Schatten's theorems on functionally defined Schur algebras

SCHATTEN’S THEOREMS ON FUNCTIONALLY

DEFINED SCHUR ALGEBRAS

PACHARA CHAISURIYA AND SING-CHEONG ONG

Received 25 January 2005 and in revised form 20 July 2005

For each triple of positive numbersp,q,r≥1 and each commutativeC∗-algebraᏮwith identity 1 and the sets(Ꮾ) of states onᏮ, the set᏿r(Ꮾ) of all matricesA=[ajk] over Ꮾsuch thatϕ[A[r]] :=[ϕ(|ajk|r

)] defines a bounded operator fromp toq for allϕ∈ s(Ꮾ) is shown to be a Banach algebra under the Schur product operation, and the norm ||A|| = |||A|||p,q,r=sup{||ϕ[A[r]]||1/r:ϕ∈s(Ꮾ)}.Schatten’s theorems about the dual of the compact operators, the trace-class operators, and the decomposition of the dual of the algebra of all bounded operators on a Hilbert space are extended to the᏿r(Ꮾ) setting.

1. Introduction

Fix p andq with 1≤p,q <∞. The space of pth power summable sequences of com-plex numbers is denoted byp, and the space of matrices which define bounded linear transformations fromp toq is denoted byᏮ(p,q). LetA=[ajk], B=[bjk] be infi-nite matrices, not necessarily inᏮ(p,q). TheSchur productA•BofAandBis defined byA•B=[ajkbjk]. Many areas in mathematics such as matrix theory, function theory, operator theory, and operator algebras have made use of results from the study of Schur product and have injected new problems in return. See [1,4,6] for further references to the related literature.

As a generalization of a result of Schur in [9] forp=2, andq=2, Bennett proved in [1, Theorem 2.2] the following theorem.

Theorem 1.1 (Bennett). If Λ=[λjk],Σ=[σjk]∈Ꮾ(

p

,q), then Λ•Σ=[λjkσjk]∈ Ꮾ(p,q), and

||Λ•Σ||p,q≤ ||Λ||p,q||Σ||p,q. (1.1)

(i.e.,Ꮾ(p,q)is a commutative Banach algebra under the Schur product operation and operator norm|| · ||p,q.)

Based on these results of Schur and Bennett, in [2] we studied algebras, under the Schur product operation, of matrices over a Banach algebra, which have the matrix of

Copyright©2005 Hindawi Publishing Corporation

the norms of the entries define bounded operators. Here we give a functional version of the generalization. This is another direction of generalization of the numerical Schur r-algebras,᏿r discussed therein.

A matrixA=[ajk], overC, is in theabsolute Schurr-algebra᏿r if theSchurrth power A[r]:=[|ajk|r

] ofAis inᏮ(p,q). Ther-normofA is defined by||A||:= |||A|||p,q,r:= ||A[r]||1/r

, where||A[r]|| = ||A[r]||p,qis the operator norm ofA

[r]

as an element ofᏮ(p,q). That||| · |||p,q,ris a norm follows, as expected, from an inequality that is analogous to the H¨older inequality.

Theorem1.2. For eachr≥1,᏿r is a Banach algebra under the Schur multiplication and the norm||| · |||p,q,r.

This is a special case of a result in [2] that will be used here. The matrices could be over a Banach algebra, and the norm is defined by the nonnegative matrix of the rth power of norms of the entries. Here we investigate the situation where the matrices are over a commutativeC∗-algebraᏮ, the norm is defined by the absolute values of linear functional values of the entries, as the linear functional runs over the set of states onᏮ.

In [7] Schatten’s theorems [8] concerning the dual of the compact operators, the trace-class operators, and the decomposition of the dual of the algebra of bounded operators on a Hilbert space have been extended to the setting of matrices of functions acting on func-tion sequence space analogous to the2sequence space. Here we extend these Schatten-type theorems to the algebra᏿r(Ꮾ) consisting of certain classes of matrices over a com-mutativeC∗-algebraᏮ, in a situation wherepandqneed not be equal to 2.

Since thepandqwill be fixed throughout our discussion, we will sometimes suppress the subscripts p,qin|| · ||p,q and write|| · ||instead, if no confusion can arise. We will occasionally use subscripts if an emphasis for clarity is warranted. We also use|| · ||to denote the norm on a Banach space, and let the context determine which one is intended.

For convenience of reference, we also record the following simple useful fact.

Lemma1.3. Let[αjk]and[βjk]be matrices over the complex fieldCsuch that|αjk| ≤βjk for all jandk. Suppose that[βjk]∈Ꮾ(p,q); that is, the matrix defines a bounded linear transformation fromptoq. Then[αjk]∈Ꮾ(p,q)and||[αjk]|| ≤ ||[βjk]||.

2. Definitions and preliminary results

We establish some of our results in a more general setting before moving on to the settings on which Schur product makes sense. Letᐄbe a Banach space with norm|| · ||, and dual spaceᐄ#. Consider the setᏹ(ᐄ) of all infinite matrices overᐄ. Let (ᐄ#)1denote the set

of all bounded linear functionals onᐄwhich have norm 1. Lets(ᐄ)⊆(ᐄ#)1be a set of

linear functionals onᐄsuch that

(i)||x|| =sup{|f(x)|:f ∈s(ᐄ)}for everyx∈ᐄ; (ii) the linear span ofs(ᐄ) is all ofX#.

One such example is s(ᐄ)=(ᐄ#)1 by the Hahn-Banach theorem and the fact that all

linear functionals inᐄ#are multiples of elements in (ᐄ#)1.

For each f ∈ᐄ#

the matrix f[A] as a linear transformation ofptoq, if it is defined. (The closed graph theorem implies that it is bounded if it is everywhere defined.) Let᏿(ᐄ)=᏿(ᐄ)_p,q be the set of matricesA=[ajk]∈ᏹ(ᐄ) such that f[A]∈Ꮾ(p,q) for all f ∈s(ᐄ). The following result provides us with a natural way of defining a norm on᏿(ᐄ).

