URL: http://www.elsevier.nl/locate/entcs/volume29.html 19 pages

Hilbert Q-modules and Nuclear Ideals in the Category of \/-semilattices with a Duality

J. Paseka1

Department of Mathematics, Masaryk University, Janackovo nam. 2a, 66295 Brno, Czech Republic

Abstract

The basics of the theory of Hilbert Q-modules are established. Using this approach to Y-semilattices with a duality i.e. Hilbert 2-modules, we describe a nuclear traced ideal and characterize nuclear maps in the respective category.

Quantales, that can be seen as algebras of finite observations, are a generalization of locales studied by Mulvey [12] with the aim of providing a constructive formulation of the foundations of quantum mechanics. They also provide a basis for posetal models of linear logic (see [6,18]). Essentially, quantales are to linear logic what locales (complete Heyting algebras) are to intuition-istic logic and complete Boolean algebras to classical logic. Quantale modules, that can be seen as algebras of finitely observable properties, provide an algebraic framework for operational, denotational and axiomatic semantics (see [1]). Nuclear ideals, a new categorical structure introduced in [4], are slight variants of compact closed categories, and very roughly, they can be viewed as models of multiplicative linear logic. If a monoidal category is traced the Geometry of Interaction programme (see [3,7]) can be used to obtain a compact closed category from it. Moreover, interaction categories proposed by Abramsky (see [2]) as a new paradigm for the semantics of computation can be conveniently treated as quantaloids, i.e., as categories enriched over the category of \/-semilattices. Our particular interest here is to develop basic theory of a new mathematical structure closely related to the above important notions and present an interesting example of a traced nuclear ideal which arises from this theory.

Since Girard in his Geometry of Interaction programme (see [6-8]) used the formalism of C*-algebras and Banach spaces and Abramsky and others (see [4]), generalizing ordinary binary relations to probabilistic relations with

1 Financial Support of the Grant Agency of the Czech Republic under the grant No. 201/99/0310 is gratefully acknowledged.

©1999 Published by Elsevier Science B. V.

an eye towards certain applications in computer science, studied the category of Hilbert spaces as an example of a tensored *-category with a traced nuclear ideal, there arises a natural idea to study a lattice-theoretical counterpart of Hilbert C*-modules. They provide (see [11]) a conceptual and mathematical framework for modern operator algebra and play a large role in noncommut-ative topology and geometry, being a noncommutative version of a vector bundle. Intuitively speaking, a Hilbert C*-module Y is defined exactly like a Hilbert space, except that Y is also a module over a C*-algebra A, the inner product is A-valued and we have a compatibility condition. Another motivation for their study comes in getting a technical underpinning for the C*-algebraic approach to quantum groups.

The organization of the paper is as follows. The first part of the paper attempts to begin the investigation of Hilbert Q-modules. The basics of the theory of Hilbert Q-modules are established. Our idea is, similarly as for Hilbert C*-modules, to generalize a \/-semilattice with a duality by allowing the inner product to take values in a (unital) involutive quantale rather than in the two-element Boolean algebra 2. Analogously as for Hilbert C*-modules the morphisms in the category of Hilbert Q-modules are adjointable maps. A special class of the so called compact operators playing a similar role as finite-rank operators is introduced. Moreover, the category of Hilbert Q-modules and adjointable maps between them is an idempotent involutive category with biproducts and an involutive quantaloid. To get insight into the structure of Hilbert Q-modules we explore in the second part of the paper the notion of the tensor product in the category of Hilbert 2-modules. We shall give its explicit description using only the order. We prove that the category of (strict) Hilbert 2-modules is a *-tensored category and, motivated by the paper [4], that the class of compact operators forms a traced nuclear ideal for (strict) Hilbert 2-modules. A characterization of nuclear maps and nuclear objects is given. Subsequently, in the last part of the paper we make some concluding remarks.

The paper is closely related to the papers [13], [15], [19] and [4] where the interested reader can find unexplained terms, notation and motivation concerning the subject. For facts concerning quantales in general we refer to [18].

1 Hilbert modules

The theory of Hilbert modules is a generalization of the theory of complete semilattices with a duality and it is the natural framework for the study of modules over an involutive quantale Q endowed with Q-valued inner products. Throughout this section, Q will be an involutive quantale (almost always unital).

Definition 1.1 Let Q be an involutive quantale, M a right (left) Q-module with a right (left) module action o (•). We say that M is a right (left) pre-Hilbert Q-module (right (left) Hilbert Q-module, strict Hilbert Q-module) if

M is equipped with a map ( —, —} : M X M —Q, called the pre-inner product (inner product), such that for all a £ Q, m}n £ M and rrii £ M, where i £ I, the conditions (l)-(4) ((l)-(5), (l)-(6)) are satisfied

(1) (to,re} • a = (to,re o a) (a ■ (m}n) = (a • to,re});

(2) \J(m^n) = (\J m^n)-, iei iei

(3) \/(m'= V

¿e/ iei

(4) (to, re}* = (re, m);

(5) ( —, m) = ( —, re} ((to, —) = (re, —)) implies to = re;

(6) (to, to) = 0 implies to = 0.

Lemma 1.2 Lei Q 6e an involutive quantale and let M be a left (right) pre-Hilbert Q-module. Then the factor module Mrh defined by the equivalence relation Rh = {(to, re) £ M X M : {m}p) = (n,p) for all p £ M} ¿s a left (right) Hilbert Q-module.

