Girard couples of quantales

Girard couples of quan tales J. M. Egger Da vid Kruml ∗ No v em ber 7, 2018 Keyw ords: Girard quan tale, qu antale , sp ectrum of op erator algebra, ∗ -autonomous functor catego ry . Abstract W e introduce the concept of a Gir ar d c ouple , which consists o f t wo (not necessarily unital) quant ales link ed b y a s trong form of du- ality . The t wo basic examples of Girard couples arise in the study of endomorphism quan tales a nd o f the spe ctra of op erator algebr as. W e construct, for a n arbitrary sup-lattice S , a Girar d quan tale whose right-sided part is isomor phic to S . 1 In tro duction Girard quan tales were introd uced by Y etter to p ro vide seman tics for a cer- tain fragmen t of n on-co mmuta tiv e linear logic known as cyclic line ar lo gic [5, 17]. Th ey are, essentia lly , quanta les with a w ell-b eha v ed n ega tion op era- tion, and therefore pla y a role among all quan tales analogous to that p la yed b y complete b o ol ean algebras among f r ames. They are also related to the m uc h older n ot ion of MV-algebr a [2, 11]. Endomorph isms quan tales Q ( S ) h a v e b een stud ied b y C. J . Mulvey and J. Wic k P elletier as qu an tales of line ar r elations [9]. Th ese quantale s admit a “v on Neumann dualit y” b et w een their righ t- and left-sided elemen ts; but this ca n n ot , in general, b e extended to arbitrary elemen ts of th e quan tale. In this pap er, we construct a pr e dual quantale C ( S ) for the endomorp h ism quan tale Q ( S ) and sho w that the pairs ( C ( S ) , Q ( S )) enjo y prop erties among all su ita ble p ai rs of qu an tales w hic h are analogous to those of a Girard quan tale. In particular, w e d efine a negation op eration which extends the existing v on Neumann dualit y . ∗ Supp orted by the M inistry of Edu cati on of th e Czec h Republic under the pro ject MSM143100 009. 1 Our constru cti on is r eminiscen t of one arising in fun ctio nal analysis, where the ideal of trace-class op erato rs on a Hilb ert space is the (Banac h space) predual of the a lgebra of all b ounded o p erators on that Hil b ert space. By considering approp r iat e top ologie s on the preceding algebras, w e can construct further fun damen tal examples of what w e shall call Gir ar d c ouples . W e also note that there is a c haracterisation of Girard couples in terms of monoidal functors which, in tur n, suggests furth er generalisations: b y considering more complex “gradings” of q u an tale structur es, or by replacing sup-lattices by ob j ects of an other ∗ -autonomous category . 2 Preliminaries W e review some of th e b asic definitions and results of quantale theory whic h will b e extensive ly used in the sequel. Details ma y b e foun d in [1, 6, 7, 8 , 10, 16]. The cat egory of complete latt ices and supremum-preserving maps will b e denoted S up ; we shall follo w the con v en tion of referr ing to ob jects and arro w s of S up as sup-lattic es and sup-homomorphisms , resp ectiv ely . The top and b ottom el emen ts of a sup -la ttice will b e denoted 1 , 0, resp ectiv ely . W e say that a sup-homomorphism is str ong if