Simple Cuntz-Pimsner rings
aa r X i v : . [ m a t h . R A ] O c t SIMPLE CUNTZ-PIMSNER RINGS
TOKE MEIER CARLSEN, EDUARD ORTEGA, AND ENRIQUE PARDO
Abstract.
Necessary and sufficient conditions for when every non-zero ideal in arelative Cuntz-Pimsner ring contains a non-zero graded ideal, when a relative Cuntz-Pimsner ring is simple, and when every ideal in a relative Cuntz-Pimsner ring is graded,are given. A “Cuntz-Krieger uniqueness theorem” for relative Cuntz-Pimsner rings isalso given and condition (L) and condition (K) for relative Cuntz-Pimsner rings areintroduced. Introduction
In [5] the two first named authors introduced the notion of a relative Cuntz-Pimsnerring O ( P,Q,ψ ) ( J ) as an algebraic analogue of (relative) Cuntz-Pimsner C ∗ -algebras (seefor example [11], [13], [7] and [9]), and showed that for instance Leavitt path algebras(see for example [1], [2] and [17]), crossed products of a ring by a single automorphism(also called a skew group ring, see for example [10] and [12]) and fractional skew monoidrings of a single corner isomorphism (see [3]) can be constructed as relative Cuntz-Pimsner rings. They also gave a complete description of the graded ideals of an arbitraryrelative Cuntz-Pimsner ring O ( P,Q,ψ ) ( J ). The purpose of this paper is to study the non-graded ideals of such a relative Cuntz-Pimsner ring O ( P,Q,ψ ) ( J ). Although we do notreach a complete description of all (graded or non-graded) ideals of O ( P,Q,ψ ) ( J ), we dofind necessary and sufficient conditions for when every non-zero ideal in O ( P,Q,ψ ) ( J )contains a non-zero graded ideal (Theorem 3.2), when O ( P,Q,ψ ) ( J ) is simple (Theorem5.3), and when every ideal in O ( P,Q,ψ ) ( J ) is graded (Theorem 6.2). We also give a “Cuntz-Krieger uniqueness theorem” for O ( P,Q,ψ ) ( J ) (Theorem 4.2) and introduce condition (L)(Definition 3.1) and condition (K) (Definition 6.1) for relative Cuntz-Pimsner rings.These results and definitions are generalizations of similar results and definitions aboutLeavitt path algebras given in [17], and analogues of similar results and definitions given Date : December 2, 2017.2000
Mathematics Subject Classification.
Primary 16D25, 16D70; Secondary 16S10, 16S35, 16S99,46L55.
Key words and phrases.
Cuntz-Pimsner rings, simplicity, invariant cycles, condition (K), condition(L), Cuntz-Krieger uniqueness, Toeplitz rings, Leavitt path algebras, crossed products, fractional skewmonoid rings.This research was supported by the NordForsk Research Network "Operator Algebras and Dynamics""(grant in the C ∗ -algebraic setting for graph C ∗ -algebras (see for example [14]), ultragraph C ∗ -algebras (see [16]), topological graph C ∗ -algebras (see [8]), and (relative) Cuntz-Kriegeralgebras of finitely aligned higher rank graphs (see for example [15]).It is worth pointing out that analogues in the C ∗ -algebraic setting of these results donot exist in the generality of this paper. It does not seem unreasonable to believe thatit should be possible to obtain such, but a different approach than the one used in thispaper seems to be needed. Contents.
Section 2 contains some preliminary results and the pivotal Proposition 2.6.In Section 3 condition (L) is introduced (Definition 3.1), and sufficient and necessaryconditions for when every non-zero ideal in O ( P,Q,ψ ) ( J ) contains a non-zero graded idealare given (Theorem 3.2). Section 4 contains the Cuntz-Krieger uniqueness theorem(Theorem 4.2). In Section 5 sufficient and necessary for when O ( P,Q,ψ ) ( J ) is simple aregiven (Theorem 5.3), and in Section 6 condition (K) is introduced (Definition 6.1), andsufficient and necessary conditions for when every ideal in O ( P,Q,ψ ) ( J ) is graded are given(Theorem 6.2). In Section 7 the case when J = 0 and O ( P,Q,ψ ) ( J ) is the Toeplitz ring T ( P,Q,ψ ) of ( P, Q, ψ ) is considered. Finally, in Section 8 and Section 9 we illustrate theresults obtained in the paper by applying them to Leavitt path algebras (Section 8),and to crossed products of a ring by a single automorphism and fractional skew monoidrings of a single corner isomorphism (Section 9), and thereby obtain characterizationsof when these algebras are simple. The characterization of when a Leavitt path algebrais simple is well-know (see [17, Theorem 6.18]), whereas the characterizations of when acrossed product of a ring by an automorphism and a fractional skew monoid ring by acorner isomorphism are simple, to the knowledge of the authors, are new.
Notation and conventions.
In this paper every ideal will be a two-sided ideal. Theset of integers will be denoted by Z , the set of positive integers will be denoted by N and the set of non-negative integers will be denoted by N .We will use the same notation as in [5] with the addition that R will always denotea fixed ring, ( P, Q, ψ ) will be a fixed R -system satisfying condition (FS) and J will bea fixed faithful and ψ -compatible ideal in R . To ease notation we will let σ , S , T and π denote ι JR , ι JP , ι JQ and π J , respectively. We will repeatedly use that since ( P, Q, ψ )satisfies condition (FS), the R -system ( P ⊗ n , Q ⊗ n , ψ n ) will for each n ∈ N also satisfycondition (FS) (see [5, Lemma 3.8]).2. Preliminaries
This section contains some preliminary results leading to Proposition 2.6, which ispivotal for the rest of the paper.
Lemma 2.1. If n ∈ N , x − n ∈ O ( P,Q,ψ ) ( J ) ( − n ) \ { } and x n ∈ O ( P,Q,ψ ) ( J ) ( n ) \ { } , thenthere is a p ∈ P ⊗ n and a q ∈ Q ⊗ n such that x − n T n ( q ) = 0 and S n ( p ) x n = 0 .Proof. Write x n as P ki =1 T n ( q i ) y i where q i ∈ Q ⊗ n and y i ∈ O ( P,Q,ψ ) ( J ) (0) for i =1 , , . . . , k . It follows from condition (FS) that there is a θ ∈ F P ⊗ n ( Q ⊗ n ) such that θq i = q i for each i = 1 , , . . . , k . It follows that S n ( p ) x n cannot be 0 for all p ∈ P ⊗ n .That x − n T n ( q ) = 0 for some q ∈ Q ⊗ n can be proved in a similar way. (cid:3) IMPLE CUNTZ-PIMSNER RINGS 3
Definition 2.2.
For an ideal I in R , let ψ − ( I ) be the ideal n x ∈ R | ψ ( px ⊗ q ) ∈ I for all q ∈ Q and all p ∈ P o , and let I [ ∞ ] be the ideal ∞ \ k =1 I [ k ] where I [ k ] is defined recursively by I [1] = I and I [ k ] = ψ − (cid:16) I [ k − (cid:17) ∩ I for k > I is an ideal in R , then QI = span { qx | q ∈ Q, x ∈ I } (see [5, Definition7.1]). Lemma 2.3.
Let x ∈ R . Then x ∈ ψ − ( I ) if and only if xq ∈ QI for all q ∈ Q .Proof. Assume first that x ∈ ψ − ( I ) and that q ∈ Q . Then it follows from condition(FS) that there are q , . . . , q m ∈ Q and p , . . . , p m ∈ P such that xq = P mi =1 q i ψ ( p i ⊗ xq ).Since each ψ ( p i ⊗ xq ) ∈ I , it follows that xq ∈ QI .Assume then that x ∈ R and xq ∈ QI for all q ∈ Q , and let q ∈ Q and p ∈ P . Thenthere are q , . . . q m ∈ Q and x , . . . , x m ∈ I such that xq = P mi =1 q i x i , from which itfollows that ψ ( px ⊗ q ) = ψ ( p ⊗ xq ) = P mi =1 ψ ( p ⊗ q i ) x i ∈ I . Thus x ∈ ψ − ( I ). (cid:3) Let us now specialise to the case where I = J . Lemma 2.4.
Let k ∈ N and x ∈ R . Then x ∈ J [ k ] if and only if σ ( x ) ∈ span { T k ( q ) S k ( p ) | q ∈ Q ⊗ k , p ∈ P ⊗ k } .Proof. We will prove the lemma by induction over k . For k = 1 the lemma follows from[5, Proposition 3.28].Assume now that k > x ∈ J [ k − if and only if σ ( x ) ∈ span { T k − ( q ) S k − ( p ) | q ∈ Q ⊗ k − , p ∈ P ⊗ k − } . We will then prove that x ∈ J [ k ] if and only if σ ( x ) ∈ span { T k ( q ) S k ( p ) | q ∈ Q ⊗ k , p ∈ P ⊗ k } for all x ∈ R . If x ∈ J [ k ] = ψ − ( J [ k − ) ∩ J , then itfollows from [5, Proposition 3.28] that there are q , . . . , q m ∈ Q and p , . . . , p m ∈ P suchthat σ ( x ) = P mi =1 T ( q i ) S ( p i ). It follows from condition (FS) that there are q ′ , . . . , q ′ n ∈ Q and p ′ , . . . , p ′ n ∈ P such that P nj =1 θ p ′ j ,q ′ j p i = p i for each i , from which it follows that σ ( x ) = m X i =1 T ( q i ) S ( p i ) = m X i =1 n X j =1 T ( q i ) S ( p i ) T ( q ′ j ) S ( p ′ j ) = n X j =1 T ( xq ′ j ) S ( p ′ j ) . It follows from Lemma 2.3 that there for each j are q j, . . . , q j,m j ∈ Q and x j, , . . . , x j,m j ∈ J [ k − such that xq j = P m j l =1 q j,l x j,l , and it then follows from the induction hypothesisthat σ ( x ) = n X j =1 T ( xq ′ j ) S ( p ′ j )= n X j =1 m j X l =1 T ( q j,l ) σ ( x j,l ) S ( p ′ j ) ∈ span { T k ( q ) S k ( p ) | q ∈ Q ⊗ k , p ∈ P ⊗ k } . Conversely, if σ ( x ) = P mi =1 T k ( q i ) S k ( p i ), then ι R ( x ) − P mi =1 ι kQ ( q i ) ι kP ( p i ) ∈ T ( J ) (cf. [5,Definition 3.15 and 3.16]), so it follows from [5, Lemma 3.21] that x ∈ J . If p ∈ P and IMPLE CUNTZ-PIMSNER RINGS 4 q ∈ Q , then σ (cid:16) ψ ( px ⊗ q ) (cid:17) = S ( p ) m X i =1 T k ( q i ) S k ( p i ) T ( q ) ∈ span { T k − ( q ′ ) S k − ( p ′ ) | q ′ ∈ Q ⊗ k − , p ′ ∈ P ⊗ k − } , which together with the induction hypothesis implies that ψ ( px ⊗ q ) ∈ J [ k − , and thusthat x ∈ ψ − ( J [ k − ) ∩ J = J [ k ] . (cid:3) Definition 2.5.
