism 1 n → m is a matrix x ∈ R m × n whose columns are n consecutive vectors of the canonical basis of R m .Then colim P ( R ) F = colim I ( R ) V n ( R ) ∗ = V ∞ ( R ) ∗ . Similarly, if p ∈ Proj ( R ) isstrictly full, then V ∞ ( pRp ) ∗ = colim P ( pRp ) F ∼ −→ colim P ( R ) F = V ∞ ( R ) ∗ . (cid:3) Consider the group completion K ( R ) ∗ := ( V ∞ ( R ) ∗ ) + Now assume that the center of R satisfies the λ -assumption 1.11. The HermitianWitt-Grothendieck group K h ( R ) is defined as the group completion of the monoid(7.2) V h ∞ ( R ) = V ∞ ( M ± R ) ∗ thus we have(7.3) K h ( R ) = K ( M ± ( R )) ∗ . The inclusion ι + : R → M ± R induces a canonical homomorphism V ∞ ( R ) ∗ →V h ∞ ( R ). We may also regard V h ∞ ( R ) as the monoid of unitary isomorphism classes ofall pairs [( P, φ )] consisting of a finitely generated projective right module equippedwith a nondegenerate Hermitian form φ . In the same fashion, the monoid V ∞ ( R ) ∗ consists of the unitary classes of those P = ( P, φ ) for which there is an n ≥ P embeds as an orthogonal direct summand of the free module of rank n equipped with the standard Hermitian form h x, y i = n X i =1 x ∗ i y i . Observe that − R if and only if there is some n ≥ h x, x i = − x ∈ R n . In this case p : R n → R n , p ( y ) = − x h x, y i is an orthogonalprojection onto xR , and R n decomposes as the orthogonal direct sum of xR and P = Ker( p ). In particular the hyperbolic module of rank 2, H ( R ), embeds as anorthogonal summand in R n +1 . If follows that if − V ∞ ( R ) ∗ → V h ∞ ( R ) is an isomorphism, as is the induced map K ( R ) ∗ → K h ( R ). Remark . Let R be a unital ∗ -algebra. The involution defines an action of Z / Z on K ( R ). With no assumptions on ℓ , for any unital ∗ -algebra R , we havea canonical forgetful map forg : K ( R ) ∗ → K ( R ) Z / Z which maps the class ofa projection modulo M-vN equivalence of projections to its class modulo Mv-N equivalence of idempotents. If E is a graph with finitely many vertices and v ∈ reg( E ), then the identity [ v ] = P s ( e )= v [ r ( v )] holds in K ( LE ) ∗ . Hence we have acanonical group homomorphismcan ′ : BF ( E ) → K ( LE ) ∗ . Next we introduce a hypothesis on ℓ that implies that can ′ is an isomorphism ontothe fixed points of K ( L ( E )) ∗ under the involution.Let Z = K ( Z ) → K ( ℓ ) be the canonical map. We have a chain complex(7.6) 0 / / Z / / K ( ℓ ) −∗ / / K ( ℓ )Recall that a sequence of abelian groups is pure exact if it is exact and remains soupon tensoring it with any abelian group. Assumption . ℓ is regular supercoherent and (7.6) is pure exact. Example 7.8.
Assumption 7.7 is satisfied when ℓ is regular supercoherent and Z → K ( ℓ ) is an isomorphism, and also when ℓ = inv( ℓ ) for some regular supercoherent ℓ such that Z → K ( ℓ ) is an isomorphism. Remark . If ℓ satisfies Assumption 7.7 then K ( LE ) Z / Z = BF ( E ), so we mayview can ′ as a homomorphism K ( LE ) Z / Z → K ( LE ) ∗ . In this case the compositeforg ◦ can ′ is the identity of K ( LE ) Z / Z .8. Strictly properly infinite ∗ -algebras, purely infinite simplegraphs, and their K invariants Let C n be the Cohn path algebra of R n (0 ≤ n ≤ ∞ ). A unital ∗ -algebra R is strictly properly infinite if there is a unital ∗ -homomorphism C → R , orequivalently, a unital ∗ -homomorphism C ∞ → R . Observe that if R is strictlyproperly infinite and φ : R → S is a unital ∗ -homomorphism, then S is strictlyproperly infinite too. In particular, if R is strictly properly infinite, then M n R and M n M ± R are strictly properly infinite for all 1 ≤ n < ∞ . A projection p ∈ R is strictly properly infinite if pRp is strictly properly infinite. Equivalently p is strictlyproperly infinite if there are nonzero orthogonal projections p , p ∈ pRp , such that p ∼ p ∼ p . Remark . The sum of orthogonal strictly properly infinite projections in a unital ∗ -algebra is again strictly properly infinite. In particular, if R contains orthogonalstrictly properly infinite projections p , . . . , p n such that P ni =1 p i = 1, then R isstrictly properly infinite.We say that a graph E is simple if it is cofinal [2, Definitions 2.9.4] and everycycle in E has an exit [2, Definitions 2.2.2]. A simple graph having at least onecycle is called purely infinite simple . If ℓ is a field, then L ( E ) is (purely infinite)simple if and only if E is [2, Theorems 2.9.1 and 3.1.10 and Lemma 2.9.6]. Example 8.2.
The graph Υ of Example 4.7 is purely infinite simple.
Lemma 8.3.
Let E be a finite graph and v ∈ sour( E ) \ sink( E ) . Let E /v be thesource elimination graph of E ( [2, Definition 6.3.26] ). Then i) The element − v ∈ L ( E ) is a strictly full projection. ii) The inclusion E /v ⊂ E induces ∗ -isomorphisms L ( E /v ) → pL ( E ) p and L ( E /v ) → pL ( E ) p . iii) E is (purely infinite) simple if and only if E /v is. LASSIFYING LEAVITT PATH ALGEBRAS 27
Proof.
Let p = 1 − v ; the argument of the proof of [4, Proposition 1.4] shows thatthe canonical homomorphism L ( E /v ) → L ( E ) induced by the inclusion, which isa ∗ -homomorphism with respect to both the standard and the twisted involution,corestricts to an isomorphism onto pL ( E ) p ; this proves ii). Because v is a source butnot a sink, we have v = P s ( e )= v epe ∗ , whence 1 = p + P s ( e )= v epe ∗ and therefore p is strictly full, proving i). If ℓ is a field, the (pure infinite) simplicity of E isequivalent to that of L ( E ), which by ii), is Morita equivalent to L ( E /v ). The proofof iii) and of the lemma is concluded by using that simplicity and purely infinitesimplicity are preserved by Morita equivalence [7, Corollary 1.7]. (cid:3) Lemma 8.4.
