Quantum triads: an algebraic approach

Quan tum triads: an algebraic approac h Da vid Kruml ∗ No ve mber 2, 201 8 Keyw ords: Quant ale, quan tale mo dule, couple of quan tales, quantum frame, quan tu m triad. Abstract A concept of quantum triad and its solution is introduced. It rep- resents a common framework for several situatio ns where we have a quantale with a r ight mo dule and a left mo dule, provided with a bi- linear inner pro duct. Examples include V an de n Bos sche quan talo ids, quantum frames, simple and Galois quantales, op era tor alg ebras, o r orthomo dular la ttices. 1 In tro d uction The la ttice of ideals of a c ommutativ e ring is a v ery useful charact eristic and pro vid es to apply top ological tec hniqu es in ring theory . When the r ing is non-comm utativ e, the t wo-sided id eals ( T ) are muc h less d escriptiv e and w e should rather consider lattices of righ t ( R ) or left ( L ) ideals, or ev en b etter all of them. The la ttices are naturally equipp ed by an asso ciativ e m u ltiplicatio n whic h distributes ov er joins and this led J. C. Mulvey to introdu ce a concept of quantale . Q uan tales arisin g from r igh t ideals were studied by many authors (see cf. [1, 3, 13], etc.). In [16] (see also [15]) G. V an den Bossc h e adv anced an idea of F. W. La wv ere to consider th e lattices L, T , R as h om-sets in a t wo -ob ject quanta loid together with a lattice Q of all su bgroups wh ic h are ∗ Supp orted by the Gran t Agency of th e Czec h Repu blic under th e gran t No. 201/06/ 0664. 1 mo dules of a cen ter: • L ! ! Q 9 9 • T y y R a a ( ∗ ) Notice that R, L are then considered as mo dules r ather than quanta les and that t he scheme pr eserv es a quite important m u ltiplicatio n L × R → T . Th is imp ortance is w ell visible when w e are d ealing with an op erator algebra A and th e ideals are realized by p ro jections — for p ro jections p, q the m ore in teresting pro du ct Apq A is obtained b y m u ltiplying left ideal Ap w ith righ t ideal q A , wh ile r igh t quanta le structur e on R giv es only pAq A wh ic h can b e calculat ed as a righ t actio n of t wo-sided Aq A on pA . This fact w as recognized b y J. Rosic k ´ y [14] and studied in a con text of quantum fr ames . The aim of this pap er is to f orm alize th e r elationship among on e- and t wo - sided ideals and to construct a quant ale subsumin g the stru cture. Drop- ping Q fr om the V an den Bossc he quan taloid but pr eserving all remaining comp ositions and asso ciativ e la w s w e obtain a basic example of what is called quantum triad ( L, T , R ). The quan tu m triads, or shortly just triads, can b e u ndersto o d as m ultiplicativ e Chu spaces but notice th at morphisms of triads (whic h are not stu died h ere) w ould arise from ring morp hisms and this is a different philosophy then th e one of Chu spaces. The fi ll-in by quantal e Q in ( ∗ ) is an instance of solution of the triad ( L, T , R ). It is sho wn that ev ery triad has t wo extremal solutions denoted b y Q 0 and Q 1 and they enclose a categ ory of all solutions. The solution Q 0 is realized by tensor pro duct R ⊗ T L while Q 1 is a generalizat