A subring A of O ( P,Q,ψ ) ( J ) has the ideal intersection property if theimplication K ∩ A = { } ⇒ K = { } holds for every ideal K in O ( P,Q,ψ ) ( J ).We of course have that O ( P,Q,ψ ) ( J ) itself has the ideal intersection property. We willin this paper study when σ ( R ) and O ( P,Q,ψ ) ( J ) (0) have the ideal intersection property.We begin with O ( P,Q,ψ ) ( J ) (0) .Let n ∈ N . Recall from [5, Section 2] that there for each p ∈ P exists a unique R -bimodule homomorphism S p : Q ⊗ n +1 → Q ⊗ n characterised by S p ( q ⊗ q n ) = ψ ( p ⊗ q ) q n for q ∈ Q and q n ∈ Q ⊗ n . Similarly, there exists for each q n ∈ Q ⊗ n an R -bimodulehomomorphism T q n : Q → Q ⊗ n +1 given by T q n ( q ) = q n ⊗ q for q ∈ Q . Notice that T n ( S p T q n ( q )) = S ( p ) T n ( q n ) T ( q ) for p ∈ P , q n ∈ Q ⊗ n and q ∈ Q . Proposition 2.6.
The following 3 conditions are equivalent:(1) The subring O ( P,Q,ψ ) ( J ) (0) does not have the ideal intersection property.(2) There is a non-zero graded ideal L k ∈ Z H ( k ) in O ( P,Q,ψ ) ( J ) , an n ∈ N and afamily ( φ k ) k ∈ Z of injective O ( P,Q,ψ ) ( J ) (0) -bimodule homomorphisms φ k : H ( k ) →O ( P,Q,ψ ) ( J ) ( k + n ) such that xφ k ( y ) = φ k + j ( xy ) and φ k ( y ) x = φ k + j ( yx ) for k, j ∈ Z , x ∈ O ( P,Q,ψ ) ( J ) ( j ) and y ∈ H ( k ) .(3) There is a non-zero ψ -invariant ideal I of R , an n ∈ N and an injective R -bimodule homomorphism η : I → Q ⊗ n such that S p T η ( x ) ( q ) = η ( ψ ( px ⊗ q )) for p ∈ P , x ∈ I and q ∈ Q , and such that I ⊆ J [ ∞ ] .Proof. (1) ⇒ (2): Let K be a non-zero ideal in O ( P,Q,ψ ) ( J ) such that K ∩O ( P,Q,ψ ) ( J ) (0) = { } . Let N be the set of n ∈ N for which there are x i ∈ O ( P,Q,ψ ) ( J ) ( i ) , i = 0 , , . . . , n with x = 0 such that P ni =0 x i ∈ K . Let P ki = j x i ∈ K where j ≤ k ∈ Z , x i ∈ O ( P,Q,ψ ) ( J ) ( i ) for i = j, j + 1 , . . . , k and x j = 0. If j = 0, then it follows from Lemma 2.1 that there is a y − j ∈ O ( P,Q,ψ ) ( J ) ( − j ) such that either y − j x j or x j y − j is non-zero. It follows that N = ∅ .Since K ∩ O ( P,Q,ψ ) ( J ) (0) = { } , it follows that 0 / ∈ N . Let n = min N . Then n ∈ N .For each k ∈ Z let H ( k ) := ( x k ∈ O ( P,Q,ψ ) ( J ) ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ∃ x k + i ∈ O ( P,Q,ψ ) ( J ) ( k + i ) , i = 1 , , . . . , n : n X i =0 x k + i ∈ K ) . If x ∈ H ( k ) and y ∈ O ( P,Q,ψ ) ( J ) ( j ) , then xy, yx ∈ H ( k + j ) . It follows that L k ∈ Z H ( k ) is agraded ideal in O ( P,Q,ψ ) ( J ), and since H (0) = { } , it must be the case that L k ∈ Z H ( k ) isnon-zero.Let k ∈ Z and let x k ∈ H ( k ) . It follows from Lemma 2.1 and the minimality of n that there is a unique x k + n ∈ O ( P,Q,ψ ) ( J ) ( k + n ) satisfying that there exist x k + i ∈O ( P,Q,ψ ) ( J ) ( k + i ) , i = 1 , , . . . , n − P ni =0 x k + i ∈ K . It also follows from IMPLE CUNTZ-PIMSNER RINGS 5
Lemma 2.1 and the minimality of n that x k + n = 0 if x k = 0. Thus there is aninjective map φ k : H ( k ) → O ( P,Q,ψ ) ( J ) ( k + n ) sending x k to x k + n . It is easy to checkthat φ k is a O ( P,Q,ψ ) ( J ) (0) -bimodule homomorphism, and that xφ k ( y ) = φ k + j ( xy ) and φ k ( y ) x = φ k + j ( yx ) when k, j ∈ Z , x ∈ O ( P,Q,ψ ) ( J ) ( j ) and y ∈ H ( k ) .(2) ⇒ (3): We will first prove that H (0) ∩ σ ( R ) = { } , so assume, for contradiction,that H (0) ∩ σ ( R ) = { } . Then it follows from [5, Lemma 3.21 and Theorem 7.27] that H (0) = span (cid:16) { T n ( q )( σ ( x ) − π (∆( x ))) S n ( p ) | n ∈ N , q ∈ Q ⊗ n , x ∈ J ′ , p ∈ P ⊗ n }∪ { σ ( x ) − π (∆( x )) | x ∈ J ′ } (cid:17) for some faithful ψ -compatible ideal J ′ of R which contains J . We claim that H (0) mustcontain a non-zero element of the form σ ( x ) − π (∆( x )), x ∈ J ′ . To see that this is thecase, let y be a non-zero element of H (0) and write it as σ ( x ) − π (∆( x )) + k X i =1 T n i ( q i ) (cid:16) σ ( x i ) − π (∆( x i )) (cid:17) S n i ( p i )where k ∈ N , x , x , . . . , x k ∈ J ′ and n i ∈ N , q i ∈ Q ⊗ n i , p i ∈ P ⊗ n i for each i ∈{ , , . . . , k } , and assume that P i ∈ M T n i ( q i ) (cid:16) σ ( x i ) − π (∆( x i )) (cid:17) S n i ( p i ) = 0 where M isthe set of those i ’s for which n i is maximal among { n , n , . . . , n k } . Let n be the maximalvalue of n i . It follows from condition (FS) that there are q ∈ Q ⊗ n and p ∈ P ⊗ n suchthat if we let x = P i ∈ M ψ n ( p ⊗ q i ) x i ψ n ( p i ⊗ q ), then σ ( x ) − π (∆( x )) = S n ( p ) X i ∈ M T n i ( q i ) (cid:16) σ ( x i ) − π (∆( x i )) (cid:17) S n i ( p i ) T n ( q ) = 0 . Since ( σ ( x ) − π (∆( x ))) T n ( q ) = 0 and ( σ ( x i ) − π (∆( x i ))) S n i ( p i ) T n ( q ) = 0 for each i / ∈ M , it follows that σ ( x ) − π (∆( x ))= S n ( p ) σ ( x ) − π (∆( x )) + k X i =1 T n i ( q i ) (cid:16) σ ( x i ) − π (∆( x i )) (cid:17) S n i ( p i ) ! T n ( q ) ∈ H (0) . Thus H (0) contains a non-zero element of the form σ ( x ) − π (∆( x )), x ∈ J ′ . If followsfrom condition (FS) that there is a p ′ ∈ P ⊗ n such that S n ( p ′ ) φ (cid:16) σ ( x ) − π (∆( x ) (cid:17) = 0 , but since S n ( p ′′ )( σ ( x ) − π (∆( x ))) = 0 for all p ′′ ∈ P ⊗ n , it follows that S n ( p ′ ) φ (cid:16) σ ( x ) − π (∆( x ) (cid:17) = φ − n (cid:16) S n ( p ′ )( σ ( x ) − π (∆( x )) (cid:17) = 0 , and we have reached a contradiction. Thus it must be the case that H (0) ∩ σ ( R ) = { } .Let I = { x ∈ R | σ ( x ) ∈ H (0) } . Then I is a non-zero ψ -invariant ideal of R . For each m ∈ N let A m = span n T n + k ( q ) S k ( p ) | k ∈ { , , . . . , m } , q ∈ Q ⊗ n + k , p ∈ P ⊗ k o ⊆ O ( P,Q,ψ ) ( J ) ( n ) and I m = { x ∈ I | φ ( σ ( x )) ∈ A m } . IMPLE CUNTZ-PIMSNER RINGS 6
Then I ⊆ I ⊆ I ⊆ . . . and each I m is a ψ -invariant two-sided ideal in R . In fact, x ∈ I m +1 , implies that ψ ( px ⊗ q ) ∈ I m for all p ∈ P and q ∈ Q . Since I is nonzero, thereexists an x = 0 and an m ∈ N such that x ∈ I m . Choose k ∈ N such that kn ≥ m .Then φ ( k − n ◦ φ ( k − n ◦ · · · ◦ φ n ◦ φ ( σ ( x )) ∈ O ( P,Q,ψ ) ( J ) ( nk ) \ { } so it follows from Lemma 2.1 that there is a p ∈ P ⊗ nk such that φ − n ◦ φ − n ◦· · ·◦ φ − ( k − n ◦ φ − kn ( S nk ( px )) = S nk ( p ) φ ( k − n ◦ φ ( k − n ◦· · ·◦ φ n ◦ φ ( σ ( x )) = 0 , from which it follows that px = 0. It follows from condition (FS) that there is a q ∈ Q ⊗ kn such that ψ kn ( px ⊗ q ) = 0. We have that ψ kn ( px ⊗ q ) ∈ I , so I = { } .Since φ ( σ ( x )) ∈ T n ( Q ⊗ n ) for every x ∈ I , and T n : Q ⊗ n → O ( P,Q,ψ ) ( J ) ( n ) is injective,we can define η : I → Q ⊗ n by, for x ∈ I , letting η ( x ) be the unique element of Q ⊗ n such that T n ( η ( x )) = φ ( σ ( x )). It is straightforward to check that η is an injective R -bimodule homomorphism, and if p ∈ P , x ∈ I and q ∈ Q , then T n (cid:18) η (cid:16) ψ ( px ⊗ q ) (cid:17)(cid:19) = φ (cid:18) σ (cid:16) ψ ( px ⊗ q ) (cid:17)(cid:19) = S ( p ) φ (cid:16) σ ( x ) (cid:17) T ( q )= S ( p ) T n (cid:16) η ( x ) (cid:17) T ( q ) = T n (cid:16) S p T η ( x ) ( q ) (cid:17) , from which it follows that η ( ψ ( px ⊗ q )) = S p T η ( x ) ( q ).If x ∈ I then it follows from condition (FS) that there are q i ∈ Q ( n ) , p i ∈ P ( n ) , i = 1 , , . . . , m such that P mi =0 θ q i ,p i η ( x ) = η ( x ). We then have that T n ( η ( x )) = T n m X i =0 θ q i ,p i η ( x ) ! = m X i =0 T n ( q i ) S n ( p i ) φ ( σ ( x )) = φ m X i =0 T n ( q i ) S n ( p i x ) ! from which it follow that σ ( x ) = P mi =0 T n ( q i ) S n ( p i x ). It now follows from Lemma 2.4that x ∈ J [ n ] ⊆ J . Since I is ψ -invariant, it follows that x ∈ J [ ∞ ] .