Let E be a finite, purely infinite simple graph. Then every vertex of E is a strictly properly infinite projection of L ( E ) and L ( E ) .Proof. If v ∈ E is in a cycle, then there is a cycle α v based at v . Let α v be aclosed path starting at v , following α v up to an exit, taking the exit, then comingback to the cycle –as is possible due to cofinality of E – and following α v again until v . Upon replacing α v and α v by their squares, if necessary, we may assume thattheir lengths are even, so that α iv = ( α iv ) ∗ for i = 1 ,
2. Then ( α iv ) ∗ α jv = δ i,j v , and v is a strictly properly infinite projection of both L ( E ) and L ( E ). If sour( E ) = ∅ ,every vertex is in a cycle, and the lemma follows. Otherwise, we can proceed bysource elimination until we arrive to a purely infinite simple graph without sources.At each step, the source eliminated is equal to a sum of projections of the form ee ∗ with e ∈ E and r ( e ) ∈ E /v . If x , x ∈ r ( e ) L ( E ) r ( e ) are orthogonal isometriesfor either of the involutions, then ex e ∗ , ex e ∗ ∈ ee ∗ L ( E ) ee ∗ are again orthogonalisometries, for the same involution. Since the sum of orthogonal strictly properlyinfinite projections is again strictly properly infinite, we get that every vertex is astrictly properly infinite projection. (cid:3) Corollary 8.5. If E is finite and purely infinite simple, then L ( E ) and L ( E ) arestrictly properly infinite.Proof. Immediate from Lemma 8.4 and Remark 8.1. (cid:3)
Let R ∈ Alg ∗ ℓ be unital and p ∈ Proj ( R ). Following [8, Section 6.11] we saythat p is very full if there exists q ∈ Proj ( R ) such that p ≥ q ∼
1. Observethat any projection equivalent to a very full one is again very full. We write
Proj f ( R ) ⊂ Proj ( R ) for the subset of very full projections. Example 8.6.
Let
R, S ∈ Alg ∗ ℓ be unital, p ∈ Proj f ( R ), q ∈ Proj f ( S ). Then p ⊗ q ∈ Proj f ( R ⊗ S ). If φ : R → S is a ∗ -homomorphism with φ (1) ∈ Proj f ( R ),then φ ( p ) ∈ Proj f ( S ). Lemma 8.7.
Let E be a finite, purely infinite simple graph. Then every vertex of E is a very full projection of L ( E ) and L ( E ) .Proof. Let v ∈ E . By Lemma 8.4 we may choose an element x w ∈ vL ( E ) v foreach w ∈ E \ sour( E ) such that x ∗ w x w ′ = δ w,w ′ v . Because E is purely infinitesimple, for every w ∈ E \ sour( E ), there is a path α w from v to w . Then(8.8) x = X w ∈ E \ sour( E ) x w α w satisfies xx ∗ ≤ v and p := x ∗ x = P w ∈ E \ sour( E ) w . By Lemma 8.4 and Remark8.1 R = pL ( E ) p is strictly properly infinite. Hence we may choose a family oforthogonal isometries R ⊃ { y e : s ( e ) ∈ sour( E ) } ∪ { y w : w ∈ E \ sour( E ) } ; y = P s ( e ) ∈ sour( E ) y e e ∗ + P w ∈ E \ sour( E ) y w w . Then yy ∗ ≤ p and y ∗ y = 1; hence p is very full and therefore v is a very full projection of L ( E ). To prove that it is also a very full projection of L ( E ) proceed as follows. Choose the x w asisometries of R = pL ( E ) p and the α w of even length; then (8.8) satisfies xx ≤ v and xx = p . Next take a family orthogonal partial isometries indexed by thevertices that are not sources and all the paths β of length 2 with s ( β ) ∈ sour( E ).Then y = P s ( β ) ∈ sour( E ) y β β ∗ + P w ∈ E \ sour( E ) y w w satisfies yy ≤ p and yy = 1. (cid:3) Let R be a unital, strictly pure infinite ∗ -algebra. Consider the set of equivalenceclasses V ( R ) ⊃ V f ( R ) = { [ p ] : p ∈ R very full } . Proposition 8.9 (cf. [8, Theorem 6.11.7]) . Let R be a unital, strictly properlyinfinite ∗ -algebra. Then the orthogonal sum makes V f ( R ) into a group, canonicallyisomorphic to K ( R ) ∗ .Proof. The argument of the proof of [8, Theorem 6.11.7] shows this. (cid:3)
Corollary 8.10.
Let R be as in Proposition 8.9. Further assume that the cen-ter of R satisfies the λ -assumption 1.11. Then V f ( M ± R ) is a group, canonicallyisomorphic to K h ( R ) .Proof. Combine Proposition 8.9 and (7.3). (cid:3)
Let
A, B ∈ Alg ∗ ℓ and let ι : B → M B be the upper left hand corner inclusion.Two ∗ -homomorphisms φ, ψ : A → B are M - ∗ -homotopic –and we write φ ∼ ∗ M ψ –if ι φ and ι ψ are ∗ -homotopic. We write [ A, B ] ∗ M for the set of M - ∗ -homotopyclasses of ∗ -homomorphisms.Let R be a strictly properly infinite ∗ -algebra. By definition, there are s , s ∈ R such that s ∗ i s j = δ i,j . Let ⊞ : R ⊕ R → R, a ⊞ b = s as ∗ + s bs ∗ . Let φ, ψ : A → R be ∗ -homomorphisms. Put(8.11) φ ⊞ ψ : A → R, ( φ ⊞ ψ )( a ) = φ ( a ) ⊞ ψ ( a ) . Lemma 8.12.