it preserves the top elemen t. The categ ory S up has a ∗ -autonomous stru cture: the tensor pr o duct of sup-lattices S and T , denoted S ⊗ T , is the f r ee sup -la ttice w ith generators { s ⊗ t | s ∈ S , t ∈ T } satisfying the rela tions _ ( s i ⊗ t ) =  _ s i  ⊗ t _ ( s ⊗ t j ) = s ⊗  _ t j  for all s, s i ∈ S, t, t j ∈ T ; th e tensor uni t is the t w o-elemen t chain 2 = { 0 , 1 } ; the dual of a sup-lattice S is simp ly its opp osite, denoted S op . W e mark elemen ts of S op with ′ whenev er the distinction from elemen ts of S is desirable. Every sup-homomorph ism f : S → T has a r ig ht adj oint f ⊣ : T → S whic h pr eserves arb itrary infi ma, and so ma y b e regarded as a sup-homomorph ism f ∗ : T op → S op , f ∗ ( x ′ ) = f ⊣ ( x ) ′ ; this is the dual of f . A quantale is a su p-lat tice Q equipp ed with an a sso ciativ e m ultiplication that distr ib utes j oins . The right adjoin ts of a · ( ) and ( ) · a are denoted ( ) ← a, a → ( ), resp ec tiv ely; they can b e computed as b elo w. b ← a = _ { c | ac ≤ b } a → b = _ { c | ca ≤ b } An ele ment r ∈ Q is said to b e right-side d if r 1 ≤ r . Simila rly , l ∈ Q is left-side d if 1 l ≤ l . A two-side d element is b oth righ t- and left-sided. Th e 2 sets of right -, left-, and tw o-sided elemen ts are den ot ed R ( Q ), L ( Q ) and T ( Q ), resp ectiv ely . Note that the left annulator a → 0 of any a ∈ Q is left- sided and that the right an nulator 0 ← a is right- sided. Thus the mappings ( ) → 0 , 0 ← ( ) establish a pseudo dualit y b et we en R ( Q ) and L ( Q ). W e write l ⊥ r ⇔ l r = 0, r ⊥ = r → 0, l ⊥ = 0 ← l . W e say th at Q is: unital if it has a neutral elemen t, i.e . , an e ∈ Q , suc h that ae = ea = a for all a ∈ Q ; se miunital if r 1 = r , 1 l = l for all r ∈ R ( Q ) , l ∈ L ( Q ); von Neumann if ⊥ is a dualit y b et wee n R ( Q ) and L ( Q ). In a semiunital quantale Q , 1 a ≥ a and a 1 ≥ a hold for eve ry a ∈ Q . A homo morphism of q uanta les is a sup-homomorphism preserving the m ultiplication. It is unital if it also preserves th e neutral elemen t. A left Q -mo dule is a sup -lattice M together w ith an act ion of Q on M whic h resp ects joins in b oth v ariables and satisfies ( ab ) m = a ( bm ) for all a, b ∈ Q, m ∈ M . R ight Q -mo dules are defined similarly and a Q -bimo dule is r equired to also satisfy ( am ) b = a ( mb ) for all a, b ∈ Q, m ∈ M . W e sa y that a left Q -mo dule M is: unital if Q is u nital and em = m for ev ery m ∈ M ; str ong if 1 Q m = 1 M for ev ery m ∈ M , m 6 = 0. The righ t adjoin ts of a · ( ) and ( ) · a on a Q -bimo dule M are also denoted ( ) ← a, a → ( ). W e also write m → n = W { a ∈ Q | am ≤ n } , m ← n = W { a ∈ Q | n a ≤ m } for m, n ∈ M . A homomorph ism of left Q -mo dules f : M → N is a sup -homomorphism whic h satisfies f ( am ) = af ( m ) for ev ery a ∈ Q, m ∈ M . Homomorphism s of r ight Q -mo dules and Q -b imodu les are defined in a similar w a y . An imp orta nt example is Q ( S ), the qu antale of e ndomorp h isms of a fix ed sup-lattice S with comp osition as m ultiplication and supr ema calculated p oin twise. Its righ t- and left-sided elements are th