Leavitt Path Algebras

Transcription

1 Leavitt Path Algebras At the crossroads of Algebra and Functional Analysis Mark Tomforde University of Houston USA

2 In its beginnings, the study of Operator Algebras received a great deal of motivation from Ring Theory. Nowadays... Both Operator Algebra and Ring Theory have great potential to influence, inspire, and explain much of the work being done in the other subject. Today I would like to tell you about a novel way in which this interplay between Operator Algebra and Ring Theory continues.

3 LEAVITT PATH ALGEBRAS AND GRAPH C -ALGEBRAS: A Brief History It is likely that when you first learned of rings you studied the examples Z, fields, matrix rings, polynomial rings These all have the Invariant Basis Number property: RR m = R R n as left R-modules implies m = n.

4 In the 1950 s Bill Leavitt showed that if R is a unital ring, then R R 1 = R R n for n > 1 if and only if there exist x 1,..., x n, x 1,..., x n R such that (1) x i x j = δ ij 1 R (2) n i=1 x i x i = 1 R. For a given field K, we let L K (1, n) denote the K-algebra generated by elements x 1,..., x n, x 1,..., x n satisfying the relations (1) and (2) above.

5 In 1977 J. Cuntz introduced a class of C - algebras (now called Cuntz algebras) generated by n nonunitry isometries. Specifically, if n > 1 the C -algebra O n is generated by isometries s 1,..., s n satisfying (1) s i s j = δ ij Id (2) n i=1 s i s i = Id. Note: L C (1, n) is isomorphic to a dense - subalgebra of O n. The Cuntz algebras have been generalized in a number of ways, including (in the late 1990 s) the graph C -algebras.

6 Definition. If E = (E 0, E 1, r, s) is a directed graph consisting of a countable set of vertices E 0, a countable set of edges E 1, and maps r, s : E 1 E 0 identifying the range and source of each edge, then C (E) is defined to be the universal C -algebra generated by mutually orthogonal projections {p v : v E 0 } and partial isometries {s e : e E 1 } with mutually orthogonal ranges that satisfy 1. s es e = p r(e) for all e E 1 2. p v = s(e)=v s e s e when 0 < s 1 (v) < 3. s e s e p s(e) for all e E 1.

7 Graph C -algebras are important and interesting for a variety of reasons: They contain a large class of C -algebras (e.g. Cuntz-Krieger algebras, AF-algebras, Kirchberg-Phillips algebras) The structure of C (E) is reflected in the graph E. They are tractable objects of study, and their invariants (e.g. K-theory, Ext, Primitive Ideal Space) can be computed.

8 Inspired by the success of graph C -algebras, Gene Abrams and Gonzalo Aranda-Pino introduced Leavitt path algebras. Definition. Given a graph E = (E 0, E 1, r, s) and a field K, we let (E 1 ) denote the set of formal symbols {e : e E 1 } (called ghost edges). The Leavitt path algebra L K (E) is the universal K-algebra generated by a set {v : v E 0 } of pairwise orthogonal idempotents, together with a set {e, e : e E 1 } of elements satisfying 1. s(e)e = er(e) = e for all e E 1 2. r(e)e = e s(e) = e for all e E 1 3. e f = δ e,f r(e) for all e, f E 1 4. v = ee when 0 < s 1 (v) <. s(e)=v

9 Basic Facts about Leavitt Path Algebras If we list the vertices E 0 = {v 1, v 2,...} and define t n := n i=1 v i, then {t n } is a set of local units for L K (E). (I.e., given a finite number of elements x 1,..., x k L K (E) there exists n N such that t n x i = x i t n = x i for all 1 i k.) L K (E) is an idempotent ring (i.e.; R R = R.) If α := e 1... e n denotes a path in E, and α := e n... e 1, then L K (E) = span{αβ : α, β are paths in E}. We may define a linear, antimultiplicative involution on L K (E) by n k=1 λ k α k β k = n k=1 λ k β k α k. Fact: There exists φ : L C (E) C (E) with φ(v) = p v and φ(e) = s e and φ(e ) = s e.

10 Differences between Leavitt Path Algebras and graph C -algebras L K (E) is not closed, and the involution on L K (E) is linear rather than conjugate linear. Ideals in L K (E) are not necessarily self-adjoint. L K (E) is an algebra over an arbitrary field K, while C (E) is an algebra over C. There is a gauge action γ : T Aut C (E) with γ z (p v ) = p v for all v E 0 and γ z (s e ) = zs e for all e E 1. There is no such action of T on L K (E).

11 Similarities between Leavitt Path Algebras and graph C -algebras C (E) L K (E) Properties of E Unital Unital finite number of vertices Simple Simple (1) Every cycle has an exit (2) No saturated hereditary sets Simple Simple (1) Every cycle has an exit and and (2) No saturated hereditary sets Purely Purely (3) Every vertex can reach a cycle Infinite Infinite AF Locally no cycles Matricial

12 A structure theorem for graph C -algebras. Recall: There is a gauge action γ : T Aut C (E) with γ z (p v ) = p v and γ z (s e ) = zs e. Theorem (Gauge-Invariant Uniqueness). Let E be a graph, and let γ : T Aut C (E) be the gauge action on C (E). Also let φ : C (E) A be a -homomorphism between C -algebras. If β : T Aut A is a gauge action on A and the following two conditions are satisfied (1) β z φ = φ γ z for all z T (2) φ(p v ) 0 for all v E 0 then φ is injective. Surprisingly, much initial work on Leavitt path algebras has been done without such a Uniqueness Theorem

13 How can a version of this theorem be obtained for Leavitt path algebras? As mentioned before, there is no natural action of T on L K (E). What about an action of K on L K (E)? This turns out to be inappropriate. The correct thing to do is look at the graded structure of L K (E). A ring R is Z-graded if it contains a collection of subgroups {R n } n Z with R = R n as left R-modules and R m R n R m+n. An ideal I R is Z-graded if I = n Z (I R n). A ring homomorphism φ : R S is Z-graded if φ(r n ) S n for all n Z.

