aa r X i v : . [ m a t h . R A ] N ov CENTERS OF CUNTZ-KRIEGER C ∗ -ALGEBRAS ADEL ALAHMADI AND HAMED ALSULAMI
Abstract.
For a finite directed graph Γ we determine the center of the Cuntz-Krieger C ∗ -algebra CK (Γ) . Definitions and Main Results
Let Γ = (
V, E, s, r ) be a directed graph that consists of a set V of vertices and aset E of edges, and two maps r : E → V and s : E → V identifying the range andthe source of each edge. A graph is finite if both sets V and E are finite. A graphis row-finite if | s − ( v ) | < ∞ for an arbitrary vertex v ∈ V. Definition 1.
If Γ is a row-finite graph, the Cuntz-Krieger C ∗ -algebra CK (Γ) isthe universal C ∗ -algebra generated ( as a C ∗ -algebra ) by V ∪ E and satisfying therelations (1) s ( e ) e = er ( e ) = e for all e ∈ E, (2) e ∗ f = δ e,f r ( f ) for all e, f ∈ E, (3) v = X e ∈ s − ( v ) ee ∗ whenever s − ( v ) = ∅ . The discrete C -subalgebra of CK (Γ) generated by V, E, E ∗ is isomorphic to theLeavitt path algebra L (Γ) of the graph Γ ( see [T2] ). We will identify L (Γ) withits image in CK (Γ) . Clearly L (Γ) is dense in CK (Γ) in the C ∗ - topology.A path is a finite sequence p = e · · · e n of edges with r ( e i ) = s ( e i +1 ) for 1 ≤ i ≤ n −
1. We consider the vertices to be paths of length zero. We let
P ath (Γ) denotethe set of all paths in the graph Γ and extend the maps r, s to P ath (Γ) as follows:for p = e · · · e n we set s ( p ) = s ( e ) , r ( p ) = r ( e n ) . For v ∈ V viewed as a path weset s ( v ) = r ( v ) = v. A vertex w is a descendant of a vertex v if there exists a path p ∈ P ath (Γ) such that s ( p ) = v , r ( p ) = w .A cycle is a path C = e · · · e n , n ≥ s ( e ) = r ( e n ) and all vertices s ( e ) , . . . , s ( e n ) are distinct. An edge e ∈ E is called an exit from the cycle C if s ( e ) ∈ { s ( e ) , · · · , s ( e n ) } , but e / ∈ { e , · · · , e n } . A cycle without an exit is called a Key words and phrases.
Cuntz-Krieger algebra, Leavitt path algebra.
N E -cycle. A subset W ⊂ V is hereditary if all descendants of an arbitrary vertex w ∈ W also lie in W. For two nonempty subsets W , W ⊂ V let E ( W , W ) denotethe set of edges { e ∈ E | s ( e ) ∈ W , r ( e ) ∈ W } . For a hereditary subset W ⊂ V the C ∗ -subalgebra of CK (Γ) generated by W, E ( W, W ) is isomorphic to the Cuntz-Krieger algebra of the graph(
W, E ( W, W ) , s | E ( W,W ) , r | E ( W,W ) ) , ( see [BPRS], [BHRS] ). We will denote it as CK ( W ) . Example 1.
Let the graph Γ be a cycle, V = { v , · · · , v n } , E = { e , · · · , e n } ,s ( e i ) = v i , ≤ i ≤ n ; r ( e i ) = v i +1 for 1 ≤ i ≤ n − , r ( e n ) = v . The algebra CK (Γ) in this case is isomorphic to the matrix algebra M n ( T ) , where T is the C ∗ -algebra of continuous functions on the unit circle. The center of CK (Γ) is generatedby the element e · · · e n + e e · · · e + · · · + e n e · · · e and is isomorphic T. For an arbitrary row-finite graph Γ if a path C = e · · · e n is a N E -cycle withthe hereditary set of vertices V ( C ) = { s ( e ) , · · · , s ( e n ) } then we denote CK ( C ) = CK ( V ( C )) ∼ = M n ( T ) . The center Z ( C ) of the subalgebra CK ( C ) of CK (Γ) isgenerated by the element z ( C ) = e · · · e n + e e · · · e n e + · · · + e n e · · · e n − . We will need some definitions and some results from [AA1], [AA2].
Definition 2.