Let A and R ∈ Alg ∗ ℓ , with R strictly properly infinite. Then (8.11) makes [ A, R ] ∗ M into an abelian monoid.Proof. The proof is the same as that of [10, Lemma 2.5]. (cid:3)
Let A and R be as in Lemma 8.12 and let φ , φ : A → R be ∗ -homomorphisms;we say that φ and φ are stably M -homotopic , and write φ ∼ sM φ , if thereexists a ∗ -homomorphism such that(8.13) φ ⊞ ψ ∼ ∗ M φ ⊞ ψ. In other words, φ ∼ sM φ means that the M -homotopy classes of φ and φ goto the same element in the group completion(8.14) [ A, R ] ∗ M → ([ A, R ] ∗ M ) + . Lemma 8.15.
Let R be a unital ∗ -algebra and let p ∈ Proj f ( R ) . Then the inclusion inc : pRp ⊂ R is an M -homotopy equivalence.Proof. Because p is very full, there exists an isometry x with xx ∗ ≤ p . Then xRx ∗ ⊂ pRp and for the corestriction φ of ad( x ) to pRp , we have inc ◦ φ = ad( x ) ∼ ∗ M id R and φ ◦ inc = ad( xp ) ∼ ∗ M id pRp . (cid:3) LASSIFYING LEAVITT PATH ALGEBRAS 29 Lifting K -maps to ∗ -algebra maps Let E be a graph, R a strictly properly infinite unital ∗ -algebra and φ : L ( E ) → R an algebra homomorphism. We say that φ has property (P) if(9.1) { φ ( ee ∗ ) : e ∈ E } ∪ { φ ( v ) : v ∈ sing( E ) } ⊂ Proj f ( R ) . Example 9.2.
Let E be a finite, purely infinite graph, let φ : L ( E ) → R be ∗ -homomorphism and put p = φ (1). By Lemma 8.7, for every element q in the lefthand side of the inclusion (9.1) there is a projection q ′ ≤ q such that q ′ ∼ p . Hence φ has property (P) if and only if p ∈ Proj f ( R ). Remark . If φ, ψ : LE → R are ∗ -homomorphisms and both have property (P),then so does their sum (8.11). Thus the subset[ LE, R ] ∗ M ⊃ [ LE, R ] PM = { [ φ ] : φ has property (P) } is a subsemigroup.In the following theorem and elsewhere, if E is a finite graph, we write[1] = [1] E = X v ∈ E [ v ] ∈ BF ( E ) . Theorem 9.4.
Let E be a countable graph and R a unital ∗ -algebra. Assume that R is strictly properly infinite. Let ξ : BF ( E ) → K ( R ) ∗ be a group homomorphismand let can ′ : BF ( E ) → K ( LE ) ∗ be the canonical map of Remark 7.5. Thenthere is a ∗ -homomorphism φ : L ( E ) → R with property (P) of (9.1) such that K ( φ ) ∗ ◦ can ′ = ξ . If furthermore E is finite and p ∈ Proj f ( R ) is such that ξ ([1]) = [ p ] , then φ can be chosen so that, in addition to the above properties, alsosatisfies φ (1) = p .Proof. Because R is strictly properly infinite by assumption, it has a sequence oforthogonal projections equivalent to 1. Hence in view of Proposition 8.9 and thecountability assumption on E , there are orthogonal very full projections { p e : e ∈ E } ∪ { p v : v ∈ sing( E ) } ⊂ Proj f ( R ) such that, in V f ( R ) = K ( R ) ∗ , ξ [ v ] = [ p v ]and ξ [ ee ∗ ] = [ p e ] for all v ∈ sing( E ) and e ∈ E . If E is row-finite, then proceedingas in the proof of [11, Theorem 3.1] we obtain a ∗ -homomorphism φ : LE → R asrequired. Also as in loc.cit. the case of general countable E reduces to the row-finitecase via desingularization. Finally if E is finite, p ∈ Proj ( R ) and ξ ([1]) = [ p ],then by what we have just seen there is a ∗ -homomorphism ψ : LE → R withproperty (P) such that K ∗ ( ψ ) ◦ can ′ = ξ . Let q = ψ (1); choose an MvN equivalence y : q ∼ −→ p . Consider the ∗ -homomorphism ad( y ) : qRq → pRp ⊂ R .Let φ := ad( y ) ◦ ψ : L ( E ) → R ; then φ has property (P) and φ (1) = p . Moreover, K ( φ ) ∗ = K ( ψ ) ∗ , hence we also have K ( φ ) ∗ ◦ can ′ = ξ . (cid:3) Remark . The proof of 9.4 uses the fact that the edges of E are partial isometriesin L ( E ); since this is no longer true in L ( E ), the proof does not work with L ( E )substituted for L ( E ). Remark . If ℓ satisfies Assumption 7.7, then BF ( E ) ∼ = K ( L ( E )) Z / Z , so in thiscase, Theorem 9.4 can be equivalently formulated with K ( L ( E )) Z / Z substitutedfor BF ( E ). Corollary 9.7.
Let E , R and ℓ be as in Theorem 9.4. Further assume that ℓ sat-isfies the λ -assumption 1.11. Let ξ : BF ( E ) → K h ( R ) be a group homomorphism.Then there is a ∗ -homomorphism ψ : L ( E ) → M ± R with property (P) such that K h ( ψ ) ◦ can = K h ( ι + ) ◦ ξ . If furthermore E is finite and ξ ([1]) = [ p ] for some p ∈ Proj f ( M ± R ) , then we can choose ψ so that, in addition to the above properties,also satisfies ψ (1) = p . Proof.
By Theorem 9.4 there is a ∗ -homomorphism ψ : LE → M ± R –which can bechosen so that ψ (1) = p – such that K ( ψ ) ∗ ◦ can ′ = ξ . Hence K h ( ψ ) ◦ can = K ( M ± ψ ) ∗ ◦ K ( ι + ) ∗ ◦ can ′ = K ( ι + ) ∗ ◦ K ( ψ ) ∗ ◦ can ′ = K ( ι + ) ∗ ◦ ξ. Let u = − ∈ M ± M ± One checks that u is unitary and that ad( u ) ◦ ι + = M ± ι + . Hence K ( ι + ) ∗ = K h ( ι + ) = K ( M ± ι + ) ∗ on K h ( R ) = K ( M ± ) ∗ , concluding the proof. (cid:3) Theorem 9.8.