ion of a quantale of endomorphisms. The results are based on a sp ecial case s tu died in [4] and further comm unication with J. Egger and th e idea of P . Resende [12] w ho constructed Q 1 for the case of Galois connections. T he quanta les Q 0 , Q 1 re- flects t w o asp ects of the qu an tization of topology — it is a non-comm utativ e in tersection represented by m ultiplication on Q 0 , and transitivit y of states represent ed b y actions of Q 1 on L and R . W e discuss pr op erties when L, T , R app ears as left-, t w o-, and r ight-sided elemen ts of a solution, th at is when the solution represents a un ique ob ject co v erin g all comp onent s of th e tr iad. A s p ecial atten tion is kept to in v olutive ( L ∼ = R ) an d Girard ( L op ∼ = R ) tr iads. In particular, solutions of a triad giv en by a complete orthomod ular la ttice r epresen ts a con tribution to topics of dynamical asp ects of quantum logic [2]. Since one can find other examples of qu an tum triads outsid e lattice the- ory , it is r easonable to w ork with a maxi mal generalit y . The author presen ts 2 here only applications in s u p-lattices an d uses an algebraic language . A catego r ical app roac h will b e p resen ted in a sep arate p ap er [8]. 2 Preliminaries Recall that a c ate gory of sup-lattic es consists of complete lattices as ob j ects and sup rema preserving maps as morphism s . Ev ery s up-lattice morph ism f : S → S ′ has an adjoint f ⊣ : S ′ → S giv en b y f ( x ) ≤ y ⇔ x ≤ f ⊣ ( y ) wh ich preserve s all infima. By d ualizing S and S ′ w e obtain a sup -lattice morphism denoted by f ∗ : ( S ′ ) op → S op . A map f : S × S ′ → S ′′ of sup -lattice s is called a bimorphism if it is a morp h ism in b oth v ariables, i.e. fixing an elemen t o f x ∈ S (or y ∈ S ′ ) w e obtain morphism f ( x, − ) : S ′ → S ′′ (or f ( − , y ) : S → S ′′ ). When the b imorphism is apparen t, an ele m en t f ( x, y ) is understo o d as a pro d ucts an d d enoted b y xy . The adjoin ts of f ( x, − ) , f ( − , y ) are r eferred as r esiduations and denoted by − ← x, y → − . A quantale Q is a sup-lattice equipp ed by an asso ciativ e bimorph ism Q × Q → Q . T he top or b ottom elemen t is den oted by 1 or 0, resp ectiv ely . The qu an tale is called unital if it admits a u nit e ∈ Q , i.e. q e = q = eq for ev ery q ∈ Q , semiunital if q ≤ q 1 ∧ 1 q for eve r y q ∈ Q , strictly two-side d if it is unital and the unit coincide with the top elemen t, Gir ar d if admits an ele m en t d ∈ Q w hic h is cyclic , i.e. q q ′ ≤ d ⇔ q ′ q ≤ d , and dualizing , i.e. q = d ← ( q → d ) = ( d ← q ) → d , for every q , q ′ ∈ Q . The elemen t q ⊥ = q → d = d ← q is regarded as a c omplement of q . A quanta le Q is called involutive if it is equip p ed by a u nary op eration ∗ of involution p ro vided th at ( q ∗ ) ∗ = q , ( q q ′ ) ∗ = ( q ′ ) ∗ q ∗ and ( W q i ) ∗ = W q ∗ i for all q , q ′ , q i ∈ Q . An elemen t q ∈ Q is said to b e right - , left- , or two-side d if q 1 ≤ q , 1 q ≤ q or b oth the inequalities hold, resp ectiv ely . The resp ectiv e sup-lattices are denoted by R ( Q ) , L ( Q ) , T ( Q ). Recall that every u nital quant ale is semi- unital