Theorem2.1. Letᐄands(ᐄ)be as above, andA=[ajk]∈᏿(ᐄ), so thatf[A]∈Ꮾ(p,q) for all f ∈s(ᐄ). Then

supf[A]:f ∈s(ᐄ)<∞. (2.1)

Proof. First we note that since each f ∈ᐄ#

is a linear combination f =n

j=1αjgj of elements g1,. . .,gn in s(ᐄ), and for the given A∈᏿(ᐄ), gj[A]∈Ꮾ(p,q), 1≤j≤n, and hence f[A]=α1(g1[A]) +···+αn(gn[A])∈Ꮾ(

p

,q). Therefore, the map᐀A: f → f[A] is a linear transformation fromᐄ# to Ꮾ(p,q). Since both the domain ᐄ# and codomainᏮ(p,q) are Banach spaces, the continuity of the map᐀A will follow if we can show that the graph of᐀A is closed. To that end, suppose that{(fn,᐀A(fn))}is a sequence in the graphᏳ(᐀A)ᐄ#⊕∞Ꮾ(p,q) of᐀Athat converges to some (f,M)∈ ᐄ#

⊕∞Ꮾ(p,q) withM=[mjk]. Then fn→ f inᐄ

#

and fn[A]→MinᏮ(p,q). Thus, for each j,k=1, 2,. . .,|fn(ajk)−mjk| ≤ ||fn[A]−M|| →0 asn→ ∞. Also the conver-gence of{fn}to f implies that|fn(ajk)−f(ajk)| →0 asn→ ∞. Therefore, f(ajk)=mjk for all j,k, and hence f[A]=M. Thus the graph of ᐀Ais closed. By the closed graph theorem,᐀Ais bounded. Therefore,

supf[A]:f ∈s(ᐄ)=sup{᐀A(f):f ∈s(ᐄ)} ≤ ||᐀A||<∞. (2.2)

Next we prove that᏿(ᐄ) is a Banach space.

Theorem2.2. The set᏿(ᐄ)is a Banach space under the usual (entrywise) scalar multipli-cation and addition, and the norm

||A||:=supf[A]:f ∈s(ᐄ), A∈᏿(ᐄ). (2.3)

Proof. First we show that the function as defined inTheorem 2.1is indeed a norm on ᏿(ᐄ). LetA,B∈᏿(ᐄ) and letf ∈s(ᐄ). Then||f[A+B]|| = ||f[A] +f[B]|| ≤ ||f[A]||+ ||f[B]|| ≤ ||A||+||B||. Therefore,||f[A+B]|| ≤ ||A||+||B||for all f ∈s(ᐄ), and hence ||A+B|| ≤ ||A||+||B||.

To prove that᏿(ᐄ) is complete, let{An}be a Cauchy sequence in᏿(ᐄ). Then for each f ∈s(ᐄ),

_f

An

−fAm=f

An−Am≤An−Am−→0 asn,m−→ ∞. (2.4)

Thus{f[An]}is a Cauchy sequence inᏮ(p,q). SinceᏮ(p,q) is complete, there exists a bounded matrixΛf ∈Ꮾ(p,q) to which{f[An]}converges inᏮ(p,q). For each j,k= 1, 2,. . .,

asn,m→ ∞. Since this is true for every f ∈s(ᐄ), the quantity||An−Am||does not de-pend on f, and||x|| =sup{|ψ(x)|:ψ∈s(ᐄ)}for allx∈ᐄ, we have

a(n)_jk−a(m)_jk= sup f∈s(ᐄ)

fa(n)_jk−a(m)_jk = sup f∈s(ᐄ)

fa(n)_jk −fa(m)_jk

≤An−Am−→0 asn,m−→0.

(2.6)

Thus the sequence{a(n)_jk}is a Cauchy sequence inᐄ. By the completeness ofᐄ, there exists anajk∈ᐄsuch thata

(n)

jk→ajkinᐄ. Thus there is anA=[ajk]∈ᏹ(ᐄ) such that {An}converges toAentrywise. It is clear from this construction that for each f ∈s(ᐄ),

f[A]=Λf. Since eachΛf∈Ꮾ(

p

,q), we haveA∈᏿(ᐄ).

We have to show that||An−A|| →0 as n→ ∞. Let >0 be given. Since{An} is a Cauchy sequence, there exists anNsuch that||An−Am||</3, for alln,m≥N. Letn≥N be arbitrarily fixed. We will show that||An−A||<. There exists an f ∈s(ᐄ) such that

_An−A<f

An−A+₃=f

An

−fA+ 3=f

An

−Λf+₃. (2.7)

Since f[An]→Λf in the norm ofᏮ(p,q), there exists aν> Nsuch that||f[Aν]−Λf||< /3. Thus

_An−A<f

An

−Λf+ 3 ≤f

An

−fAν+fAν−Λf+ 3 <An−Aν+₃+₃ <.

(2.8)

Therefore,᏿(ᐄ) is complete.

For a given matrixA=[ajk], denote byAn the matrix whose (j,k) entry isajk for

1≤j,k≤nand 0 otherwise.

Proposition2.3. LetA=[ajk]∈᏿(ᐄ). Then||An|| ||A||asn→ ∞.

Proof. Letξ= {ξk}∞_k=1∈

p

;f ∈s(ᐄ); andn∈N. Denoteξ(n)= {ξ1,ξ2,. . .,ξn, 0,. . .}.Then

fAn

ξ=

_n

j=1

n

k=1

fajk

ξk

q1/q

≤

_n+1

j=1

n

k=1

fajk

ξk

q1/q

=fA(n+1)

ξ(n)≤fA(n+1)p,qξ

(n)

p≤A(n+1)||ξ||.

(2.9)

So

_f

An≤A(n+1)n. (2.10)

Since this is true for every f ∈s(ᐄ),

_A

n

Let>0. There is an f ∈s(ᐄ) such that||f[A]|| ≥ ||A|| −/4. Since f[A] is inᏮ(p, q), there is a unit vectorξ= {ξk}∞_k=1∈

p

such that

_f[A]ξ

q= _∞ j=1 ∞ k=1

fajk

ξk

q1/q

≥f[A]−

4. (2.12)

Thus there exists ann0such that

n0 j=1 ∞ k=1

fajk

ξk

q1/q

≥f[A]− 4−

12=f[A]−

3. (2.13)

For the finite sum, there is ann1such thatn1≥n0and that

fA(n_{1 )}

ξ=

n1 j=1 n1 k=1

fajk

ξk

q1/q

≥ n0 j=1 n1 k=1

fajk

ξk

q1/q

≥f[A]− 3−

6 =f[A]− 2.

(2.14)

So, forn≥n1,

_A

n

≥A(n1 )

_≥f

A(n1 )

_≥f

A(n1 )

ξ≥f[A]− 2 ≥ ||A|| −

4−

2>||A|| −.

(2.15)

Therefore,||An|| ||A||, as asserted.