ose of the form ρ x ( y ) = ( x, y 6 = 0 , 0 , y = 0 , λ x ( y ) = ( 1 , y  x, 0 , y ≤ x, hence T Q ( S ) = 2 ; Q ( S ) is von Neum ann b ecause ρ ⊥ x = λ x , λ ⊥ x = ρ x . More- o ver Q ( S ) is simple [14 , 12] and eve ry elemen t α ∈ Q ( S ) can b e expressed as b elo w. α = ^ x ∈ S ( ρ α ( x ) ∨ λ x ) = ^ x ∈ S ( ρ x ∨ λ α ⊣ ( x ) ) An elemen t d of a quan tale Q is ca lled: cyclic if ab ≤ d ⇔ ba ≤ d for all a, b ∈ Q ; dualizing if d ← ( a → d ) = ( d ← a ) → d = a for ev er y a ∈ Q . Q is called Gir ar d if it has a cyclic dualizing element. In th at case we write a ⊥ = a → d = d ← a . 3 The sp e ctrum of a C*-algebra A , denoted Max A , is the sup-lattice of all linear s ubspaces of A w h ic h are closed with resp ect to the n orm top ol ogy . It is a qu an tale with r espect to the multiplica tion ab = cl { AB | A ∈ a, B ∈ b } . The r ig ht- an d left-sided elemen ts of Max A are, resp ectiv ely , the closed righ t and left ideals of A . Giv en a v on Neumann algebra M ⊆ B ( H ), one ma y consider either the we ak sp e ctrum M ax w M or the ultr awe ak sp e ctrum Max σw M ; the former consists of all linear subspaces of M which are closed with resp ect to the w eak (op erato r) top olo gy, the latt er of those which are closed w ith resp ect to the somewhat finer ultra weak top ology. The we ak sp ectrum of a von Neumann algebra is a v on Neumann quanta le [13]; it follo ws that the same is true f or ultra w eak sp ectra, whic h are b etter su ited to our purp oses. [It is well -kno wn that an ideal is ultraw eakly closed if and only if it is w eakly closed.] W e recall that a functional B ( H ) → C is ultraw eakly con tin uous if and only if it h as the form P ∞ i =1 x i ( φ i , ( ) · ψ i ) for some orthonormal families φ i , ψ i ∈ H and coefficient s x i ∈ C suc h that P | x i | < ∞ . Moreo ve r, a subspace of B ( H ) is ultra we akly closed if and on ly if it is the in tersection of the kernels of some set of u ltra w eakly con tinuous fun ct ionals. An elemen t C ∈ B ( H ) is s ai d to b e tr ac e-c la ss if k C k 1 = ∞ X i =1 ( φ i , √ C ∗ C φ i ) < ∞ for some orthonormal basis φ i of H . T he set of all trace-class elemen ts is an ideal in B ( H ) and is denoted C 1 ( H ). Th e num b er k C k 1 do es not dep end on the c hosen basis and defines a norm on C 1 ( H ). The k k 1 -closed subspaces form a sp ectrum Max 1 C 1 ( H ). The more general Schatten class C p ( H ) for p ≥ 1 is defined as a set of all elements C ∈ B ( H ) such that k C k p = ∞ X i =1 ( φ i , √ C ∗ C φ i ) p ! 1 /p < ∞ ; subspaces closed in the k k p -norm form a sp ectrum, Max p C p ( H ). Giv en a f amily of Hilb ert spaces H i , w e can think of t he algebra Q B ( H i ) as that subalgebra of B ( L H i ) consisting of those op erators for whic h H i are in v ariant s u bspaces; similarly , L C 1 ( H i ) = C 1 ( L H i ) ∩ Q B ( H i ). Th e induced to p ologies on Q B ( H i ) ⊆ B ( L H i ), L C 1 ( H i ) ⊆ C 1 ( L H i ) allo w us to defin e sp ectra Max σw Q B ( H i ), Max 1 L C 1 ( H i ). 4 The algebra of n × n complex matrices, whic h is isomorphic to B ( C n ), is denoted M n C . 