Dong LIXiaoman CHENShengzhi XU

1 Introduction

The description of systemsS=(H;E1,E2,···,En)ofnsubspacesHi(i=1,···,n),of a Hilbert spaceH,which can be finite or infinite dimensional,up to an isomorphism or the unitary equivalence,is famous as the multi-space theory,and the classification of these systems is a subject which attracts many mathematicians’attention.In a finite dimensional space,the classification of indecomposable systems ofnsubspaces forn=1,2 and 3 is simple.Jordan blocks give indecomposable systems of 4 subspaces.But there exist many other kinds of indecomposable systems of 4 subspaces.Therefore,it was surprising that Gelfand and Ponomarev[1]gave a complete classification of indecomposable systems offour subspaces in a finite dimensional space.

In this paper,we generalize this theory to the case ofA-modules,whereAis an involutive algebra,and we construct a group,called Φ-group,whichis a generalization of theK-group and gives more information of the algebraAthan theK-theory.This group,which can be regarded as the multi-operator edition of theK-group,has essential relations with the problem of classification of systems ofnsubspaces whenA=.

2 Preliminaries

We first recall the basic notations of systems ofnsubspaces.

LetHbe a Hilbert space andE1,···,Enbensubspaces inH.Then we say thatS=(H;E1,···,En)is a system ofnsubspaces inHor annsubspace system inH.LetT=(K;F1,···,Fn)be another system ofnsubspaces in a Hilbert spaceK.Thenφ:S→Tis a homomorphism ifφ:H→Kis a bounded linear operator satisfying thatφ(Ei)⊆Fifori=1,···,n.Moreover,φis an isomorphism if it is an invertible linear operator andφ(Ei)=Fifori=1,···,n.We say that systemsSandTare isomorphic if there exists an isomorphismφ:S→T.And ifφis moreover a unitary operator,we say that the two systems are unitarily equivalent.

There are notations about direct sum and indecomposable systems(see[2]),and the main work on multi-subspace systems is about the classification.Many problems of linear algebra can be reduced to the classification of the systems of subspaces in a finite dimensional vector space.In a finite dimensional space,the classification of indecomposable systems ofnsubspaces forn=1,2 and 3 is simple.Gelfand and Ponomarev[1]gave a complete classification of indecomposable systems offour subspaces in a finite dimensional space.

Proposition 2.1(see[3])Let H be a Hilbert space and S=(H;E)be a system of onesubspace.Then S=(H;E)is indecomposable if and only if

Let S=(H;E1,E2)be a system of two subspace in a Hilbert space H.Then S is indecomposable if and only if S is isomorphic to one of the following four commutative systems:

Gelfand and Ponomarev[1]claimed that there exist only nine finite-dimensional indecomposable systems of three subspaces.But we do not know whether there exists an infinitedimensional transitive system of three subspaces.

Proposition 2.2(see[1])Let S=(H;E1,E2,E3)be an indecomposable system of three subspaces.If H is finite-dimensional,then S is isomorphic to one of the following nine systems:

One of the main problems to tackle is the classification of indecomposable systemsS=(H;E1,E2,E3,E4)offour subspaces in a Hilbert spaceH.In the case whenHis finitedimensional,Gelfand and Ponomarev completely classified indecomposable systems and gave a complete list of them in[1].

Now we generalize the former definition to the term of(right)modules.

Definition 2.1Given an involutive algebra A,let H be a finitely generated free A-module and E1,···,Enbe n finitely generated projective submodules of H.Then we say that S=(H;E1,···,En)is a system of n-submodules in H.

Let T=(H′;F1,···,Fn)be another system of n-submodules.Then φ:S→T is called a homomorphism if φ:H→H′is a module map satisfying that φ(Ei)⊆Fifor i=1,···,n.And φ:S→T is called an isomorphism if φ:H→H′is an isomorphism satisfying that φ(Ei)=Fifor i=1,···,n.We say that system S and T are isomorphic if there exists an isomorphism φ:S→T.