Assume that ℓ satisfies Assumption 7.7. Let E and F be graphswith finitely many vertices and let ξ : BF ( E ) → BF ( F ) be a group homomorphism.Assume that F is purely infinite simple. Then there is a ∗ -homomorphism φ : L ( E ) → L ( F ) with property ( P ) such that K ( φ ) Z / Z = ξ ; if ξ ([1] E ) = [1] F then φ can be chosen to be unital.Proof. Let η = can ′ ◦ ξ : BF ( E ) → K ( L ( F )) ∗ . By Theorem 9.4, there is a ∗ -homomorphism φ : L ( E ) → L ( F ) such that K ( φ ) ∗ ◦ can ′ = η , and we can choose φ unital if ξ is. Since ℓ satisfies Assumption 7.7, we may identify BF ( E ) and BF ( F )with K ( L ( E )) Z / Z and K ( L ( F )) Z / Z , and we have a commutative diagram K ( L ( E )) Z / Z ξ (cid:15) (cid:15) can ′ / / K ( L ( E )) ∗ forg / / K ( φ ) ∗ (cid:15) (cid:15) K ( L ( E )) Z / Z K ( φ ) Z / Z (cid:15) (cid:15) K ( L ( F )) Z / Z can ′ / / K ( L ( F )) ∗ forg / / K ( L ( F )) Z / Z By Remark 7.5 the rows compose to identity maps. (cid:3)
In the next lemma and elsewhere, if E is a finite graph, we write DL ( E ) for the diagonal subalgebra of L ( E ), DL ( E ) = ( M v ∈ sink( E ) ℓv ) ⊕ ( M e ∈ E ℓee ∗ ) . The following lemma will be used later on, in the proof of Theorem 13.7.
Lemma 9.9.
Let E be finite graph, R ∈ Alg ∗ ℓ strictly properly infinite and φ, ψ : L ( E ) → R ∗ -homomorphisms with property (P). If K ( φ ) ∗ ◦ can ′ = K ( ψ ) ∗ ◦ can ′ then there exists a ∗ -homomorphism ψ ′ : L ( E ) → R with property (P) such that K ( φ ) ∗ ◦ can ′ = K ( ψ ′ ) ∗ ◦ can ′ , ψ ′ ∼ ∗ M ψ and ψ ′| DL ( E ) = φ | DL ( E ) . The same istrue with L ( E ) substituted for L ( E ) .Proof. Put ξ = K ( φ ) ∗ ◦ can ′ . Let p = φ (1), q = ψ (1) and inc p : pRp ⊂ R andinc q : qRq ⊂ R the inclusions. By Proposition 8.9 there is a partial isometry x ∈ R which implements and M vN equivalence x : q ∼ −→ p . Let ψ = inc p ad( x ) ψ ; then ψ (1) = p , K ( ψ ) ∗ can ′ = ξ and ψ ∼ ∗ M ψ , by the ∗ -analogue of [10, Lemma2.3]. By the argument of [11, Proposition 3.5], there is a partial isometry y ∈ pRp such that ψ ′ := inc p ad( y ) ψ agrees with φ on DL ( E ). Another application of the ∗ -analogue of [10, Lemma 2.3] finishes the proof. (cid:3) LASSIFYING LEAVITT PATH ALGEBRAS 31
Unitary K of a strictly properly infinite ∗ -algebra Let R be a unital ∗ -algebra. For n ≥
1, write U n R = U ( M n R ) for the group ofunitary elements; set U ∞ ( R ) = colim n U n ( R ). Put(10.1) K ( R ) ∗ = U ∞ ( R ) ab , K h ( R ) = K ( M ± R ) ∗ . Lemma 10.2.
Let R be a unital ∗ -algebra and p ∈ R a strictly full projection.Then the inclusion pRp → R induces an isomorphism K ( pRp ) ∗ ∼ −→ K ( R ) ∗ .Proof. The proof is similar to that of Lemma 7.1 once we observe that K ( R ) ∗ =colim P ( R ) U ( pRp ) ab . (cid:3) Remark . If R ∈ Alg ∗ ℓ and − R , then K ( R ) ∗ → K h ( R ) is anisomorphism, by the same argument as used in the paragraph after Lemma 7.1 toestablish the analogous statement for K ∗ and K h . Proposition 10.4.
Let R be a strictly properly infinite ∗ -algebra. Let N ( R ) ⊳ U ( R ) be the smallest normal subgroup containing the subset { u − ( xux ∗ + 1 − xx ∗ ) : u ∈ U ( R ) , x ∗ x = 1 } . Then K ( R ) ∗ = U ( R ) / N ( R ) .Proof. We keep the notation as in the proof of Lemma 10.2. Because R is strictlyproperly infinite, the full subcategory I of P ( R ) whose only object is the identityelement 1 ∈ R is cofinal. Hence K ( R ) ∗ = colim I U ( R ) ab = U ( R ) / [ U ( R ) : U ( R )] · N ( R ) . It remains to show that N ( R ) ⊃ [ U ( R ) , U ( R )]. Let s , s be orthogonal isometriesand let u, v ∈ U ( R ). Modulo N ( R ), uv ≡ ( s us ∗ + (1 − s s ∗ )) · ( s vs ∗ + (1 − s s ∗ ))= s us ∗ + s vs ∗ + 1 − X i =1 s i s ∗ i =( s vs ∗ + (1 − s s ∗ ))( s us ∗ + (1 − s s ∗ )) ≡ vu. (cid:3) Put KV ( R ) ∗ = Coker(ev − ev : K ( R [ t ]) ∗ → K ( R ) ∗ ) , KV h ( R ) ∗ = KV ( M ± R ) ∗ . Let U ( R ) = { u ∈ U ( R ) | ( ∃ U ∈ U ( R [ t ])) U (0) = 1 , U (1) = u } , U n ( R ) = U ( M n R ) ( n ≥ . Lemma 10.5.
Let R ∈ Alg ∗ ℓ be strictly properly infinite. Then KV ( R ) ∗ = U ( R ) / U ( R ) ∩ U ( R ) . Proof.