(3) ⇒ (1): Let K be the ideal in O ( P,Q,ψ ) ( J ) generated by { σ ( x ) − T n ( η ( x )) | x ∈ I } .Clearly, K is non-zero, so we just have to prove that K ∩ O ( P,Q,ψ ) ( J ) (0) = { } . Usingcondition (FS) and the properties of η , one can show that if p ∈ P , x ∈ I and q ∈ Q ,then S ( p ) (cid:16) σ ( x ) − T n ( η ( x )) (cid:17) ∈ span n(cid:16) σ ( x ′ ) − T n ( η ( x ′ )) (cid:17) S ( p ′ ) (cid:12)(cid:12)(cid:12) x ′ ∈ I , p ′ ∈ P o and (cid:16) σ ( x ) − T n ( η ( x )) (cid:17) T ( q ) ∈ span n T ( q ′ )( σ ( x ′ ) − T n ( η ( x ′ ))) (cid:12)(cid:12)(cid:12) q ′ ∈ Q, x ′ ∈ I o . It follows that K = span (cid:18)n T k ( q ) (cid:16) σ ( x ) − T n ( η ( x )) (cid:17) (cid:12)(cid:12)(cid:12) k ∈ N , q ∈ Q ⊗ k , x ∈ I o ∪ n T k ( q ) (cid:16) σ ( x ) − T n ( η ( x )) (cid:17) S l ( p ) (cid:12)(cid:12)(cid:12) k, l ∈ N , q ∈ Q ⊗ k , x ∈ I , p ∈ P ⊗ l o ∪ n σ ( x ) − T n ( η ( x )) (cid:12)(cid:12)(cid:12) x ∈ I o ∪ n T k ( q ) (cid:16) σ ( x ) − T n ( η ( x )) (cid:17) (cid:12)(cid:12)(cid:12) l ∈ N , x ∈ I , p ∈ P ⊗ l o(cid:19) , so to show that K ∩ O ( P,Q,ψ ) ( J ) (0) = { } , it sufficies to show the following 3 things: IMPLE CUNTZ-PIMSNER RINGS 7 (i) if l ∈ N , A is a finite subset of { ( k, q, x, p ) | k ∈ N , q ∈ Q ⊗ l + k , x ∈ I , p ∈ P ⊗ k } and B is a finite subset of { ( q, x ) | q ∈ Q ⊗ l , x ∈ I } , then X ( k,q,x,p ) ∈ A T l + k ( q ) σ ( x ) S k ( p ) + X ( q,x ) ∈ B T l ( q ) σ ( x ) = 0if and only if X ( k,q,x,p ) ∈ A T l + k ( q ) T n ( η ( x )) S k ( p ) + X ( q,x ) ∈ B T l ( q ) T n ( η ( x )) = 0 , (ii) if A is a finite subset of { ( k, q, x, p ) | k ∈ N , q ∈ Q ⊗ k , x ∈ I , p ∈ P ⊗ k } and x ∈ I , then X ( k,q,x,p ) ∈ A T k ( q ) σ ( x ) S k ( p ) + σ ( x ) = 0if and only if X ( k,q,x,p ) ∈ A T k ( q ) T n ( η ( x )) S k ( p ) + T n ( η ( x )) = 0 , (iii) if l ∈ N , A is a finite subset of { ( k, q, x, p ) | k ∈ N , q ∈ Q ⊗ k , x ∈ I , p ∈ P ⊗ l + k } and B is a finite subset of { ( x, p ) | x ∈ I , p ∈ P ⊗ l + k } , then X ( k,q,x,p ) ∈ A T k ( q ) σ ( x ) S l + k ( p ) + X ( q,x ) ∈ B σ ( x ) S l + k ( p ) = 0if and only if X ( k,q,x,p ) ∈ A T k ( q ) T n ( η ( x )) S l + k ( p ) + X ( x,p ) ∈ B T n ( η ( x )) S l ( p ) = 0 . We will just prove (i). The other two claims can be proved in a similar way.To prove (i), notice first that if x ∈ I and k ∈ N , then, since I ⊆ J [ ∞ ] ⊆ J [ k ] , it followsfrom Lemma 2.4 that there are q , . . . , q m ∈ Q ⊗ k and p , . . . , p m ∈ P ⊗ k such that σ ( x ) = P mi =1 T k ( q i ) S k ( p i ). It follows from condition (FS) that there are q ′ , . . . , q ′ r , q ′′ , . . . , q ′′ s ∈ Q ⊗ k and p ′ , . . . , p ′ r , p ′′ , . . . , p ′′ s ∈ P ⊗ k such that σ ( x ) = m X i =1 T k ( q i ) S k ( p i ) = r X j =1 s X l =1 m X i =1 T k ( q ′ j ) S k ( p ′ j ) T k ( q i ) S k ( p i ) T k ( q ′′ l ) S k ( p ′′ l )= r X j =1 s X l =1 T k ( q ′ j ) S k ( p ′ j ) σ ( x ) T k ( q ′′ l ) S k ( p ′′ l ) = r X j =1 s X l =1 T k ( q ′ j ) σ ( ψ k ( p ′ j x ⊗ q ′′ l )) S k ( p ′′ l ) . Since I is ψ -invariant, it follows that each ψ k ( p ′ j x ⊗ q ′′ l ) ∈ I and thus that T n ( η ( x )) = r X j =1 s X l =1 T k ( q ′ j ) T n ( η ( ψ k ( p ′ j x ⊗ q ′′ l ))) S k ( p ′′ l ) . Thus it sufficies to show that if k, l ∈ N and C is a finite subset of { ( q, x, p ) | q ∈ Q l + k , x ∈ I , p ∈ P ⊗ k } , then it is the case that P ( q,x,p ) ∈ C T l + k ( q ) σ ( x ) S k ( p ) = 0 if andonly if P ( q,x,p ) ∈ C T l + k ( q ) T n ( η ( x )) S k ( p ) = 0, and that can be done using condition (FS)and the properties of η . (cid:3) IMPLE CUNTZ-PIMSNER RINGS 8 Condition (L)
In this section condition (L) is introduced (Definition 3.1) and sufficient and necessaryconditions for when every non-zero ideal in O ( P,Q,ψ ) ( J ) contains a non-zero graded ideal(Theorem 3.2) are given. Definition 3.1.
We say that a ψ -invariant ideal I in R is an ψ -invariant cycle if thereexist n ∈ N and an injective R -bimodule homomorphism η : I → Q ⊗ n such that S p T η ( x ) ( q ) = η ( ψ ( px ⊗ q )) for p ∈ P , x ∈ I and q ∈ Q , and we say that J satisfies condition (L) with respect to the R -system ( P, Q, ψ ) if there are no non-zero ψ -invariantcycles I in R such that I ⊆ J [ ∞ ] .We will often, when it is clear from the context which R -system ( P, Q, ψ ) we areworking with, simply call a ψ -invariant cycle for an invariant cycle, and say that J satisfies condition (L) instead of saying that it satisfies condition (L) with respect to( P, Q, ψ ).Recall that if ( S ′ , T ′ , σ ′′ B ) is a covariant representation of ( P, Q, ψ ), then J ( S ′ ,T ′ ,σ ′ ,B ) is defined to be the ideal { x ∈ R | σ ′ ( x ) ∈ π T ′ ,S ′ ( F P ( Q ) } (see [5, Definition 3.23]). Theorem 3.2.
The following 4 conditions are equivalent:(1) The ideal J satisfies condition (L).(2) The subring O ( P,Q,ψ ) ( J ) (0) has the ideal intersection property.(3) Every non-zero ideal in O ( P,Q,ψ ) ( J ) contains a non-zero graded ideal.(4) If ( S ′ , T ′ , σ ′′ B ) is an injective covariant representation of ( P, Q, ψ ) and J = J ( S ′ ,T ′ ,σ ′ ,B ) , then the ring homomorphism η J ( S ′ ,T ′ ,σ ′ ,B ) : O ( P,Q,ψ ) ( J ) → B from [5,Theorem 3.29 (ii)] is injective.Proof. (1) ⇔ (2) follows from Proposition 2.6.(2) ⇒ (3): Let K be a non-zero ideal in O ( P,Q,ψ ) ( J ). Then K ∩ O ( P,Q,ψ ) ( J ) (0) = { } by assumption, and it follows from [5, Lemma 3.35] that the ideal H generated by K ∩ O ( P,Q,ψ ) ( J ) (0) is graded. Since H is obviously contained in K , this proves (3).(3) ⇒ (2): Let K be a non-zero ideal in O ( P,Q,ψ ) ( J ). By assumption there is anon-zero graded ideal H such that H ⊆ K . It follows from [5, Lemma 3.35] that H ∩O ( P,Q,ψ ) ( J ) (0) = { } , so also K ∩O ( P,Q,ψ ) ( J ) (0) = { } , which proves that O ( P,Q,ψ ) ( J ) (0) has the ideal intersection property.(2) ⇒ (4): Let H be the ideal in O ( P,Q,ψ ) ( J ) generated by ker η J ( S ′ ,T ′ ,σ ′ ,B ) ∩O ( P,Q,ψ ) ( J ) (0) ,and let ℘ : O ( P,Q,ψ ) ( J ) → O ( P,Q,ψ ) ( J ) /H be the quotient map. Then ( ℘ ◦ S, ℘ ◦ T, ℘ ◦ σ, O ( P,Q,ψ ) ( J ) /H ) is a surjective covariant representation of ( P, Q, ψ ). It fol-lows from [5, Lemma 3.35] that H is graded, from which it follows that the repre-sentation ( ℘ ◦ S, ℘ ◦ T, ℘ ◦ σ, O ( P,Q,ψ ) ( J ) /H ) is graded (see [5, Definition 3.20]). Since H ⊆ ker η J ( S ′ ,T ′ ,σ ′ ,B ) , it follows that there is a ring homomorphism φ : O ( P,Q,ψ ) ( J ) /H → B such that φ ◦ ℘ = η J ( S ′ ,T ′ ,σ ′ ,B ) and φ ◦ ℘ ◦ S = S ′ , φ ◦ ℘ ◦ T = T ′ and φ ◦ ℘ ◦ σ = σ ′ .Since ( S ′ , T ′ , σ ′ , B ) is an injective representation, it follows that also ( ℘ ◦ S, ℘ ◦ T, ℘ ◦ σ, O ( P,Q,ψ ) ( J ) /H ) is injective. It follows from [5, Remark 3.13] that J ⊆ J ( ℘ ◦ S,℘ ◦ T,℘ ◦ σ, O ( P,Q,ψ ) ( J ) /H ) ⊆ J ( S ′ ,T ′ ,σ ′ ,B ) = J. IMPLE CUNTZ-PIMSNER RINGS 9
Thus J ( ℘ ◦ S,℘ ◦ T,℘ ◦ σ, O ( P,Q,ψ ) ( J ) /H ) = J , and it follows from [5, Theorem 3.29] that ℘ isan isomorphism, and thus that ker η J ( S ′ ,T ′ ,σ ′ ,B ) ∩ O ( P,Q,ψ ) ( J ) (0) = { } . It follows byassumption that ker η J ( S ′ ,T ′ ,σ ′ ,B ) = { } , and thus that η J ( S ′ ,T ′ ,σ ′ ,B ) is injective.(4) ⇒ (2): Let K be an ideal in O ( P,Q,ψ ) ( J ) such that K ∩ O ( P,Q,ψ ) ( J ) (0) = { } ,and let ℘ : O ( P,Q,ψ ) ( J ) → O ( P,Q,ψ ) ( J ) /K be the quotient map. Then ( ℘ ◦ S, ℘ ◦ T, ℘ ◦ σ, O ( P,Q,ψ ) ( J ) /K ) is a surjective covariant representation of ( P, Q, ψ ). Since σ ( R ) and π T,S ( F P ( Q )) are subsets of O ( P,Q,ψ ) ( J ) (0) and K ∩ O ( P,Q,ψ ) ( J ) (0) = { } , it follows from[5, Proposition 3.28] that J ( ℘ ◦ S,℘ ◦ T,℘ ◦ σ, O ( P,Q,ψ ) ( J ) /H ) = J ( S,T,σ, O ( P,Q,ψ ) ( J )) = J. Thus ℘ = η ( ℘ ◦ S,℘ ◦ T ℘ ◦ σ, O ( P,Q,ψ ) ( J ) /K ) is injective by assumption, and K = { } which provesthat O ( P,Q,ψ ) ( J ) (0) has the ideal intersection property. (cid:3) The Cuntz-Krieger uniqueness theorem
In this Section the
Cuntz-Krieger uniqueness property is defined (Definition 4.1), andthe
Cuntz-Krieger uniqueness result is proved (Theorem 4.2).