Let S=(H;E1,···,En)and T=(H′;F1,···,Fn)be two systems of n submodules in the module H.Then their direct sum S⊕T is defined by

Similar to the typical systems ofnsubspaces,we also have the notation of indecomposability and irreducibility.

Let us introduce an important kind ofA-module,and thusAn,which denotes the direct sumA⊕A···⊕Aofncopies ofA,Anbecomes a module overAwith the module action defined by

We are mainly interested in the system ofn-submodules inH,whichis of this type,denoted by(Am;E1,E2,···,En),where eachEkis a finitely generated projectively submodule ofAm.Then eachEkis isomorphic topkAmfor some projectionpkonAm.Then we can write the system ofnsubmodules in the form:(p1,p2,···,pn),where eachpiis a projection,namely

In this paper,we construct an Abelian group,namely Φ-group,for systems ofnoperators(p1,p2,···,pn)to generalize the classicalK-group.The Φ-group for systems of one operator is just the classicalK-group,and whenn≥2,the Φ-group contains theK-group as a direct summand,and hence we can see that Φ(A)contains more information ofAthan that ofK(A).

We mainly describe the Φ-group for systems ofnoperators whenn=2,and it only has some of the propositions of the classicalK-theory.We compute the Φ-group for multi-operators when the operators have some relations.In fact,to compute the Φ-group is the process to describe the structure of multi-operators up to unitary equivalence.

Finally,we remark that the Φ-group has a relationship with the problem of the classification of systems ofnsubspaces.The Φ-group can be regarded as a classification theory for systems ofnsubspaces up to unitary equivalence whenA=.

We firstly discuss the case when the involutive algebraAis unital.Since every finite size projection onAnis in fact a matrixx∈Mn(A)such thatx2=xandx∗=x,then every system of one submodule corresponds to such a fixed matrixx.Therefore,the Grothendiek group of stable isomorphism classes of the systems of one submodule is nothing but theK-groupK0(A).

In what follows,we will mainly discuss the systems of two submodules,and thus pairs of projections.

3 Systems of two Submodules and the Group Φ0(A)

In this section,we define the Φ-group for systems of two submodules,and thus pairs of projections onAn,n≥1,for a given involutive algebraA.We begin the procedure from the unital case.

Definition 3.1Given a unital involutive algebra A,let X be the set of all the pairs of projections(p1,p2)and∼be the smallest equivalence relation on X,such that

(1)(p1,p2)∼(p1⊕0m,p2⊕0n)for any m,n in;

(2)(p1,p2)∼u(p1,p2)u∗for any unitary u;

(3)(p1,p2)∼(q1,q2)if there exists a pair(r1,r2)in X and a unitary u such that

Remark 3.1By(1)of this definition,we can assume that any pair of projections has the same size.The check of(2)is similar to the check of(3),and the former is much easier.Hence,from now on,we assume that any pair of projections has the same size,and we use(3)to check if two pairs of projections are equivalent.

Let[(p1,p2)]denote the equivalence class of(p1,p2),and we denote the set of the class by

We define an addition on Ω2by

The addition is well-defined as follows.

We firstly fix(q1,q2).If,by Remark 3.1,we only check(3)in De finition 3.1.Thus there exist a pair(v1,v2)∈Xand a unitaryusuch that

Then

Using the procedure of changing orders,we have two unitaryxandysuch that

Hence

and thus

Next,we fixand supposing thatwith the same procedure,we can prove that

Next we give an important proposition as follows.

Proposition 3.1Ω2is an Abelian semi-group with cancellation.

ProofSince the propositions of association and commutation are obvious,we only need to check the cancellation.If

then

By Remark 3.1,there exists a pair(s1,s2)and a unitaryusuch that

Hence[(p1,p2)]=[(q1,q2)].

In the end,we note that this simi-group has zero element.Since(p1,p2)∼(p1⊕0n,p2⊕0n),we can see directly that

for any system(p1,p2)and any natural numbern,and therefore[(0n,0n)]is the zero element for any natural numbern.