With ᐄand s(ᐄ) as above and for 1≤r <∞, let ᏿r

(ᐄ) denote the set of all ma-tricesA=[ajk]∈ᏹ(ᐄ) with the property that f[A][r]=[|f(ajk)|r

]∈Ꮾ(p,q) for all f ∈s(ᐄ).

Theorem2.4. For eachA∈᏿r

(ᐄ), it holds that

supf[A][r]:f ∈s(ᐄ)<∞. (2.16)

Proof. ByTheorem 1.2, the set of all matricesΛ=[λjk] overCwith bounded absolute Schurrth power (Λ[r]=[|λ|r

]∈Ꮾ(p,q)) is a Banach algebra. SinceX#is the linear span ofs(ᐄ), for the fixedA=[ajk]∈᏿

r

(ᐄ), the map᐀A:f →f[A] is a linear map fromᐄ

#

into᏿r. To show that᐀A is a closed map, let{(fn,fn[A])}be a sequence in the graph Ᏻ(᐀A)⊆ᐄ#

⊕∞᏿r

of᐀Aconverging to some (f,Λ)∈ᐄ

#

⊕∞᏿r

. Then fn→f inᐄ

#

and fn[A]→Λin᏿r. LetΛ=[λjk]. We then have, for each (j,k)∈N×N,

fn

ajk

−→fajk

, fn

ajk

−→λjk asn−→ ∞. (2.17)

Therefore, f[A]=Λand᐀Ahas a closed graph. Sinceᐄ

#

and᏿rare Banach spaces,᐀A is bounded by the closed graph theorem. Thus

supf[A]_᏿r :f ∈s(ᐄ)

Since||f[A]||᏿r = |||f[A]|||_p,q,r= ||f[A][r]||1/r, we have

supf[A][r]:f ∈s(ᐄ)<∞. (2.19)

For eachA∈᏿r

(ᐄ), define||A||or|||A|||p,q,rby

||A||:= |||A|||p,q,r:=supf[A]p,q,r:f ∈s(ᐄ)

=supf[A][r]

1/r

:f ∈s(ᐄ). (2.20)

The preceding theorem guarantees that || · ||is a function defined on ᏿r(ᐄ). We will prove that it is a norm inTheorem 2.6. Using an argument similar to that inProposition 2.3, we can show that this function has the same monotone property.

Proposition2.5. LetA=[ajk]∈᏿r

(ᐄ). Then||An|| ||A||asn→ ∞.

Proof. This is just a routine adaptation of the proof ofProposition 2.3, therefore omitted.

Theorem2.6. Forr≥1, the function|| · ||defined in (2.20) is a norm on the space᏿r(ᐄ), and᏿r(ᐄ)is a Banach space under this norm and the usual addition and scalar multiplica-tion.

Proof. To see that|| · ||is indeed a norm, letA,B∈᏿r

(ᐄ) andf ∈s(ᐄ). Then byTheorem 1.2,

f[A+B][r]=f[A] +f[B][r]=f[A] +f[B]r_p,q,r ≤f[A]_p,q,r+f[B]p,q,r

r

(by the triangle inequality for the norm on᏿r)

≤||A||+||B||r.

(2.21)

Since f ∈s(ᐄ) is arbitrary, we have

||A+B|| ≤ ||A||+||B||, (2.22)

as asserted. Let{An=[a

(n)

jk]}∞n=1be a Cauchy sequence in᏿

r

(ᐄ). Then for each (j,k)∈N×N,

a(n)_jk−a(m)_jk=supfa(n)_jk−a(m)_jk : f ∈s(ᐄ) ≤supfAn−Amp,q,r:f ∈s(ᐄ)

=An−Am−→0 asn,m−→ ∞.

(2.23)

Thus each{a(n)_jk}∞n=1 is a Cauchy sequence inᐄ. By the completeness of ᐄ, there is an

ajk∈ᐄto which{a

(n)

jk}

∞

n=1converges. LetA=[ajk]. For each f ∈s(ᐄ),{f[An]}

∞

n=1⊆᏿

is a Cauchy sequence. By the completeness of᏿r, there is aΛf ∈᏿

r

such that||f[An]− Λf|| →0 in᏿r. Since f(a

(n)

jk)→f(ajk) for all j,k, asn→ ∞, f[A]=Λf ∈᏿r. As this is true for all f ∈s(ᐄ), we haveA∈᏿r

(ᐄ). To see that||An−A|| →0, we first note that for eachν∈N, we have, by the finiteness ofνandLemma 1.3,

_An

ν

−Aν≤

a(n)_jk−ajk

r

1≤j,k≤ν 1/r

p,q−→0 asn−→ ∞. (2.24) Let>0. There is anN∈Nsuch that

_{An−An+l< ∀n≥N,∀l≥1. (2.25)}

Then, byProposition 2.5,

An_ν −An+l_ν=An−An+l_ν ≤An−An+l< ∀n≥N,∀l≥1. (2.26)

Taking limit asl→ ∞and using (2.24), we have

An_ν

−Aν

≤ ∀n≥N. (2.27)

Sinceν∈Nis arbitrary, we have, byProposition 2.5,

_{An−A≤ ∀n≥N; (2.28)}

thus||An−A|| →0 asn→ ∞.

3. Schur algebras over commutativeC∗-algebras

In this section, we fix a commutativeC∗-algebraᏮwith identity 1 and the sets(Ꮾ) of states onᏮ, that is, the set of positive linear functionals of norm 1 onᏮ. By the Gelfand-Naimark theorem, Ꮾ is isometrically∗-isomorphic to the function algebra C(X) for some compact HausdorffspaceX. We will treatᏮasC(X). Sinces(Ꮾ) contains all eval-uation linear functionalsϕx(a)=a(x) forx∈X, anda∈Ꮾ, we have||a|| =sup{|ϕ(a)|: ϕ∈s(Ꮾ)}, for alla∈Ꮾ.