Definition 4.1.
We say that the ideal J has the Cuntz-Krieger uniqueness property with respect to the R -system ( P, Q, ψ ) if the following holds:If ( S , T , σ , B ) and ( S , T , σ , B ) are two injective covariant representations of( P, Q, ψ ) and they are both Cuntz-Pimsner invariant relative to J , then there is a ringisomorphism φ between Rh S , T , σ i and Rh S , T , σ i such that φ ◦ σ = σ , φ ◦ S = S and φ ◦ T = T .We will often, when it is clear from the context which R -system ( P, Q, ψ ) we areworking with, simply say that J has the Cuntz-Krieger uniqueness property instead ofsaying that it has the Cuntz-Krieger uniqueness property with respect to ( P, Q, ψ ).Recall from [5, Definition 4.6] that J is said to be a maximal ψ -compatible ideal if J = J ′ for any faithful ψ -compatible ideal J ′ in R satisfying J ⊆ J ′ . Theorem 4.2.
The following 5 conditions are equivalent:(1) The ideal J has the Cuntz-Krieger uniqueness property.(2) If ( S ′ , T ′ , σ ′ , B ) is an injective covariant representation of ( P, Q, ψ ) which isCuntz-Pimsner invariant relative to J , then the ring homomorphism η J ( S ′ ,T ′ ,σ ′ ,B ) : O ( P,Q,ψ ) ( J ) → B from [5, Theorem 3.18] is injective.(3) The subring σ ( R ) has the ideal intersection property.(4) The subring O ( P,Q,ψ ) ( J ) (0) has the ideal intersection property, and J is a maximal ψ -compatible ideal.(5) The ideal J satisfies condition (L) and is a maximal ψ -compatible ideal.Proof. (1) ⇒ (2): The ring homomorphism η J ( S ′ ,T ′ ,σ ′ ,B ) : O ( P,Q,ψ ) ( J ) → B is the uniquering homomorphism from O ( P,Q,ψ ) ( J ) to B such that η J ( S ′ ,T ′ ,σ ′ ,B ) ◦ σ = σ ′ , η J ( S ′ ,T ′ ,σ ′ ,B ) ◦ S = S ′ and η J ( S ′ ,T ′ ,σ ′ ,B ) ◦ T = T ′ , so it follows by assumption that η J ( S ′ ,T ′ ,σ ′ ,B ) is injective.(2) ⇒ (1): If ( S , T , σ , B ) and ( S , T , σ , B ) are two injective covariant represen-tations of ( P, Q, ψ ) and there are both Cuntz-Pimsner invariant relative to J , then φ = η J ( S ,T ,σ ,B ) ◦ ( η J ( S ,T ,σ ,B ) ) − is a ring isomorphism between Rh S , T , σ i and Rh S , T , σ i such that φ ◦ σ = σ , φ ◦ S = S and φ ◦ T = T . IMPLE CUNTZ-PIMSNER RINGS 10 (2) ⇒ (3): Let K be an ideal in O ( P,Q,ψ ) ( J ) such that K ∩ σ ( R ) = { } , andlet ℘ : O ( P,Q,ψ ) ( J ) → O ( P,Q,ψ ) ( J ) /K be the quotient map. Then ( ℘ ◦ S, ℘ ◦ T, ℘ ◦ σ, O ( P,Q,ψ ) ( J ) /K ) is an injective and surjective covariant representation of ( P, Q, ψ )which is Cuntz-Pimsner invariant relative to J . It follows by assumption that ℘ = η ( ℘ ◦ S,℘ ◦ T ℘ ◦ σ, O ( P,Q,ψ ) ( J ) /K ) is injective. Thus K = { } , which proves that σ ( R ) has theideal intersection property.(3) ⇒ (4): Since σ ( R ) ⊆ O ( P,Q,ψ ) ( J ) (0) , it follows that O ( P,Q,ψ ) ( J ) (0) has the idealintersection property if σ ( R ) has. If J is not a maximal ψ -invariant ideal, then thereexists a ψ -compatible ideal J ′ such that J ( J ′ . It follows from [5, Remark 4.1] that ρ J ( T ( J ′ )) then would be a non-zero ideal in O ( P,Q,ψ ) ( J ) with a zero intersection with σ ( R ), which would mean that σ ( R ) does not have the ideal intersection property. Thusit must be the case that J is a maximal ψ -invariant ideal.(4) ⇒ (2): Since J is a maximal ψ -compatible ideal by assumption, it follows that J ( S ′ ,T ′ ,σ ′ ,B ) = J . Thus it follows from Theorem 3.2 that η J ( S ′ ,T ′ ,σ ′ ,B ) is injective.(4) ⇔ (5) follows from Theorem 3.2. (cid:3) Simplicity of O ( P,Q,ψ ) ( J )In this section sufficient and necessary conditions for when O ( P,Q,ψ ) ( J ) is simple aregiven (Theorem 5.3). Definition 5.1.
We say that J is a super maximal ψ -compatible ideal if the only T -pairs( I, J ′ ) of ( P.Q, ψ ) which satisfies that J ⊆ J ′ , are (0 , J ) and ( R, R ).Since (0 , J ′ ) is a T -pair of ( P.Q, ψ ) for any any faithful ψ -compatible ideal J ′ in R ,it follows that if J is a super maximal ψ -compatible ideal, then it is also a maximal ψ -compatible ideal. Remark . It follows from [5, Theorem 7.27] that J is a super maximal ψ -compatibleideal if and only if the only graded ideals in O ( P,Q,ψ ) ( J ) are { } and O ( P,Q,ψ ) ( J ). Theorem 5.3.
The following 5 conditions are equivalent:(1) The ring O ( P,Q,ψ ) ( J ) is simple.(2) The subring σ ( R ) has the ideal intersection property and J is a super maximal ψ -compatible ideal.(3) The subring O ( P,Q,ψ ) ( J ) (0) has the ideal intersection property and J is a supermaximal ψ -compatible ideal.(4) The ideal J satisfies condition (L) and is a super maximal ψ -compatible ideal.(5) If ( S ′ , T ′ , σ ′ , B ) is a non-zero covariant representation of ( P, Q, ψ ) which is Cuntz-Pimsner invariant relative to J , then the ring homomorphism η J ( S ′ ,T ′ ,σ ′ ,B ) : O ( P,Q,ψ ) ( J ) → B from [5, Theorem 3.18] is injective.Proof. (1) ⇒ (2): If O ( P,Q,ψ ) ( J ) is simple, then clearly σ ( R ) has the ideal intersectionproperty. If ( I, J ′ ) is a T -pair of ( P, Q, ψ ) different from (0 , J ), then it follows from [5,Theorem 7.27] that H J ( I,J ′ ) is a non-zero ideal in O ( P,Q,ψ ) ( J ). If O ( P,Q,ψ ) ( J ) is simple,then that would imply that H J ( I,J ′ ) = O ( P,Q,ψ ) ( J ) and thus ( I, J ′ ) = ( R, R ) from whichit follows that J is a super maximal ψ -compatible ideal. IMPLE CUNTZ-PIMSNER RINGS 11 (2) ⇔ (3) and (3) ⇔ (4) follow from Theorem 4.2 and the fact that J is a maximal ψ -compatible ideal if it is a super maximal ψ -compatible ideal.(2) ⇒ (5): It follows from [5, Proposition 7.8] that ( I ( S ′ ,T ′ ,σ ′ ,B ) , J ( S ′ ,T ′ ,σ ′ ,B ) ) is a T -pair.Since ( S ′ , T ′ , σ ′ , B ) is Cuntz-Pimsner invariant relative to J , it follows from [5, Remark3.25] that J ⊆ J ( S ′ ,T ′ ,σ ′ ,B ) , and since ( S ′ , T ′ , σ ′ , B ) is non-zero, it follows from [5, Theorem7.11] that ( I ( S ′ ,T ′ ,σ ′ ,B ) , J ( S ′ ,T ′ ,σ ′ ,B ) ) = ( R, R ). Thus ( I ( S ′ ,T ′ ,σ ′ ,B ) , J ( S ′ ,T ′ ,σ ′ ,B ) ) = (0 , J ) whichimplies that ( S ′ , T ′ , σ ′ , B ) is an injective representation. It then follows from Theorem4.2 that η J ( S ′ ,T ′ ,σ ′ ,B ) is injective.(5) ⇒ (1): Let K be a proper ideal in O ( P,Q,ψ ) ( J ), and let ℘ : O ( P,Q,ψ ) ( J ) →O ( P,Q,ψ ) ( J ) /K be the quotient map. Then ( ℘ ◦ S, ℘ ◦ T, ℘ ◦ σ, O ( P,Q,ψ ) ( J ) /K ) is a surjec-tive covariant representation of ( P, Q, ψ ) which is Cuntz-Pimsner invariant relative to J .It follows by assumption that ℘ = η ( ℘ ◦ S,℘ ◦ T ℘ ◦ σ, O ( P,Q,ψ ) ( J ) /K ) is injective. Thus K = { } which proves that O ( P,Q,ψ ) ( J ) is simple. (cid:3) Condition (K)
In this section condition (K) is introduced (Definition 6.1), and sufficient and necessaryconditions for when every ideal in O ( P,Q,ψ ) ( J ) is graded are given (Theorem 6.2).Recall from [5, Section 7] that if I is a ψ -invariant ideal in R , then R I = R/I , Q I = Q/QI and I P = P/IP , and ℘ I denote the corresponding quotient map. Recall also thatthere is an R I -bimodule homomorphism ψ I : I P ⊗ Q I → R I given by ψ I ( ℘ I ( p ) ⊗ ℘ I ( q )) = ℘ I ( ψ ( p ⊗ q )). The triple ( I P, Q I , ψ I ) is then an R I -system satisfying condition (FS) (see[5, Lemma 7.4]). When ( I, J ′ ) is a T -pair, then J ′ I denote the faithful ψ I -compatibleideal ℘ I ( J ′ ) in R I . Definition 6.1.
