SCHATTEN’S THEOREMS ON FUNCTIONALLY DEFINED SCHUR ALGEBRAS

For each triple of positive numbers p,q,r≥1 and each commutative C*-algebra ℬ with identity 1 and the set s(ℬ) of states on ℬ, the set a#x1D4AE;r(ℬ) of all matrices A=[ajk] over ℬ such that ϕ[A[r]]:=[ϕ(|ajk|r)] defines a bounded operator from lp to lq for all ϕ∈s(ℬ) 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(ℬ)}. 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 a#x1D4AE;r(ℬ) setting.


Introduction
Fix p and q with 1 ≤ p, q < ∞.The space of pth power summable sequences of complex numbers is denoted by p , and the space of matrices which define bounded linear transformations from p to q is denoted by Ꮾ( p , q ).Let A = [a jk ], B = [b jk ] be infinite matrices, not necessarily in Ꮾ( p , q ).The Schur product A • B of A and B is defined by A • B = [a jk b jk ].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] for p = 2, and q = 2, Bennett proved in [1, Theorem 2.2] the following theorem.
(1.1) (i.e., Ꮾ( p , q ) is a commutative Banach algebra under the Schur product operation and operator norm || • || p,q .)2176 Schatten's theorems on Schur algebras 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 matrix A = [a jk ], over C, is in the absolute Schur r-algebra r if the Schur rth power ).The r-norm of A is defined by ||A|| := |||A||| p,q,r := 1/r , where ||A [r] || = ||A [r] || p,q is the operator norm of A [r] as an element of Ꮾ( p , q ).That ||| • ||| p,q,r is a norm follows, as expected, from an inequality that is analogous to the Hölder inequality.
Theorem 1.2.For each r ≥ 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 commutative C * -algebra Ꮾ, 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 on Ꮾ.
In [7] Schatten's theorems [8] concerning the dual of the compact operators, the traceclass 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 function sequence space analogous to the 2 sequence space.Here we extend these Schattentype theorems to the algebra r (Ꮾ) consisting of certain classes of matrices over a commutative C * -algebra Ꮾ, in a situation where p and q need not be equal to 2. Since the p and q will be fixed throughout our discussion, we will sometimes suppress the subscripts p, q in || • || 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.
Lemma 1.3.Let [α jk ] and [β jk ] be matrices over the complex field C such that |α jk | ≤ β jk for all j and k.Suppose that [β jk ] ∈ Ꮾ( p , q ); that is, the matrix defines a bounded linear transformation from

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.Let ᐄ be a Banach space with norm || • ||, and dual space ᐄ # .Consider the set ᏹ(ᐄ) of all infinite matrices over ᐄ.Let (ᐄ # ) 1 denote the set of all bounded linear functionals on ᐄ which have norm 1.Let s(ᐄ) ⊆ (ᐄ # ) 1 be a set of linear functionals on ᐄ such that (i (ii) the linear span of s(ᐄ) is all of X # .One such example is s(ᐄ) = (ᐄ # ) 1 by the Hahn-Banach theorem and the fact that all linear functionals in ᐄ # are multiples of elements in (ᐄ # ) 1 .For each f ∈ ᐄ # , and each matrix A = [a jk ] ∈ ᏹ(ᐄ), denote by f [A] = [ f (a jk )] the complex matrix whose ( j,k) entry is f (a jk ).For fixed p and q with 1 ≤ p, q < ∞, regard P. Chaisuriya and S.-C.Ong 2177 the matrix f [A] as a linear transformation of p to q , 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 matrices The following result provides us with a natural way of defining a norm on (ᐄ).
Theorem 2.1.Let ᐄ and s(ᐄ) be as above, and (2.1) Proof.First we note that since each f ∈ ᐄ # is a linear combination f = n j=1 α j g j of elements g 1 ,...,g n in s(ᐄ), and for the given . Therefore, the map . Since both the domain ᐄ # and codomain Ꮾ( 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 (2.2) Next we prove that (ᐄ) is a Banach space.
Theorem 2.2.The set (ᐄ) is a Banach space under the usual (entrywise) scalar multiplication and addition, and the norm Proof.First we show that the function as defined in Theorem 2.1 is indeed a norm on (ᐄ).Let A,B ∈ (ᐄ) and let f ∈ s(ᐄ).
To prove that (ᐄ) is complete, let {A n } be a Cauchy sequence in (ᐄ).Then for each (2.4) (2.6) Thus the sequence {a jk } is a Cauchy sequence in ᐄ.By the completeness of ᐄ, there exists an a jk ∈ ᐄ such that a Let n ≥ N be arbitrarily fixed.We will show that ||A n − A|| < .There exists an f ∈ s(ᐄ) such that (2.8) Therefore, (ᐄ) is complete.
For a given matrix A = [a jk ], denote by A n the matrix whose ( j,k) entry is a jk for 1 ≤ j,k ≤ n and 0 otherwise. (2.9) Since this is true for every f ∈ s(ᐄ), (2.12) Thus there exists an n 0 such that For the finite sum, there is an n 1 such that n 1 ≥ n 0 and that (2.15) Therefore, ||A n || ||A||, as asserted.
With ᐄ and s(ᐄ) as above and for 1 ≤ r < ∞, let r (ᐄ) denote the set of all matrices A = [a jk ] ∈ ᏹ(ᐄ) with the property that f (2.16) Proof.By Theorem 1.2, the set of all matrices Λ = [λ jk ] over C with bounded absolute Schur rth power (Λ (2.17) Therefore, f [A] = Λ and -A has a closed graph.Since ᐄ # and r are Banach spaces, -A is bounded by the closed graph theorem.Thus (2.18)