As in the scalar case, define theSchur product (also known asHadamard product, or entrywise product) of two matricesA=[ajk],B=[bjk] inᏹ(Ꮾ) (the set of all matrices overᏮ) byA•B=[ajkbjk]. The productajkbjk is the algebra product defined in Ꮾ. Schur product of this form with entries in the algebra of bounded linear operators on a Hilbert space was first studied in [6]. For eachϕ∈s(Ꮾ), and for eachA=[ajk]∈ᏹ(Ꮾ), ϕ[A]=[ϕ(ajk)] is the scalar matrix obtained by applyingϕto each entry ofA. Let᏿(Ꮾ) denote the set of all matricesA∈ᏹ(Ꮾ) with the property thatϕ[A] defines a bounded linear transformation fromp toq for everyϕ∈s(Ꮾ). Denote byᏮ#the Banach space dual ofᏮ(the space of bounded linear functionals onᏮ). Since each f ∈Ꮾ#

is a linear combination of at most four states (see [5, Corollary 4.3.7, page 260]), it then follows fromTheorem 2.1that

(where the norm||ϕ[A]||is the operator norm ofϕ[A] as an element inᏮ(p,q)). Define the norm ofA∈᏿(Ꮾ) by||A|| =sup{||ϕ[A]||:ϕ∈s(Ꮾ)}.

The following proposition follows immediately fromTheorem 2.2.

Proposition3.1. For a commutativeC∗-algebraᏮwith state spaces(Ꮾ),᏿(Ꮾ)is a Ba-nach space under the norm|| · ||and the usual addition and scalar multiplication.

We do not know, however, whether᏿(Ꮾ) is a Banach algebra under the Schur multi-plication operation.

We now turn our attention to functional analog of Schurr-algebras. As before, we fix a commutativeC∗-algebraᏮwith the sets(Ꮾ) of states. For each real numberr≥1, A=[ajk]∈ᏹ(Ꮾ), denote byA[r]=[|ajk|r

], where|x| =√x∗_{xforx∈Ꮾ. Forϕ∈s(Ꮾ),}

denote byϕ[A[r]] :=[ϕ(|ajk|r

)]. Let᏿r(Ꮾ) denote the set of all matricesA∈ᏹ(Ꮾ) such that ϕ[A[r]]∈Ꮾ(p,q), that is, the numerical matrixϕ[A[r]] defines a bounded linear transformation fromptoqfor everyϕ∈s(Ꮾ).

We show that for eachA∈᏿r

(Ꮾ),

||A||:= |||A|||p,q,r:=sup

ϕA[r]1/r:ϕ∈s(Ꮾ)

<∞, (3.2)

and that this is indeed a norm on᏿r(Ꮾ). Furthermore, we show that᏿r(Ꮾ) is in fact a Banach algebra under the Schur multiplication and this norm. As a suitable adaptation of the argument used in the proof ofProposition 2.3, we have the following.

Proposition3.2. LetA∈᏿r

(Ꮾ). Then||An|| ||A||asn→ ∞.

We also need this simple observation.

Lemma3.3 (Minkowski’s inequality for linear functionals). Letϕ∈s(Ꮾ). Then

ϕ|a+b|r1/r

≤ϕ|a|r1/r

+ϕ|b|r1/r

∀a,b∈Ꮾ. (3.3)

Proof. SinceᏮ=C(X) for some compact HausdorffspaceX, the givenϕ∈s(Ꮾ) has an integral representationϕ(a)=Xa dµϕ for alla∈Ꮾand some measureµϕ on X. The Minkowski inequality forµϕis exactly the asserted inequality. Theorem3.4. The function|| · ||defined in (3.2) above is a norm on᏿r(Ꮾ); and᏿r(Ꮾ)is a Banach algebra under the Schur product operation and this norm.

Proof. LetA=[ajk]∈᏿r(Ꮾ). Since each f ∈Ꮾ#

is a linear combination of at most four states, the map f → f[A[r]] is a linear transformation fromᏮ# toᏮ(p,q). Using argu-ments similar to that used in the proof ofTheorem 2.1, we have

supϕA[r]:ϕ∈s(Ꮾ)<∞. (3.4)

To see that (3.2) indeed defines a norm, let A=[ajk], B=[bjk]∈᏿r(Ꮾ). Then ϕ[A[r]][1/r]=[(ϕ(|ajk|r

))1/r]∈᏿r

for allϕ∈s(Ꮾ). Using the norm on᏿r, we have

ϕ(A+B)[r]

1/r

=ϕajk+bjk

r1/r r

1/r

≤ϕajk

r1/r

+ϕbjk

r1/r r

1/r

by Lemmas1.3and3.3

≤ϕ|ajk|r1/r

+ϕ|bjk|r1/r

(by triangle inequality for the norm on᏿r) ≤ ||A||+||B||.

(3.5)

Since this is true for allϕ∈s(Ꮾ), we have||A+B|| ≤ ||A||+||B||.

For the completeness, pick a Cauchy sequence{A(n)=[a(n)_jk]}in᏿r(Ꮾ). For eachϕ∈ s(Ꮾ),

ϕa(n)_jk−a(m)_jk

r 1/r

≤ϕA(n)−A(m)

[r]

1/r

≤A(n)−A(m)−→0. (3.6)

Since||a|| =sup_ϕ∈s(Ꮾ)|ϕ(a)|for alla∈Ꮾ,

a(n)_jk−a(m)_jk−→0 asn,m−→ ∞. (3.7)

SinceᏮis complete, there is anajk∈Ꮾsuch thata

(n)

jk→ajk. LetA=[ajk]. We show that A∈᏿r

(Ꮾ) andA(n)→A.

For a fixedν∈N, sinceϕ(|a(n)_jk−ajk|r

)≤ ||a(n)_jk−ajk||r

for allϕ∈s(Ꮾ) and all (j,k)∈

N×N, we have, byLemma 1.3,

A(n)_ν −Aν ≤a

(n)

jk−ajk

r

1≤j,k≤ν

1/r−→0 asn−→ ∞. (3.8)

Let>0. There is an N such that ||A(n)−A(m)||< for alln,m≥N. Then, for a fixed

ν∈N, byProposition 3.2,

A(n)_ν

−A

(m)

ν

=A(n)−A(m) ν

≤A(n)−A(m)< ∀m,n≥N. (3.9)

Taking limit asm→ ∞, we have||A(n)_ν −Aν|| ≤for alln≥N. Since this is true for allν∈

N, we have, byProposition 3.2,||A(n)−A|| ≤ for alln≥N. Thus||A|| ≤ ||A−A(N)||+ ||A(N)|| ≤+||A(N)||<∞, and henceA∈᏿r

To see the submultiplicativity of the norm, letA,B∈᏿r

(Ꮾ); and letϕ∈s(Ꮾ). Then ϕ[A[r]],ϕ[B[r]]∈Ꮾ(p,q). For each (j,k),

_ajkbjkr

=ajk

r

bjk

r

≤ajk

r

bjk

r

≤Arbjk

r

. (3.10)

Thus

ϕajkbjk

r

=ϕajk

r

bjk

r

≤Arϕbjk

r

. (3.11)

Therefore, byLemma 1.3,

ϕ(A•B)[r]1/r=ϕajkbjk

r1/r

≤Arϕ|bjk|r1/r

p,q

= ||A||ϕB[r]1/r

p,q≤ ||A||||B||.