We say that the ideal J satisfies condition (K) with respect to the R -system ( P, Q, ψ ) if J ′ I satisfies condition (L) with respect to the R I -system ( I P, Q I , ψ I )whenever ( I, J ′ ) is a T -pair of ( P, Q, ψ ) such that J ⊆ J ′ .We will often, when it is clear from the context which R -system ( P, Q, ψ ) we areworking with, simply say that J satisfies condition (K) instead of saying that it satisfiescondition (K) with respect to ( P, Q, ψ ). Theorem 6.2.
The following 3 conditions are equivalent:(1) Every ideal of O ( P,Q,ψ ) ( J ) is graded.(2) The ideal J satisfies condition (K).(3) If ( S ′ , T ′ , σ ′ , B ) is a covariant representation of ( P, Q, ψ ) which is Cuntz-Pimsnerinvariant relative to J , and ( I, J ′ ) = ω ( S ′ ,T ′ ,σ ′ ,B ) , then the ring homomorphism η ( I,J ′ )( S ′ ,T ′ ,σ ′ ,B ) : O ( I P,Q I ,ψ I ) ( J ′ I ) → B from [5, Theorem 7.11 (ii)] is injective.Proof. (1) ⇒ (2): Let ω = ( I, J ′ ) be a T -pair of ( P, Q, ψ ) such that J ⊆ J ′ and let H be a non-zero ideal in O ( I P,Q I ,ψ I ) ( J ′ I ). Recall from [5, Page 36] that there is a covariantrepresentation ( ι ωP , ι ωQ , ι ωR , O ( I P,Q I ,ψ I ) ( J ′ I )) such that ι ωP = ι J ′ II P ◦ ℘ I , ι ωQ = ι J ′ I Q I ◦ ℘ I and ι ωR = ι J ′ I R I ◦ ℘ I . It follows from [5, Remark 3.25 and Theorem 3.29] that there is a surjectivegraded ring homomorphism φ : O ( P,Q,ψ ) ( J ) → O ( I P,Q I ,ψ I ) ( J ′ I ) which intertwines the tworepresentations ( S, T, σ, O ( P,Q,ψ ) ( J )) and ( ι ωP , ι ωQ , ι ωR , O ( I P,Q I ,ψ I ) ( J ′ I )). We then have that φ − ( H ) is an ideal in O ( P,Q,ψ ) ( J ). Thus φ − ( H ) is graded by assumption. It follows that IMPLE CUNTZ-PIMSNER RINGS 12 also H is graded. It therefore follows from Theorem 3.2 that J ′ I satisfies condition (L)with respect to the R I -system ( I P, Q I , ψ I ). This proves that J satisfies condition (K).(2) ⇒ (3): It follows from [5, Lemma 7.10] that there is an injective covariant represen-tation ( S I , T I , σ I , B ) of ( I P, Q I , ψ I ) such that S I = S ′ ◦ ℘ I , T I = T ′ ◦ ℘ I and σ I = σ ′ ◦ ℘ I .Since π T I ,S I ( F I P ( Q I )) = π T ′ ,S ′ ( F P ( Q )), it follows that J ( S I ,T I ,σ I ,B ) = ℘ I ( J ( S ′ ,T ′ ,σ ′ ,B ) ) = ℘ I ( J ′ ) = J ′ I . It therefore follows from Theorem 3.2 that η ( I,J ′ )( S ′ ,T ′ ,σ ′ ,B ) = η J ′ ( S I ,T I ,σ I ,B ) isinjective.(3) ⇒ (1): Let H be an ideal in O ( P,Q,ψ ) ( J ) and let ℘ : O ( P,Q,ψ ) ( J ) → O ( P,Q,ψ ) ( J ) /H be the quotient map. Then ( ℘ ◦ S, ℘ ◦ T, ℘ ◦ σ, O ( P,Q,ψ ) ( J ) /H ) is a covariant representationwhich is Cuntz-Pimsner invariant relative to J . Let ( I, J ′ ) = ω ( ℘ ◦ S,℘ ◦ T,℘ ◦ σ, O ( P,Q,ψ ) ( J ) /H ) .Then η ( I,J ′ )( ℘ ◦ S,℘ ◦ T,℘ ◦ σ, O ( P,Q,ψ ) ( J ) /H ) is injective by assumption. Since ⊕ n ∈ Z O ( n )( I P,Q I ,ψ I ) ( J ′ I ) is a Z -grading of O ( I P,Q I ,ψ I ) ( J ′ I ), it follows that ⊕ n ∈ Z ℘ ( O ( n )( P,Q,ψ ) ( J )) = ⊕ n ∈ Z η ( I,J ′ )( ℘ ◦ S,℘ ◦ T,℘ ◦ σ, O ( P,Q,ψ ) ( J ) /H ) ( O ( n )( I P,Q I ,ψ I ) ( J ′ I ))is a Z -grading of O ( P,Q,ψ ) ( J ) /H . Thus H is graded. (cid:3) Remark . It follows from the above theorem that if J satisfies condition (K), then [5,Theorem 7.27] gives a bijective correspondence between the set of all ideals of O ( P,Q,ψ ) ( J )and the set of T -pairs ( I, J ′ ) of ( P, Q, ψ ) satisfying J ⊆ J ′ .7. Toeplitz rings
When J = { } , then O ( P,Q,ψ ) ( J ) is the Toeplitz ring T ( P,Q,ψ ) and J automaticallysatisfies condition (L). Thus the following 3 corollaries follow from Theorem 3.2, Theorem4.2 and Theorem 5.3, respectively. Corollary 7.1. If ( S ′ , T ′ , σ ′ , B ) is an injective covariant representation of ( P, Q, ψ ) ,then the ring homomorphism η ( S ′ ,T ′ ,σ ′ ,B ) : T ( P,Q,ψ ) → B from [5, Theorem 1.7] is injectiveif and only if J ( S ′ ,T ′ ,σ ′ ,B ) = { } . Corollary 7.2.
Assume that there are no non-zero faithful ψ -compatible ideals of R . If ( S , T , σ , B ) and ( S , T , σ , B ) are two injective covariant representations of ( P, Q, ψ ) ,then there is a ring isomorphism φ between Rh S , T , σ i and Rh S , T , σ i such that φ ◦ σ = σ , φ ◦ S = S and φ ◦ T = T . Corollary 7.3.
The Toeplitz ring T ( P,Q,ψ ) is simple if and only if (0 , and ( R, R ) arethe only T -pairs of ( P, Q, ψ ) . Leavitt path algebras
We will in this section show how we can recover from the results obtained in thispaper Theorem 6.8, Corollary 6.10, Theorem 6.16, Corollary 6.17 and Theorem 6.18 of[17] and obtain an algebraic analogue of [6, Theorem 4.1].Let ( E , E , r, s ) be a directed graph (ie. E and E are sets and r and s are mapsfrom E to E ) and let F be a field. When n is a positive integer, then we let E n bethe set { ( e , e , . . . , e n ) ∈ E × E × · · · × E | r ( e i ) = s ( e i +1 ) for i = 1 , , . . . , n − } .For α = ( e , e , . . . , e n ) ∈ E n we define s ( α ) to be s ( e ) and r ( α ) to be r ( e n ). For each v ∈ E we let vE n denote the set { α ∈ E n | s ( α ) = v } and we let E n v denote theset { α ∈ E n | r ( α ) = v } . A closed path is an α ∈ E n such that r ( α ) = s ( α ). The IMPLE CUNTZ-PIMSNER RINGS 13 element s ( α ) is called the base of α . A closed path α = ( e , e , . . . , e n ) is said to be simple if s ( e i ) = s ( e ) for each i = 2 , , . . . , n , and to have an exit if | s ( e i ) E | > i ∈ { , , . . . , n } .Following [5, Example 5.8] we define R be the ring ⊕ v ∈ E R v where each R v is a copyof F ; we let Q be R -bimodule ⊕ e ∈ E Q e where each Q e is a copy of F and the left andthe right multiplication are defined by X e ∈ E q e e · X v ∈ E r v v = X e ∈ E q e r r ( e ) e X v ∈ E r v v · X e ∈ E q e e = X e ∈ E r s ( e ) q e e where v denotes the unit of R v , e denotes the unit of Q e , and { r v } v ∈ E and { q e } e ∈ E are families of elements from F with only a finite number of non-zero elements; we let P be the R -bimodule ⊕ e ∈ E P e where each P e is a copy of F and the left and the rightmultiplication are defined by X e ∈ E p e e · X v ∈ E r v v = X e ∈ E p e r s ( e ) e X v ∈ E r v v · X e ∈ E p e e = X e ∈ E r r ( e ) p e e where e denotes the unit of P e , and { r v } v ∈ E and { p e } e ∈ E are families of elements from F with only a finite number of non-zero elements; and we define ψ : P ⊗ R Q → R to bethe R -bimodule homomorphism given by X e ∈ E p e e ⊗ X e ∈ E q e e X v ∈ E X e ∈ E v p e q e v , then ( P, Q, ψ ) is an R -system. Recall also that if we let J be the ideal span F { v | v ∈ E , < | vE | < ∞} ⊆ R , then J is a maximal faithful ψ -compatible ideal and O ( P,Q,ψ ) ( J ) is isomorphic to the Leavitt path algebra of ( E , E ) (see for example [1, 2]and [17]). It is straightforward to check that J [ n ] = span F { v | v ∈ E , < | vE n | < ∞} for each n ∈ N from which it follows that J [ ∞ ] = span F { v | v ∈ E , < | vE n | < ∞ for all n ∈ N } .Suppose that I is a non-zero ψ -invariant cycle and let η : I → Q ⊗ n be an injective R -bimodule homomorphism satisfying S p T η ( x ) ( q ) = η ( ψ ( px ⊗ q )) for p ∈ P , x ∈ I and q ∈ Q . We will prove that it follows that ( E , E , r, s ) has a closed path without an exit.We can, and will, identify Q ⊗ n with the R -bimodule ⊕ α ∈ E n Q α where each Q α is a copyof F and the left