Proof. We shall define an operator ju '■ M —> M by iff (to) = V{n ^ M : (to, re) £ i?//}. Then evidently to < iff (to), (to, i//(to)) £ RH, iff (iff (to)) = 3H{jn)i m < n implies re = to V re i.e. (iff (re), iff (to) V iff (re)) £ Rh i.e. iff (re) > iff (to) V iff (re) > iff (to) for all to, re £ M. Moreover, if a £ Q and p £ M, we have that {m}p) = (iff (to), p) i.e. a • {m}p) = a • (iff (to), p) i.e. (a*TO,p) = (a«iff(m),p) i.e. (a • to, a • j'//(to)) £ Rh i.e. a «iff(to) < iff (a • to). So we have proved that j'// is a module nucleus on M i.e. Mrh is a factor module on M. Note that, for all to, re £ M, (m,n) = (iff (to),iff(re)). We shall now define the inner product ( —, —}// on Mrh by (to, re}^ = (to, re) for all to, re £ Mrh. Let us prove the conditions (l)-(5). We have a - (to, re}// = a • (to, re) = (a* to, re) = (a* //to, re}// for all a £ Q, to, re £ Mrh. Similarly, \JieI{mi,n)H = \fieI(mi,n) = (\fieImi,n) = (jH{\/'ieImi),n)H for all TO8,re £ Mrh7 i £ I. By the same procedure we have \JieI{n} m^n = (re, i//( Vie/ mi))ii- Evidently, (m}n)*H = (to, re}* = (re, to} = (re, to}// for all to, re £ Mrh. Now, let us assume that (to,—}// = (re,—}//. Then, for all p £ m, we have that (to, jh{p))h = {n, 3h{p))h i-e- we have {m}p) = (n,p) i.e. to = iff (to) = i//(re) = re. □

Lemma 1.3 Lei M be a \j-semilattice. Then the following conditions are equivalent:

(i) M is a \J-semilattice with a duality '.

(ii) M is a left (right) Hilbert 2-module; here 2 is a complete lattice {0, e} equipped with a non-trivial multiplication (• = A) such that the involution on it is the identity.

Proof. It is evident (we put to oe = to = e • to and (to, re} = 0 iff to < re'). □

Corollary 1.4 Let M be a \J-semilattice. Then the following conditions are equivalent:

(i) M is an ortholattice with an orthocomplement '.

(ii) M is a left (right) strict Hilbert 2-module.

Proof, (i) =>- (ii): Let (to, to) = 0. Then to < to' i.e. to = to A to < to' A to = 0 i.e. to = 0.

(ii) =>- (i): We have (to A to', to A to') < (to, to') = 0 i.e. to A to' = 0. □

A basic example of a right (left) Hilbert module over an involutive unital quantale Q is M = Q with the standard inner product (to, n) = to* • n and the module action to oa = to • a (ö • to = to • a*); another example is any complete right (left) ideal of Q with the same inner product.

Note that, if M is both left and right Hilbert module we shall speak about a Hilbert w-bimodule and we have that a*m = mo a*. Note that any right (left) Hilbert module gives rise to a Hilbert w-bimodule and conversely.

Lemma 1.5 Let Q be an involutive quantale, M a \J-semilattice. Then the following is equivalent:

(i) M is a right Hilbert Q-module.

(ii) M is a left Hilbert Q-module.

(iii) M is a Hilbert Q-w-bimodule.

Proof, (i) =>- (ii): Let us define, for all a £ Q and all to £ M, a*m = mo a*. Then, for all a £ Q and all m,n £ M, we have a ■ (m}n) = (a • (m}n))** = ((n, m)-a*)* = (n, to o a*)* = (to o a*, n) = (a • to, n) i.e. the property (1) from the definition 1.1 is satisfied. The properties (2)-(4) from 1.1 are evident, (ii) (i): The proof dualizes the proof of (i) (ii). (i) (iii): It is evident. □

Since the theory of \/-semilattices with a duality (complete ortholattices) and Y~preserving maps is based on the use of duality (othocomplementation), it is clear that there will be obstacles to develop an analogous theory for Hilbert Q-modules. Nevertheless, it is useful to use \/-semilattices with a duality (complete ortholattices) as a guide, adding extra conditions when necessary to obtain a working theory for Hilbert Q-modules. With this in mind and motivated by theory of Hilbert C*-modules, we now introduce an important class of operators on Hilbert Q-modules.

Definition 1.6 Let Q be an involutive quantale, f : M —> N a map between right (left) Hilbert Q-modules. We say that a map f* : N —> M is a *-adjoint to / and f is adjointable if

(f(m),n) = (mj*(n)} 4

for all m £ M, n £ N. Note that the *-adjoint to f is uniquely determined by 1.1, property (5). The set of all adjointable maps from M to N is denoted by Aq(M}N).

Let us point out that, similarly as for Hilbert C*-modules, the existence of a *-adjoint is not automatic, unlike for \/-semilattices with a duality (complete ortholattices).

Lemma 1.7 Let Q be a (unital) quantale, M}N}P right (left) Hilbert Q-modules, f £ Aq(M} N), h £ Aq(N} P). Then we have

f preserves arbitrary suprema;

f = r;

f is a right (left) module homomorphism; hof(E Aq(M} P) and (h o f)* = f* o h*; id m = id *M;

Aq(M, N) is a \J-subsemilattice of NM; Aq(M,N)*AQ(N,M);

Aq(M} M) is an involutive quantale.

I1 (ii (iii

(vi (vii fviii

Proof, i. Let S C M. Then, for all n £ N, we have (f(\J S),n) =

{ysj*(n)} = yseS{sJ*(n)} = ya€S(f(s),n) = <V,es/00,n> i-e. Vsesm = f(\J S) from 1.1 property (5).

ii. Let m £ M, n £ N arbitrary. Then (/(m),n) = (m,/*(n)} = (f*(n), m}* = (n, f**(m)}* = (f**(m), n) = i.e. f**(m) = f(m) for all m £ M.

iii. Let a £ Q. Then, for all n £ N} (n, f(m o a)) = (n, f**(m <> a)) = (/*(n), mo a) = (/*(n), m)-a = (n, f**(m))-a = (n, f**(m) oa) = (n, f(m) o a) i.e. f(m)oa = f(moa).

iv. Let m £ M, p £ P. Then we have ((h o f)(m),p) = (h(f(m))}p) = (/(m), h*(p)} = (m, f*(h*(p))} = (m, (f* o h*)(p)}.

v. Let m,n £ M. We have (id= {m}n) = (m}idM(n)) i.e. id m = id *M;

vi. Let T C Aq(M, N). Then(\/T(m),n) = (\/geTg(m),n) = \/geT(g(m),n) =

V3eT(m' 9*(n)) = (m> VgeT9*(n)) m ^ n £ N i.e. the map V^gT^*

is a *-adjoint to \JgeTg i.e. \JgeT9 £ Aq(M,N).

vii. Note that we have bijective mappings *M-N : Aq(M,N) —> Aq(N, M), *n,m . M) —y Aq(M} N) that preserve arbitrary suprema such that *M,N 0 *N,M = \dAQ(NM) and o *m,n = \dAQ(M,N).

viii. Evidently, Aq(M, M) is a subquantale of the quantale MM. By ii, iv and vi we have that it is an involutive quantale. □

Lemma 1.8 Let M be a \J-semilattice with a duality, f : M —> M a map. Then the following conditions are equivalent:

(i) / is a \J-semilattice morphism.

