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Leavitt Path Algebras

At the crossroads of Algebra andFunctional Analysis

Mark Tomforde

University of Houston

USA

In its beginnings, the study of Operator Alge-

bras received a great deal of motivation from

Ring Theory.

Nowadays . . .

“Both Operator Algebra and Ring Theory have

great potential to influence, inspire, and ex-

plain 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.

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 prop-

erty:

RRm ∼= RRn as left R-modules implies m = n.

In the 1950’s Bill Leavitt showed that if R is

a unital ring, then RR1 ∼= RRn for n > 1 if and

only if there exist x1, . . . , xn, x∗1, . . . , x∗n ∈ R such

that

(1) x∗i xj = δij1R

(2)n∑

i=1

xix∗i = 1R.

For a given field K, we let LK(1, n) denote the

K-algebra generated by elements

x1, . . . , xn, x∗1, . . . , x∗n

satisfying the relations (1) and (2) above.

In 1977 J. Cuntz introduced a class of C∗-algebras (now called Cuntz algebras) gener-

ated by n nonunitry isometries. Specifically,

if n > 1 the C∗-algebra On is generated by

isometries s1, . . . , sn satisfying

(1) s∗i sj = δij Id

(2)n∑

i=1

sis∗i = Id.

Note: LC(1, n) is isomorphic to a dense ∗-subalgebra of On.

The Cuntz algebras have been generalized in a

number of ways, including (in the late 1990’s)

the graph C∗-algebras.

Definition. If E = (E0, E1, r, s) is a directed

graph consisting of a countable set of vertices

E0, a countable set of edges E1, and maps

r, s : E1 → E0 identifying the range and source

of each edge, then C∗(E) is defined to be the

universal C∗-algebra generated by mutually or-

thogonal projections {pv : v ∈ E0} and partial

isometries {se : e ∈ E1} with mutually orthog-

onal ranges that satisfy

1. s∗ese = pr(e) for all e ∈ E1

2. pv =∑

s(e)=v

ses∗e when 0 < |s−1(v)| < ∞

3. ses∗e ≤ ps(e) for all e ∈ E1.

Graph C∗-algebras are important and interest-

ing 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.

Inspired by the success of graph C∗-algebras,Gene Abrams and Gonzalo Aranda-Pino intro-duced Leavitt path algebras.

Definition. Given a graph E = (E0, E1, r, s)and a field K, we let (E1)∗ denote the setof formal symbols {e∗ : e ∈ E1} (called ghostedges). The Leavitt path algebra LK(E) is theuniversal K-algebra generated by a set {v : v ∈E0} of pairwise orthogonal idempotents, to-gether with a set {e, e∗ : e ∈ E1} of elementssatisfying

1. s(e)e = er(e) = e for all e ∈ E1

2. r(e)e∗ = e∗s(e) = e∗ for all e ∈ E1

3. e∗f = δe,f r(e) for all e, f ∈ E1

4. v =∑

s(e)=v

ee∗ when 0 < |s−1(v)| < ∞.

Basic Facts about Leavitt Path Algebras

• If we list the vertices E0 = {v1, v2, . . .} anddefine tn :=

∑ni=1 vi, then {tn} is a set of local

units for LK(E). (I.e., given a finite number ofelements x1, . . . , xk ∈ LK(E) there exists n ∈ Nsuch that tnxi = xitn = xi for all 1 ≤ i ≤ k.)

• LK(E) is an idempotent ring (i.e.; R R = R.)

• If α := e1 . . . en denotes a path in E, andα∗ := e∗n . . . e∗1, then

LK(E) = span{αβ∗ : α, β are paths in E}.

• We may define a linear, antimultiplicative in-volution on LK(E) by

n∑k=1

λkαkβ∗k =n∑

k=1

λkβkα∗k.

• Fact: There exists φ : LC(E) → C∗(E) withφ(v) = pv and φ(e) = se and φ(e∗) = s∗e.

Differences between Leavitt Path Algebras

and graph C∗-algebras

• LK(E) is not closed, and the involution on

LK(E) is linear rather than conjugate linear.

• Ideals in LK(E) are not necessarily self-adjoint.

• LK(E) is an algebra over an arbitrary field

K, while C∗(E) is an algebra over C.

• There is a gauge action γ : T → AutC∗(E)

with γz(pv) = pv for all v ∈ E0 and γz(se) = zse

for all e ∈ E1. There is no such action of T on

LK(E).

Similarities between Leavitt Path Algebras

and graph C∗-algebras

C∗(E) LK(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 exitand and (2) No saturated hereditary sets

Purely Purely (3) Every vertex can reach a cycleInfinite Infinite

AF Locally no cyclesMatricial

A structure theorem for graph C∗-algebras.

Recall: There is a gauge action γ : T → AutC∗(E)with γz(pv) = pv and γz(se) = zse.

Theorem (Gauge-Invariant Uniqueness). LetE be a graph, and let γ : T → AutC∗(E) be thegauge action on C∗(E). Also let φ : C∗(E) → A

be a ∗-homomorphism between C∗-algebras. Ifβ : T → AutA is a gauge action on A and thefollowing two conditions are satisfied

(1) βz ◦ φ = φ ◦ γz for all z ∈ T

(2) φ(pv) 6= 0 for all v ∈ E0

then φ is injective.