and the right multiplication are defined by X α ∈ E n q α α ! · X v ∈ E r v v = X α ∈ E n q α r r ( α ) α X v ∈ E r v v · X α ∈ E n q α α ! = X α ∈ E n r s ( α ) q α α IMPLE CUNTZ-PIMSNER RINGS 14 where α denote the unit of Q α , and { r v } v ∈ E and { q α } α ∈ E n are families of elements of F with only a finite number of non-zero elements. Likewise, we identify P ⊗ n with the R -bimodule ⊕ α ∈ E n P α where each P α is a copy of F and the left and the right multiplicationare defined by X α ∈ E n p α α ! · X v ∈ E r v v = X α ∈ E n p α r r ( α ) α X v ∈ E r v v · X α ∈ E n p α α ! = X α ∈ E n r s ( α ) p α α where α denote the unit of P α , and { r v } v ∈ E and { p α } α ∈ E n are families of elements of F with only a finite number of non-zero elements. We then have that ψ n : P ⊗ n ⊗ Q ⊗ n → R is given by X α ∈ E n p α α ! ⊗ X α ∈ E n q α α ! X v ∈ E X α ∈ E n v p α q α ! v . Let H be the set { v ∈ E | v ∈ I } . It follows from the ψ -invariance of I that H ishereditary (that is, whenever e ∈ E with s ( e ) ∈ H , then r ( e ) ∈ H ). Let v ∈ H . Then η ( v ) = P α ∈ K f α α for some non-empty finite subset K ⊆ E n and non-zero elements f α ∈ F, α ∈ K . Since v η ( v ) v = η ( v v v ) = η ( v ), it follows that r ( α ) = s ( α ) = v for each α ∈ K . Let α ∈ E n with r ( α ) = s ( α ) = v . Since ψ n (cid:16) α ⊗ η ( v ) (cid:17) α = η (cid:16) ψ n ( α v ⊗ α ) (cid:17) = η ( v ) , it follows that K ⊆ { α } . Hence it must be the case that there is exactly one α v ∈ E n with r ( α ) = s ( α ) = v , and that K consists of this element. Thus there is for each v ∈ H a unique α v ∈ E n with r ( α ) = s ( α ) = v and η ( v ) = f α v α v for some f α v ∈ F \ { } .Let v ∈ H , let α v = ( e , e , . . . , e n ) and assume that there is an e ′ ∈ E \ { e } with s ( e ) = v . Then η ( r ( e ) ) = η (cid:16) ψ ( e v ⊗ e ) (cid:17) = S e T η ( v ) e = f α v S e T αv e = 0which contradicts the fact that η is injective. Thus, for each v ∈ H it is the case that vE = { e } where e is the initial part of α v . It follows that every v ∈ H is the base ofa closed path which has no exit. In particular, ( E , E , r, s ) has a closed path which hasno exit.On the other hand, it is straightforward to check that if α v = ( e , e , . . . , e n ) is a closedpath without an exit, then H = { s ( e i ) | i ∈ { , , . . . , n } is a hereditary subset of E , I = span F { v | v ∈ H } is contained in J [ ∞ ] and is a ψ -invariant ideal in R , and the F -linear map η : I → Q ⊗ n given by s ( e i ) ( e i ,e i +1 ,...,e n ,e ,e ,...,e i − ) for i ∈ { , , . . . , n } is aninjective R -bimodule homomorphism η : I → Q ⊗ n satisfying S p T η ( x ) ( q ) = η ( ψ ( px ⊗ q ))for p ∈ P , x ∈ I and q ∈ Q . Thus J satisfies condition (L) if and only every closedpath in ( E , E , r, s ) has an exit (cf. [17, Definition 6.3]). We therefore recover [17,Theorem 6.8 and Corollary 6.10] from Theorem 4.2. By combining [17, Theorem 5.7and Proposition 6.12] and [5, Example 7.31] with the above characterization of when J satisfies condition (L), one sees that J satisfies condition (K) if and only if every v ∈ E is either the base of no closed path or the base of at least two simple closed paths(cf. [17, Definition 6.11]). We therefore recover [17, Theorem 6.16 and Corollary 6.17] IMPLE CUNTZ-PIMSNER RINGS 15 from Theorem 6.2 and Remark 6.3. Finally, it follows from [17, Theorem 5.7] (cf. [5,Example 7.31]) and Remark 5.2 that J is super maximal if and only if the only saturatedhereditary subsets of E are ∅ and E , thus we recover [17, Theorem 6.18] from Theorem5.3 and the above characterization of when J satisfies condition (L).We will end this subsection by using Corollary 7.2 to give a uniqueness theorem forthe Toeplitz ring T ( P,Q,ψ ) = O ( P,Q,ψ ) (0). Definition 8.1.
Let E = ( E , E , r, s ) be a directed graph, let F be a field and B an F -algebra. A Toeplitz-Cuntz-Krieger E -family in B consists of a family { p v | v ∈ E } of pairwise orthogonal idempotents in B together with a family { x e , y e | e ∈ E } ofelements in B satisfying the following relations(1) p s ( e ) x e = x e = x e p r ( e ) for e ∈ E ,(2) p r ( e ) y e = y e = y e p s ( e ) for e ∈ E ,(3) y e x f = δ e,f p r ( e ) for e, f ∈ E ,where δ e,f denotes the Kronecker’s delta function. Theorem 8.2.
Let E = ( E , E , r, s ) be a directed graph and let F be a field. Let R and ( P, Q, ψ ) be as defined above and let ( S, T, σ, T ( P,Q,ψ ) ) be the Toeplitz representationof ( P, Q, ψ ) . Then { σ ( v ) | v ∈ E } together with { T ( e ) , S ( e ) | e ∈ E } is a Toeplitz-Cuntz-Krieger E -family. If B is an F -algebra and { p v | v ∈ E } together with { x e , y e | e ∈ E } is a Toeplitz-Cuntz-Krieger E -family, then there exists a unique F -algebrahomomorphism η : T ( P,Q,ψ ) → B satisfying η ( σ ( v )) = p v for v ∈ E , and η ( T ( e )) = x e and η ( S ( e )) = y e for e ∈ E . The homomorphism η is injective if and only if p v = 0 for each v ∈ E and p v = P e ∈ vE x e y e for v ∈ E with < | vE | < ∞ .Proof. That T ( P,Q,ψ ) is an F -algebra and that { σ ( v ) | v ∈ E } ∪ { T ( e ) , S ( e ) | e ∈ E } is a Toeplitz-Cuntz-Krieger E -family is proved in [5, Example 1.10]. It is also proved in[5, Example 1.10] that if B is an F -algebra and { p v | v ∈ E } together with { x e , y e | e ∈ E } is a Toeplitz-Cuntz-Krieger E -family, then there is a covariant representation( S ′ , T ′ , σ ′ , B ) of ( P, Q, ψ ) such that S ′ ( λ e ) = λy e and T ′ ( λ e ) = λx e for e ∈ E and λ ∈ F , and σ ′ ( λ v ) = λp v for v ∈ E and λ ∈ F . It then follows from [5, Theorem 1.7]that there is a ring homomorphism η : T ( P,Q,ψ ) → B such that η ( σ ( λ v )) = σ ′ ( λ v ) = λp v for v ∈ E and λ ∈ F , and η ( T ( λ e )) = T ′ ( λ e ) = λx e and η ( S ( λ e )) = S ′ ( λ e ) = λy e for e ∈ E and λ ∈ F . It follows that η is a F -algebra homomorphism and that η ( σ ( v )) = p v for v ∈ E , and η ( T ( e )) = x e and η ( S ( e )) = y e for e ∈ E . Since T ( P,Q,ψ ) is generated, as an F -algebra, by { σ ( v ) | v ∈ E } ∪ { T ( e ) , S ( e ) | e ∈ E } ,there cannot be any other F -algebra homomorphism from T ( P,Q,ψ ) to B which for every v ∈ E maps σ ( v ) to p v and for any e ∈ E maps T ( e ) to x e and S ( e ) to y e .The map σ is injective by [5, Theorem 1.7]. It follows that if η is injective, then p v = 0for each v ∈ E . Assume that p v = 0 for each v ∈ E . Since R = ⊕ v ∈ E R v where each R v is a copy of F , it follows that σ ′ is injective. Thus it follows from Corollary 7.2 that η is injective if and only if J ( S ′ ,T ′ ,σ ′ ,B ) = 0. It follows from [5, Lemma 3.24] that J ( S ′ ,T ′ ,σ ′ ,B ) = n r ∈ ∆ − ( F P ( Q )) | σ ′ ( r ) = π T ′ ,S ′ (∆( r )) o . It is proved in [5, Example 5.8] that∆ − ( F P ( Q )) = span F { v | < | vE | < ∞} , IMPLE CUNTZ-PIMSNER RINGS 16 and is straightforward to check that ∆( v ) = P e ∈ vE θ e , e if v ∈ ∆ − ( F P ( Q )). Itfollows that J ( S ′ ,T ′ ,σ ′ ,B ) = span F (cid:26) v (cid:12)(cid:12)(cid:12) < | vE | < ∞ , p v = X e ∈ vE x e y e (cid:27) . Thus η is injective if and only if p v = 0 for each v ∈ E and p v = P e ∈ vE x e y e for v ∈ E with 0 < | vE | < ∞ . (cid:3) Theorem 8.2 is the algebraic analogue of [6, Theorem 4.1].9.