and in a s emiunital quant ale it holds that r 1 = r , 1 l = l for all r ∈ R ( Q ) , l ∈ L ( Q ). Since 1 ∗ = 1 in an y inv olutiv e quan tale, the inv olution pro vid es a sup-lattice isomorphism b etw een R ( Q ) and L ( Q ). A sup-lattice morphism f : Q → K b etw een quan tales Q, K is called a (involutive) qu antale morphism if it preserv es the multiplica tion (and the in volutio n ), i.e. f ( qq ′ ) = f ( q ) f ( q ′ ) (and f ( q ∗ ) = f ( q ) ∗ ). The morph ism f is called str ong if it preserv es the top elemen t, i.e. f (1 Q ) = 1 K . It f ollo w s easily that a strong morphism preserve also right- and left-sided elemen ts. The t wo -element sup-lattice 2 = { 0 , 1 } , as well as an y frame, will b e 3 regarded as a unital (in vol u tiv e) qu an tale with m ultiplication ∧ (and trivial in volutio n ). A su p-lattice M is called a left Q -mo dule if there is a b imorphism Q × M → M associativ e with the quan tale m ultiplication, i.e. ( qq ′ ) m = q ( q ′ m ) for ev ery q , q ′ ∈ Q, m ∈ M . M is said to b e u n ital wh en Q is un ital and em = m for ev ery m ∈ M . Right mo dules are defined in an analogous wa y . M is called a ( Q, Q ′ ) -bimo dule for qu antale s Q, Q ′ if it is left Q -mo dule, righ t Q ′ - mo dule and it holds that ( q m ) q ′ = q ( mq ′ ) f or ev ery q ∈ Q, m ∈ M , q ′ ∈ Q ′ . Notice that ev ery quan tale Q is automatically ( Q, Q )-bimo dule. Categories of left Q -mo d u les, righ t Q ′ -mo dules and ( Q, Q ′ )-bimo dules are denoted by Q -Mo d , Mod- Q, Q -Mod- Q ′ , resp ectiv ely . A sup-lattice morph ism f : M → M ′ b et ween tw o left Q -mo d ules is called a Q -mo dule morphism if f ( q m ) = q f ( m ) for every q ∈ Q, m ∈ M , etc. Quant ales C , Q together w ith a morp hism φ : C → Q are called a c ouple of quantales if C is a ( Q, Q )-bimo dule, φ is a ( Q, Q )-bimo dule morph ism and it holds that cc ′ = φ ( c ) c ′ = cφ ( c ′ ) for ev ery c, c ′ ∈ C . A couple C φ → Q is said to b e unital if Q is unital and C is a un ital ( Q, Q )-mod ule, Gir ar d if C admits a cyclic du alizing elemen t d , bu t no w with resp ect to the bimo d ule actions, i.e. q c ≤ d ⇔ cq ≤ d , q = d ← ( q → d ) = ( d ← q ) → d , where the r esiduations are calculated as adj oin ts of q − , − q : C → C , and c = d ← ( c → d ) = ( d ← c ) → d with residuations adjoint to c − , − c : Q → C , for all q ∈ Q, c ∈ C . 3 T riads and solutions 3.1 Definit ion. A (quantum) triad consists of the follo wing data: • quantale T , • right T -mo du le R , • left T -mo du le L , • and bimorphism L × R → T compatible with the mo d ule actions, i.e., there are f our b imorphisms, r eferred as (TT, R T, T L, LR), satisfying all the five reasonable asso ciativ e laws (TTT, TTL, R TT, LR T, TLR). 3.2 Definit ion. A quan tale Q is s aid to b e a solution of triad ( L, T , R ) if • R is a ( Q, T )-bimo du le, 4 • L is a ( T , Q )-bimo dule, • there is a compatible bimorp hism R × L → Q, ( r , l ) 7→ rl , whic h means that we add fur th er four bimorp hisms (QQ, QR, LQ, R L ) and require all the remaining asso ciativ e laws (QQQ , LQ Q, QQR, TLQ, QR T, QRL, RLQ , R TL, LQR, RL R, LRL) for scheme ( ∗ ). 