By Proposition 10.4, we have KV ( R ) ∗ = U ( R ) / N ( R ) · ( U ( R ) ∩ U ∞ ( R ) ) . It follows from (the ∗ -analogue of) [10, Lemma 2.3] that if u ∈ U ( R ) and x ∈ R is such that x ∗ x = 1, then there exists U ( t ) ∈ U ( R [ t ]) such that U (0) = u and U (1) = 1 − xx ∗ + xux ∗ . In particular, N ( R ) ⊂ U ( R ) ∩ U ( R ) . It remains toshow that U ( R ) ⊃ U ( R ) ∩ U ∞ ( R ) . Let u ∈ U ( R ) and suppose that for some n ≥ U ( t ) ∈ U n ( R [ t ]) such that U (0) = 1 and U (1) = u ⊕ n − .Because R is strictly properly infinite, there exists x = ( x , . . . , x n ) ∈ R × n suchthat xx ∗ = 1 ∈ R and x ∗ x = 1 n ∈ M n R . Then V ( t ) = 1 − xx ∗ + xU ( t ) x ∗ ∈ U ( R [ t ])satisfies V (0) = 1 and V (1) = 1 − xx ∗ + x ( u ⊕ n − ) x ∗ = 1 − x x ∗ + x ux ∗ . By the first part of the proof, V (1) is connected to u by a path W ( t ) ∈ U ( R [ t ]);hence W ( t ) V (1 − t ) − connects 1 to u . (cid:3) Poincar´e duality
Let E be a finite graph. The dual graph E t is the graph with E it = E i for i = 0 , s t = r and r t = s . Write e t for an edge e ∈ E regarded as an edge of E t . The purpose of this section is to prove Theorem 11.2,which is an algebraic version of a similar result for graph C ∗ -algebras [15]. Firstwe need the following lemma. Lemma 11.1.
Let π : R → S be a surjective, unital homomorphism of ∗ -algebras,set I = Ker( π ) and let ∂ : K h ( S ) → K h ( I ) be the connecting map. Let u ∈ U n ( S ) .Assume that there exists a partial isometry ˆ u ∈ M n R such that π (ˆ u ) = u . Then ∂ι + [ u ] = ι + ([1 − ˆ u ∗ ˆ u ] − [1 − ˆ u ˆ u ∗ ]) .Proof. For every pair of elements ˆ u, ˆ v ∈ M n R such that π (ˆ u ) = u and π (ˆ v ) = u ∗ ,we can lift diag( u, u ∗ ) ∈ U ( M n S ) to an elementary matrix h = h (ˆ u, ˆ v ) ∈ E n R ; aformula for this matrix is given in [9, Formula (17)]. One checks that if ˆ u is a partialisometry, then h (ˆ u ) := h (ˆ u, ˆ u ∗ ) ∈ U n R . Thus if p ∈ M n R is the identity matrix, wehave ∂ ( ι + [ u ]) = ι + ([ad( h (ˆ u ))( p )] − [ p ]), and one computes that [ad( h (ˆ u ))( p )] − [ p ] =[1 − ˆ u ∗ ˆ u ] − [1 − ˆ u ˆ u ∗ ] ∈ K ( I ) ∗ . (cid:3) Theorem 11.2.
Let E be a finite graph without sinks or sources. Then the functors − ⊗ Ω L ( E t ) and − ⊗ Ω L ( E t ) : kk h → kk h are right adjoint to the functors LE ⊗ − and LE ⊗ − : kk h → kk h . Thus for every R, S ∈ Alg ∗ ℓ there are naturalisomorphisms of KH h ( ℓ ) -modules kk h ( R ⊗ LE, S ) ∼ −→ kk h ( R, S ⊗ L ( E t )) ,kk h ( R ⊗ LE, S ) ∼ −→ kk h ( R, S ⊗ L ( E t )) . Proof.
Let P be the set of finite paths in E and P ≥ ⊂ P the subset of paths ofpositive length; let X = ( P ≥ ) + be the pointed set obtained by adding a basepoint • . Let π : Γ X → Σ X be the projection. For each e ∈ E , put ρ ( e t ) = π ( X α ∈P s ( e ) ǫ αe,α ) , ρ ( e ) = π ( X α ∈P r ( e ) ǫ eα,α ) . One checks (as in [15, Proposition 4.2]) that the assignments e t ρ ( e t ) and e ρ ( e ) extend to unital ∗ -homomorphisms ρ : L ( E t ) → Σ X and ρ : L ( E ) → Σ X and that ρ ( a ) and ρ ( b ) commute for every a ∈ L ( E t ) and b ∈ L ( E ). Hence for E = L ( E t ) ⊗ L ( E ) we have a ∗ -homomorphism ρ : E → Σ X , which defines a class κ = j h ( ρ )[+1] ∈ kk h (Ω L ( E t ) ⊗ L ( E ) , ℓ ) = kk h − ( L ( E t ) ⊗ L ( E ) , ℓ ). Hence we havea homomorphism of KH h ( ℓ )-modules(11.3) kk h ( R, S ⊗ L ( E t )) → kk h ( R ⊗ L ( E ) , S ) , ξ ( S ⊗ κ ) ◦ ( ξ ⊗ L ( E )) . Consider the elements p = P v ∈ E v ⊗ v and u = P e ∈ E e ⊗ e ∗ t of E ′ = L ( E ) ⊗ L ( E t ).One checks that u ∈ U ( p E ′ p ). Hence u = u + 1 − p ∈ U ( E ′ ); write ∇ for the imageof the class [ u ] ∈ K ( E ′ ) ∗ under the composite of canonical maps K ∗ ( E ′ ) → K h ( E ′ ) → KH h ( E ′ ) = kk h ( ℓ, E ′ ) . We have another KH h ( ℓ )-linear homomorphism(11.4) kk h ( R ⊗ L ( E ) , S ) → kk h ( R, S ⊗ L ( E t )) , η (( η ⊗ L ( E t ))[+1]) ◦ ( R ⊗ ∇ ) . LASSIFYING LEAVITT PATH ALGEBRAS 33
To prove that (11.3) and (11.4) are isomorphisms, it suffices to establish the fol-lowing identities ( κ ⊗ L ( E t ))[+1] ◦ ( L ( E t ) ⊗ ∇ ) = − id j h ( L ( E t )) , (11.5) ( L ( E ) ⊗ κ )[+1] ◦ ( ∇ ⊗ L ( E )) = − id j h ( L ( E )) . Consider the cone extension M X / / Γ X / / / / Σ X . Tensor this extension on the left with L ( E t ); the first identity of (11.5) boils downto the assertion that the index map KH h ( LE t ⊗ Σ X ) → KH ( LE t ⊗ M X ) ∼ = KH h ( LE t ) maps the class of P e ρ (1 LE t ⊗ e ) ⊗ e ∗ t to minus the class of ι + (1); thesecond identity is the analogous statement for the index of the class of P e e ⊗ ρ ( e ∗ t ⊗ L ( E ) ). Both are straightforward using Lemma 11.1. Thus the first adjunction ofthe theorem is proved. Observe that the homomorphisms ρ and ρ defined aboveare also ∗ -homomorphisms for the involutions of L ( E ), L ( E t ) and Σ X . Notice alsothat the elements p and v are a projection and a unitary also for the involution of L ( E ) ⊗ L ( E t ). Hence the identities (11.5) also prove the second adjunction of thetheorem. (cid:3) Corollary 11.6.