3 Girard coup le s 1 Definition. A c ouple (of qu an tales) consists of t wo quan tales C, Q to- gether with a c oupling map φ : C → Q such that C is also a Q -bimodu le, φ is a Q -bimod ule h omomorp hism, and φ ( c 1 ) c 2 = c 1 φ ( c 2 ) = c 1 c 2 ( ∗ ) holds for all c 1 , c 2 ∈ C . Assume that C φ → Q is a couple. An elemen t d ∈ C is said to b e cyclic if ac ≤ d ⇔ ca ≤ d for all a ∈ Q, c ∈ C . T he elemen t d is said to b e dualizing i f d ← ( a → d ) = ( d ← a ) → d = a for all a ∈ Q and d ← ( c → d ) = ( d ← c ) → d = c for all c ∈ C . In the case where d is b ot h cyclic and dualizing we write a ⊥ c ⇔ ac ≤ d ⇔ ca ≤ d for a ∈ Q, c ∈ C and a ⊥ = a → d = d ← a = W { c ∈ C | a ⊥ c } , c ⊥ = c → d = d ← c = W { a ∈ A | a ⊥ c } . A couple C φ → Q is said to b e: str ong if φ is strong; uni ta l if Q is a unital quan tale and C is a unital Q -bimo dule; Gir ar d if it has a cyclic dualizing elemen t. 2 Example. (1) Q id → Q is clearly a strong couple for any quantale Q . It is unital, or Girard, if an d only if Q is unital, or Girard, r espectiv ely . (2) Giv en an arbitrary u nital quantale Q , we can construct a Girard couple Q op 0 → Q as follo w s: Q op is equ ip p ed w ith the zero multiplica tion and the Q -bimo dule structure giv en b y ac = ( a → c ′ ) ′ , ca = ( c ′ ← a ) ′ ; 0 is the constantly zero m ap; the cyclic dualising elemen t is e ′ ∈ Q op . [Recall that we u se ′ to distingu ish element s of Q and Q op .] Ind ee d, a ⊥ = a → e ′ = W { c | ca ≤ e ′ } = W { c | c ′ ← a ≥ e } = W { c | a ≤ c ′ } = a ′ . Th is couple is clearly n ot strong unless Q = { 0 } . (3) If C j φ j → Q j are couples, then so is Q j C j ( φ j ) j → Q j Q j . Moreo v er, it is strong, u nital, or Girard, if and only if eac h comp onen t is so. (4) Let R b e a (un ita l) ring, I a tw o-sided ideal, and Sub R , Sub I the quan tales of their additiv e sub groups. Then Sub I ⊆ Su b R is a (unital) couple. 3 Prop osition. L et C φ → Q b e a c ouple. Then 5 (1) a ( c 1 c 2 ) = ( ac 1 ) c 2 , ( c 1 c 2 ) a = c 1 ( c 2 a ) and ( c 1 a ) c 2 = c 1 ( ac 2 ) for al l a ∈ Q, c 1 , c 2 ∈ C . (2) φ : C → Q is a quantale homom orphism. Pr o of. (1) Using ( ∗ ) t wice, and the fact that C is a Q -bimo dule, we obtain a ( c 1 c 2 ) = a ( c 1 φ ( c 2 )) = ( ac 1 ) φ ( c 2 ) = ( ac 1 ) c 2 . T he p roofs of the other t w o equations are similar. (2) Using ( ∗ ) and th e fact that φ is left Q -mo dule homomorph ism, we ha ve φ ( c 1 c 2 ) = φ ( φ ( c 1 ) c 2 ) = φ ( c 1 ) φ ( c 2 ). 4 Remark. Let J denote th e t w o-elemen t c hain, no w r ega rded not as an ob ject of S up b ut as a monoidal category in its o wn righ t (with ⊗ = ∧ ), and let ! : 0 → 1 denote the un ique non-iden tity morp hism of J . Then monoidal functors F : J → S up are in bijectiv e corresp ondence with unital couples of quantal es C φ → Q . [By w ay of comparison, recall that a unital quantale is equiv alen t to a monoid in S up whic h, in turn , is equ iv alen t to a monoidal functor T → S up where T is the terminal ca tegory .] The corresp ondence is giv en b y C = F 0 , Q = F 1 , φ = F ! . The multipli- c ation natur al tr ansformation of F en co mpasses all four binary