(ii) f is acljointable on the left (right) Hilbert 2-module M,

Proof. Evidently, (ii)^=^(i). Let us show that (i)^=^(ii). Assume that / is a Y-semilattice morphism i.e. it is an element of the simple involutive quantale Q(M) (see [16]). It is enough to check that the involution in Q(M) is the *-adjoint to /. Note that the involution is defined by ' o o'; here /h is the right adjoint to /. We have, for all to, n £ M, 0 = (/(to), n) iff /(to) < n' iff to < /h(n') iff to' > /h(n')' iff f*(n) < to' iff (f*(n), to) = 0 iff (to, f*(n)} = 0 i.e. (f(m),n) = (to, f*(n)) i.e. / £ A2(M,M). □

Proposition 1.9 Let Q be an involutive quantale. Then the category Hilbg (SHilbg) of right (strict) Hilbert Q-modules and adjointable maps between them is an involutive category.

Proof. It follows immediately from the lemma 1.7. □

Proposition 1.10 Let Q be an involutive quantale. Then

(i) The null object of Hilbg (SHilbg) is the one-element lattice O = {0} with the unique inner product i.e. (0,0) = 0q.

(ii) Let Mk}k £ J be right (strict) Hilbert Q-modules. Then the cartesian product Mk is a biproduct in the category HilbQ (SHilbg).

Proof, i. It is evident that O is initial in the category Hilbg^S'Hilbg). Namely, for any right (strict) Hilbert Q-module M we have a unique \/-preserving map 0m : O —> M such that (0m(0),to) = 0q = (0,0^(to)}, where 0^(to) = 0 for all to £ M. Then O is terminal i.e. it is a null object.

ii. First, let us show that the cartesian product flfceJ ^k a right (strict) Hilbert Q-module. Evidently, it is a right Q-module. Let us define the obvious inner product on flfceJ ^k as f°U°ws:

((mk)kej,(nk)kej) = \J {mk,nk)Mk ke-J

for all (mk)kej} (nk)kej £ flfceJ ^k' Note that the properties (l)-(4) from the definition 1.1 are evidently satisfied. Let us check the property (5).

Now, let us put Pj((mk)keJ) = m3 f°r aU (mk)kej £ YlkeJ Mk} j £ J and

I m, for j = k

Pk{lj{mj)) = I

( 0Mk for j ^ k.

for all TTij £ Mj and all j, A; £ J. Then pj o ij = idj^f and pk o ij = 0M Mk for

Assume that ( — }(mk)kej) = ( — }(nk)kej). Then, for all j £ J and all q £ Mj, we have (q,rrij) = (iJ(q), (mk)keJ) = (iJ(q), (nk)keJ) = {q,n3) i.e. rij = m3 for all j £ J i.e. (mk)keJ = (nk)keJ.

Moreover, for all (mk)keJ £ YlkeJ Mk and n3 £ Mj, we have (p3((mk)keJ), n3) = {mji nj) = {m3,n3)y\/kej_{3}{mk^Mk) = ({mk)keJ, i3{n3)). Evidently, UkeJ Mk is strict whenever Mk,k £ J are strict. □

Corollary 1.11 LetQ be a commutative involutive quantale, M}N left (right) Hilbert Q-modules. Then Aq(M, N) is a left (right) Hilbert Q-submodule of NM.

Proof. It is enough to check that, for all a £ Q and all / £ Aq(M, N), a* g £ Aq(M,N). We have, for all to £ M, n £ N, (a»f(m),n) = a ■ (f(m),n) = a • (to, f*(n)} = ((to, f*(n)}* ■ a*)* = (a* ■ <m, /*(n))*)* = (a* ■ (f*(n), m})* = (a* • f*(n), m}* = (m, a* • f*(n)). □

Definition 1.12 A quantaloid ([18]) is a category K, enriched over the category CSem of complete \J-semilattices and \J-preserving maps. An involutive quantaloid is an involutive category which is a quantaloid and satisfies

'A//1 V/Theorem 1.13 LetQ be an involutive quantale. The category TiilbQ (ST-Lilbg) of right (strict) Hilbert Q-modules and adjointable maps between them is an idempotent involutive category with biproducts. Moreover, it is an involutive quantaloid.

Proof. The existence of biproducts follows from propositions 1.9 and 1.10. Let us show that TiilbQ is idempotent. Note that, for a right Hilbert Q-module M, we have AJM{m) = (m)3ej and V'lI(mk)kej = \JkeJmk for all to £ M and all (mk)keJ £ f]keJ Mk. Then, for all to £ M, (VJM o AJM)(m) = VJM(/\JM(m)) = VJM((m)keJ) = \fkeJm = to = idM(m). The rest follows from the lemma 1.7. □

Lemma 1.14 Let Q be a unital involutive quantale with the standard inner-product. Then Aq(Q, Q) = {ig(a) : a £ Q} — Q; here jg(a)(&) = a ■ b.

Proof. Let g £ Aq(Q,Q), b,c £ Q. Then we have (c,g(b)} = (g*(c),b}. Let us put c = e. Then gib) = e* ■ gib) = g*(e)* ■ b i.e. gib) = jg(a)(&), where a = g*(e)*. Conversely, we have to show that, for all a £ Q, jg(a) £ Aq(Q} Q). Namely, we have (jg(a)(6), c) = (a • b)* ■ c = b* ■ (a* ■ c) = (6,jg(a*)(c)) for all b,c £ Q. □

We shall sometimes write instead of Jg(a). Motivated by the preceding lemma we have

Lemma 1.15 Let M be a right Hilbert Q-module, Q an involutive quantale with the standard inner product, to £ M. Then the map to~ : Q —> M defined by a i—> mo a has a *-adjoint to* : M —> Q defined by n ^ (m,n). Moreover, if Q is unital then Aq(M, Q) = {to* : to £ M} = M and Aq(Q, M) = {to~ : to £ M} = M.