Let W ⊂ V be a nonempty subset. We say that a path p = e · · · e n , e i ∈ E, is an arrival path in W if r ( p ) ∈ W, and { s ( e ) , · · · , s ( e n ) } * W. In other words, r ( p ) is the first vertex on p that lies in W. In particular, every vertex w ∈ W, viewed as a path of zero length, is an arrival path in W. Let
Arr ( W ) bethe set of all arrival paths in W. Definition 3.
A hereditary set W ⊆ V is called finitary if | Arr ( W ) | < ∞ .If W is a hereditary finitary subset of V then e ( W ) = X p ∈ Arr ( W ) pp ∗ is a centralidempotent in L (Γ) and, hence in CK (Γ), see [AA1],[AA2]. If C is a N E -cycle in Γwith the hereditary set of vertices V ( C ) then we will denote Arr ( C ) = Arr ( V ( C )) . If the set V ( C ) is finitary then we will say that the cycle is finitary. In this casefor an arbitrary element z ∈ Z ( C ) the sum X p ∈ Arr ( C ) pzp ∗ lies in the center of L (Γ)and CK (Γ) , see [AA2]. ENTERS OF CUNTZ-KRIEGER C ∗ -ALGEBRAS 3 Theorem 1.
Let Γ be a finite graph. The center Z ( CK (Γ)) is spanned by: (i) central idempotents e ( W ) , where W runs over all nonempty hereditary fini-tary subsets of V ; (ii) subspaces { X p ∈ Arr ( C ) pzp ∗ | z ∈ Z ( C ) } , where C runs over all finitary N E -cycles of Γ . Corollary 1. Z ( CK (Γ)) is the closure of Z ( L (Γ)) . Corollary 2.
The center Z ( CK (Γ)) is isomorphic to a finite direct sum C ⊕ · · · ⊕ C ⊕ T ⊕ · · · ⊕ T. In [CGBMGSMSH] it was shown that the center of a prime Cuntz-Krieger C ∗ -algebra is equal to C · W $ W the central idem-potents e ( W ) , e ( W ) may be equal.To make the statement of the Theorem more precise we will consider annihilatorhereditary subsets.Let W be a nonempty subset of V. Consider the subset W ⊥ = { v ∈ V | the vertex v does not have descendants in W } . For empty set we let ∅ ⊥ = V. It is easy to see that W ⊥ is always a hereditarysubset of V. If W ⊆ W ⊆ V then W ⊥ ⊇ W ⊥ , ( W ⊥ ) ⊥ ⊇ W, (( W ⊥ ) ⊥ ) ⊥ = W ⊥ . Definition 4.
We will refer to W ⊥ , W ⊆ V as annihilator hereditary subsets.In [AA1] it was proved that ( W ⊥ ) ⊥ is the largest hereditary subset of V suchthat every vertex in it has a descendant in W and that e ( W ) = e (( W ⊥ ) ⊥ ) . Hence in the part (i) of the Theorem we can let W run over nonempty finitaryhereditary annihilator subsets of V. In fact there is a 1-1 correspondence ( and a Boolean algebra isomorphism )between finitary hereditary annihilator subsets of V and central idempotents of L (Γ) and CK (Γ) . Example 2.
Let Γ = v v . The set { v } is hereditary, but not finitary.Thus there are no proper finitary hereditary subsets and Z ( L (Γ)) = C · . ADEL ALAHMADI AND HAMED ALSULAMI
Example 3.
Let Γ = v v v v . The set { v , v , v } is hereditary and finitary,but ( { v , v , v } ⊥ ) ⊥ = V. Thus there are no proper finitary hereditary annihilatorsubsets and again Z ( L (Γ)) = C · . Example 4.
Let Γ = v v v v v . The only finitary hereditary annihilatorsubsets are { v } and { v , v , v } . Hence Z ( L (Γ)) = T ⊕ C . Closed Ideals
Let W be a hereditary subset of V. The ideal I ( W ) of the Leavitt path algebragenerated by the set W is the C -span of the set { pq ∗ | p, q ∈ P ath (Γ) , r ( p ) = r ( q ) ∈ W } . The closure I ( W ) of the ideal W is the closed ideal of the C ∗ -algebra CK (Γ)generated by the set W. In this section we will use induction on | V | to prove that for a proper hereditarysubset W ⊂ V central elements from Z ( CK (Γ)) ∩ I ( W ) are of the form predictedby the Theorem.Given paths p, q ∈ P ath (Γ) we say q is a continuation of the path p if there existsa path p ′ ∈ P ath (Γ) such that q = pp ′ . In this case the path p is called a beginning of the path q. We will often use the following well known fact: if p, q ∈ P ath (Γ), p ∗ q = 0 , then one of the paths p, q is a continuation of the other one.Remark that if W is a hereditary subset of V and p, q ∈ Arr ( W ) , p = q, thennone of the paths p, q is a continuation of the other one.For a path p ∈ P ath (Γ) we consider the idempotent e p = pp ∗ . Lemma 1.