op erations of the couple ( e.g. , its (1 , 0)-compon ent, F 1 ⊗ F 0 → F 1 ⊗ 0 = F 0 , corresp ond s to the left action of Q on C ); its natur ali t y is equiv alen t to the restrictions placed on φ ; the p en tagon whic h its required to s at isfy s u mmarises all the asso cia tivit y conditions which a couple s atisfies, including those of P roposi- tion 3(1). S imila rly , the unit arr ow of F , which m ust hav e th e form 2 → F 1 , pic ks out an elemen t of Q ; the triangles wh ic h it is requ ired to satisfy assert not only that this b e a u nit for Q but also that it act as a u nit on C . A v ery abstract approac h to dualising elemen ts, whic h can b e applied to a m uch larger class of monoidal f unctors, is discussed in a parallel pap er [4]. Muc h of what follo ws for couples of qu antale s remains tru e in the more general s et ting. 5 Prop osition. L et C φ → Q b e a Gir ar d c ouple. Then φ is self-adjoint, i.e. φ ∗ = φ . Pr o of. The assertion follo ws f rom c 1 ≤ φ ⊣ ( c ⊥ 2 ) ⇔ φ ( c 1 ) ≤ c ⊥ 2 ⇔ φ ( c 1 ) c 2 = c 1 φ ( c 2 ) ≤ d ⇔ c 1 ≤ φ ( c 2 ) ⊥ . 6 Remark. Giv en a Girard coup le C φ → Q , one can defi n e a ⊔ b = ( b ⊥ a ⊥ ) ⊥ for a, b ∈ C ∪ Q . Th e four resultan t op erations all corresp ond to th e mul- tiplic ative join alias p ar of linear logic. Collectiv ely , they giv e Q op φ ∗ → C op 6 the structure of a Girard coup le, with neutral el emen t d ′ and cycli c dualis- ing elemen t e ′ ; by the pr evio us prop osition, this is isomorphic, as a Girard couple, to C φ → Q . 7 Prop osition. L e t C φ → Q b e a str ong c ouple of semiunital quantales. Then φ is an isomorp hism on right- and left- side d elements. Pr o of. W e prov e the righ t-sided case. Let r ∈ R ( Q ) , s ∈ R ( C ). Then φ ( r 1 C ) = r φ (1 C ) = r 1 Q = r and φ ( s )1 C = s 1 C = s , hence φ | R ( C ) and ( ) · 1 C are mutually inv erse sup-homomorp hisms. 8 Prop osition. A Gir ar d c ouple C φ → Q is unital; if i t is also str ong, then b oth C and Q ar e von Neumann quantales. Pr o of. Let d ∈ C b e a cyclic dualizing elemen t. All the equalities of [16, Prop osition 6.1.2] can b e easily adapted for Girard couples; in particular, e = d ⊥ is a unit for Q . No w assume that r ≤ d for some r ∈ R ( C ); then r 1 C = r 1 Q ≤ d , and hence r ≤ 1 ⊥ Q = 0 C . That is, the only right- or left-sided elemen t b elo w d is 0. It follo w s that, for all pairs r ∈ R ( C ) , l ∈ L ( Q ), and f or all pairs r ∈ R ( Q ) , l ∈ L ( C ), al r ≤ d ⇔ l r = 0. Th us l r = l φ ( r ) = φ ( l ) r for r ∈ R ( C ) , l ∈ L ( C ) and Prop ositio n 7 ent ail that C is vo n Neumann . The previous prop osition also entail s l r = 0 Q ⇔ φ ⊣ ( l ) φ ⊣ ( r ) = φ ⊣ (0 Q ) = 0 C for r ∈ R ( Q ) , l ∈ L ( Q ); hence Q is also v on Neumann. 9 C orollary . Every Gir ar d quantale is von Ne umann and the Gir ar d duality extends the von Neumann duality. 10 Theorem. L e t S b e a sup-lattic e. Then the assignment ( x ⊗ y ′ )( u ⊗ v ′ ) = ( 0 if u ≤ y , x ⊗ v ′ otherwise defines a quantale structur e on S ⊗ S op which wil l b e denote d C ( S ) . The assignment φ ( x ⊗ y ′ ) = ρ x λ y defines a str ong Gir ar d c ouple C ( S ) φ → Q ( S ) with a cyclic dual izing element d = _ x ∈ S ( x ⊗ x ′ ) . 