Proof. Let a G Q, n G M. Then (ra~(a),re) = (m o a,n}** = (n,moa)* = a* • (n, to}* = a* • (to, n) = a* ■ m*(n) = (a, ra*(re)). Let g £ M). Then,

for all a G Q, re G M, we have (g(a)}n} = (a,g*(n)}. Let us put a = e. Then g(e)*(re) = (g(e)}n} = (e,g*(re)) = e* • <7* (re) = <7* (re). So we have that g = (/(e)~. The rest is evident. □

We now introduce a class of operators analogous to the finite-rank operators on a Hilbert space.

Definition 1.16 Let Q be an involutive quantale, M}N right (left) Hilbert Q-modules. For to G M and n G N, define a fr-operator Qm,n '■ N —M by

®m,n(p) = mo (n,p) = (p,n) • to = (ra~ o n*)(p)

for allp G iV. H^e shall denote by ICq(N} M) i/ie sub-\j-semilattice ofAq(N} M) generated by the set {Qm,n '■ m G M, re G iV} and the elements of ICq(N} M) we will call compact operators. Note that {Qm,n{p)i q) = (re,p)*(TO, g).

Lemma 1.17 Lei Q be an involutive quantale, M}N}P right (left) Hilbert Q-modules, m}v G M, n,u G A", g G P); h G *4<g(-P, AQ. Then we

(i) ~~

(ll) ©m,ra O ©u,"u 0m O (n,u),v ®m,» O (u,n) ©{u,ri) • m,v ©m,{ra,-u) • v;

(iii) g o Qm,n = 03(m)jn;

(iv) 0mj„ o h = 0mij,.(B).

Proof, i. Let x G M, y £ N. Then we have (0mjn(y), x) = (m~(re*(y)), x) = (n*(y),TO*(x)} = (y,ra~(m*(a;))) = {y,Qn.m(x)).

ii. Let x G M. Then we have (0mjn o 0Mj„)(i) = ©m,ra(©u,^(®)) = o n* o o v*)(x) = ((to~ o n* o u~) o v*)(x) = (m~ o (n* o o

First, let us check that (ra~ o n* o u~)(a) = 6mi„(ti)oa. We have to~ o n* o u~(a) = (ra~ o n*)(u~(a)) = (ra~ o n*)(u o«) = o a since 0mjn is

a module homomorphism. This gives us that Qm,n 0 ©u,^ = ©m o (n,u),v

Applying the preceding observation on the *-adjoint to n* o o v* we have that o u* o n~ = 0^,u(re)* i.e. (n* o o v*)(x) = 0VjU(n)~ (x). We obtain that Qmjn O QUjV — 0m,» O (ii,n)'

iii. First, let us check that g o to~ = g(m)~. We have, for all a G Q, (g o m~)(a) = g(m~(a)) = g(moa) = g(m) o a = g(ra)~(a) since g is a module homomorphism.

Now, we have go©m n = go(m~on*) = (gom~)on* = g(m)~on* = 03(m)jn.

iv. Similarly, we have n*oh = (n*oh)** = (h*on~)* = (/i*(n)~)* = h*(n)*. Then we compute 0mjn o h = (m~ o n*) o h = to~ o (n* o h) = to~ o h*(n)* =

0m,/i*(n)*' 1=1

Corollary 1.18 Lei Q 6e an involutive quantale, M a right Hilbert Q-module. Then k ,)\ \l. \l ! ¿s an involutive subquantale of Aq(M} M).

Corollary 1.19 LetQ be an involutive quantale with a standard inner product, M, N, P right Hilbert Q-modules, a}b G Q, m,n G N, g G Aq(M,P), h G Aq(P} N). Then we have

(i) ob- = (a ■ b)-;

(ii) to* o n~ = (to, n) —

iii) 0 to* = (a • to)*

iv) to~ o a~" = (to o a)n

(v) to* o h = h*(m)*;

vi) 9 0 TO~ =

Proof. It follows from the proof of the lemma 1.17. □

Proposition 1.20 LetQ be a commutative involutive quantale, N}M (strict) Hilbert Q-modules. Then ICq(N} M) is a (strict) Hilbert Q-module.

Proof. First, let us define the action • : Q x Kq(N,M) Kq(N,M) by a»0mi„ = ®a.m,,n for all a G Q, m G M,n G N. Evidently, ICq(N} M) is a Q-module.

Now, we denote by Tq(N}M) = {Qm,n '■ n G N,m G M}. Then we shall define a map a : Tq(N, M) X Tq(N, M) A Q by <j(0ijS, 0„iU) = (s, u) ■ (t, v)} for all 5, u G N, t,v G M. This gives us a map a : )Cq(N, M)xICq(N} M) Q defined by a(d, f) = \J a(QtjiSj, Qyi,Xi) for all / = Vi d =

We have, for all /, d G /Cq(5', T), / = Vi = d = Vj^t^sj =

\Ji®Zl,wn Xi,Sj,uk,wi G iV, yl,tJ,vk,zi G M,

= V, (Vj (sj,• ij,y.-) = V, (V/ xi) • y,-) = v»,/ ■ Vi) = V/ V,-z«■ ° y«'))

= V/ Vfc uk o "fc}} = Vfc,/ '

= V k,l a(®vk,uk,®zi,wi) = a(\/l®zi,wn\/k®vk,uk)

i.e. our definition of a was correct.