Let W be a nonempty hereditary subset of V let a ∈ I ( W ) , let ǫ > . Then the set { p ∈ Arr ( W ) | k e p a k ≥ ǫ } is finite. ENTERS OF CUNTZ-KRIEGER C ∗ -ALGEBRAS 5 Proof.
The ideal I ( W ) generated by the set W in the Leavitt path algebra L (Γ) isdense in I ( W ) . Choose an element b ∈ I ( W ) , b = X p,q ∈ Arr ( W ) pb p,q q ∗ , b p,q ∈ L ( W ) , such that k a − b k < ǫ. If p ′ ∈ Arr ( W ) is an arrival path in W that is differentfrom all the paths involved in the decomposition of the element b then p ′∗ b = 0 and e p ′ b = 0 . Now k e p ′ a k = k e p ′ ( a − b ) k ≤ k e p ′ kk a − b k < ǫ, which proves the Lemma. (cid:3) Lemma 2.
Let a ∈ CK (Γ) , v ∈ V, a ∈ vCK (Γ) v ; p ∈ P ath (Γ) , r ( p ) = v. Then k pap ∗ k = k a k . Proof.
We have k pap ∗ k ≤ k p k · k a k · k p ∗ k = k a k . On the other hand, a = p ∗ ( pap ∗ ) p, k a k ≤ k p ∗ k · k pap ∗ k · k p k = k pap ∗ k , which proves the Lemma. (cid:3) Lemma 3.
Let W be a nonempty hereditary subset of V let z ∈ Z ( CK (Γ) ∩ I ( W ) . For an arbitrary vertex w ∈ W if wz = 0 then the set { p ∈ Arr ( W ) | r ( p ) = w } isfinite.Proof. Let p ∈ Arr ( W ) , r ( p ) = w, zw = 0 . We have e p z = pp ∗ z = pzp ∗ = p ( wzw ) p ∗ . By Lemma 2 k e p z k = k wz k > . Now it remains to refer to Lemma1. (cid:3)
Let e Z denote the sum of all subspaces C e ( W ) , where W runs over nonemptyfinitary hereditary subsets of V, and all subspaces { X p ∈ Arr ( C ) pzp ∗ | z ∈ Z ( C ) } , where C runs over finitary N E -cycles of Γ . Our aim is to show that Z ( CK (Γ)) = e Z. Let’s use induction on the number of vertices. In other words, let’s assume thatfor a graph with < | V | vertices the assertion of the Theorem is true. Lemma 4.
Let W be a proper hereditary subset of V. Then Z ( CK (Γ) ∩ I ( W ) ⊆ e Z. ADEL ALAHMADI AND HAMED ALSULAMI
Proof.
Let 0 = z ∈ Z ( CK (Γ) ∩ I ( W ) . Consider the element z = z ( X w ∈ W w ) ∈ Z ( CK ( W )) . If z = 0 then zW = (0) , zI ( W ) = (0) , z = 0 , which contradictssemiprimeness of CK (Γ) ( see [BPRS],[BHRS]).By the induction assumption there exist disjoint hereditary finitary ( in W )cycles C , · · · , C r and hereditary finitary (again in W ) subsets W , · · · , W k ⊂ W such that z = r X i =1 α i X p ∈ Arr W ( C i ) pa i p ∗ + k X j =1 β j X q ∈ Arr W ( W j ) qq ∗ ; α i , β j ∈ C , a i ∈ Z ( C i ) . The notations
Arr W ( C i ) , Arr W ( W j ) are used to stress that arrival paths areconsidered in the graph ( W, E ( W, W )) . The fact that the hereditary finitary subsets V ( C i ) , W j can be assumed disjointfollows from the description of the Boolean algebra of finitary hereditary subsets in[AA2].If α i = 0 then for arbitrary vertex w ∈ V ( C i ) we have z w = zw = 0 . Hence byLemma 3 there are only finitely many paths p ∈ Arr ( W ) such that r ( p ) = w. Hence V ( C i ) is a finitary subset of V. Similarly, if β j = 0 then W j is a finitary subset in V. Consider the central element z ′ = r X i =1 α i X p ∈ Arr ( C i ) pa i p ∗ + k X j =1 β j e ( W j ) ∈ Z ( CK (Γ)) . We have z ′ ( X w ∈ W w ) = z = z ( X w ∈ W w ) . Hence, ( z − z ′ )( X w ∈ W w ) = (0) , ( z − z ′ ) I ( W ) =(0) , ( z − z ′ ) = 0 . Again by semiprimeness of CK (Γ) we conclude that z = z ′ , whichproves the Lemma. (cid:3) Proof of the Theorem
Definition 5.