Then it is convenient to give our main definition.

Definition 3.2Φ(A):=the Grothendieck ofΩ2.

Remark 3.2We can only consider the commutative systems of two submodules,and using the same procedure we get the Φ-group,denoted by Φc(A)whichis a subgroup of Φ(A).In Section 5,we will compute some examples of Φc(A)for differentA.

The functor Φ for unital involutive algebrasLetAandBbe unital involutive algebras,and letφ:A→Bbe a∗-homomorphism.Associate toφa group homomorphism Φ(φ):Φ(A)→Φ(B)as follows.φextends to a∗-homomorphismφ:Mn(A)→Mn(B)for eachn.A unital∗-homomorphism maps projections to projections and unitaries to unitaries.Then we can define Φ(φ):Φ(A)→Φ(B)by Φ(φ)[(p1,p2)]=[(φ(p1),φ(p2))].It is easy to check that it is a group homomorphism,and therefore we get the following proposition.

Proposition 3.2(Functoriality of Φ for Unital Involutive Algebras)

(i)For each unital involutive algebra A,Φ(idA)=idΦ(A).

(ii)If A,B and C are unital involutive algebras,and if φ:A→B and ψ:B→C are∗-homomorphisms,then

The non-unital caseIfAis an involutive algebra,unital or non-unital,being its unitalization,then

is a short exact sequence,and we define

Remark 3.3In Section 4 of this paper,we will compute Φ()and we will see that it is not trivial.

Proposition 3.3Φ0is a covariant functor from the category of involutive algebras to the category of Abelian groups.

ProofThe proofis similar to the case of the usualK-theory.Let:A→Bbe a homomorphism between involutive algebrasAandB,and define:→by

Then there is a commutative diagram

We get the diagram

Thenand by the commutativity of the above diagram,we see thatsince Φ is a covariant functor from the category of involutive algebras to the category of Abelian groups,so is Φ0.

4 Propositions of the Group Φ0(A)

The standard picture of the group Φ0(A)IfAis an involutive algebra,unital or non-unital,eAbeing its unitalization,then

is a short exact sequence.Then

[π(p1,p2)]=[π(q1,q2)]implies that there exists a pair of projections(x,y)and a unitaryusuch that

Sinceu,xandyare scalar matrices,we have

As[(p1⊕x,p2⊕y)]−[(q1⊕x,q2⊕y)]=[(p1,p2)]−[(q1,q2)],we replacep1withp1⊕x,p2withp2⊕y,q1withq1⊕xandq2withq2⊕y,then we have

As[u∗(p1,p2)u]=[(p1,p2)],we replacep1withu∗p1uandp2withu∗p2u,and then we have

Conversely,given[(p1,p2)]−[(q1,q2)]∈Φ(eA)such thatπ(p1,p2)=π(q1,q2),and then obviously we getπ∗([(p1,p2)]−[(q1,q2)])=0,and hence

Then we get the standard picture of Φ0(A):

By the standard picture of Φ0(A),we can demonstrate thatfor the unital involutive algebraA.This is an important fact since it ensures that we can denote by Φ0(A)whetherAis unital or not.

Proposition 4.1(Direct Sums)For every pair of involutive algebras A and B,we have

ProofLetιA:A→A⊕BandιB:B→A⊕Bbe the canonical inclusion maps,and they induce a homomorphism

which maps(g,h)in Φ0(A)⊕Φ0(B)to Φ0(ιA)(g)+ Φ0(ιB)(h).We have the commutative diagram

By 5-lemma,we only have to show the case whenAandBare both unital.This is obvious,since every element in the matrixMn(A⊕B)is of the form(a,b)fora∈Aandb∈B,and the product and addition of matrices happen on each component independently.

In theK-theory,the Morita invarianceis a well-known result.The key point of the proofis to show the unital case,and thuswhenAis a unital involutive algebra,and the general case is got by 5-lemma(see[4]).Unfortunately in the case of Φ0-group,since the functor Φ0does not preserve the split exact sequence,we can not use 5-lemma.But we can still show the unital case in a direct way.