19)
For each A ∈ r (ᐄ), define ||A|| or |||A||| p,q,r by (2.20) The preceding theorem guarantees that || • || is a function defined on r (ᐄ).We will prove that it is a norm in Theorem 2.6.Using an argument similar to that in Proposition 2.3, we can show that this function has the same monotone property.
Proof.This is just a routine adaptation of the proof of Proposition 2.3, therefore omitted.
Theorem 2.6.For r ≥ 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 multiplication.
(by the triangle inequality for the norm on r ) Chaisuriya As this is true for all f ∈ s(ᐄ), we have A ∈ r (ᐄ).To see that ||A n − A|| → 0, we first note that for each ν ∈ N, we have, by the finiteness of ν and Lemma 1.3, (2.24) Let > 0. There is an N ∈ N such that Then, by Proposition 2.5, Taking limit as l → ∞ and using (2.24), we have Since ν ∈ N is arbitrary, we have, by Proposition 2.5, (2.28)

Schur algebras over commutative C * -algebras
In this section, we fix a commutative C * -algebra Ꮾ with identity 1 and the set s(Ꮾ) of states on Ꮾ, that is, the set of positive linear functionals of norm 1 on Ꮾ.By the Gelfand-Naimark theorem, Ꮾ is isometrically * -isomorphic to the function algebra C(X) for some compact Hausdorff space X.We will treat Ꮾ as C(X).Since s(Ꮾ) contains all evaluation linear functionals ϕ x (a) = a(x) for x ∈ X, and a ∈ Ꮾ, we have ||a|| = sup{|ϕ(a)| : ϕ ∈ s(Ꮾ)}, for all a ∈ Ꮾ.
As in the scalar case, define the Schur product (also known as Hadamard product, or entrywise product) of two matrices The product a jk b jk 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 each ϕ ∈ s(Ꮾ), and for each A = [a jk ] ∈ ᏹ(Ꮾ), ϕ[A] = [ϕ(a jk )] is the scalar matrix obtained by applying ϕ to each entry of A. Let (Ꮾ) denote the set of all matrices A ∈ ᏹ(Ꮾ) with the property that ϕ[A] defines a bounded linear transformation from p to q for every ϕ ∈ s(Ꮾ).Denote by Ꮾ # the Banach space dual of Ꮾ (the space of bounded linear functionals on Ꮾ).Since each f ∈ Ꮾ # is a linear combination of at most four states (see [ The following proposition follows immediately from Theorem 2.2. Proposition 3.1.For a commutative C * -algebra Ꮾ with state space s(Ꮾ), (Ꮾ) is a Banach 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 multiplication operation.
We now turn our attention to functional analog of Schur r-algebras.As before, we fix a commutative C * -algebra Ꮾ with the set s(Ꮾ) of states.For each real number r ≥ 1, , that is, the numerical matrix ϕ[A [r] ] defines a bounded linear transformation from p to q for every ϕ ∈ s(Ꮾ).We show that for each A ∈ r (Ꮾ), 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 of Proposition 2.3, we have the following.
We also need this simple observation.