A vertex v ∈ V is called a sink if s − ( v ) = ∅ . Definition 6.
A hereditary subset W ⊂ V is called saturated if for an arbitrarynon-sink vertex v ∈ V the inclusion r ( s − ( v )) ⊆ W implies v ∈ W. Definition 7. If W is hereditary subset then we define the saturation of W to be thesmallest saturated hereditary subset c W that contains W. In this case I ( W ) = I ( c W ) . ENTERS OF CUNTZ-KRIEGER C ∗ -ALGEBRAS 7 Definition 8. If W is a hereditary saturated subset of V then the graph Γ /W =( V \ W, E ( V, V \ W )) is called the factor graph of Γ modulo W. We have CK (Γ) /I ( W ) ∼ = CK (Γ /W ) ( see[T1] ). In [AAP], [T2] it was provedthat the following 3 statements are equivalent:1) the Cuntz-Krieger C ∗ -algebra CK (Γ) is simple,2) the Leavitt path algebra L (Γ) is simple,3) (i) V does not have proper hereditary saturated subsets, (ii) every cyclehas an exit.We call a graph satisfying the condition 3) simple.The following lemma is well known. Still we prove it for the sake of completeness. Lemma 5.
Let Γ be a graph such that V does not have proper hereditary subsets.Then Γ is either simple or a cycle.Proof. If Γ is not simple then Γ contains a
N E -cycle C. The set of vertices V ( C )is hereditary subset of V. In view of our assumption V ( C ) = V, which proves theLemma. (cid:3) Let U = { α ∈ C | | α | = 1 } be the unit circle in C . Let E ′ be a subset of the set E of edges. For an arbitrary α ∈ U the mapping g E ′ ( α ) such that g E ′ ( α ) : v v, v ∈ V ; g E ′ ( α ) : e αe, e ∗ αe, e ∈ E ′ ; g E ′ ( α ) : e e, e ∗ e ∗ , e ∈ E \ E ′ , extendsto an automorphism g E ′ ( α ) of the C ∗ -algebra CK (Γ) . Denote G E ′ = { g E ′ ( α ) , α ∈ U } ≤ Aut ( CK (Γ)) . The group G E is called the gauge group of the C ∗ -algebra CK (Γ) . An ideal of CK (Γ) is called gauge invariant if it is invariant with respectto the group G E . In [BPRS], [BHRS] it is proved that a nonzero closed gauge invariant ideal of CK (Γ) has a nonempty intersection with V. Lemma 6.
Let the graph Γ be a cycle, Γ = (
V, E ) , V = { v , · · · , v d } , E = { e , · · · , e d } ,s ( e i ) = v i for ≤ i ≤ d ; r ( e i ) = v i +1 for ≤ i ≤ d − , r ( e d ) = v . Then the centralelements from Z ( CK (Γ)) that are fixed by all g E ( α ) , α ∈ U , are scalars.Proof. The center of CK (Γ) is isomorphic to the algebra of continuous function T = { f : U → C } , the corresponding action of G E on T is ( g E ( α ) f )( u ) = f ( α d u ) . Now,if f ( u ) = f ( α d u ) for all α, u ∈ U then f is a constant function, which proves theLemma. (cid:3) ADEL ALAHMADI AND HAMED ALSULAMI
Lemma 7.
Let W be a hereditary saturated subset of V such that Γ /W is a cycle, E ( V \ W, W ) = ∅ , Γ /W = { v , · · · , v d } . Then d X i =1 v i CK (Γ) v i ! ∩ Z ( CK (Γ)) = (0) . Proof.