(3.12)

Sinceϕ∈s(Ꮾ) is arbitrary,A•B∈᏿r

(Ꮾ) and

||A•B|| ≤ ||A||||B||. (3.13)

This completes the proof.

The following simple observation will be used to prove a H¨older inequality for the norm||| · |||p,q,r.

Lemma3.5 (H¨older’s inequality for positive linear functionals). Leta,b∈Ꮾ. Then, for eachϕ∈s(Ꮾ),r∈(1,∞), andr∗satisfying1/r+ 1/r∗=1,

ϕ|ab|≤ϕ|a|r1/r

ϕ|b|r∗ 1/(r

∗₎

. (3.14)

Proof. As in the proof ofLemma 3.3,ϕhas an integral representation, and the asserted inequality is just the usual H¨older inequality, written in functional form.

Theorem3.6 (H¨older’s inequality). LetA=[ajk]andB=[bjk]be matrices with entries inᏮ. Then, forr∈(1,∞)andr∗satisfying1/r+ 1/r∗=1,

|||A•B|||p,q,1≤ |||A|||p,q,r· |||B|||p,q,r∗. (3.15)

Proof. If the right-hand side is∞, there is nothing to prove. So suppose that both fac-tors on the right are finite and nonzero. Letϕ∈s(Ꮾ) andξ= {ξk}

∞

k=1∈

p

. Write|ξ| = {|ξk|}

∞

k=1. UsingLemma 3.5and the usual H¨older inequality, we have

ϕ(A•B)[1] ξ

q = _∞ j=1 ∞ k=1

ϕajkbjkξk

q1/q

≤

_∞

j=1 _∞

k=1

ϕajkbjkξk

q1/q

≤ _∞ j=1 _∞ k=1

ϕajk

r1/r

ξk

1/r

ϕbjk

r∗1/(r

∗₎

_ξk1/(r∗) q1/q

(byLemma 3.5)

≤ _∞ j=1 _∞ k=1

ϕajk

r ξk q/r ∞ k=1

ϕbjk

r∗

_ξk

q/(r∗)1/q

≤

_∞

j=1 _∞

k=1

ϕajk

r

ξk

q1/(qr)

∞

j=1 _∞

k=1

ϕbjk

r∗

_ξk

q1/(qr∗)

=ϕA[r]|ξ|1/r

q ·

ϕB[r

∗_]

|ξ|1/(r ∗₎

q

≤ϕA[r]

1/r

p,q·|ξ|

1/r

pϕ

B[r

∗_] 1/(r

∗₎

p,q ·|ξ|

1/(r∗)

p

≤ |||A|||p,q,r|||B|||p,q,r∗|||ξ|||p= |||A|||p,q,r|||B|||p,q,r∗||ξ||p.

(3.16)

Therefore,

_ϕ[A•B]

p,q,1≤ |||A|||p,q,r|||B|||p,q,r∗. (3.17)

Sinceϕ∈s(Ꮾ) is arbitrary, we have, as asserted,

|||A•B|||p,q,1≤ |||A|||p,q,r|||B|||p,q,r∗. (3.18) Here is an analogue of the relationship betweenpand its dual space.

Theorem3.7. LetB=[bjk]be a matrix with entries inᏮand1< r <∞. ThenB∈᏿

r

(Ꮾ)

if and only ifA•B∈᏿1

(Ꮾ)for allA∈᏿r∗

(Ꮾ), where1/r+ 1/r∗=1. Moreover, whenever applied,

|||B|||p,q,r=sup

|||A•B|||p,q,1:A∈᏿

r∗

(Ꮾ),|||A|||p,q,r∗≤1

Proof. Sufficiency follows from the preceding Theorem 3.6. For necessity, define Φ: ᏿r∗

(Ꮾ)→᏿1(Ꮾ) by Φ(A)=A•B for all A∈᏿r∗

(Ꮾ). We show that the graph ofΦ is closed. SupposeAn=[a

(n)

jk]→A=[ajk] in᏿

r∗

(Ꮾ) andAn•B→C=[cjk] in᏿

1

(Ꮾ). Then for all (j,k),

a(n)_jk−→ajk, a(n)_jkbjk−→cjk asn−→ ∞. (3.20)

Therefore,ajkbjk=cjk. Since this is true for all (j,k), we have the graph ofΦis closed andΦis a bounded linear transformation, by the closed graph theorem.

By the same argument, we see also thatB[1]=[|bjk|] has the same property asB, and B[1] defines a bounded linear transformationΦ_B[1] from᏿

r∗

(Ꮾ) to᏿1(Ꮾ). For eachn∈

N, (B[r∗])n =[|bjk| r∗

]n is in᏿ 1

(Ꮾ), as it has only finitely many nonzero entries, and

ϕ[(B[r∗])[1]]=ϕ[B[r∗]] for allϕ∈s(Ꮾ). Thus

B[r]_n

p,q,1≤

B[r−1]_n

•B

[1]

p,q,1≤ _Φ

B[1]

B[r−1]_n

p,q,r∗. (3.21)

It is just a matter of writing out the definitions to see that||ΦB[1]|| = ||ΦB||, and that

|||(B[r])n|||p,q,1= |||Bn||| r

p,q,rand|||(B

[r−1]

)n|||p,q,r∗ = |||Bn||| r/r∗

p,q,r. Therefore, the preceding inequality is equivalent to

|||B|||p,q,r≤ΦB[1]

=ΦB<∞. (3.22)

ThusB∈᏿r

(Ꮾ). We also have

|||B|||p,q,r≤ ||Φ|| =sup|||A•B|||p,q,1:A∈᏿

r∗

(Ꮾ), |||A|||p,q,r∗ ≤1

≤ |||B|||p,q,r,

(3.23)

by H¨older’s inequality,Theorem 3.6. Therefore, all inequalities reduce to equalities.

Arguments similar to those used in [2] can be used to proof the following.