Let E and S be as in Theorem 11.2. Assume that S is unitaland contains a central element x such that xx ∗ = − . There there is a naturalisomorphism of KH h ( ℓ ) -modules kk h ( L ( E ) , S ) ∼ −→ kk h ( L ( E ) , S ) . Proof.
Using Theorem 11.2 at the first and third steps and Example 2.3.3 at thesecond step, we obtain natural isomorphisms kk h ( L ( E ) , S ) ∼ −→ KH h ( S ⊗ L ( E t )) ∼ −→ KH h ( S ⊗ L ( E t )) ∼ −→ kk h ( L ( E ) , S ) . (cid:3) Universal coefficient theorem
Let E be a graph. Put BF ∨ ( E ) = Coker( I t − σA E ) , BF ∨ ( E ) = BF ∨ ( E ) ⊗ Z [ σ ] Z . Let R ∈ Alg ∗ ℓ ; consider the mapsev : kk h ( LE, R ) → hom( BF ( E ) , KH h ( R )) , (12.1) ev : kk h ( L ( E ) , R ) → hom( BF ( E ) , KH h ( R )) ξ ξ = KH h ( ξ ) ◦ can . Theorem 12.2.
Let E be a graph such that | E | < ∞ and let R ∈ Alg ∗ ℓ . Then themaps (12.1) are surjective and fit into exact sequences → KH h ( R ) ⊗ BF ∨ ( E ) → kk ( L ( E ) , R ) → hom( BF ( E ) , KH h ( R )) → , (12.3)0 → KH h ( R ) ⊗ Z [ σ ] BF ∨ ( E ) → kk ( L ( E ) , R ) → hom Z [ σ ] ( BF ( E ) , KH h ( R )) → . Proof.
Applying kk h ( − , R ) to the triangle of Theorem 3.5, and using that for anyfinite set X ,(12.4) kk h ( ℓ X , R ) ∼ = hom( Z X , KH h ( R )) = hom Z [ σ ] ( Z [ σ ] X , KH h ( R ))one obtains exact sequences as in the theorem. We claim that the surjectionstherein agree with the evaluation maps (12.1). For the graph consisting of a singlevertex and no edges, the Leavitt path algebra is just ℓ , and the map (12.1) isprecisely the isomorphism of [14, Proposition 4.1]. Using the isomorphism (12.4) for X ∈ { E , reg( E ) } and taking the kernel of the map induced by I − A tE yieldsthe surjection in (12.3); by naturality, this agrees also with (12.1), proving theclaim. (cid:3) Lemma 12.5.
Let E and R be as in Theorem 11.2. Then the following diagrams,where the horizontal maps are as in Theorem 12.2, the slanted ones as in Theorem4.2 and the vertical ones as in Theorem 11.2, commute. BF ( E ) ⊗ KH h ( R ) 4 . ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ . / / kk h ( LE, R ) ≀ . (cid:15) (cid:15) KH h ( R ⊗ L ( E t )) BF ( E ) ⊗ Z [ σ ] KH h ( R )4 . ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ . / / kk h ( LE, R ) ≀ . (cid:15) (cid:15) KH h ( R ⊗ L ( E t )) Proof.
For v ∈ E , the slanted arrows map an element [ v ] ⊗ ξ to the cup product[ v ] ∪ ξ . Consider the algebra extension E of L ( E ) ⊗ L ( E t ) which results by applying ⊗ L ( E t ) to the Cohn extension (2.5.4). The composite of the horizontal and verticalarrows in the first diagram is the cup product with the index η = ∂ E ([ u ]) ∈ KH h ( ℓ E ⊗ L ( E t )) of the KH h -class of the element u appearing in the proof ofTheorem 11.2. A straightforward calculation, using Lemma 11.1, shows that η =[ P v ∈ E χ v ⊗ v ]. It follows that the first diagram commutes. The same argument,substituting L ( E ) and L ( E t ) for L ( E ) and L ( E t ) and the extension (2.5.5) for(2.5.4), proves that also the second diagram commutes. (cid:3) Lifting kk h -maps to algebra maps Let E be a finite graph such that sink( E ) = ∅ . Set L ( E ) = L ( E ) , L ( E ) = L ( E ) . For ǫ ∈ { , } , let φ : L ǫ ( E ) → R be a unital ∗ -algebra homomorphism with R strictly properly infinite. Assume that φ has property (P) of (9.1). Set R φ = { x ∈ R : φ ( ee ∗ ) x = xφ ( ee ∗ ) , for all e ∈ E } (13.1) = ⊕ e ∈ E φ ( ee ∗ ) Rφ ( ee ∗ ) . Because φ has property (P), φ ( ee ∗ ) ∈ Proj f ( R ) for all e ∈ E , whence the inclusion φ ( ee ∗ ) Rφ ( ee ∗ ) ⊂ R induces an isomorphism in K ∗ , by Lemma 10.2. It follows thatthe direct sum R φ ⊂ R E of those inclusions induces an isomorphism(13.2) K ( R φ ) ∗ = M e ∈ E K ( φ ( ee ∗ ) Rφ ( ee ∗ )) ∗ ∼ −→ ( K ( R ) ∗ ) E . Let f : X → Y be a map between finite sets and M an abelian group. We write f ∗ : Z X → Z Y , f ∗ ( χ x ) = χ f ( x ) ; we shall abuse notation and also write f ∗ for f ∗ ⊗ M . In particular the source map s : E → E induces a homomorphism s ∗ : K ( R ) E → K ( R ) E . Consider the composite(13.3) ∂ : K ( R φ ) ∗ ∼ = ( K ( R ) ∗ ) E s ∗ −→ ( K ( R ) ∗ ) E → K h ( R ) E → kk h ( L ǫ ( E ) , R ) LASSIFYING LEAVITT PATH ALGEBRAS 35
Let(13.4) u = ( u e ) e ∈ E ∈ U ( R φ ) = M e ∈ E U ( φ ( ee ∗ ) Rφ ( ee ∗ ));consider the ∗ -homomorphism(13.5) φ u : L ǫ ( E ) → R, φ u ( e ) = uφ ( e ) . Lemma 13.6.