3.3 Example. (1) Let Q b e a quantale . Then ( L ( Q ) , T ( Q ) , R ( Q )) is a triad and Q is a solution. (2) Let A b e a ring. As mentio n ed in I n tro d uction we assu m e L, T , R to b e sup -lattic es of left-, t wo- , and righ t-sided ideals. As solution we can consider a quan tale of all add itiv e subgroups of A or a qu an tale of those subgroups that are mo d ules o ver th e cen ter of A . (3) When A is a C *-alge b ra, it is p ossible to consider only ideals closed in norm top ology . Then sp ectrum Max A consisting of all closed subs p aces [10] is a solution. (4) Let S b e a sup -lattice. Putting xy = ( 0 , y ≤ x, 1 , y  x for x, y ∈ S we obtain a bilinear map S op × S → 2 . Since ev ery sup-lattice is a unital 2 -mo dule, w e get a triad ( S op , 2 , S ). Quanta le Q ( S ) of all sup- lattice endomorphisms [11] of S and quanta le C ( S ) = S ⊗ S op are clearly solutions of th e triad [4]. (5) Let H b e a Hilb ert sp ace. Th en left ideals of op er ator algebra B ( H ) closed in n ormal top ology , as w ell as those r igh t ideals, can b e ident ifi ed with closed subspaces of H . The sup-lattice is denoted by L ( H ). Th e only close d t wo -sid ed ideals are { 0 } and A , h ence we obtain a triad ( L ( H ) , 2 , L ( H )) whic h is a sp ecial case of (4). Except Q ( L ( H )) , C ( L ( H )) there are also solutions Max B ( H ) , Max σw B ( H ) and Max 1 C ( H ) (see [4]). (6) Let S, S ′ b e sup-lattices and f : S → S ′ , g : S ′ → S Galois connec- tions, i.e. f ( x ) ≤ y ⇔ x ≥ g ( y ) for ev ery x ∈ S, y ∈ S ′ . In that c ase w e write x ⊥ y and put xy = 0, or 1 otherwise. W e h a ve obtained a triad ( S, 2 , S ′ ). Galois quantale Q = { ( α, β ) ∈ Q ( S ) ⊗ Q ( S ′ ) | α ( x ) ⊥ y ⇔ x ⊥ β ( y ) } con- structed in [12] is a s olution of the triad. Notice that (4) is a sp ecial case for d u alit y f : S → S op , g : S op → S . (7) F o r a quantum frame F (see [14]) we consider triad ( F , ˜ F , F ) where ˜ F is a frame of tw o-sided elements of F , and actions are defin ed by xz = z x = x ∧ z , (R T, TL) xy = x ◦ y (LR) 5 for x, y ∈ F, z ∈ ˜ F . As ele ments of a qu antum frame represen t q-op en se ts in quan tized top ology (see [5]), a solution of th e triad provides a “dynamical logic of quan tised topology”. In con trast to other candidates like Q ( F ), solutions resp ect the un derlying classical top ology repr esented by t wo -sid ed elemen ts (central q-op en sets). (8) A sp ecial instance of (7) (and generalizati on of (5)) is a complete orthomo dular lattice M . The quantum frame structure defin ed by J. Rosic k ´ y yields a triad ( M , Z ( M ) , M ) w h ere Z ( M ) is a cen ter of M . Notice that x ◦ y can b e calculate d also as | x ˙ ∧ y | , i.e. a cen tral co ver of sk ew meet (also kn own as Sasaki pr o jection φ x ( y ) = x ˙ ∧ y = ( x ∨ y ⊥ ) ∧ y ), and x ˙ ∧ y coincide with x ∧ y whenever x or y is cen tral. This su ggests f u rther examp les emerging from skew op erations. Let us reca ll that R ⊗ L is cal cu lated as a free su p-lattice on R × L factorized b y congruen ce generated by r elations _ ( r i , l ) ∼  _ r i , l  , _ ( r , l i ) ∼  r , _ l i  for all r, r i ∈ R , l, l i ∈ L . Ev ery elemen t of R ⊗ L repr esentable by some ( r , l ) is called a pur e tensor and denoted by r ⊗ l . 