Surprisingly, much initial work on Leavitt pathalgebras has been done without such a Unique-ness Theorem

How can a version of this theorem be obtained

for Leavitt path algebras?

As mentioned before, there is no natural action

of T on LK(E).

What about an action of K∗ on LK(E)? This

turns out to be inappropriate.

———————————————————

The correct thing to do is look at the graded

structure of LK(E).

A ring R is Z-graded if it contains a collection

of subgroups {Rn}n∈Z with R =⊕

Rn as left

R-modules and RmRn ⊆ Rm+n.

An ideal I / R is Z-graded if I =⊕

n∈Z(I ∩Rn).

A ring homomorphism φ : R → S is Z-graded if

φ(Rn) ⊆ Sn for all n ∈ Z.

For a path α = e1 . . . en, let |α| := n denote thelength of |α|.

For n ∈ Z, let

LK(E)n := span{αβ∗ : α and β are paths

with |α| − |β| = n}.

This gives a Z-grading on LK(E).

We can prove the following “Graded Unique-ness Theorem”.

Theorem. (T) Let E be a graph, and let LK(E)have the Z-grading described above. Also letφ : LK(E) → R be a homomorphism betweenrings. If the following two conditions are sat-isfied

(1) R has a Z-grading and φ(LK(E)n) ⊆ Rn

(2) φ(v) 6= 0 for all v ∈ E0

then φ is injective.

This Graded Uniqueness Theorem allows us toprove that the graded ideals in LK(E) corre-spond to pairs (H, S) where H is a saturatedhereditary subset of vertices and S is a subsetof breaking vertices.

Furthermore, if E is row-finite (there are novertices that emit an infinite number of edges)then there are no breaking vertices and themap

H 7→ IH := 〈{v : v ∈ H}〉

is a lattice isomorphism from saturated heredi-tary subsets of vertices in E onto graded idealsof LK(E).

Also, in this case, we have

LK(E)/IH∼= LK(E \H)

and

IH is Morita equivalent to LK(EH).

Moreover, all of these proofs are very simi-

lar to the proofs for gauge-invariant ideals in

C∗(E) — one simply uses the Graded Unique-

ness Theorem in place of the Gauge-Invariant

Uniqueness Theorem

———————————————————

Also, we can use the Graded Uniqueness The-

orem to show the map φ : LC(E) → C∗(E)

with φ(v) = pv and φ(e) = se and φ(e∗) = s∗e is

injective.

We have

imφ = span{sαs∗β : α, β are paths in E}

and if we use the gauge action γ : T → AutC∗(E)

to define

An := {a ∈ imφ :∫T

z−nγz(a) dz = a}

then imφ =⊕

n∈Z An is graded and φ is a

graded homomorphism.

Theorem. (T) If φ : LC(E) → C∗(E) is the

natural map with φ(v) = pv and φ(e) = se and

φ(e∗) = s∗e, then φ is injective. Hence we may

view LC(E) as a dense ∗-subalgebra of C∗(E).

Moreover, if we make this identification then

the map

I 7→ I

is an isomorphsim from the lattice of graded

ideals of LC(E) onto the gauge-invaraint ideals

of C∗(E).

This correspondence I 7→ I does not hold for

general ideals.

Example: Let E be the graph

.��

then C∗(E) ∼= C(T) and LC(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 6= 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 ∩ LC(E) = {0}.Thus I is not the closure of any ideal in LC(E).

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, thenI is generated as an ideal by I ∩ LK(E)0.

Step 2: Define Fk := span{αβ∗ : |α| = |β| = k}.Then

F0 ⊆ F1 ⊆ F2 ⊆ . . .

and LK(E)0 =⋃∞

k=0Fk.

For αβ∗, γδ∗ ∈ Fk we have

(αβ∗)(γδ∗) =

αδ∗ if β = γ

0 otherwise.

so {αβ∗ : |α| = |β| = k} is a set of matrix unitsfor Fk. Thus Fk is isomorphic to Mn(K) orM∞(K), and Fk is simple.

(The non-row-finite case is harder.)

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(pv) 6= 0

for all v ∈ E0, then φ is injective.

A version of this can be proven for Leavitt path

algebras.

Theorem. (T) Let E be a graph in which ev-

ery cycle has an exit. If φ : LK(E) → R is a

homomorphism between rings with the prop-

erty that φ(v) 6= 0 for all v ∈ E0, then φ is

injective.

Using the Cuntz-Krieger Uniqueness Theorem

for Leavitt path algebras we can show:

Proposition. The Leavitt path algebra LK(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.

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 Leav-

itt 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 LK(E) has

stable rank 2, but C∗(E) has stable rank 1.)

• 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

Mm(On) and On when gcd(m, n) = 1.

• C∗-algebra results have also contributed to

Leavitt path algebra results. Currently Anh,

Abrams, and Pardo are working to develop an

algebraic version of the Kirchberg-Phillips Clas-

sification Theorem for certain purely infinite

simple rings.