Assume that E is purely infinite simple and that R is strictly prop-erly infinite. Then j h ( φ u ) = j h ( φ ) + ∂ ([ u ])) Proof.
Let n = | sour( E ) | ; we shall prove the lemma by induction on n . First weassume that n = 0. By Lemma 12.5, the composite of ∂ with the isomorphism(11.4) sends [ u ] to the class in KH h ( R ⊗ L ǫ ( E t )) of the element Y e ∈ E ((1 − φ ( ee ∗ ) + u e ) ⊗ s ( e ) + X v = s ( e ) ⊗ v ) . Let ψ : L ǫ ( E ) → R be a ∗ -homomorphism such that φ ( ee ∗ ) = ψ ( ee ∗ ) for all e ∈ E .Then (11.4) sends j h ( ψ ) to the KH h -class of ξ ( ψ ) := 1 ⊗ − X v ∈ E φ ( v ) ⊗ v + X f ∈ E ψ ( f ) ⊗ f ∗ t . We shall abuse notation and write u e for the element of L f ∈ E U ( φ ( ee ∗ ) Rφ ( ee ∗ ))whose f -coordinate is δ f,e u e φ ( f f ∗ ). One checks that((1 − φ ( ee ∗ ) + u e ) ⊗ s ( e ) + X v = s ( e ) ⊗ v ) ξ ( ψ ) = ξ ( ψ u e ) . Starting with ψ = φ and applying the identity above repeatedly we obtain that theisomorphism (11.4) sends the two sides of the identity of the lemma to the sameelement of KH h ( R ⊗ L ǫ ( E t )). This concludes the proof of the case n = 0. Nextlet n ≥
0, assume that the lemma holds for graphs with at most n sources, and let E be a finite graph without sinks and with n + 1 sources. Let v ∈ sour( E ) and let F = E \ v be the source elimination graph. Because E is purely infinite simple byassumption, Lemma 8.4 implies that the the corner embedding L ǫ ( E \ v ) → L ǫ ( E )of Lemma 8.3 induced by the inclusion E \ v ⊂ E is kk h -equivalence. Let φ ′ be therestriction of φ to L ǫ ( E \ v ). The inclusion E \ v ⊂ E induces a commutative diagram K h ( R φ ) / / inc ∗ (cid:15) (cid:15) kk h ( L ǫ ( E ) , R ) ≀ (cid:15) (cid:15) K h ( R φ ′ ) / / kk h ( L ǫ ( E \ v ) , R )By induction, the composite of the vertical map on the left with the horizontal mapat the bottom sends the class of u = ( u e ) to j h ( φ ′ inc ∗ ( u ) ) − j h ( φ ′ ). Observe that φ ′ inc ∗ u is the restriction of φ u to L ǫ ( E \ v ). Because the vertical map on the rightof the diagram above is an isomorphism, it follows that the top horizontal arrowmaps [ u ] j h ( φ u ) − j h ( φ ). (cid:3) Theorem 13.7.
Let E be a finite, purely infinite simple graph and R ∈ Alg ∗ ℓ a K h -regular, strictly properly infinite ∗ -algebra over a ring ℓ satisying the λ -assumption.Assume that − is positive in R . Then the map (13.8) j h : [ L ( E ) , R ] PM → kk h ( L ( E ) , R ) is a semigroup isomorphism. For each p ∈ Proj f ( R ) , we have ev − ([ p ]) = j h ( { [ φ ] : φ (1) = p } ) . Proof.
Let ξ ∈ kk h ( L ( E ) , R ). Because by assumption, − R , themap K ( R ) ∗ → K h ( R ) is an isomorphism. Hence by Theorem 9.4 there is a ∗ -homomorphism φ : L ( E ) → R with property (P) such that K h ( φ ) ◦ can = ev( ξ ).Let p = φ (1) and inc : pRp ⊂ R the inclusion. By Lemma 8.15, there exists ξ ′ ∈ kk h ( L ( E ) , pRp ) such that inc ∗ ( ξ ′ ) = ξ . By Lemma 13.6, there exists u ∈U (( pRp ) φ ) such that ξ ′ = j h ( φ u ). Hence ξ = j h (inc ◦ φ u ) and the map of thetheorem is surjective. Next let φ, ψ : LE → R be ∗ -homomorphisms such that j h ( φ ) = j h ( ψ ). Then K h ( φ ) ◦ can = K h ( ψ ) ◦ can, so by Lemma 9.9, we may assumethat ψ | DL ( E ) = φ | DL ( E ) . Thus there is p ∈ V f ( R ) with ψ (1) = φ (1) = p ; by Lemma8.15, we may assume that p = 1. For each e ∈ E , let u e = ψ ( e ) φ ( e ∗ ) ∈ U ( φ ( ee ∗ ) Rφ ( ee ∗ )) . Put u = ( u e ) ∈ L e ∈ E U ( φ ( ee ∗ ) Rφ ( ee ∗ )) = U ( R φ ). A calculation shows that ψ = φ u . Hence ∂ ([ u ]) = 0, by Lemma 13.6. Let λ : R φ → R φ , λ ( a ) = P e ∈ E φ ( e ) aφ ( e ∗ ).Because R is K h -regular by assumption, KH h ( R ) = KV h ( R ). Let B = B E be asin (3.7); one checks that the following diagram commutes KV h ( R φ ) ≀ (cid:15) (cid:15) λ / / KV h ( R φ ) ≀ (cid:15) (cid:15) KV h ( R ) E B t / / KV h ( R ) E . Thus identifying [ u ] = P e ∈ E [ u e ], and using that ∂ ([ u ]) = 0, it follows that thereexists ν ∈ U ( R φ ) such that [ u ] = [ νλ ( ν ) − ]. Hence by Lemma 10.5 there is U ( t ) ∈U ( R φ [ t ]) with U (0) = νλ ( ν ) − , U (1) = u , and thus for the inclusion inc : R φ ⊂ R ,we have ψ = φ u ∼ ∗ M φ νλ ( ν ) − = inc ad( ν ) ◦ φ ∼ ∗ M φ. This proves the first assertion of the theorem. Next let p ∈ Proj f ( R ) and let ξ ∈ kk h ( LE, R ) such that ev( ξ ) = p . By Theorem 9.4 there is a ∗ -homomorphism φ : LE → R with property (P) such that φ (1) = p and ev( j h ( φ )) = ev( ξ ), and byLemma 13.6 there is u ∈ U ( R φ ) such that j h ( φ u ) = ξ . This finishes the proof, since φ u (1) = φ (1) = p . (cid:3) In the next corollary we refer to the stable M -homotopy relation ∼ sM definedin (8.13). Corollary 13.9.