Proposition 4.2(Morita Invariance)Let A be a unital involutive algebra and let n be a natural number.ThenΦ(A)is isomorphic toΦ(Mn(A)).

ProofWe will show that the∗-homomorphism

induces an isomorphismwithwherep1,p2are the sizes ofmand(λA)mis the∗-homomorphisminduced byλA.

We should check that this definition is well given.If[(q1,q2)]=[(p1,p2)],then by Remark 3.2,there exists a pair(r1,r2)and a unitaryusuch thatWithout lost of generalization,we assume thatpi,qi,ri(i=1,2)are all of sizem.

Let{e1,e2,···,e2mn}be the standard basis for2mn,and letvbe a permutation unitary inM2mn(C)that fulfills

Then

and

Since

then

Therefore,

For each natural numberk,letγk:Mk(Mn(A))→Mkn(A)be the∗-isomorphism given by viewing each element ofMk(Mn(A))as one big matrix inMkn(A).Defineβ:Φ(Mn(A))→Φ(A)byβ[(p1,p2)]=[γk(p1,p2)]forp1,p2∈Mk(Mn(A)).

We should show that this definition is well given.In fact,given a pair(q1,q2)such that(q1,q2)∼(p1,p2),by Remark 3.1,there exists a pair(x,y)and a unitaryusuch thatu∗(p1⊕x,p2⊕y)u=(q1⊕x,q2⊕y).Suppose thatpi,qiare inMk(Mn(A))andx,yare inMl(Mn(A)),souis inMk+l(Mn(A)).Then

Sinceγk+luis also a unitary element,we have

and thus

We claim thatβis the inverse toα.To prove this claim it suffices to show that

where(λA)mis the∗-homomorphisminduced byλA.We prove the second claim,and the proof of the first claim is similar.

Let{e1,e2,···,ekn}be the standard basis forkn,and letube a permutation unitary inMkn()that fulfills

Then

Therefore

The direct system and the direct limit

Recall that the direct limit(A,φi)of the direct system of involutive algebras

is characterized by

Theorem 4.1 Suppose that(A,φi)is the direct limit of the direct system of involutivealgebrasand thenis a direct systemof Abelian groups with a direct limit

ProofWe have a diagram

Since the direct limit preserves exactness,by 5-lemma,we may assume that allAiandAare unital and thatφiandφjipreserve units.It suffices to show thatWe prove it by two steps.

(1)

For any projectionsp,q∈Mn(A),there arei,j∈Jandpi∈Mn(Ai),qj∈Mn(Aj)such thatThusfor somek≥i,and thenφkiis a projection inMn(Ak).Similarly,there is somel≥jsuch thatφlj(qj)is a projection inMn(Al).Lett≥k,l.Thenφti(pi)andφtj(qj)are projections inMn(At)such that

(2)

LettingThen by Remark 3.1 there are some unitaryuinMn(A)and some projectionsx,y∈Mn(A)such that

There exists ajsuch thatu=φj(uj),x=φj(xj),y=φj(yj),whereujis a unitary,and thus

Letk≥i,j,and then

Enlargekif necessary,and we can get

Therefore,

and thus

The relationship between Φ-groups and K-groupsWe firstly study the unital case.Suppose thatAis a unital involutive algebra,andXis the corresponding set in Definition 3.1.If we only consider the subset ofXconsisting of pairs with the form(p,0),we get a direct summand of Φ(A)and thus{[(p,0)]−[(q,0)]:(p,0),(q,0)∈X},which obviously is isomorphic to the typicalK(A).Similarly,we have another direct summand{[(0,p)]−[(0,q)]:(0,p),(0,q)∈X}whichis also isomorphic toK(A).Hence,we get that

and Φ(A)/(K(A)⊕K(A))is an Abelian group.