Evidently, for all f,ge K.Q(S, T), f = \Jt Qyi,Xi, g = Vj @

rrij G S,

yi7n3 G T, <r( —,/) = —,(/) implies f = g. Namely, for all s G »*? and all

f G T, <*,/(*)) = <f,(V,-©«,*,)(*)> = (tiViQnM) = (tiViisiXi) .y,-> =

\/i(s,Xi)-(t,yi) = \J3{s,m3)-{t,n3) = (t,\J.{s,mj) • n3^ = (t, (Vj0^^) («)) = i.e. /(s) = g(s) for all s G S i.e. f = g. Moreover, if / = Vi ©2/w

g = Vj ©rxj.mj, a e Q then

v(f,9)* = ^(Vi ©3«,*i> Vj ©»V = V.-,j

= • (yi,nj))* = Vi,j • (x^rrij)*

= V,-,j K-,y.-> • = V,-,j ®«-> • K-,y.->

= V,\j ^(©nj.mj, ®yt,xt) = ^(Vj ©nj.mj, V,- = ^(flS /)

« • ^(/,30 = « • ^(Vi Vj ©rxj.mj) = a • Vi,j ©rxj.mj) = a • Vi,j mj) ' = Vi,j mj) ' a ' uj)

= V,-,j rn3) ■ (a • yi, rij) = Vij ^(Qa.K^,0«,,™,) = 9a.yi,Xi, Vj ©rxj.mj) = ^(Vi a • Vj ©rxj.mj)

= V • • V ©nj.mj) = v(a»f,g).

Now, let fk = Vu ®yik,=0ik, g = Vj Then

\lkaUk,g) = Vfc Vtfc Vj =

Similarly, we have <r(g,/fc) = <r(g, /fc). Altogether, <7 defines an inner-product on )Cq(N, M). □

Under a further assumption one can completely characterize Aq(M, M). A Hilbert Q-module is called self-dual when every module morphism <£> : M —y Q is of the form tp(m) = (n, to) for some n £ M i.e. <£> = n*.

Proposition 1.21 In a self-dual Hilbert Q-module M, Aq(M, M) coincides with the \j-semilattice Hq(M} M) of all module homomorphisms on M.

Proof. In view of the lemma 1.7 we have that Aq(M, M) C /HQ{M, M). So we only need to show that a given module homomorphism / £ Hq(M} M) is adjointable. Indeed, for fixed n £ M define tpfjn : M —> Q by Lpj^n(m) : = (m,/(n)}. By self-duality this must equal (g(m)}n) for some g(m) £ M. So the map g : M —> M is by definition /*. □

Definition 1.22 Let M, N be Hilbert Q-modules, f : M ^ N a module homomorphism. We say that f is inner-product preserving if, for all TOi,to2 £ M, (TOI,to2) = (/(mi),/(m2)}.

Note that any inner-product preserving homomorphism is injective. Namely, /(m) = f(n) implies (m,p) = (f(m)J(p)) = (f(n)J(p)) = (n,p) for all p i.e. m = n.

Proposition 1.23 Let M,N be Hilbert Q-modules, f : M —N a module homomorphism. Then, for the following conditions are equivalent:

(i) f is an adjointable map such that f* o f = id,M,

(ii) f is an adjointable inner-product preserving map.

Proof. Let us assume that m,n £ M. Then we have (m,n) =

<m, (/*o/)(«)> = </M,/(«)>■

Let us prove that f*of = idjvi- Let to, n £ M. Then (n, (/* o /)(to)} = (nj*(f(m))) = (f(n)J(m)) = (n,m) = i.e. to = (f* o f)(m) for all to £ M. □

Definition 1.24 Let M,N be Hilbert Q-modules, f : M —N an adjointable map. Then f is said to be unitary if

(7) t of = idM, fof*= idjv.

Lemma 1.25 Let M}N be Hilbert Q-modules, f : M —> N a module homomorphism. Then the following conditions are equivalent:

(i) / is a surjective (bijective) inner-product preserving map,

(ii) / is unitary.

Proof. (i)<^=^(ii) It follows immediately from 1.23. □ 2 Tensor products and nuclear ideals of Hilbert 2-modules

We study the tensor product S®hT of (strict) Hilbert 2-modules (i.e. objects are Y-semilattices with a duality and morphisms are \/-preserving maps). We prove the existence of a tensor product in the respective category. Let us recall the following standard definitions.

Definition 2.1 Let S,T and U are (strict) Hilbert 2-modules. A function f : S X T —> U is a bimorphism in the category of (strict) Hilbert 2-modules if the functions gs : T —> U defined by gs(t) = f(s,t) and ht : S —> U defined by ht(s) = f(s,t) are morphisms of (strict) Hilbert 2-modules for each s £ S and t £ T.

Definition 2.2 Let S and T be (strict) Hilbert 2-modules. A (strict) Hilbert 2-module S (E>h t is a tensor product of S and T in the category of (strict) Hilbert 2-modules if there exists a canonical bimorphism es,T '■ S xT —> S(£)hT such that for any (strict) Hilbert 2-module U and any bimorphism g : S xT —> U there is a unique morphism of (strict) Hilbert 2-modules h : S (£)h T —> U satisfying g = h o es,T-

Note that then f(S X T) generates S ®h T.

Lemma 2.3 Let S be a pre-Hilbert 2-module. If we define an operator ' : S —> S by s' = \/{t £ S : (s,t) = 0} for all s £ S we have, for all to £ S, jiiijn) = rri" and to' = jH(m').

Proof. Note first that ' is a Galois connection on S. Namely, for all m,n £ S, to < n' iff (to, n) = 0 iff (n, to} = 0 iff n < to'. Evidently, m < n implies n' < to', to < to". Moreover (to, to") £ Rh i.e. to" < j#(to). Conversely, (to, to'} = 0 i.e. (j#(to), to'} = 0 i.e. to" > j#(to). We then have to' = to'" = Mm'). □

Lemma 2.4 Lei S and T be Hilbert 2-modules. Then the tensor product S (£)ST in the category of \j-semilattices is a pre-Hilbert 2-module.

Proof. Note that, since S (£)s T is a \/-semilattice, it is a left 2-module. Its elements are called C-ideals and (X)s denotes the least C-ideal containing X. First, let us define a map [ —, — ] : V(S X T) X V(S X T) —> 2 as follows:

[X,Y] := \/{(s1,s2)s ■ (tut2) : (s^U) £ X,(s2,t2) £ Y}

for all X,Y £ V(S x T). We have, for all X, Y £ V(S x T), [X,Y]=[(X}S,Y]. Namely, let us put Z = {(s,t) £ S X T : [{(M)},Y] < [X,Y]}- Evidently, X c z, {0} x T C x {0} c z. Let (sA,t) £ Z. Then, for all (u,v) £ Y, we have (sx,u)s ■ (t,v)T < [X, Y] i.e. (\/XeAsx,u)s ■ (t,v)T = \/XeA(s\,u)s ■ (t,v)T < [X,Y] i.e. (\JXeAsx,t) £ Z. Similarly, if (s,tx) £ Z, A £ A then (s,\/XeAh)ez.