Proposition3.8. For1≤r < r<∞, andᏮas above,᏿r(Ꮾ)᏿r(Ꮾ). Proof. LetA=[ajk]∈᏿r

(Ꮾ). Then for each (j,k), we have

_ajk=ajkr1/r

=

sup ϕ∈s(Ꮾ)

ϕajk

r1/r

≤ |||A|||p,q,r. (3.24)

Choose a suitable constantα, with 0< α≤1, such thatαA has all entries bounded in norm by 1. Then forr < randϕ∈s(Ꮾ), we have, for each (j,k)∈N×N,

αrϕ|ajk|r

≤αrϕajk

r

that is, each entry of ϕ[(αA)[r]] is bounded by the corresponding entry ofϕ[(αA)[r]]. Thus

α|||A|||p,q,r= |||αA|||p,q,r≤ |||αA|||p,q,r=α|||A|||p,q,r. (3.26)

So that|||A|||p,q,r≤ |||A|||_p,q,r, and hence᏿r(Ꮾ)⊆᏿r

(Ꮾ). To see that the inclusion is proper, we take a sequence {αk}of nonnegative numbers which is inr but not inr (more explicitly, takeαk=[1/k]1/r). Then the matrixAwith the first column, the sequence {(αk)

1/q

1}(1 the identity ofᏮ), and all other columns 0 is in᏿r(Ꮾ) but not in᏿r(Ꮾ).

4. Dual spaces

We consider in this section an analogue of compact and trace-class operators on a Hilbert space, in the Schur algebras considered in the preceding sections. Analogues of Schatten’s trace duality theorems will be proved in this setting. LetᏮbe a unital commutativeC∗ -algebra with the sets(Ꮾ) of states as in the preceding section. Letᏹ0 be the set of all

infinite matrices with entries inᏮhaving only finitely many nonzero entries. Denote by ᏷r

(Ꮾ) the closure in᏿r(Ꮾ) ofᏹ0. That is,

᏷r

(Ꮾ)=A∈᏿r

(Ꮾ) :∀>0∃A0∈ᏹ0such thatA0−A<

, (4.1)

where the norm|| · ||is the norm ||| · |||p,q,r of ᏿

r

(Ꮾ). With the norm inherited from ᏿r

(Ꮾ),᏷r(Ꮾ) is a Banach space. We will identify the dual of᏷r(Ꮾ), in analogy with the fact that the dual of the compact operators on a Hilbert space is the trace-class operators. Denote by (Ꮽ᏿) the space of matrices over the complex fieldCthat are absolutely summable. This is just the space1(N×N). Therefore, it is a Banach space with the1 norm; that is, a matrixS=[sjk] overCis in (Ꮽ᏿) if and only if

||S||(Ꮽ᏿):=

∞

j,k=1

_{sjk<∞. (4.2)}

As an analogue of the trace-class operators on a Hilbert space, we consider the space ᏹ(᏿r(Ꮾ), (Ꮽ᏿)) defined as follows:

ᏹ᏿r

(Ꮾ), (Ꮽ᏿):=

Φ=ϕjk:ϕjk∈Ꮾ

#

, ∞

j,k=1 _ϕjk

ajk<∞ ∀A=ajk∈᏿

r

(Ꮾ)

.

(4.3)

Thus a matrixΦof functionals is inᏹ(᏿r(Ꮾ), (Ꮽ᏿)) if and only if it “Schur multiplies” each matrix in᏿r(Ꮾ) to a matrix in (Ꮽ᏿). EachΦ∈ᏹ(᏿r(Ꮾ), (Ꮽ᏿)) defines a bounded linear transformation by theSchur multiplicationbyΦ, that is, by the closed graph theo-rem,Φ•[ajk]=[ϕjk(ajk)] from the Banach space᏿r(Ꮾ) to the Banach space (Ꮽ᏿) is a bounded linear transformation. Therefore, it has an operator norm

||Φ||ᏹ(᏿r(Ꮾ),(Ꮽ᏿)):=sup

||Φ•A||(Ꮽ᏿):A∈᏿

r

Theorem4.1. The spaceᏹ(᏿r(Ꮾ), (Ꮽ᏿))equipped with the norm defined above is a Ba-nach space.

Proof. Since ᏹ(᏿r(Ꮾ), (Ꮽ᏿))⊆Ꮾ(᏿r(Ꮾ), (Ꮽ᏿)), the space of bounded linear trans-formations from᏿r(Ꮾ) to (Ꮽ᏿), it suffices to show that it is closed. To that end, sup-pose that{Ψn=[ψ

(n)

jk]}is a sequence in the spaceᏹ(᏿

r

(Ꮾ), (Ꮽ᏿)) such thatΨn→T∈ Ꮾ(᏿r(Ꮾ), (Ꮽ᏿)). Then each{ψ(n)_jk}∞n=1is a Cauchy sequence inᏮ

#

, therefore converges to someψjk∈Ꮾ

#

. LetΨ=[ψjk]. We show thatΨ=T. Let>0. There is anNsuch that

_{Ψn−Ψn+l< ∀n≥N,l≥1. (4.5)}

LetA∈᏿r

(Ꮾ); andm∈N. Then

Ψn•A−Ψn+l•A

m

(Ꮽ᏿)=

Ψn•Am−Ψn+l•Am

(Ꮽ᏿)

≤Am≤||A|| ∀n≥N,l≥1.

(4.6)

Taking limits asl→ ∞, we have

Ψn•A−Ψ•A

m

(Ꮽ᏿)≤||A|| ∀n≥N,∀A∈᏿

r

(Ꮾ). (4.7)

Now asm→ ∞, we obtain

_{Ψn•A−Ψ•A}

(Ꮽ᏿)≤||A|| ∀n≥N,∀A∈᏿

r

(Ꮾ). (4.8)

So we see thatΨn→Ψ. Therefore,T(A)=Ψ•Afor allA∈᏿

r

(Ꮾ).