14 For a path α = e 1... e n, let α := n denote the length of α. For n Z, let L K (E) n := span{αβ : α and β are paths with α β = n}. This gives a Z-grading on L K (E). We can prove the following Graded Uniqueness Theorem. Theorem. (T) Let E be a graph, and let L K (E) have the Z-grading described above. Also let φ : L K (E) R be a homomorphism between rings. If the following two conditions are satisfied (1) R has a Z-grading and φ(l K (E) n ) R n (2) φ(v) 0 for all v E 0 then φ is injective.

15 This Graded Uniqueness Theorem allows us to prove that the graded ideals in L K (E) correspond to pairs (H, S) where H is a saturated hereditary subset of vertices and S is a subset of breaking vertices. Furthermore, if E is row-finite (there are no vertices that emit an infinite number of edges) then there are no breaking vertices and the map H I H := {v : v H} is a lattice isomorphism from saturated hereditary subsets of vertices in E onto graded ideals of L K (E). Also, in this case, we have and L K (E)/I H = LK (E \ H) I H is Morita equivalent to L K (E H ).

16 Moreover, all of these proofs are very similar to the proofs for gauge-invariant ideals in C (E) one simply uses the Graded Uniqueness Theorem in place of the Gauge-Invariant Uniqueness Theorem Also, we can use the Graded Uniqueness Theorem to show the map φ : L C (E) C (E) with φ(v) = p v and φ(e) = s e and φ(e ) = s e is injective.

17 We have im φ = span{s α s β : α, β are paths in E} and if we use the gauge action γ : T Aut C (E) to define A n := {a im φ : T z n γ z (a) dz = a} then im φ = n Z A n is graded and φ is a graded homomorphism.

18 Theorem. (T) If φ : L C (E) C (E) is the natural map with φ(v) = p v and φ(e) = s e and φ(e ) = s e, then φ is injective. Hence we may view L C (E) as a dense -subalgebra of C (E). Moreover, if we make this identification then the map I I is an isomorphsim from the lattice of graded ideals of L C (E) onto the gauge-invaraint ideals of C (E).

19 This correspondence I I does not hold for general ideals. Example: Let E be the graph then C (E) = C(T) and L C (E) = C[z, z 1 ].. One can find an element a C[z, z 1 ] such that I := a is not self-adjoint. If we let J = a, then I J but I = J. Also, if we choose any infinite closed subset K T and let I := {f C(T) : f K = 0}, then I is an ideal in C (E), but I L C (E) = {0}. Thus I is not the closure of any ideal in L C (E).

20 The proof of the Graded Uniqueness Theorem (in the row-finite case)... Let I := ker φ. Step 1: Show that if I is a graded ideal, then I is generated as an ideal by I L K (E) 0. Step 2: Define F k := span{αβ : α = β = k}. Then F 0 F 1 F 2... and L K (E) 0 = k=0 F k. For αβ, γδ F k we have (αβ )(γδ ) = αδ if β = γ 0 otherwise. so {αβ : α = β = k} is a set of matrix units for F k. Thus F k is isomorphic to M n (K) or M (K), and F k is simple. (The non-row-finite case is harder.)

21 Another uniqueness theorem... Theorem (Cuntz-Krieger Uniqueness). Let E be a graph in which every cycle has an exit. If φ : C (E) A is a -homomorphism between C -algebras with the property that φ(p v ) 0 for all v E 0, then φ is injective. A version of this can be proven for Leavitt path algebras. Theorem. (T) Let E be a graph in which every cycle has an exit. If φ : L K (E) R is a homomorphism between rings with the property that φ(v) 0 for all v E 0, then φ is injective.

22 Using the Cuntz-Krieger Uniqueness Theorem for Leavitt path algebras we can show: Proposition. The Leavitt path algebra L K (E) is simple if an only if E satisfies (1) Every cycle in E has an exit (2) There are no proper, nontrivial saturated hereditary subsets in E. The proof of this result is similar to the proof of the corresponding result for graph C -algebras.

23 Let s look again at the relationship between graph C -algebras and Leavitt path algebras. Results for graph C -algebras have guided the investigation of Leavitt path algebras. (And, now with the Uniqueness Theorems, some of the graph algebra proofs can be used for Leavitt path algebras with only slight modifications.) Many similar results hold for graph C -algebras and Leavitt path algebras, but there are some results that are different. (E.g. It has been shown there is a graph E such that L K (E) has stable rank 2, but C (E) has stable rank 1.)

24 Leavitt path algebras have been used to prove results about graph C -algebras. Leavitt path algebra results were used to: (1) show graph C -algebras satisfy the stable weak cancellation property. (2) construct explicit isomorphisms between M m (O n ) and O n when gcd(m, n) = 1. C -algebra results have also contributed to Leavitt path algebra results. Currently Ánh, Abrams, and Pardo are working to develop an algebraic version of the Kirchberg-Phillips Classification Theorem for certain purely infinite simple rings.