Lemma 3.3 (Minkowski's inequality for linear functionals). Let ϕ ∈ s(Ꮾ).
Then Proof.Since Ꮾ = C(X) for some compact Hausdorff space X, the given ϕ ∈ s(Ꮾ) has an integral representation ϕ(a) = X adµ ϕ for all a ∈ Ꮾ and some measure µ ϕ on X.The Minkowski inequality for µ ϕ is exactly the asserted inequality.
Theorem 3.4.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.
# is a linear combination of at most four states, the map f → f [A [r] ] is a linear transformation from Ꮾ # to Ꮾ( p , q ).Using arguments similar to that used in the proof of Theorem 2.1, we have Using the norm on r , we have Since Ꮾ is complete, there is an a jk ∈ Ꮾ such that a jk − a jk || r for all ϕ ∈ s(Ꮾ) and all (j,k) ∈ N × N, we have, by Lemma 1.3, (3.9) Taking limit as m → ∞, we have ||A

.14)
Proof.As in the proof of Lemma 3.3, ϕ has an integral representation, and the asserted inequality is just the usual Hölder inequality, written in functional form.
(3.16) Therefore, is arbitrary, we have, as asserted, Here is an analogue of the relationship between p and its dual space.
Proof.Since ᏹ( r (Ꮾ),(Ꮽ)) ⊆ Ꮾ( r (Ꮾ),(Ꮽ)), the space of bounded linear transformations from r (Ꮾ) to (Ꮽ), it suffices to show that it is closed.To that end, sup- There is an N such that Let A ∈ r (Ꮾ); and m ∈ N. Then Taking limits as l → ∞, we have Now as m → ∞, we obtain So we see that Ψ n → Ψ.Therefore, Let Ψ ∈ ( r (Ꮾ)) # .For each ( j,k), define a linear functional on Ꮾ as follows.For b ∈ Ꮾ, let A b, j,k be the matrix whose ( j,k) entry is b and all others 0. Put (4.9) Then ψ jk is a bounded linear functional on Ꮾ.
Since Ꮾ may 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 that Ꮾ must be a dual space as well.We therefore assume, from this point on, that Ꮾ is the dual of some Banach space Ꮾ # .We can then consider the space )). Arguments similar to those used in the proof of Theorem 4.1 can be used to prove that ᏹ 0 # ( r (Ꮾ),(Ꮽ)) is also a Banach space.Since the predual of B( 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, analogously, the space ᏹ # ( r (Ꮾ),(ᏭS)) as the space of all matrices B ∈ ᏹ 0 # ( r (Ꮾ),(ᏭB)) such jk ]} α∈Λ be a σ-Cauchy net in 1 .We show that for each (µ,ν) ∈ N × N, the net of the (µ,ν)-entries (4.28) For a fixed n ≥ N, (A α ) n → A n in the weak * topology σ.Taking limit in β, we have Since this is true for all n ≥ N, we may take the limit as n → ∞ to obtain We claim that the norm on ᏹ # is the same as that on r (Ꮾ), and ᏹ # = r (Ꮾ).For each A ∈ 1 , ||φ A || ≤ ||A|| ≤ 1.Thus 1 is contained in the closed unit ball of ᏹ # .Suppose this inclusion is proper.Then there is A 0 ∈ ᏹ # with ||A 0 || ≤ 1 such that A 0 ∈ 1 .Since 1 is weak * closed and convex, by [3, Theorem V 2.10, page 417], there is a weak * continuous linear functional on ᏹ # , that is, an element M ∈ ᏹ, such that (n)ν − A ν || ≤ for all n ≥ N. Since this is true for all ν ∈ N, we have, by Proposition 3.2,||A (n) − A|| ≤ for all n ≥ N. Thus ||A|| ≤ ||A − A (N) || + ||A (N) || ≤ + ||A (N)|| < ∞, and hence A ∈ r (Ꮾ), and also A (n) → A.