7 Pr o of. The giv en binary op eration is clearly associativ e and distr ibutiv e on generators of S ⊗ S op . F or example, ( x ⊗ _ y ′ i )( u ⊗ v ′ ) = ( x ⊗ ( ^ y i ) ′ )( u ⊗ v ′ ) = ( 0 i f ∀ i u ≤ y i x ⊗ v ′ otherwise = _ ( x ⊗ y ′ i )( u ⊗ v ′ ) . Th us, b y the definition of ⊗ , it extends to all element s of S ⊗ S op . φ to o is clearly a w ell-defined sup-homomorp h ism since it is “bilinear” on generators. It is str ong b ecause φ (1 C ( S ) ) = φ (1 ⊗ 0 ′ ) = ρ 1 λ 0 = 1 Q ( S ) . S is a left Q ( S )-mod u le with action αx = α ( x ) and S op is a right Q ( S )- mo dule with action y ′ α = α ⊣ ( y ) ′ . Th erefore S ⊗ S op carries Q ( S )-bimo dule structure. The axiom ( ∗ ) is obtained as follo ws φ ( x ⊗ y ′ )( u ⊗ v ′ ) = ρ x λ y ( u ) ⊗ v ′ = ( 0 i f u ≤ y , x ⊗ v ′ otherwise = ( x ⊗ y ′ )( u ⊗ v ′ ) and sym metrica lly for ( x ⊗ y ′ ) φ ( u ⊗ v ′ ). Sin ce the op erations of Q ( S ) are giv en p oin t w ise, the remainin g axioms of a couple are eviden t. The du alit y C ( S ) ∼ = Q ( S ) op w as p ro v en in [6] and is given by ( λ x ∨ ρ y ) ⊥ = x ⊗ y ′ . Namely , for α ∈ Q ( S ) , c ∈ C ( S ) w e hav e α ⊥ c wh en x ⊗ y ′ ≤ c implies that α ≤ λ x ∨ ρ y . W e will sho w that d is a cyc lic dualizing elemen t of the couple C ( S ) φ → Q ( S ). W e can s ee that ( λ x ∨ ρ y )( u ⊗ v ′ ) ≤ d ⇔ u ≤ x and y ≤ v ⇔ ( u ⊗ v ′ )( λ x ∨ ρ y ) ≤ d whenev er ( λ x ∨ ρ y ) 6 = 1 , ( u ⊗ v ′ ) 6 = 0 an d 1( u ⊗ v ′ ) ≤ d ⇔ u ⊗ v ′ = 0 ⇔ ( u ⊗ v ′ )1 ≤ d. Hence ( x ⊗ y ′ ) → d = d ← ( x ⊗ y ′ ) = λ x ∨ ρ y and ( λ x ∨ ρ y ) → d = x ⊗ y ′ = d ← ( λ x ∨ ρ y ) . 8 Consequent ly , α → d = _ x ∈ S ( α ⊣ ( x ) ⊗ x ′ ) = ^ x ∈ S ( λ α ⊣ ( x ) ∨ ρ x ! ⊥ = α ⊥ and similarly d ← α = _ x ∈ S ( α ( x ) ⊗ x ′ ) = α ⊥ for eve ry α ∈ Q ( S ). The inv erse dualit y ⊥ : C ( S ) → Q ( S ) follo ws d irect ly from a general p rop er ty of adjoin ts ( W c i ) → d = V ( c i → d ). Thus d is a cyclic du aliz ing ele men t of C ( S ) φ → Q ( S ). W e remark that the morphism φ : C ( S ) → Q ( S ) is an instance of a mix map [3]. G. N. Raney [15] pr o v ed that this φ is an isomorp h ism if and only if S satisfies c omplete distributivity : ^ j ∈ J _ k ∈ K α j k = _ f ∈ K J ^ j ∈ J α j f ( j ) . W e obtain th e follo win g statemen t wh ic h h as already b een mentio ned in [8 ]. 11 Corollary . Q ( S ) i s a Gir ar d quantale if and only if S is c ompletely distributive. 12 Theorem. L et H b e a Hilb ert sp ac e. Then the assignment φ ( c ) = cl σw ( c ) ( i.e. the ultr awe ak closur e) defines a Gir ar d c ouple Max 1 C 1 ( H ) φ → Max σw B ( H ) with a cyclic dual izing element d = { C ∈ C 1 ( H ) | tr C = 0 } . Pr o of. The basic idea is that ( A, C ) 7→ tr( AC ) = tr( C A ) is a bilinear form on B ( H ) × C 1 ( H ) → C whic h is con tinuous in eac h v ariable with resp ect to to the appropriate top ology . It is kn o w n [7 ] that the ultra we akly con tinuous fun ct ionals on B ( H ) are of the form tr( C · ( )) for some C ∈ C 1 ( H ), and con v ersely , the k k 1 - norm con tin uous fu nctional s on C 1 ( H ) are of the form tr( A · ( )) for some A ∈ B ( H ). In th e sp ectra, op erators corresp ond to atoms and functionals to co atoms. More precisely , we w ork with one-dimensional subspaces and k er n els of fun ctio nals. Ev ery closed subs p ace (in th e top olog y considered) can th en b e obtained as a join of atoms or meet of coatoms and it is known 9 that the families of atoms and coatoms separate eac h other. F rom this fact it follo ws that the assignm ent a ⊥ c ⇔ ( ∀ A ∈ a, C ∈ c tr( AC ) = 0) admits a du al it y b et w een Max σw B ( H ) and Max 1 C 1 ( H ). Moreo v er, the trace is sy m metric on operators and thus also on sub spaces, i.e. d is cyclic and from the dualit y it follo w s that d is dualizing. W e obtain a ⊥ = ^ { k er tr( A · ( )) | A ∈ a } . Max 1 C 1 ( H ) is a Max σw B ( H )-bimo dule since C 1 ( H ) is a t w o-sided ideal in B ( H ) and b oth m ultiplications B ( H ) × C 1 ( H ) → C 1 ( H ), C 1 ( H ) × B ( H ) → C 1 ( H ) are con tinuous. The ultra w eak top ology is w eak er than the k k 1 -norm top olog y , th us it defines a cl osure on Max 1 C 1 ( H ). F rom con tinuit y it follo ws again that cl σw ( ac ) = a cl σw ( c ) and h ence φ is a bimo dule h omo morphism . It is strong b ecause cl σw C 1 ( H ) = B ( H ). 13 Corollary . The sp e ctrum Max M n C is a Gir ar d q uanta le. Pr o of. On a finite-dimensional Hilb ert space all op erators are trace-class and the norm, k k 1 -norm, and u ltra w eak top ologie s coincide, hence φ is an isomorphism. 14 Proposition. L et H i b e a family of Hilb ert sp ac es. Then Max 1 L C 1 ( H i ) φ → Max σw Q B ( H i ) is a str ong Gir ar d c ouple. In p articular, the sp e ctrum Max A of a finite- dim ensional C*-algebr a A is a Gir ar d quantale. Pr o of. Since Q B ( H i ) ⊆ B ( L H i ), L C 1 ( H i ) ⊆ C 1 ( L H i ) are closed sub- algebras, w e can correctly restrict φ : Max 1 C 1 ( L H i ) → Max σw B ( L H i ) to φ | Max 1 L C 1 ( H i ) : Max 1 L C 1 ( H i ) → Max σw Q B ( H i ). Then all calcu- lations are m ade with resp ect to the inv arian t su bspaces H i and { C ∈ L C 1 ( H i ) | tr( C ) = 0 } p ro vides a dualit y b et wee n elemen ts of L C 1 ( H i ) and ultraw eakly con tinuous fu n ctio nals r estricte d to Q B ( H i ). It is true that cl σw ( L C 1 ( H i )) = Q B ( H i ). T he r est follo ws from Theorem 12 . Finite-dimensional C*-algebras are of th e form Q k i =1 M n i C = L k i =1 M n i C , hence th e assertio n. 15 Theorem. L et C φ → Q b e a Gir ar d c ouple. Then φ factors thr ough a Gir ar d quantale G , i.e. ther e ar e qu ant ale homomorphisms γ : C → G and 10 α : G → Q such that φ = αγ . Mor e over, C γ → G is a c ouple and the G -mo dule actions ar e g iven by r estricting sc alars along α : g c = α ( g ) c, cg = cα ( g ) for al l g ∈ G, c ∈ C . If φ is str ong then γ , α c an b e chosen to b e str ong. Conse que ntly, R ( C ) ∼ = R ( G ) ∼ = R ( Q ) , L ( C ) ∼ = L ( G ) ∼ = L ( Q ) . Pr o of. W e follo w th e idea of [16, T heorem 6.1.3]. Let G = { ( a, c ) ∈ A × C | φ ( c ) ≤ a } with joins giv en compon ent wise and multiplica tion ( a 1 , c 1 )( a 2 , c 2 ) = ( a 1 a 2 , a 1 c 2 ∨ c 1 a 2 ). F rom definition of a couple