For the general involutive algebraA,unital or not unital,by the standard picture of Φ0(A)andK0(A),we also have that

5 The Computation of Φ-Groups

The computation of Φ-groups is in fact the description of the structure theory of pairs up to unitary equivalence,and it is a subproblem to study the pairs of self-adjoint operators.Even for a pair of projections acting on Hilbert spaces,the problem of describing,up to unitary equivalence,irreducible(undecomposible)pairs without any relation is very difficult.For example,letp,qbe projections acting on2.To simplify the problem,we fixpasandqas any projection on.By the equationswe see thatqhas the forms oforwhere 0≤ λ ≤1,0≤θ<2π.Consider the unitary matrixuwhichis commutative withp,and thusuis of the formwhere 0≤α,β<2π.Acting on(p,q)by this kind ofu,we get

where 0≤λ≤1,0≤θ,α,β<2πandqis of the nontrivial kind.Then for differentλ1andλ2in[0,1],

and

can not be unitarily isomorphic.Therefore,the unitarily isomorphism classes for the kind of pairs(p,q),where

is of the cardℵ.So we only consider the pairs of projections that satisfy an algebraic relation.

Next,we give the general theory of self-adjiont operators by[5].

LetHbe a separable complex(finite or infinite-dimensional)Hilbert space.We consider the pairsAandBof self-adjiont operators which are solutions of the equation

whereα,β1,β2,γ,δ,ε,χ∈C.Suppose that

So we can write this equation as

whereα,β,η,γ,δ,ε,χ∈R,[A,B]=AB−BAis the commutator,and{A,B}=AB+BAis the anticommutator.We also have that

By using an affine change of variables,(5.1)can be divided into four groups:

(a)Wild relations:0=0 orA2=I.

(b)Binormal relations:A2+B2=IorA2=B2orA2−B2=I.

(c)Lie algebras and their non-linear transformations:[A,B]=0 oror

(d)Quantum relations:

In what follows,we study each of these groups of the relations for projectionsAandB.

(a)Wild relations.The relation 0=0 means thatAandBdo not satisfy any relation.For projectionA,A2=ImeansA=I,and thereforeAB=BA.

(b)Binormal relations.For projectionsAandB,the relationA2+B2=Iimplies thatA=I−B,and henceAB=BA.For the relationA2=B2,A=Bfor projectionsAandB,and thereforeAB=BA.The third relationA2−B2=Iholds only whenA=IandB=0,which also implies thatAB=BA.

(c)Lie algebras and their non-linear transformation.In this group of relations,the first six relations are partial cases of the relation

whereP2(A)is a real quadratic polynomial.Even for bounded self-adjoint pairs,(5.2)implies that[A,B]=0 by Proposition 1.19 in[5].For the last relationwe can transform it into[A,(A2+B)]=i(A2+B),and then it is converted to the form in(5.2),so we get[A,(A2+B2)]=0,and thus[A,B]=0.

(d)Quantum relations.By Proposition 1.13 in[5],the pair of bounded self-adjoint operatorsA,Bsatisfiesand thenA=B=0,soAB=BA.For the second relationwe can transform it into the relation[A,A+B]=iq(A+B)+iIfor a pair of projectionsAandB,and then it becomes the form of(5.2)in case(c),so we have[A,B]=0.For the next two relations,[A,B]=0 also holds for bounded self-adjoint operatorsAandB.

In summary,given a pair of projections which satisfy the relation(5.1),they either have no relation,or satisfy

We consider the case where the pairs of projections have commutative relations.Thus we consider the commutative systems and give some examples of the computations of Φc(A).It is not hard to see that Φc(A),as a subgroup of Φ(A),inherits all the propositions of Φ(A).

Example 5.1Φc(A)forA=,Mn(),C(S1).