For every class [ φ ] ∈ [ L ( E ) , R ] ∗ M there exists a unique class [ φ P ] ∈ [ L ( E ) , R ] PM such that φ ∼ sM φ P . The map [ L ( E ) , R ] ∗ M → [ L ( E ) , R ] PM , [ φ ] [ φ P ] is the group completion (8.14) . Theorem 13.10.
Let E , R and ℓ be as in Theorem 13.7. Then the map (13.11) j h : [ L ( E ) , R ] PM → kk h ( L ( E ) , R ) is a semigroup monomorphism. For each p ∈ Proj f ( R ) , we have ev − ([ p ]) = j h ( { [ φ ] : φ (1) = p } ) . If furthermore sour( E ) = ∅ and R contains a central element x such that xx ∗ = − ,then (13.11) is an isomorphism.Proof. The proof of the injectivity part of Theorem 13.7 applies also when L ( E ) issubstituted for L ( E ); this gives the first assertion. To prove the second assertion,begin by observing that the bijection θ x of Example 2.3.3 passes down to a homo-morphism between the monoids of M - ∗ homotopy classes. Hence it induces a map LASSIFYING LEAVITT PATH ALGEBRAS 37 [ L ( E ) , R ] PM → [ L ( E ) , R ] PM , which, together with the isomorphism of Corollary11.6 and the maps (13.8) and (13.11), fits into a diagram[ L ( E ) , R ] PM (13.11) / / θ x ≀ (cid:15) (cid:15) kk h ( L ( E ) , R ) ≀ (11.6) (cid:15) (cid:15) [ L ( E ) , R ] PM ∼ (13.8) / / kk h ( L ( E ) , R )A straightforward calculation shows that the diagram above commutes; this finishesthe proof. (cid:3) Classification theorems
Theorem 14.1.
Let E and F be purely infinite finite graphs. Assume that ℓ satisfies the λ -assumption 1.11 and that L ( E ) and L ( F ) are K h -regular. Let ξ : BF ( E ) ∼ −→ BF ( F ) be an isomorphism and let ev be as in (6.3) . Then thereare ∗ -homomorphisms φ : L ( E ) → M ± L ( F ) and ψ : L ( F ) → M ± L ( E ) with pro-perty (P) such that ev( j h ( φ )) = K h ( ι + ) ◦ can ◦ ξ , ev( j h ( ψ )) = K h ( ι + ) ◦ can ◦ ξ − , M ± ( ψ ) ◦ φ ∼ sM ι : LE → M ± M ± L ( E ) and M ± ( φ ) ◦ ψ ∼ sM ι : L ( F ) → M ± M ± L ( F ) .Proof. Because E and F are purely infinite simple, sink( E ) = sink( F ) = ∅ . Hence ξ lifts to an isomorphism ξ : j h ( LE ) ∼ = j h ( LF ) such that ev( ξ ) = can ◦ ξ , byTheorem 6.11. By Theorem 13.7, there are ∗ -homomorphisms φ : L ( E ) → M ± L ( F )and ψ : L ( F ) → M ± L ( E ) with property (P) such that j h ( φ ) = j h ( ι + ) ξ and j h ( ψ ) = j h ( ι + ) ξ − . Omitting j h for ease of notation, we have constructed the followingcommutative diagram in kk h LE φ / / ξ ●●●●●●●●● M ± LF M ± ψ / / M ± M ± LELF ξ − & & ▼▼▼▼▼▼▼▼▼▼▼ ψ / / ι + O O M ± LE ι + O O LE. ι + O O Hence j h ( ι + ι + ) = j h (( M ± ψ ) φ ). By Examples 8.6 and 9.2, ( M ± ψ ) φ has property(P); hence ι ∼ sM ( M ± ψ ) φ , by Corollary 13.9. Similarly, ι ∼ sM ( M ± φ ) ψ . (cid:3) Theorem 14.2.
Let E , F , ξ and ℓ be as in Theorem 14.1. Further assumethat − is positive in ℓ . Then there exist ∗ -homomorphisms φ : LE → LF and ψ : LF → LE with property (P) such that ev( j h ( φ )) = ξ , ev( j h ( ψ )) = ξ − , ψ ◦ φ ∼ ∗ M id LE and φ ◦ ψ ∼ ∗ M id LF . If furthermore ξ ([1] E ) = [1] F , then φ and ψ can be chosen to be unital.Proof. The proof is essentially the same as in Theorem 14.2; the only difference isthat because − LE and LF , one can apply Theorem 6.11 directly,without going through M ± LF and M ± LE . (cid:3) Example 14.3.
The hypothesis of Theorem 14.1 are satisfied, for example, when ℓ is regular supercoherent and 2 is invertible in ℓ ; if in addition − ℓ , then also Theorem 14.2 applies. For another example, if ℓ is a commutative,regular supercoherent ring, then ℓ = inv( ℓ ) satisfies the hypothesis of Theorem14.2. Indeed, (1 , − , − ∗ = (1 , − − ,
1) = ( − , − . In the next theorem we call an idempotent p of a unital ring R very full if( p, p ) ∈ inv( R ) is a very full projection. Observe that any very full projection ofinv( R ) is of the form ( p, p ) for some very full idempotent p . Theorem 14.4.
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