Consider the set W ′ = { w ∈ W | E ( V \ W, W ) CK (Γ) w = (0) } . The set W ′ is hereditary and saturated. Moreover, d X i =1 v i CK (Γ) v i ! ∩ I ( W ′ ) = (0) . Indeed, we only need to notice that if p ∈ P ath (Γ) , s ( p ) = v i and r ( p ) ∈ W ′ then p = 0 . Let p = e · · · e n , e i ∈ E. At least one edge e j , ≤ j ≤ n, lies in E ( V \ W, W ) . This implies the claim. Factoring out I ( W ′ ) we can assume that W ′ = ∅ . Let 0 = z ∈ d X i =1 v i CK (Γ) v i ! ∩ Z ( CK (Γ)) . For an arbitrary edge e ∈ E ( V \ W, W ) we have ze = ez. Hence ze = zer ( e ) = ezr ( e ) = 0 . Consider the ideal J = { a ∈ CK (Γ) | E ( V \ W, W ) CK (Γ) a = aCK (Γ) E ( V \ W, W ) = (0) } of the algebra CK (Γ) . The element z lies in J. The ideal J is gauge invariant. Henceby [BPRS], [BHRS] J ∩ V = ∅ . A vertex v i , ≤ i ≤ d, can not lie in J because v i CK (Γ) E ( V \ W, W ) = (0) . On the other hand J ∩ W ⊆ W ′ = ∅ , a contradiction,which proves the Lemma. (cid:3) Lemma 8.
Let W be a hereditary saturated subset of V such that Γ /W is a cycleand E ( V \ W, W ) = ∅ . Then Z ( CK (Γ)) ⊆ C · CK (Γ) + I ( W ) . Proof.
As in the Lemma 7 we assume that V \ W = { v , · · · , v d } , E ′ = E \ E ( V, W ) = { e , · · · , e d } ; s ( e i ) = v i , ≤ i ≤ d, r ( e i ) = v i +1 for 1 ≤ i ≤ d − ,r ( e d ) = v . Consider the action of the group G E ′ on CK (Γ) . Let z ∈ Z ( CK (Γ)) ,z = a + b, a = d X i =1 v i zv i , b = X w ∈ W wzw. Since every element from G E ′ fixes CK ( W )it follows that g ( b ) = b for all g ∈ G E ′ . Now g ( z ) = g ( a ) + b, g ( z ) − z = g ( a ) − a ∈ d X i =1 v i CK (Γ) v i ! ∩ Z ( CK (Γ)) = (0) by Lemma 7. we proved that an arbitraryelement of Z ( CK (Γ)) is fixed by G E ′ . The ideal I ( W ) is invariant with respect to G E ′ . Hence the group G E ′ acts on CK (Γ) /I ( W ) ∼ = CK (Γ /W ) as the full gaugegroup of automorphisms. The image of the central element z in Z ( CK (Γ /W ))is fixed by G Γ /W . Hence by Lemma 6 it is scalar. This finishes the proof of theLemma. (cid:3)
ENTERS OF CUNTZ-KRIEGER C ∗ -ALGEBRAS 9 Proof of Theorem 1. If V does not contain proper hereditary subsets then by Lemma5 Γ is either simple or a cycle. If the C ∗ -algebra CK (Γ) is simple then Z ( CK (Γ)) = C · C is a cycle then Z ( CK (Γ)) ∼ = T and the assertion of the Theorem is againtrue.Let now W be a maximal proper hereditary subset of V. The saturation c W isequal to V or the set W is saturated. In the first case CK (Γ) = I ( W ) and it sufficesto refer to Lemma 4. Suppose now that the set W is saturated. The graph Γ /W does not have proper hereditary subsets. Again by Lemma 5 the graph Γ /W iseither simple or a cycle. If Γ /W is simple then Z ( CK (Γ /W )) = C · , which implies Z ( CK (Γ) ⊆ C · I ( W ) , which together with Lemma 4 implies the Theorem. IfΓ /W is a cycle then by Lemma 8 we again have Z ( CK (Γ) ⊆ C · I ( W ) and theTheorem follows. (cid:3) Acknowledgement
This project was funded by the Deanship of Scientific Research (DSR), KingAbdulaziz University, under Grant No. (27-130-36-HiCi). The authors, therefore,acknowledge technical and financial support of KAU.
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