LetΨ∈(᏿r(Ꮾ))#. For each (j,k), define a linear functional onᏮas follows. Forb∈Ꮾ, letAb,j,kbe the matrix whose (j,k) entry isband all others 0. Put

ψjk(b)=ΨAb,j,k. (4.9)

Thenψjkis a bounded linear functional onᏮ. PutBΨ=[ψjk]. Each matrixΦ=[ϕjk]∈ᏹ(᏿

r

(Ꮾ), (Ꮽ᏿)) defines a linear functional

Φ(A)=

∞

j,k=1

ϕjk

ajk

, A=ajk

∈᏿r

(Ꮾ). (4.10)

Theorem4.2. (1)LetΨ∈(᏿r(Ꮾ))#; and letψjkandBΨbe as defined above. Then

BΨ∈ᏹ᏿r(Ꮾ), (Ꮽ᏿), Ψ−BΨ∈᏷r(Ꮾ)⊥. (4.11)

Proof. (1) LetA=[ajk]∈᏿r(Ꮾ). Forz∈C, let sgn(z)=z/|z|ifz=0, and sgn(0)=0. Let ¯A=[sgn(ψjk(ajk))ajk]. Then ¯A∈᏿

r

(Ꮾ), and for eachn,

n

j,k=1 _ψjk

ajk=Ψ

_¯

An(Ꮽ᏿)≤ ||Ψ||A¯n≤ ||Ψ||||A¯|| ≤ ||Ψ||||A||. (4.12)

So

∞

j,k=1 _ψjk

ajk≤ ||Ψ||||A||<∞; (4.13)

andBΨ•A∈(Ꮽ᏿) for allA∈᏿r

(Ꮾ); that is,BΨ∈ᏹ(᏿r(Ꮾ), (Ꮽ᏿)). ForA=[ajk]∈᏷

r

(Ꮾ), we have for eachn,

BΨAn

=ΨAn

(4.14)

by linearity. Since||An −A|| →0 asn→ ∞, by the continuity of both functionals, we have

BΨ(A)=Ψ(A) ∀A∈᏷r(Ꮾ). (4.15)

(2) LetΨ∈(᏷r(Ꮾ))#. For eachA=[ajk]∈᏷r

(Ꮾ) such that||A|| ≤1,

_Ψ(A)=BΨ(A)=

∞

j,k=1

ψjk

ajk

≤BΨ_ᏹ(᏿r_{(Ꮾ),(Ꮽ᏿))}, (4.16)

by the definition of the norm onᏹ(᏿r(Ꮾ), (Ꮽ᏿)). Therefore,

||Ψ|| ≤BΨᏹ(᏿r(Ꮾ),(Ꮽ᏿)). (4.17)

Let>0. There is anA=[ajk]∈᏿r

(Ꮾ) such that||A|| ≤1 and

_BΨ

ᏹ(᏿r_{(Ꮾ),(Ꮽ᏿))}−

3< ∞

j,k=1 _ψjk

ajk. (4.18)

By the absolute convergence of the series, there is anNsuch that

N

j,k=1 _ψjk

ajk> ∞

j,k=1 _ψjk

ajk−₃>BΨᏹ(᏿r(Ꮾ),(Ꮽ᏿))

−2

3 . (4.19)

LetAbe the matrix whose (j,k) entry is (sgn(ψjk(ajk))ajk) for j,k=1, 2,. . .,N and all others 0. ThenA∈᏷r

(Ꮾ),||||A≤1, and

Ψ()A= N

j,k=1 _ψjk

ajk>BΨᏹ(᏿r_{(Ꮾ),(Ꮽ᏿))}−2

Therefore, we have

||Ψ||>BΨᏹ(᏿r_{(Ꮾ),(Ꮽ᏿))}−. (4.21)

Sinceis arbitrary, we have equality of the norms.

A linear functionalΦon᏿r(Ꮾ) issingularifΦ(A)=0 for allA∈᏷r

(Ꮾ). Recall that for eachΨ∈᏿r(Ꮾ)#, there corresponds a bounded linear functionalBΨon᏿r(Ꮾ). De-note

᏿r

(Ꮾ)#_a:= BΨ:Ψ∈᏿r(Ꮾ)#. (4.22)

Theorem 4.3. The set᏿r(Ꮾ)#_s consisting of all singular linear functionals together with the zero functional on᏿r(Ꮾ)is a nontrivial closed subspace of the dual᏿r(Ꮾ)#of᏿r(Ꮾ). Furthermore,

᏿r

(Ꮾ)#=᏿r(Ꮾ)#a⊕᏿

r

(Ꮾ)#s. (4.23)

Proof. Since᏷r(Ꮾ) is a nontrivial closed subspace of᏿r(Ꮾ), the Hahn-Banach theorem ensures that᏿r(Ꮾ)#sis a nonempty proper subset of᏿

r

(Ꮾ)#. The preceding theorem shows that ᏿r(Ꮾ)#=᏿r(Ꮾ)#a+᏿

r

(Ꮾ)#s. Let Ψ=[ψjk]∈ ᏿r

(Ꮾ)#_a ᏿r(Ꮾ)_s#. Then Ψ(K)=0 for allK∈᏷r

(Ꮾ). Let b∈Ꮾ; and let Ab,j,k be the matrix whose (j,k) entry isband all others 0. Then for each (j,k),

ψjk(b)=ΨAb,j,k=0. (4.24)

Therefore,Ψ=0.

SinceᏮmay not be the dual of any normed space, we cannot expect᏿r(Ꮾ) to be a dual space. For if it were, then it would not be hard to see thatᏮmust be a dual space as well. We therefore assume, from this point on, thatᏮis the dual of some Banach space Ꮾ#. We can then consider the space

ᏹ0

#

᏿r

(Ꮾ), (Ꮽ᏿)=

B=bjk:bjk∈Ꮾ#,

∞

j,k=1 _ajk

bjk<∞,∀A=ajk∈᏿

r

(Ꮾ)

(4.25)

with the norm

||B|| =sup ∞

j,k=1

ajk

bjk

:A=

ajk

∈᏿r

(Ꮾ),|||A|||p,q,r≤1

, (4.26)

forB=[bjk]∈ᏹ#0(᏿

r

(Ꮾ), (Ꮽ᏿)).

Arguments similar to those used in the proof ofTheorem 4.1can be used to prove that ᏹ0

#(᏿

r

(Ꮾ), (Ꮽ᏿)) is also a Banach space.

Since the predual ofB(2) is the trace-class operators, which is the class of matrices that are the trace norm limits of their upper left-hand corner truncations, we define, analo-gously, the spaceᏹ#(᏿

r

(Ꮾ), (ᏭS)) as the space of all matricesB∈ᏹ0

#(᏿

r

that||B−B_n

|| →0 asn→ ∞.It is not hard to see thatᏹ#(᏿ r

(Ꮾ), (ᏭS)) is a closed sub-space ofᏹ0#(᏿

r

(Ꮾ), (ᏭS)) under the norm defined above. For brevity of notation, denote byᏹthe spaceᏹ#(᏿

r

(Ꮾ), (ᏭS)) with the induced norm. Thenᏹis a Banach space. We show that᏿r(Ꮾ) is the dual ofᏹ.