So we have that Z £ S ®s T, X C Z and [X, Y] < [(X)s, Y] <[Z,Y] < [X, Y]. Note that, since 2 is a commutative quantale, we have [X, Y] = [ Y, X] i.e. [X,Y] =[X,(Y)S].

We shall define a pre-inner product ( —, —} : (S (£)s T) X (S (£)s T) —> 2 as follows:

(/, J} :=[/,/]

for all I, J £ S ®s T. Let us check the conditions (l)-(4). Let a £ 2, I,J,J\ e S ®ST, A £ A. We have a • (/, J} = a ■ \J{{slls2)s • (ti,t2)T :

(si,ti) £ /, (¿2? t2) £ j} = v{a'('si5's2}t-(^i,^2}t : (-si, ti) £ /, (¿2? t2) £ j} = v{(a*55i,s2}5 • (¿1, : (-si, ti) £ I,(s2,t2) £ j} = vii^^x • :

(p,t i) £ a»I,(s2,t2) £ «/} = (a*/, J} and \JXeA{J\,J) = \J XeA\/{{su s2)s • (ti,t2}t : (si,ti) £ J\,(s2,t2) £ j} = V{(si5's2}t • : (¿a^i) £ «/a, A £

A, (s2,t2) £ j} = v{(si>s2>t • (ii,*2>t : (si,ii) g Vaga ja,(s2,t2) £ j} = (Vaga ./)• Similarly, we have \/aga = («/, vaga Using the fact

that (X}Y) = (Y,X) and the involution on 2 is the identity map, we have

(X,Y)* = (Y,X). □

Theorem 2.5 Lei S and T be (strict) Hilbert 2-modules. Then the factor (strict) Hilbert 2-module S (E>h T = (S (£)s T)rh is a tensor product in the category of (strict) Hilbert 2-modules and the tensor product is unique up to isomorphism of Hilbert 2-modules.

Proof. First, let us show that, for all s £ S, (s (g)s 1)' = s' 1 i.e. ((s, l))s' = ((s',l)}s. Let us put Z = {(u,u) £ S X T : [(s, 1), (it, u)] = 0}. Evidently, Z is an element of S (£)s T. Then (u,v) £ Z iff ((s,u)s = 0 or (l}v)T = 0) iff

(u < s' or V = 0) iff ((m,u) < (s',1) or {u,v) < (1,0)) iff {u,v) G l))s. Then ((s', l))s = Z = {(u, v) (E S XT :[(s} 1), (u, u)] = 0} = {(u, v) G S X T :

[<(S,1)>„ («,«)] =0} = ((3,l))s'.

Now, we shall prove that s ®s 1 is a j^-fixed element i.e. ((s,l))s = jH(((s, l))s) for all 5 G S. Evidently, ((s, l))s C jH(((s, l))s). Let (u, v) G S X T. Then [ (it, u), ((s, l))s] =0iff[(u,i;),jii(((5,l))s)] = 0 i.e. [ (s', 1), jH(((s, l))s)] Oi.e. jH(((s,l))3)C((s',l))s'= ((s,l))3.

Similarly, we have, for all t G T, ((1 ,*))/ = ((l,t'))s and ((l,t))s =

M{(ht))s).

Now, let 5 G S, t G T arbitrary. Then (s <g)s 1 \/H 1 <g)s t)' = s' ®s t'. Namely, we have (s ®s 1 Vff 1 (g)s t)' = (({(s, 1)})S ({(1 ,*)}),)' = ({(5,1)})/ AH

({(1,0})/ = <{(*', 1)}), n ({(l,i')}>. = ({(*',*)})• =

We shall now define the canonical map / : S X T —S (£)h T by f(s,t) = s®st e S ®HT. Now, let s G S. Then, for all T' C T, we have f(s, V T') = s®s\/r = \/SzaT{s®at : t G T'} = ySQHT{s®st: t G T'} = Vs®ht{/(M) : t G T1} i.e. hs : T —S (E)h T is \/-preserving. Similarly, for all t G T, the map gt : S —S (£)h T is \/-preserving. So we have proved that / is the canonical bimorphism.

Let U be any Hilbert 2-module and let g : S X T —> U be any bimorphism. There is a unique morphism of \/-semilattices <p : S (£)ST —U such that g(s, t) = <p{s®at). We define a map ip* :U S ®aT by ip*(n) = (^H(n'))'. We then have, for all to G S (£)s T and all n G /7, {(f(m)} n)jj = 0 iff tp(m) < n' iff to < (JcH(n/) iff ((JcH(n'))/ < to' iff <p*(n) < to' iff [to, ip*(n)\ = 0. Since <p*(n) G (£)h T by 2.3 we have a map ip* : U —S (£)h T defined by ip*(n) = f*(n) for all n G U. Let us assume ip(m) = f(m) for all to G S (£)h T. Then we have (ip(m)}n)u = 0 iff (ip(m),n)u = 0 iff [m,t^*(n)] = 0 iff (m}tp*(n)) = 0 i.e. ?/> and ip* are adjointable maps in the category of Hilbert 2-modules. Moreover, g(s,t) = cp(s (g)s t) = tp(s t) = ^(f(s,t)) for all (s,t) G S X T and ip is uniquely determined. Finally, since the set {s (x)s t : (s,t) G S X T} generates the \/ —semilattice S (£)ST it generates also the Hilbert 2-module S (£)h T. This shows that Hilbert 2-module S (£)h T and the canonical bimorphism / satisfy the conditions of the definition of a tensor product. The uniqueness of a tensor product is clear from its definition as a solution of a universal problem. Evidently, if S and T are strict Hilbert 2-modules we have that S (£)h T is also strict. □

Definition 2.6 Let S,T be Hilbert 2-modules, X C S x T. We define X+ = {(m,u) G S xT : (V(s,i) G X)(s < u) or (t < u)}, X" = {(i/,u) G S X T : (V(s,t) G X)(s > u) or (t > v)}.