3.4 L e mma. L et ( L, T , R ) b e an triad in S up . L et R ⊗ T L b e a sup-lattic e quotient of R ⊗ L via ( r t ) ⊗ l ∼ r ⊗ ( tl ) assume d f or any r ∈ R, l ∈ L, t ∈ T . L et Q 0 b e define d as R ⊗ T L with op er ations ( r ⊗ l ) r ′ = r ( l r ′ ) , (QR) l ( r ⊗ l ′ ) = ( l r ) l ′ , (LQ) ( r ⊗ l )( r ′ ⊗ l ′ ) = ( r ( l r ′ )) ⊗ l ′ , (QQ) r l = r ⊗ l (RL) for r, r ′ ∈ R, l, l ′ ∈ L . Then Q 0 is a solution of ( L, T , R ) . Pr o of. Since p ure te n sors are generators of R ⊗ L w hic h is “b if r ee” on R × L , the assignments extend to all elemen ts in a u n ique wa y . Correctness follo ws from definition of R ⊗ T L . All the asso ciativ e laws can b e prov ed only for pure tensors and the pro of is straightforw ard. 3.5 L e mma. L et ( L, T , R ) b e an triad in S up . P ut Q 1 = { ( α, β ) ∈ T -Mo d( L, L ) × Mo d - T ( R, R ) | α ( l ) r = l β ( r ) } and ( α, β ) r = β ( r ) , (QR) l ( α, β ) = α ( l ) , (LQ) ( α, β )( α ′ , β ′ ) = ( α ′ α, β β ′ ) , (QQ) r l = (( − r ) l , r ( l − )) . (RL) Then Q 1 is a solution of ( L, T , R ) . 6 Pr o of. F or ev ery l ∈ L, r ∈ R, ( α, β ) , ( α ′ , β ′ ) ∈ Q 1 w e ha ve ( α ′ α )( l ) r = α ( l ) β ′ ( r ) = ( β β ′ )( r ), hence ( α, β )( α ′ , β ′ ) ∈ Q 1 . The asso ciativ e la w (QQQ) eviden tly holds and R, L are Q 1 -mo dules (QQR, LQ Q ). Since elemen ts of Q 1 are formed by T -mo dule morp hisms, R , L are also bimo dules (QR T,TLQ). Elemen ts of the form r l b elongs to Q 1 thanks to (TL R, TTL, LR T, R TT), and consequen tly ( R L R, LRL, R T L) hold. (LQR ) foll ows b y defi n ition: ( l ( α, β )) r = α ( l ) r = lβ ( r ) = l (( α, β ) r ). Finally , (( α, β ) r ) l = β ( r ) l = (( − β ( r )) l, β ( r )( l − )) = (( α ( − ) r ) l , β ( r ( l − ))) = ( α, β )( r l ) giv es (QRL) and similarly w e wo uld pro ve (RLQ). 3.6 Definition. Let C φ → Q b e a couple. A quantal e K together w ith quan tale morp h isms φ 0 : C → K, φ 1 : K → Q suc h that φ 1 φ 0 = φ is called a c ouple factorizatio n if the K -bimo du le structure on C ob tained by restricting scalars along φ 1 mak es φ 0 a coupling map. Namely , it h olds that φ 0 ( φ 1 ( k ) c ) = kφ 0 ( c ) , φ 0 ( cφ 1 ( k )) = φ 0 ( c ) k for all c ∈ C, k ∈ K . 3.7 Theorem. L et Q b e a solution of tr iad ( L, T , R ) . The a ssignment φ ( r ⊗ l ) = (( − r ) l, r ( l − )) determines a unital c ouple Q 0 φ → Q 1 , and Q is a solution of the triad i ff ther e is a c ouple factorization Q 0 φ 0 → Q φ 1 → Q 1 . Pr o of. F rom (R TL, LRL, T LR, RLR, LR T) it follo w s that whenever Q is a solution then φ 0 : Q 0 → Q giv en by φ 0 ( r ⊗ l ) = r l is a correctly defined quan tale morp hism and together with (LQQ, QQR, QRL, RLQ) it d eter- mines a couple with actions q ( r ⊗ l ) = ( q r ) ⊗ l and ( r ⊗ l ) q = r ⊗ ( l q ). In particular φ : Q 0 → Q 1 is a coup le and it is un ital since ( id L , id R ) ∈ Q 1 . F urther, if Q is a solution then (LQQ, QQR , TLQ, QR T, LQR) yield that φ 1 ( q ) = ( − q , q − ) defines a qu an tale morp hism φ 1 : Q → Q 1 . Clearly , φ = φ 1 φ 0 and φ 0 ( φ 1 ( q )( r ⊗ l )) = φ 0 (( φ 1 ( q ) r ) ⊗ l ) = ( q r ) l = q ( r l ) = q φ 0 ( r ⊗ l ) and s im ilarly φ 0 (( r ⊗ l ) φ 1 ( q )) = φ 0 ( r ⊗ l ) q , hen ce φ = φ 1 φ 0 is a coup le fac- torizatio n . Con versely , f or a couple factorizatio n φ = φ 1 φ 0 w e pu t lq = l φ 1 ( q ) , (LQ) q r = φ 1 ( q ) r , (QR) r l = φ 0 ( r ⊗ l ) . (RL) Then (TLQ, QR T, LQR) follo w immediatelly , (LQ Q , QQ R ) hold since φ 1 is a quantal e m orp hism, (LRL, RLR, R TL) since φ 0 is a coupling map, and (QRL, RLQ) sin ce φ = φ 1 φ 0 is a coup le factorizatio n . 7 3.8 Definit ion. A triad ( L, T , R ) is called str ong if l ≤ ( l 1 R )1 L and r ≤ 1 R (1 L r ) for ev ery l ∈ L, r ∈ R , unital if T is a unital quanta le and L, R are unital T -mo dules, and strict if it is strong, un ital, and 1 L 1 R = e T . 3.9 Remark. The triad in Example 3.3 (1) is strict iff Q is s emiunital. Examples (2 – 8) provide str ict triads. 3.10 Prop osition. (1) φ : Q 0 → Q 1 is str ong iff ( L, T , R ) is str ong. (2) If ( L, T , R ) is strict, then L ( Q 0 ) ∼ = L ( Q 1 ) ∼ = L, R ( Q 0 ) ∼ = R ( Q 1 ) ∼ = R (as mo dules over any solution) and T ( Q 0 ) ∼ = T ( Q 1 ) ∼ = T (as quantales). In p articular, T is strictly two-side d. Pr o of. (1) If φ is strong, then r = e Q 1 r ≤ 1 Q 1 r = φ (1 Q 0 ) r = 1 Q 0 r = (1 R ⊗ 1 L ) r = 1 R 1 L r an d similarly for l ∈ L . Con versely , let ( L, T , R ) b e strong. Then for r , r ′ ∈ R w e ha ve r ≤ 1 R 1 L r ′ ⇒ 1 R 1 L r ≤ (1 R 1 L ) 2 r ′ = 1 R 1 L r ′ . Since 1 Q 1 acts as a T -modu le endomorphism, 1 Q 1 (1 R 1 L r ) = 1 Q 1 (1 R )1 L r = 1 R 1 L r . W e hav e 1 Q 1 r = V r ≤ 1 R 1 L r ′ 1 R 1 L r ′ = 1 R 1 L r = (1 R ⊗ 1 L ) r = 1 Q 0 r . In a similar w a y w e pro ve that l 1 Q 1 = l 1 Q 0 , hence φ (1 Q 0 ) = 1 Q 1 . (2) Due to [4], φ is an isomorp hism on righ t- and left-sided elemen ts. Assume that ρ ∈ R ( Q 1 ) an d pu t r = ρ 1 R . Then ρ = ρ 1 Q 1 yields that ρ acts as ρ 1 Q 1 = ρ 1 R 1 L = r 1 L on b oth L an d R . Since ( L, T , R ) is strict, w e can reco ver r = r e T = r 1 L 1 R . On the other han d , ( − r 1 L , r 1 L − ) = φ ( r ⊗ 1 L ) ∈ Q 1 is righ t-sided b ecause r 1 L 1 R 1 L = r 1 L . Similarly we chec k left-sided elemen ts. Ev ery elemen t t ∈ T ca n b e asso ciated w ith ( − 1 R t 1 L , 1 R t 1 L − ) whic h is clearly b oth right- and left-sided in Q 1 and from 1 R t 1 L can b e reco vered as t = 1 L 1 R t 1 L 1 R . Con v ersely , ev ery τ ∈ T ( Q ) yields t = 1 L τ 1 R and then acts as 1 R t 1 L . F or t, t ′ ∈ T we h a ve 1 R t 1 L 1 R t ′ 1 L = 1 R tt ′ 1 L , hence we ha ve obtained a qu an tale isomorphism . 3.11 Remark. Recall th at a quan tale Q is faithful [11] if ∀ l ∈ L ( Q ) , r ∈ R ( Q ) l q = l q ′ and q r = q ′ r im p lies that q = q ′ for eve ry q , q ′ ∈ Q . In the case of a strict triad we ha ve obtained that Q 0 is generated by its righ t- an d left-sided elemen ts while Q 1 is faithful. 4 Cen tral, in vo lutiv e, and Girard triads 4.1 Definit ion. A triad ( L, T , R ) is called c entr al if LR is a subset of the cen ter Z ( T ) of T , i.e. l r t = tl r for every l ∈ L, t ∈ T , r ∈ R . 8 The follo wing statemen t exhibits analogs of an em b ed ding of a cen ter and of a trace in op erator algebras. 