Theorem4.4. Under the standing assumption thatᏮhas a predualᏮ#, and the notations

defined above, the dual ofᏹis isometrically isomorphic to᏿r(Ꮾ)(as Banach spaces).

Proof. By the definition ofᏹ, it is clear that eachA=[ajk]∈᏿r(Ꮾ) can be used to define a bounded linear functionalφAonᏹas

φA(M)=

∞

j,k=1

ajk

mjk

∀M=mjk

∈ᏹ. (4.27)

Notice that by the definition (4.26) of the norm onᏹ, we also have|φA(M)| ≤ ||A||||M||.

Thus||φA|| ≤ ||A||. Therefore,᏿r

(Ꮾ) can be regarded as a subspace ofᏹ#. Denote by ᏿1 the closed unit ball of᏿

r

(Ꮾ). Then᏿1 separates points inᏹ. We show that᏿1 is

complete in the weak∗ topology,σ, it inherits fromᏹ#. To this end, let{Aα=[aα_jk]}α∈Λ be aσ-Cauchy net in᏿1. We show that for each (µ,ν)∈N×N, the net of the (µ,ν)-entries

{aα_µν}of{Aα}is a weak∗Cauchy net inᏮ. Letm∈Ꮾ#. ChooseMto be the matrix whose

(µ,ν)-entry ismand zero for all others. Then we see thatM∈ᏹand{φ_Aα(M)=a α

µν(m)} is a Cauchy net inC. Thus{aα_µν}αis a weak∗ Cauchy net inᏮ. Since||Aα|| ≤1 for allα and||aα_µν|| ≤ ||Aα||,{aα_µν}α∈Λ is a weak∗ Cauchy net in the closed unit ball ofᏮ. Since the closed unit ball is weak∗ compact, by the Alaoglu theorem, there isaµν∈Ꮾsuch that

aα_µν→aµνin the weak∗ topology. LetA=[ajk]. We show thatA∈᏿1and thatAα→Ain

the weak∗ topology ofᏹ#induced on᏿1. Let>0; and letM=[mjk]∈ᏹ. There is an

α0such that|φ_Aα(M)−φ

Aβ(M)|</2 for allα,βα0. By definition ofᏹ, there is anN

such that||M−M_n

||</4 for alln≥N. Thus, forn≥Nandα,βα0, we have

φ(Aα)

n

−

φ_(Aβ

)

n

(M)

=φ_Aα−φ

Aβ Mn

≤φ_Aα−φ_Aβ M n

−M+

φ_Aα−φ_Aβ (M)

<φ_Aα −φ_AβM n

−M+

2 ≤Aα−Aβ

4+ 2≤.

(4.28)

For a fixedn≥N, (Aα)_n →A_n in the weak∗topologyσ. Taking limit inβ, we have

φ(Aα₎ n

−

φA

n

(M)

≤ ∀n≥n0,αα0. (4.29)

Since this is true for alln≥N, we may take the limit asn→ ∞to obtain

_φ

Aα−φA

(M)≤ ∀αα0. (4.30)

We claim that the norm onᏹ#is the same as that on᏿r(Ꮾ), andᏹ#=᏿r(Ꮾ). For each A∈᏿1,||φA|| ≤ ||A|| ≤1. Thus᏿1is contained in the closed unit ball ofᏹ

#

. Suppose this inclusion is proper. Then there isA0∈ᏹ

#

with||A0|| ≤1 such thatA0∈᏿1. Since᏿1is

weak∗closed and convex, by [3, Theorem V 2.10, page 417], there is a weak∗ continuous linear functional onᏹ#, that is, an elementM∈ᏹ, such that

᏾eφM(A)≤c−< c≤᏾eφMA0

∀A∈᏿1, (4.31)

for some constantscand>0.

For eachA∈᏿1, letA=ζA, whereζ∈Cis chosen such that|ζ| =1 andφM(A)= |φM(A)|. SinceA∈᏿1for eachA∈᏿1, we have, by the definition (4.26) of the norm on

ᏹ,

||M|| =supφM(A)=᏾e

φM A

:A∈᏿1

≤c−< c≤᏾eφM(A0)

≤ ||M||, (4.32)

a contradiction. Therefore,᏿1is the unit ball ofᏹ

#

. LetA∈᏿r

(Ꮾ). As a linear functionalφAonᏹ,

_{φA=supφA(M):M∈ᏹ,||M|| ≤1}

≤ ||A||. (4.33)

If||φA|| =1, thenAis in the unit ball ofᏹ#, which is just᏿1. Thus||A|| ≤1≤ ||φA|| ≤

||A||, and hence||A|| = ||φA||.

Acknowledgments

We would like to thank the referee for the many invaluable suggestions which significantly improved the paper. This research was supported by grants from the Thailand Research Fund. It was done while I was on a sabbatical leave from Central Michigan University. The hospitality of Pachara Chaisuriya and the Department of Mathematics of Mahidol University during my visit is gratefully acknowledged.

References

[1] G. Bennett,Schur multipliers, Duke Math. J.44(1977), no. 3, 603–639.

[2] P. Chaisuriya and S.-C. Ong,Absolute Schur algebras and unbounded matrices, SIAM J. Matrix Anal. Appl.20(1999), no. 3, 596–605.

[3] N. Dunford and J. T. Schwartz,Linear Operators. Part I: General Theory, John Wiley & Sons, New York, 1957.

[4] R. A. Horn and C. R. Johnson,Topics in Matrix Analysis, Cambridge University Press, Cam-bridge, 1991.

[5] R. V. Kadison and J. R. Ringrose,Fundamentals of the Theory of Operator Algebras. Vol. I. Ele-mentary Theory, Pure and Applied Mathematics, vol. 100, Academic Press, New York, 1983. [6] L. Livshits,Generalized Schur product for matrices with operator entries, Ph.D. thesis, University

of Toronto, Ontario, 1991.

[7] L. Livshits, S.-C. Ong, and S.-W. Wang,Schur algebras over function algebras, Houston J. Math.

30(2004), no. 4, 1195–1217.

[9] J. Schur,Bemerkungen Theorie der beschr¨anken Bilinearformen mit unendlich vielen Ver¨ander lichen, J. Reine Angew. Math.140(1911), 1–28 (German).

Pachara Chaisuriya: Department of Mathematics, Faculty of Science, Mahidol University, Rama VI Road, Bangkok 10400, Thailand

E-mail address:[email protected]

Sing-Cheong Ong: Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, USA