Recall that is well known ([17], [10]) that the maps X i—> X+ and X > X~ form a Galois connection from the power set of S X T to itself. Moreover, we have an explicit description of S (£)h T using only the order on S and T.

Lemma 2.7 Let S,T be Hilbert 2-modules, X C S xT. Then X+~ = X" = MX).

Proof. Note that, for all Y C S X T we have (c,d) G Y' iff for all (e,/) G Y we have [{(c, d)},{(e, f)}]=0 iff e < d or / < d! for all (e,/) G Y iff (d,d!) G Y+. Similarly, (a, b) G F" iff (a, 6) G {(e',/') : (e,/) G Y}' iff

^ A(e,/)eFNl/lVl®ii /)•

This gives us that, for all (s,t) eSxT, (s,t) G X+" iff (s,t) G {(e',/') : (e, /) G X+}' iff (s, t) G {(c, d) : (d, dr) G X+}' iff (s, t) G {(c, d) : (c, d) G X'}' iff (s,t)eX". □

Lemma 2.8 Let S be a (strict) Hilbert 2-module. Then we have a morphism of Hilbert 2-modules Tr : S ®h S —> 2 such that (s,t) = Tr(s (E)h t) for all s,t G S i.e. \/t{st}tt} = Tr(\/isi (E)h U).

Proof. It follows immediately from the fact that ( —,—} : S X S —> 2 is a bimorphism of Hilbert 2-modules. □

Note that, similarly as in [5], we can extend the operation (E)h to morph-isms. We may also easily check that the tensor product in the category of (strict) Hilbert 2-modules is both associative and commutative and we have the unit 1 = 2. Recall that the notation cs,t- S (£)h T —> T (E)h S will be used for the symmetry operator.

It is straightforward to see that:

Theorem 2.9 The category of (strict) Hilbert 2-modules is a symmetric mo-noidal category.

Proof. The proof is simply a checking the necessary diagrams. □

Theorem 2.10 The category of (strict) Hilbert 2-modules is a *-tensored category.

Proof. By the theorem 2.9 the category is symmetric monoidal and by the theorem 1.9 it is an involutive category. Assume that / : A —> B and g : C —> D are Y~preserving maps. Let us prove that (/ (E)h g)* = f* <E>if g*■ We have, for all a G A, b G B, c G C, d G D,

{a (g>H 6, (/ (g>H g)*(c (g>H d)) = ((/ ®H g)(a (g)Hb)}c ®H d)

= {f (a) g(b)}c(g)H d)

= (/(«)>c) • {g{b),d)

= (a,r(c))-(b,g*(d)) = (a®Hb,f*(c) ®Hg*(d)) = (a ®„ b, (f* ®H g*)(c ®H d)) .

This gives us that (/ ®H g)*(c ®H d) = (f* ®H g*)(c ®h d) i.e. (f ®H g)* = f* ®H g*■ As a conjugation, we will take the identity. Evidently, A = A = A, A®H B = A®H B = A®H B = A®H~B and 2 = 2 = 2. Moreover, since for

any Y —preserving map / : 2 —> 2 we have either / = 0 or / = id2 i.e. /* = 0

or f* = id2 i.e. / = /*, then

2---► 2 2---► 2

rsj rsj | rsj rsj

2-► 2 2-► 2.

Hence the category of (strict) Hilbert 2-modules is *-tensored. □

Lemma 2.11 For all objects S,T G we pwi Af(S,T) = /C2(S,T) Ç

^(S1, T). H^e will refer to the union of these subsets as Af. The class Af is closed under composition with arbitrary morphisms of (strict) Hilbert 2-modules, closed under (E>h, closed under ()*; and the conjugate functor.

Proof. It is enough to check that all the above properties are preserved on generators i.e. mappings of the form ©tjS. By the lemma 1.17 we have that, for all (strict) Hilbert 2-modules S, T, U} V and for all g G A2(T, V), h G A2(U, S) s G S, t G T, we have that g o 0is = ©3(t),s, ©t,s oh = Qt,h*(s) and ©*,s = ©M i.e. Af is closed under composition with arbitrary morphisms of (strict) Hilbert 2-modules and under under ( )*.

Now, let S} T, U, V be any (strict) Hilbert 2-modules, s G S1, i £ T, u (H U and v G V. Then, for all x G S} y G /7, we have (©t,s ®v,u)(x y) = ©i,S(®) <£>if ®v,u(y) = (X1 S) • t {y, u) • V = ({x, s) ■ {y, u)) • (t ®H v) = (x (g)H y, s (g)H u) • (t (g)H v) = ®t®Hv,s®Hu{x (g)H y) i.e. Af is closed under ®H. Since the conjugate functor is the identity the lemma is proved. □

Lemma 2.12 Let S,T be Hilbert 2-modules. We define a mapping U:S ®h T N(S, T) by U(s ®h t)(p) = (p, s) • t, where s ®H t G S ®H T, p G S. Then Af(S}T) is a (strict) Hilbert 2-module and U is a unitary isomorphism of S ®HT onto AT{S,T).

Proof. Evidently, Af(S} T) is a (strict) Hilbert 2-module by the proposition 1.20 and the map u : S X T —Af(S,T) defined by u(s,t) = QijS is a bi-morphism of Hilbert 2-modules. This gives us that U is correctly defined. Moreover, since Af(S} T) is generated by the elements of the form ©ijS the morphism U of Hilbert 2-modules is onto. Note that, for all s,u G S} t,v G T, we have {s (g)H t}u (g)H v)S0hT = (s,u)s- {t,v)T = (Qt,s,Qv,u) = (U (s (x) h t), U (u (x) h v)) i-e. U is inner-product preserving. By the lemma 1.25, U is a unitary isomorphism. □

Corollary 2.13 Let S and T be (strict) Hilbert 2-modules. Then the (strict) Hilbert 2-module Af(S}T) is a tensor product in the category of (strict) Hilbert 2-modules. Moreover, we have a morphism of Hilbert 2-modules Tr : AT{S, S) 2 such that (s,t) = Tr(Qt,s) for all s,teS.

Note that Tr(©ijS) = e iff ©ijS o 0i s = 0i s iff s ^ t'. Let us put zs = \/{®t,s : 5, t £ S, s < t'}. Then, for all / £ J\f(S, S), Tr(f) = 0 iff f < zs.