For commutative projectionsp1,p2inMn(),they can be simultaneously diagonalized,and thus there is a unitaryuinMn()such thatandare diagonal matrices and the elements in the diagonal are 0 or 1 since bothandare projections.Then we get a couple(diag(i1,i2,···,in),diag(j1,j2,···,jn))withisandjtbeing 0 or 1 for 1≤s,t≤n.Therefore,each couple of(is,js)is an element of the set{(1,0),(0,1),(1,1)}.Although the relative position of the couples(is,js)may change for different unitary matrices,the number of times each element in the set{(1,0),(0,1),(1,1)}appears will be unchanged,and thus if(1,0),(0,1)and(1,1)appearn1,n2,n3times respectively in the set{(i1,j1),(i2,j2),···(in,jn)},thenn1,n2,n3are constant for different unitary transformations.

In fact,if there are two unitary matricesu1andu2such thatcorrespond to(n1,n2,n3)and(m1,m2,m3)respectively,and that(n1,n2,n3)≠(m1,m2,m3),thenandwill have different ranks,soandhave different ranks,whichis impossible since unitary transformation dose not change a matix’s rank.Then we can give a definition of the rank for commutative pairs of projections,whichis a generalization of the rank of one matrix.

Definition 5.1(Rank for Commutative Pairs of Projections)Given two projections p,q∈Mn(),suppose that u(p,q)u∗is the simultaneously diagonalized form.Let r1,r2,r3be the times(1,0),(0,1),(1,1)appearing in the diagonalized form respectively,and then we call the triple(r1,r2,r3)the rank of the pair(p,q).

In what follows,we claim that the rank(r1,r2,r3)is invariant under the equivalent relation∼.

Proposition 5.1Suppose that(r1,r2,r3)is the rank of a pair of projections p1,p2∈Mn(),and then(r1,r2,r3)is invariant under the equivalence relation∼.

ProofWe show the case(3)in Definition 3.1.

Give another pair of projectionsq1,q2∈Mn(C)for which there exists a unitaryuand a pair of projections(x,y)such that

Hence the pair(p1⊕x,p2⊕y)and the pair(q1⊕x,q2⊕y)have the same rank(k1,k2,k3).Suppose that(p1,p2),(q1,q2),(x,y)can be diagonalized by unitary matricesu1,u2,u3respectively,and then(p1⊕x,p2⊕y)can be diagonalized by the unitary(u1⊕u3)and(q1⊕x,q2⊕y)can be diagonalized by the unitary(u2⊕u3).If the rank of(q1,q2)is(s1,s2,s3)and the rank of(x,y)is(l1,l2,l3),then we have

Since the addition is the canonical one for vectors,we have that

By Proposition 5.1,we have

and hence

By Proposition 4.2 for the commutative case,we have

Since two commutative matrices inMn(C(S1))can also be diagonalized to the constant matrices simultaneously,we have

Remark 5.1By the same procedure,we can also construct the Φ-theory fornsubmodules,denoted by Φn(A),and we can also compute the Φ-group for the commutative systems for Hilbert subspaces,which in general,is

The relationship between the Φ-group and the problem of classification of systems of n-subspaces

In the computation of Φ-groups,we see that it is the process to describe the unitary equivalence class for multi-operator,and we should find the irreducible form of the operators as a base for the Φ-group.WhenA=,it is the problem of classification of systems ofn-subspaces in a finite dimensional Hilbert space which we have introduced in the second section of this paper.In fact Φ()describes the stable unitary equivalence class for the systems ofnsubspaces in a finite dimensional Hilbert space,but in the case offinite dimensions,the stable unitary equivalence class is almost the unitary equivalence class.

Whenn=1,there is only one direct summand that corresponds to the nontrivial indecomposable systems of one subspaces up to unitary equivalence,namely

Whenn=2,there are three direct summands that correspond to the non-trivial indecomposable commutative systems of two subspaces up to unitary equivalence,namely

Whenthere are seven direct summands that correspond to the non-trivial indecomposable commutative systems of three subspaces up to unitary equivalence,namely

AcknowledgementsThe authors would like to thank all the members of the functional analysis seminar for the inspiring discussions,and they would also like to sincerely thank the referees for the suggestions on improving the paper.

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