w e easily c h ec k that A × C is a quan tale. G is clearly closed under joins an d from φ ( a 1 c 2 ∨ c 1 a 2 ) = a 1 φ ( c 2 ) ∨ φ ( c 1 ) a 2 ≤ a 1 a 2 it follo ws th at it is a strong sub quant ale of A × C . Put γ ( c ) = ( φ ( c ) , c ) , α ( a, c ) = a . The pro jection α is evidently a strong quan tale homomorphism. Act ions c 1 ( a, c 2 ) = c 1 γ ( a, C 2 ) = c 1 a , ( a, c 1 ) c 2 = γ ( a, c 1 ) c 2 = ac 2 define a G -bimo dule stru ctur e on C . Then γ ( c 1 ( a, c 2 )) = γ ( c 1 a ) = ( φ ( c 1 a ) , c 1 a ) = ( φ ( c 1 ) a, c 1 a ) = γ ( c 1 )( a, c 2 ) b ecause φ ( c 1 ) c 2 = c 1 c 2 ≤ ac 2 . Similarly we can c hec k the other side and h ence γ is a G - bimo dule homomorphism. F rom the prop erties of φ it follo w s that C γ → G is also a couple. The largest elemen t of G is (1 Q , 1 C ), th us γ is strong whenev er φ is so. Finally , (1 , d ) is a cyclic elemen t of G and ( e, 0) is a u nit. In deed, we ha ve ( a 1 , c 1 )( a 2 , c 2 ) ≤ (1 , d ) ⇔ a 1 c 2 ∨ c 1 a 2 ≤ d ⇔ a 1 ⊥ c 2 and c 1 ⊥ a 2 whic h yields th e a wa ited dualit y ( a, c ) ⊥ = ( c ⊥ , a ⊥ ). The rest follo ws from Prop osition 7. 16 Corollary . F or every sup-lattic e S ther e exists a Gir ar d quantale G ( S ) with RG ( S ) ∼ = S, LG ( S ) ∼ = S op . 17 Remark. (1) Rosen thal’s Girard quanta le Q × Q op ([16, Theorem 6.1.3]) arises as our G for the zero Girard couple Q op 0 → Q of Example 2(2). (2) The multiplicatio n in G ( S ) can b e interpreted as a con volutio n pro d- uct: let a = ( a 0 , a 1 ) , b = ( b 0 , b 1 ) ∈ G ( S ), then ( ab ) i = _ j ∧ k ≤ i a j b k for b oth i ∈ { 0 , 1 } . 18 Example. It is p ossible to meld all the sp ectra (Max p C p ( H )) p ∈ [1 , ∞ ] (where Max ∞ C ∞ H := Max σw B ( H )) in to a sin gle monoidal fu nctor F : [0 , 1] → S up , thus extending the framew ork of Remark 4 . 11 Here [0 , 1] is regarded as a thin monoidal catego ry with the Lukasiewicz multiplic ation i & L j = max { 0 , i + j − 1 } so that if p, q , r ∈ [1 , ∞ ] satisfy 1 r = min { 1 , 1 p + 1 q } ( † ) and if f denotes the b ijec tion [1 , ∞ ] → [0 , 1] given by f ( p ) = 1 − 1 p , f − 1 ( i ) = 1 1 − i , then f ( p )& L f ( q ) = f ( r ). The functor F is give n by F i = Max f − 1 ( i ) C f − 1 ( i ) ( H ) and F ( i → j ) = cl f − 1 ( j ) ( i.e. the k k f − 1 ( j ) -norm closure), and its multiplica tion natural tr an s - formation by the well- kno wn fact that A ∈ C p ( H ) , B ∈ C q ( H ) implies AB ∈ C r ( H ) where r is determined b y ( † ). Moreo ve r it is p ossible to construct a single Girard quanta le G f r om this data, an alogous to that constru cted in Theorem 15: G =    ( a i ) ∈ Y i ∈ [0 , 1] Max f − 1 ( i ) C f − 1 ( i ) ( H )       cl f − 1 ( j ) a i ⊆ a j for i ≤ j    , ( ab ) i = _ j & L k ≤ i cl f − 1 ( i ) ( a j b k ) . 19 Op e n problems. W e hav e shown that the sp ectra of the op erator alge- bras B ( H ) (and th eir pro ducts) together with the sp ectra of their preduals from Girard couples. It is natural to ask whether ou r results can b e general- ized for all W*-alg ebras, includin g the id eas of the p revious example. 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