Crossed products of a ring by an automorphism and fractional skewmonoid rings of a corner isomorphism
We will in this section use Theorem 5.3 to give a characterization of when the fractionalskew monoid ring of a ring isomorphism is simple (Corollary 9.8), and when the crossedproduct of a ring by an automorphism is simple (Corollary 9.9).A ring R has local units if given any finite set F ⊆ R there exists an idempotent e ∈ R such that er = re = r for every r ∈ F , in other words, the set of all idempotents of R , Idem( R ), is a directed system (with order e ≤ f if and only if ef = f e = e ) and R = S e ∈ Idem( R ) eRe .Let R be a ring with local units and let α : R → R be an injective ring homomorphismsuch that α ( R ) Rα ( R ) ⊆ α ( R ) (notice this is equivalent to α ( R ) Rα ( R ) = α ( R ) since R has local units). Recall from [5, Example 5.6] that if P is the R -bimodule which isequal to span { r α ( r ) | r , r , ∈ R } as a set, has the additive structure it inherits from R , and has the left and right actions given by r · p = rp and p · r = pα ( r ) for r ∈ R and p ∈ P ; Q is the R -bimodule which is equal to span { α ( r ) r | r , r ∈ R } as aset, has the additive structure it inherits from R , and has the left and right given by r · q = α ( r ) q and q · r = qr for r ∈ R and q ∈ Q ; and ψ : P ⊗ Q → R is the R -bimodulehomomorphism given by p ⊗ q pq , then ( P, Q, ψ ) is an R -system. Recall also that R is a uniquely maximal, faithful, ψ -compatible ideal and that if α is an automorphism,then O ( P,Q,ψ ) ( R ) is isomorphic to the crossed product R × α Z of R by α . If R is unital,and we let e = α (1) (where 1 denotes the unit of R ), then e is an idempotent and α ( R ) = α ( R ) Rα ( R ) = eRe . It follows from [5, Example 5.7] that we in this case havethat O ( P,Q,ψ ) ( R ) is isomorphic to the fractional skew monoid ring R [ t + , t − ; α ] that Ara,González-Barroso, Goodearl and Pardo have constructed in [3]. We will use these factstogether with Theorem 5.3 to give a characterization of when the crossed product R × α Z is simple and when the fractional skew monoid ring R [ t + , t − ; α ] is simple, but first weintroduce some notions and results that we will use for this.Unless otherwise stated, α will just be assumed to be an injective ring homomorphismsuch that α ( R ) Rα ( R ) ⊆ α ( R ). We let ( P, Q, ψ ) be the R -system defined above. Usingthat R has local units, it is not difficult to see that for n ∈ N , the R -bimodule P ⊗ n is isomorphic to the R -bimodule which is equal to span { r α n ( r ) | r , r , ∈ R } as aset, has the additive structure it inherits from R , and has the left and right actionsgiven by r · p = rp and p · r = pα n ( r ), respectively. Likewise, Q ⊗ n is isomorphic tothe R -bimodule which is equal to span { α n ( r ) r | r , r ∈ R } as a set, has the additivestructure it inherits from R and has the left and right given by r · q = α n ( r ) q and q · r = qr ,respectively. We will simply identify P ⊗ n and Q ⊗ n with these two R -bimodules. We IMPLE CUNTZ-PIMSNER RINGS 17 will use a · to indicate the left and right actions of R on P ⊗ n and Q ⊗ n to distinguishthese actions from the ordinary multiplication in R . It is straightforward to check thatif q ∈ Q , q n ∈ Q ⊗ n and p ∈ P , then S p T q n ( q ) = α n ( p ) α ( q n ) q . Let ( S, T, σ, O ( P,Q,ψ ) ( R ))denote the Cuntz-Pimsner representation of ( P, Q, ψ ) relative to R . Then S n ( p n ) σ ( r ) = S n ( p n α n ( r )), σ ( r ) S n ( p n ) = S n ( rp n ), S n ( p n ) S n ′ ( p ′ n ′ ) = S n + n ′ ( p n α n ( p ′ n ′ )), T n ( q n ) σ ( r ) = T n ( q n r ), σ ( r ) T n ( q n ) = T n ( α n ( r ) q n ), T n ( q n ) T n ′ ( q ′ n ′ ) = T n + n ′ ( α n ′ ( q n ) q ′ n ′ ), S n ( p ) T n ( q ) = σ ( pq ) and T n ( q ) S n ( p ) = σ ( α − n ( q n p n )) for n, n ′ ∈ N , p n ∈ P n , r ∈ R , n ′ ∈ P ⊗ n ′ , q n ∈ Q ⊗ n and q n ′ ∈ Q ⊗ n ′ where p n , p n ′ , q n and q n ′ are considered as elements of R and the multiplication of R is used. It follows that O ( P,Q,ψ ) ( R ) (0) = σ ( R ), and that O ( P,Q,ψ ) ( R ) ( n ) = T n ( Q ⊗ n ) and O ( P,Q,ψ ) ( R ) ( − n ) = S n ( P ⊗ n ) for n ∈ N . We say that anideal I of R is strongly α -invariant if α ( I ) ⊆ I and α ( R ) Iα ( R ) ⊆ α ( I ) (this is equivalentto α ( R ) Iα ( R ) = α ( I ) since R has local units). Proposition 9.1.
Let R be a ring with local units, α : R → R an injective ring ho-momorphism satisfying α ( R ) Rα ( R ) ⊆ α ( R ) , and let ( P, Q, ψ ) be the R -system definedabove. Then there is a bijective correspondence between graded ideals of O ( P,Q,ψ ) ( R ) andstrongly α -invariant ideals of R .Proof. For each strongly α -invariant ideal I in R , let H I be the ideal in O ( P,Q,ψ ) ( R )generated by σ ( I ); and let for each graded ideal H in O ( P,Q,ψ ) ( R ), I H = { x ∈ R | σ ( x ) ∈ H } . We will show that H I is a graded ideal in O ( P,Q,ψ ) ( R ), that I H is a strongly α -invariant ideal in R , and that I H I = I and H I H = H for all strongly α -invariantideals I in R and all graded ideals H in O ( P,Q,ψ ) ( R ). This will establish the bijectivecorrespondence between the graded ideals of O ( P,Q,ψ ) ( R ) and the strongly α -invariantideals of R .Let I be a strongly α -invariant ideal in R . It is not difficult to check that if we let H (0) = σ ( I ) and for each n ∈ N let H ( n ) = span { T n ( α n ( r ) x ) | r ∈ R, x ∈ I } and H ( − n ) = span { S n ( xα n ( r )) | x ∈ I, r ∈ R } , then ⊕ n ∈ Z H ( n ) is an ideal in O ( P,Q,ψ ) ( R ).Since ⊕ n ∈ Z H ( n ) contains σ ( I ) and itself must be contained in any ideal which contains σ ( I ), it must be the case that H I = ⊕ n ∈ Z H ( n ) . It follows that H I is graded and that I H I = I .Let H be a graded ideal in O ( P,Q,ψ ) ( R ). It is clear that I H is an ideal in R . Assumethat x ∈ I H . Choose idempotens e , e ∈ R such that e α ( x ) e = α ( x ) and e xe = x .Then σ ( α ( x )) = S ( e α ( e )) σ ( x ) T ( α ( e ) e ) ∈ H, so α ( x ) ∈ I H . Assume then that r , r ∈ R . Choose idempotents f , f ∈ R such that f α ( r ) f = α ( r ) and f α ( r ) = α ( r ). Then σ ( α − ( α ( r ) xα ( r ))) = T ( α ( r ) f ) σ ( x ) S ( f α ( r )) ∈ H, so α ( r ) xα ( r ) ∈ α ( I H ). This shows that I H is a strongly α -invariant ideal in R . Since O ( P,Q,ψ ) ( R ) (0) = σ ( R ), it follows from [5, Lemma 3.35] that H is generated by σ ( I H ).Thus H = H I H . (cid:3) By combining the above result with Remark 5.2 we get the following characterizationof when R is a super maximal ψ -compatible ideal. IMPLE CUNTZ-PIMSNER RINGS 18
Corollary 9.2.
Let R be a ring with local units, α : R → R an injective ring homomor-phism satisfying α ( R ) Rα ( R ) ⊆ α ( R ) , and let ( P, Q, ψ ) be the R -system defined above.Then the following three conditions are equivalent:(1) The ring R is a super maximal ψ -compatible ideal.(2) The only graded ideals in O ( P,Q,ψ ) ( R ) are { } and O ( P,Q,ψ ) ( R ) .(3) The only strongly α -invariant ideals in R are { } and R . We next introduce the multiplier ring of R (see for example [4]). A double centralizeron R is a pair ( f, g ) where f : R → R is a right R -module homomorphism and g : R → R is a left R -module homomorphism satisfying r f ( r ) = g ( r ) r for all r , r ∈ R . The multiplier ring of R is the ring M ( R ) of all double centralizers on R with additiondefined by ( f , g ) + ( f , g ) = ( f + f , g + g ) and product defined by ( f , g )( f , g ) =( f ◦ f , g ◦ g ). Notice that (Id R , Id R ) is a unit of M ( R ). There is a ring homomorphism ι : R → M ( R ) given by ι ( r ) = ( f r , g r ) where f r ( s ) = rs and g r ( s ) = sr for r, s ∈ R .Since R has local units, ι is injective. We will therefore simple regard R as a subringof M ( R ). We then have that if u = ( f, g ) ∈ M ( R ) and r ∈ R , then ur = f ( r ) and ru = g ( r ). It follows that R is an ideal in M ( R ). Notice that R = M ( R ) if and only if R is unital. Definition 9.3.
Let n ∈ N and let R be a ring with local units. A ring homomorphism α : R → R is said to be inner with periodicity n if there exist u, v ∈ M ( R ) such that vu = 1 (where 1 denotes the unit of M ( R )), and α n ( r ) = urv and α ( ur ) = uα ( r ) for all r ∈ R . If α is not inner of any periodicity, then it is said to be outer . Remark . Notice that if α is an automorphism and u, v are as above, then v is theinverse of u .In [4] the authors introduce a topology on M ( R ) in the following way. A net ( x λ ) λ ∈ Λ of elements of M ( R ) converges strictly to an a element x ∈ M ( R ) if there for every r ∈ R exists λ ∈ Λ such that ( x λ − x ) r = r ( x λ − x ) = 0 for λ ≥ λ . Since R has localunits, a net in M ( R ) can at most converges strictly to one element. Such an elementwill, if it exists, be called the strict limit of the net. A net ( x λ ) λ ∈ Λ is Cauchy if therefor every r ∈ R exists λ ∈ Λ such that r ( x λ − x µ ) = ( x λ − x µ ) r = 0 for λ, µ ≥ λ .It is shown in [4, Proposition 1.6] that if R has local units, then every Cauchy net in M ( R ) converges strictly, and that every element of M ( R ) is the strict limit of a net ofelements of R .A net ( r λ ) λ ∈ Λ of elements of R that converges to the unit of M ( R ) is called an approximate unit for R . Notice that in case R has local units we can construct anapproximate unit ( e λ ) λ ∈ Λ consisting of idempotents simple by letting Λ be the directedset of finite subsets of R ordered by inclusion, and then for every λ ∈ Λ choosing anidempotent e λ such that e λ r = re λ = r for every r ∈ λ . Definition 9.5.
Let R be a ring with local units. A ring homomorphism α : R → R is said to be strict if there exists an approximate unit ( e λ ) λ ∈ Λ for R consisting ofidempotents such that ( α ( e λ )) λ ∈ Λ converges strictly. Remark . Notice that if α is an automorphism, then it is strict (since ( α ( e λ )) λ ∈ Λ converges strictly to the unit in that case). Notice also that if R is unital, then everyring homomorphism α : R → R is automatically strict (because the net consisting ofjust 1 is an approximate unit in that case). IMPLE CUNTZ-PIMSNER RINGS 19
Proposition 9.7.