Lemma 2.14 Let S be a Hilbert 2-module. Then f £ A2(2, 51) iff f = es /or some s £ SV /¿ere

0 if a = 0,

e.,(a) = ;

5 if a = e.

Moreover, S is unitary isomorphic to A2(2,S). Proof. It is evident. □

Theorem 2.15 The class of compact operators forms a traced nuclear ideal for (strict) Hilbert 2-modules.

Proof. Let S and T be Hilbert 2-modules. By the lemma 2.14, it is evident that A2(2, S(S)hT) = S®hT. So the morphism U, defined in the lemma 2.12, has an inverse unitary map V such that V gives us the transpose operator by the prescription 6(f) = £y(/), for all / £ J\f(S,T). It only remains to check the equations. First, let us check the preservation of tensored *-structure defined in [4]. Let / : S T, g : M N be nuclear. Then V(f) ®H V(g) = (ids(x)#cy,M®.ffidjv)o V(f®ng) by a straightforward computation. This gives us the property (a). Let us prove (b). Since, V(0*s) = V(©Sji) = t -s = cs,t(-s ®h t) = (c o V)(Qt,s) and V is uniquely determined by its action on generators, the property (b) is verified.

Now, let us prove the naturality of 9. We have, for all s £ S,t £ T, 0t,s ^ eS(g)fft ^ ef(s)®Hg(t) by the prescription A2(2,f ®H g) and 0ijS i-» g ° 0f,s 0 f* = ®g(t),f(s) ^ ef(s)®Hg(t) by the prescription 9 o J\f(f*,g) i.e. the required diagram commutes.

To prove compactness of A/", it is enough to check the diagram from the definition (see [4]) for / = Qw>s : S —T and g = Qm,t '■ T —M, s £ S, t,w £ T, m £ M. Then, for all p £ S, we have g(f(p)) = Qm,{w,t).s(p) = {(p,s) '

0 if (s,p) = 0 or (t,w) = 0 it, w)) • m = { . For the other part of the diagram:

m otherwise

0 0 if {s,p)= 0 or (t,w) = 0 p 1-» e®Hp t®Hm®Hp m®Ht®Hp { ,

m®u c 1—y m otherwise

since 9(Q*wJ*(t ®Hp)< 0 iff {{s,p) = 0 or (t,w) = 0).

Now, we shall show that Af is traced. Let f : S T, g : T S be nuclear maps i.e. / = \Ji ©tijSi, g = V • , Uj £ S, Vj £ T. Note that

f(e) = Vi ®H ti, g*(e) = Vj uj Vj. Then,

((9°f)(e),e) = (f(e),g*(e)) = \fi,j(si®Hti,uj®Hvj) = VM' (Si, Uj) ' (U, Vj) = V, ( Vj Vj) • Mj,

= Tr(yt Qg{ti)tai) = Tr(g o V,. 0,^) = Tr^ o /).

Similarly, let /' : S1 U, g' : U —>■ S be nuclear maps such that g' o /' = go f. Then o /')(e), = Tr(g' o f) = Tr(g o f) = ^(g o /)(e), i.e.

g' o /' = g o f. □

Proposition 2.16 Let S,T be Hilbert 2-modules, f : S —T a morphism of Hilbert 2-modules. Then f is nuclear iff for all u £ S, v £ T we have (u1, v) £ T(/) '"j v < /(u) implies v = f(u); here T(/) = {(V, f(x)) : x £ S1}.

Proof. Note that 0ijS < / iff for all a; £ S (x < s' or t < f(x)) iff (s,t) £ Y(/)~. Then we have, / is nuclear iff / = \/{®t,s • ^ Now

assume that / is nuclear. Let (u',v) £ T(/) v < /(u). Then, for all (s,t) £ T(/)", s < u' or t < v i.e. V{0m(u) : (M) e T(/)"} < u i-e-f(u) < v < f(u) i.e. /(u) = 17.

Conversely, let for all u £ S, v £ T we have (u',v) £ T(/) '"jU < /(u) implies v = f(u). Let u £ S. We put \/{©ijS(u) : (s,t) £ Y(/)~} = t>. Then v < /(u). Moreover, if (s,t) £ T(/)_ and s ^ u' then t = ©ijS(u) < v i.e. (u',v) £ T(/)-+ i.e. t; = f(u) i.e. / = V{©m : (M) £ T(/)"}.

Note that a \/-preserving map f : S T, S,T being \/-semilattices, is tight (see [20], [21]) iff / = V Eb for suitable a £ S, b £ T; here

0 if x < a

El(x) = {

b otherwise.

Theorem 2.17 Let S,T be (strict) Hilbert 2-modules, f : S —T a morphism of Hilbert 2-modules. Then f is nuclear iff it is tight.

Proof. / is nuclear iff / = V{©t,s : (M) £ ?(/)"} = \/{E? : (M) £ T(/)"} iff/is tight. □

Theorem 2.18 Let S be a Hilbert 2-module. Then S is nuclear iff it is completely distributive.

Proof. We have by [9], S is completely distributive iff ids is tight iff ids is nuclear iff S is nuclear. □

Corollary 2.19 Let S be a strict Hilbert 2-module. Then S is nuclear iff it is a complete Boolean algebra.

3 Conclusion and future work

We have introduced Hilbert Q-modules, their morphisms and compact operators on them. We have illustrated this notions in the simplest case of Hilbert 2-modules getting a *-tensored category with a traced nuclear ideal. It would of course be interesting to have more general examples of Hilbert Q-modules having this nuclear ideal structure.

There are several topics in our approach we would like to study in more detail. We hope that after some effort using the proposition 1.20 we can show that any category of Hilbert modules over a commutative quantale is a *-tensored category with a traced nuclear ideal.

Another possibility is to study tensor products of involutive quantales using the representation of involutive quantales via Hilbert 2-modules (see [14]) similarly as for C*-algebras. Having a good tensor product of involutive quantales it would be of interest to develop a notion of a Hopf involutive quantale and study related questions.

Acknowledgement

We want to thank here the anonymous referee for many valuable and perceptive comments that allow highly improve the final version of this paper. Paul Taylor's diagram macros are acknowledged.

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