4.2 Prop osition. L et ( L, T , R ) b e a c entr al triad. Then assignment t 7→ ( t − , − t ) defines a quantale morphism ζ : T → Q 1 and assignment r ⊗ l 7→ lr a sup- lattic e morphism τ : Q 0 → T . Elements of ζ ( T ) ar e c entr al in Q 1 and elements of τ ⊣ ( T ) ar e cyclic for any c ouple of solutions Q 0 φ 0 → Q . Pr o of. F or ev ery t ∈ T an d r ∈ R, l ∈ L w e hav e tl r = l r t , hence ζ ( t ) ∈ Q 1 . F o r t, t ′ ∈ T w e get ζ ( t ) ζ ( t ′ ) = (( t − ) ◦ ( t ′ − ) , ( − t ′ ) ◦ ( − t ) = ( tt ′ − , − tt ′ ) = ζ ( tt ′ ). Finally , for ( α, β ) ∈ Q 1 w e ha ve ζ ( t )( α, β ) = ( α ( t − ) , β ( − ) t ) = ( tα ( − ) , β ( − t )) = ( α, β ) ζ ( t ). Since l r is cen tral for ev ery l ∈ L, r ∈ R , τ ( r t ⊗ l ) = l r t = t lr = τ ( r ⊗ tl ) yields correctness of τ . F or q ∈ Q w e ha ve τ ( q ( r ⊗ l )) = τ (( q r ) ⊗ l ) = lq r ≤ t ⇔ τ (( r ⊗ l ) q ) ≤ t , i.e. τ ⊣ ( t ) is cyclic. 4.3 Definit ion. An triad ( L, T , R ) is called involutive if T is in vo lu tiv e and there is an isomorph ism L ∗ ∼ = R suc h that ( tl ) ∗ = l ∗ t ∗ , ( r t ) ∗ = t ∗ r ∗ , ( lr ) ∗ = r ∗ l ∗ for all r ∈ R, l ∈ L, t ∈ T . A solution Q of in volutiv e triad ( L, T , R ) is called involutive if Q is an in volutiv e quanta le and ( q r ) ∗ = r ∗ q ∗ , ( lq ) ∗ = q ∗ l ∗ , ( r l ) ∗ = l ∗ r ∗ for all q ∈ Q, r ∈ R, l ∈ L . A couple C φ → Q is s aid to b e involutive if φ is an inv olutive morph ism of inv olutiv e quan tales and ( q c ) ∗ = c ∗ q ∗ for ev ery q ∈ Q, c ∈ C . A couple factorizat ion C φ 0 → K φ 1 → Q is called involutive if K is an inv olutive quant ale and φ 0 , φ 1 are inv olutiv e m orphisms. 4.4 Theorem. If ( L, T , R ) is an involutive triad, then Q 0 φ → Q 1 is an involutive c ouple. Involutive solutions c orr esp ond to involutive c ouple fac- torizations of φ . Pr o of. On Q 0 put ( r ⊗ l ) ∗ = l ∗ ⊗ r ∗ . On Q 1 put ( α, β ) ∗ = ( ¯ β , ¯ α ) where ¯ α ( r ) = α ( r ∗ ) ∗ and ¯ β ( l ) = β ( l ∗ ) ∗ . Th e rest follo ws str aigh tforw ard s. 9 A triad ( L, T , R ) is said to b e Gir ar d if T admits a cyclic elemen t d T suc h that assignmen t r ≤ l ⊥ ( ⇔ l ≤ r ⊥ ) ⇔ lr ≤ d T pro vid es a dualit y L op ⊥ ∼ = R . 4.5 Theorem. If ( L, T , R ) i s a Gir ar d triad, then Q 1 ∼ = T -Mo d( L, L ) ∼ = Mo d- T ( R, R ) and φ : Q 0 → Q 1 is a Gir ar d c ouple. Pr o of. Let ( α, β ) ∈ Q 1 . Then α ( r ) ≤ r ′ ⇔ ( r ′ ) ⊥ α ( r ) = β (( r ′ ) ⊥ ) r ≤ d T ⇔ r ≤ β (( r ′ ) ⊥ ) ⊥ . W e ha ve prov ed that β = α ∗ and h ence the assertion. Put d Q = W lr ≤ d T r ⊗ l . F or l ∈ L, t ∈ T , r ∈ R we ha ve l r t ≤ d T ⇔ tlr ≤ d T b y cyclicit y , hence r ⊗ l ≤ d Q ⇔ lr ≤ d T , regardless how the p u re tensor r ⊗ l is represent ed in R ⊗ T L . No w p ut π l ′ l = W { α ∈ T -Mo d( L, L ) | α ( l ) ≤ l ′ } and observ e th at α = ^ α ( l ) ≤ l ′ π l ′ l for ev ery α ∈ T -Mo d( L, L ). Then α ( tl ) ≤ r ⊥ ⇔ α ( tl ) r = tα ( l ) r ≤ d T ⇔ α ( l )( r t ) ≤ d T ⇔ α ( l ) ≤ ( r t ) ⊥ yields that π ( rt ) ⊥ l = π r ⊥ tl and th us pure tensors of R ⊗ T L are duals of π s. Sin ce eleme nts of T -Mo d( L, L ) preserve arb itrary suprema, we ded uce that they are, as the meets of π s, organized d u ally to joins