Let R be a ring with local units, α : R → R an injective ring ho-momorphism satisfying α ( R ) Rα ( R ) ⊆ α ( R ) , and let ( P, Q, ψ ) be the R -system definedabove. Consider the following three conditions:(1) There exists an n ∈ N such that the homomorphism α is inner with periodicity n .(2) The ring R is a ψ -invariant cycle.(3) The ring R does not satisfy condition (L) with respect to ( P, Q, ψ ) .Then (1) implies (2) , and (2) implies (3) . If in addition R is a super maximal ψ -compatible ideal, and α n is strict for every n ∈ N , then (3) implies (1) and the threeconditions are equivalent.Proof. (1) ⇒ (2): Let u and v be elements in M ( R ) such that vu = 1, and urv = α n ( r )and α ( ux ) = uα ( x ) for all r ∈ R . Define η : R → R by η ( r ) = ur . Let r ∈ R . Choose e ∈ R such that er = r . Then we have that η ( r ) = ur = uer = uevur = α n ( e ) ur . Thisshows that η ( R ) ⊆ Q ⊗ n . It is clear that η is additive and injective. Let r , r , ∈ R . Then η ( r r ) = ur r = η ( r ) r and η ( r r ) = ur r = α n ( r ) ur = α n ( r ) η ( r ), which showsthat η is an R -bimodule homomorphism from R to Q ⊗ n . Let p ∈ P , r ∈ R and q ∈ Q .Then we have that η ( ψ ( p · r ⊗ q )) = η ( pα ( r ) q ) = upα ( r ) q = α n ( p ) uα ( r ) q = α n ( p ) α ( ur ) q = α n ( p ) α ( η ( r )) q = S p T η ( r ) ( q ) . Thus R is a ψ -invariant cycle.(2) ⇒ (3): It is easy to see that ψ − ( R ) = R from which it follows that R [ ∞ ] = R .Thus, if R is a ψ -invariant cycle, then R does not satisfy condition (L) with respect to( P, Q, ψ ).(3) ⇒ (1): Assume that R does not satisfy condition (L) with respect to ( P, Q, ψ ).It then follows from Proposition 2.6 that there is a non-zero graded ideal L k ∈ Z H ( k ) in O ( P,Q,ψ ) ( R ), an n ∈ N and a family ( φ k ) k ∈ Z of injective O ( P,Q,ψ ) ( R ) (0) -bimodule ho-momorphisms φ k : H ( k ) → O ( P,Q,ψ ) ( R ) ( k + n ) such that xφ k ( y ) = φ k + j ( xy ) and φ k ( y ) x = φ k + j ( yx ) for k, j ∈ Z , x ∈ O ( P,Q,ψ ) ( R ) ( j ) and y ∈ H ( k ) . Notice that also L k ∈ Z φ k − n ( H ( k − n ) )is a non-zero graded ideal in O ( P,Q,ψ ) ( R ). If R is a super maximal ψ -compatible ideal,then it follows from Corollary 9.2 that L k ∈ Z H ( k ) = L k ∈ Z φ k − n ( H ( k − n ) ) = O ( P,Q,ψ ) ( R )from which it follows that H (0) = φ − n ( H ( − n ) ) = O ( P,Q,ψ ) ( R ) (0) = σ ( R ), φ ( H (0) ) = O ( P,Q,ψ ) ( R ) ( n ) = T n ( Q ⊗ n ) and H ( − n ) = O ( P,Q,ψ ) ( R ) ( − n ) = S n ( P ⊗ n ). Suppose in additionthat α n is strict, and let ( e λ ) λ ∈ Λ be an approximate unit for R consisting of idempo-tents such that ( α ( e λ )) λ ∈ Λ converges strictly. Since T n and φ − n are injective, and Q ⊗ n and P ⊗ n are subsets of R , there exists for each λ ∈ Λ a unique u λ ∈ R such that T n ( u λ ) = φ ( σ ( e λ )) and a unique v λ ∈ R such that φ − n ( S n ( v λ )) = σ ( e λ ). Notice that T n ( u λ ) = φ ( σ ( e λ )) = φ ( σ ( e λ e λ )) = σ ( e λ ) φ ( σ ( e λ )) = σ ( e λ ) T n ( u λ ) = T n ( α n ( e λ ) u λ ) . It follows that α n ( e λ ) u λ = u λ . If λ, λ ∈ Λ and e λ e λ = e λ , then T n ( α n ( e λ ) u λ ) = σ ( e λ ) T n ( u λ ) = σ ( e λ ) φ ( σ ( e λ ))= φ ( σ ( e λ e λ )) = φ ( σ ( e λ )) = T n ( u λ ) , from which it follows that α n ( e λ ) u λ = u λ . Let r ∈ R . Choose λ , λ , λ ∈ Λ such that rα n ( e λ ) = rα n ( e λ ) for λ ≥ λ , e λ e λ = e λ for λ ≥ λ , and e λ r = r for λ ≥ λ . If IMPLE CUNTZ-PIMSNER RINGS 20 λ ≥ λ , λ , λ , then ru λ = rα n ( e λ ) u λ = rα n ( e λ ) u λ = ru λ , and T n ( u λ r ) = T n ( u λ ) σ ( r ) = φ ( σ ( e λ )) σ ( r ) = φ ( σ ( e λ r )) = φ ( σ ( r )) . This shows that ( u λ ) λ ∈ Λ is Cauchy and hence converges strictly to an element u ∈ M ( R ).One can by a similar method show that ( v λ ) λ ∈ Λ converges strictly to an element v ∈M ( R ).Let λ ∈ Λ. Then σ ( v λ u λ ) = S n ( v λ ) T n ( u λ ) = S n ( v λ ) φ ( σ ( e λ )) = φ − n ( S n ( v λ ) σ ( e λ ))= φ − n ( S n ( v λ )) σ ( e λ ) = σ ( e λ ) σ ( e λ ) = σ ( e λ ) , from which it follow that v λ u λ = e λ . Thus vu = 1.Let r ∈ R . Choose λ ∈ Λ such that re λ = e λ r = r for λ ≥ λ . If λ ≥ λ , then T n ( α n ( r ) u λ ) = σ ( r ) φ ( σ ( e λ )) = φ ( σ ( re λ )) = φ ( σ ( e λ r )) = φ ( σ ( e λ )) σ ( r ) = T n ( u λ r ) . It follows that α n ( r ) u = ur and thus that urv = α n ( r ).Let r ∈ R . Choose λ ∈ Λ such that e λ r = r and e λ α ( r ) = α ( r ) for λ ≥ λ . If λ ≥ λ then T n ( α ( u λ r )) = T n (cid:16) α n +1 ( e λ ) α ( u λ ) α ( r ) (cid:17) = S ( α ( e λ )) T n ( u λ ) T ( α ( r ))= S ( α ( e λ )) φ ( σ ( e λ )) T ( α ( r )) = φ (cid:16) S ( α ( e λ )) (cid:17) σ ( e λ ) T ( α ( r ))= φ (cid:16) σ ( α ( e λ e λ r )) (cid:17) = φ (cid:16) σ ( α ( r )) (cid:17) = φ (cid:16) σ ( e λ α ( r )) (cid:17) = φ ( σ ( e λ )) σ ( α ( r ))= T n ( u λ ) σ ( α ( r )) = T n ( u λ α ( r )) , from which it follows that α ( u λ r ) = u λ α ( r ). Thus α ( ur ) = uα ( r ).Hence α is inner with periodicity n in this case. (cid:3) By combining Theorem 5.3 and Corollary 9.2 with Remark 9.6, Proposition 9.7, andthe fact that O ( P,Q,ψ ) ( R ) is isomorphic to the crossed product R × α Z of R by α when α is an automorphism, and to the fractional skew monoid ring R [ t + , t − ; α ] when R isunital and α is an injective homomorphism such that α ( R ) = eRe for some idempotent e ∈ R , we get the following two corollaries. Corollary 9.8.
Let R be a unital ring and let α : R → R be an injective ring homo-morphism such that α ( R ) = eRe for some idempotent e ∈ R . Then the following twostatements are equivalent:(1) The fractional skew monoid ring R [ t + , t − ; α ] is simple.(2) The homomorphism α is outer and the only strongly α -invariant ideals in R are { } and R . Corollary 9.9.
Let R be a ring with local units and let α : R → R be a ring automor-phism. Then the following two statements are equivalent:(1) The crossed product R × α Z is simple.(2) The automorphism α is outer and the only strongly α -invariant ideals in R are { } and R . IMPLE CUNTZ-PIMSNER RINGS 21
We end by noticing that when α is an automorphism, the condition of α being outeris equivalent with the seemingly stronger, and perhaps more familiar, condition that α is strongly outer . Definition 9.10.
Let n ∈ N and let R be a ring with local units and α : R → R a ringautomorphism. If there exists an invertible element u ∈ M ( R ) such that α n ( r ) = uru − for all r ∈ R , then α is said to be weakly inner with periodicity n . If α is not weaklyinner of any periodicity, then it is said to be strongly outer . Proposition 9.11.
Let R be a ring with local units and let α : R → R a ring automor-phism. Then α is outer if and only if it is strongly outer.Proof. It follows from Remark 9.4 that if α is strongly outer, then it is also outer.Suppose that α is not strongly outer. Then there exist n ∈ N and an invertibleelement u ∈ M ( R ) such that α n ( r ) = uru − for all r ∈ R . If x = ( f, g ) ∈ M ( R )where ( f, g ) is a double centralizer, then we let ˆ α ( x ) denote the double centralizer ( α ◦ f ◦ α − , α ◦ g ◦ α − ). It is easy to check that x ˆ α ( x ) defines an automorphism ˆ α of M ( R ) and that ˆ α n ( x ) = uxu − for all x ∈ M ( R ). In particular ˆ α n ( u ) = uuu − = u andˆ α n ( u − ) = uu − u − = u − . Let u ′ = u ˆ α ( u ) . . . ˆ α n − ( u ) and v ′ = ˆ α n − ( u − ) . . . ˆ α ( u − ) u − . Then v ′ u ′ = 1. If r ∈ R , then α ( u ′ r ) = ˆ α ( u ′ ) α ( r ) = ˆ α (cid:16) u ˆ α ( u ) . . . ˆ α n − ( u ) (cid:17) α ( r ) = ˆ α ( u ) ˆ α ( u ) . . . ˆ α n ( u ) α ( r )= ˆ α n +1 ( u ) ˆ α n +2 ( u ) . . . ˆ α n ( u ) α ( r ) = ˆ α n (cid:16) ˆ α ( u ) ˆ α ( u ) . . . ˆ α n ( u ) (cid:17) α ( r )= u ˆ α ( u ) ˆ α ( u ) . . . ˆ α n ( u ) u − α ( r ) = u ˆ α ( u ) ˆ α ( u ) . . . ˆ α n − ( u ) uu − α ( r ) = u ′ α ( r )and u ′ rv ′ = u ˆ α ( u ) . . . ˆ α n − ( u ) r ˆ α n − ( u − ) . . . ˆ α ( u − ) u − = ˆ α n (cid:16) ˆ α ( u ) . . . ˆ α n − ( u ) r ˆ α n − ( u − ) . . . ˆ α ( u − ) (cid:17) = ˆ α ( u ) . . . ˆ α n − ( u ) α n ( r ) ˆ α n − ( u − ) . . . ˆ α ( u − )= ˆ α ( u ) . . . ˆ α n − ( u ) α n +1 ( r ) ˆ α n − ( u − ) . . . ˆ α ( u − )...= ˆ α ( u ) α ( n − n ( r ) ˆ α ( u − ) = α n ( r ) . Thus α is inner with periodicity n and is therefore not outer. (cid:3) Acknowledgments
Part of this work was done during visits of the third author to the Institut for Matem-atik og Datalogi, Syddansk Universitet and to the Institut for Matematiske Fag, Køben-havns Universitet (Denmark). The third author thanks both host centers for their kindhospitality.
IMPLE CUNTZ-PIMSNER RINGS 22
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Department of Mathematical Sciences, NTNU, NO-7491 Trondheim, Norway
E-mail address : [email protected]
Department of Mathematical Sciences, NTNU, NO-7491 Trondheim, Norway
E-mail address : [email protected] Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, Campusde Puerto Real, 11510 Puerto Real (Cádiz), Spain.
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