of p ure tensors and hence Q 1 ∼ = T -Mo d( L, L ) is dual to Q 0 . (Th is fact w as ment ioned b y A. Joy al and M. Tierney [6] f or comm utativ e T .) Finally , ( r ⊗ l )( α, α ∗ ) = r ⊗ α ( l ) ≤ d Q ⇔ α ( l ) r ≤ d T ⇔ α ( l ) ≤ r ⊥ ⇔ α ≤ π r ⊥ l = ( r ⊗ l ) ⊥ establishes the Girard d u alit y . Recall f rom [9] that a quan tale Q is said to b e strictly faithful if ( ∀ l ∈ L ( Q ) , r ∈ R ( Q ) l q r = l q ′ r ) ⇒ q = q ′ for all q , q ′ ∈ Q and from [7] that it is called distributive if ( r ∨ q ) ∧ ( l ∨ q ) = r l ∨ q for all q , q ′ ∈ Q, r ∈ R ( Q ) , l ∈ L ( Q ). 4.6 Prop osition. L et ( L, T , R ) b e a strict Gir ar d triad. Then Q 1 is strictly faithful and if T is distributive, then Q 1 is distributive. Pr o of. By 3.10 we can iden tify L with L ( Q ) and R with R ( Q ). If r  r ′ then r ⊥ r ≤ d T but r ⊥ r ′  d T , hence ( ∀ l ∈ L l r = l r ′ ) ⇒ r = r ′ for ev ery r , r ′ ∈ R . Thus from lq r = l q ′ r f or all r ∈ R, l ∈ L w e derive q r = q ′ r for all r ∈ R . By a similar argum ent one can also derive l q = l q ′ for all l ∈ L . W e 10 ha ve that q and q ′ are n ot distinguished either on L or on R and thus they are equ al. Assume now that T is distr ib utiv e and l ′ ( r l ∨ q ) r ′ ≤ t for l , l ′ ∈ L, r, r ′ ∈ R, q ∈ Q , th u s l ′ r l r ′ ≤ t and l ′ q r ′ ≤ t . Then l ′ (( r ∨ q ) ∧ ( l ∨ q )) r ′ ≤ ( l ′ r r ′ ∨ l ′ q r ′ ) ∧ ( l ′ lr ′ ∨ l ′ q r ′ ) ≤ ( l ′ r ∨ t ) ∧ ( l r ′ ∨ t ) = l ′ r l r ′ ∨ t = t . F rom strict faithfulness we obtain ( r ∨ q ) ∧ ( l ∨ q ) ≤ r l ∨ q . The con verse inequ alit y alw a ys holds. 4.7 Remark. Since T is a frame in examples (3 – 8) of 3.3, the triads are cen tral and solutions Q 1 are d istributiv e. In volutiv e triads arise fr om in volutiv e r ings (in particular C*-alge b ras), self-dual sup-lattices, symmetric Galois conn ections, and quantum frames. T riad ( S op , 2 , S ) is Girard for ev ery sup -lattice S and φ : Q 0 → Q 1 is the Girard couple stud ied in [4]. More generally , the triad ( M , Z ( M ) , M ) fr om 3.3 (8) is Girard. An y W*-alge b ra p ro vides a Gi r ard tria d of ideals clo s ed in normal topol- ogy . The quanta le Q 1 obtained from non -atomistic W*-alge bra is d istribu- tiv e b ut non-spatial (b ecause it d o es not ha ve enough maximal right-sided elemen ts) and represent s a natural non-comm utativ e analogy of a p oin tfr ee lo cale. W*-algebras with a non-trivial center pr o duce examples of strictly faithful quantale s (with idemp oten t righ t- and left-sided elemen ts) whic h a r e not simple (see [11]). References [1] Borceux, F., Rosick ´ y, J., a nd V an den Bos sche, G. Quan- tales and C *-algebras. Journal of the L ondon M athematic al So c iety 40 (1989 ), 398–404. [2] Coecke, B., Moore, D. J ., and Wilce, A . Op erational quant u m logic: an o ve r view. In Curr ent R ese ar ch in Op e r ational Quantum L o gi c (2000 ), Kluw er, pp. 1–36. [3] Coniglio, M . E., a n d Miraglia, F. Modules in the catego r y of shea ve s ov er qu antale s . A nnals of P u r e and Applie d L o gic 108 (2001), 103–1 36. [4] Egger, J., and Kruml, D. Girard couples of quanta les. 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