Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 31, 2006, 287–295

THE TRACE CLASS IS A Q-ALGEBRA

David Pérez-García

Universidad Rey Juan Carlos, Área de Matemática Aplicada, ESCET Departamental II, ES-28933 Móstoles (Madrid), Spain; david.perez.garcia@urjc.es

Abstract. We use an old multilinear generalization of Grothendieck’s inequality (for which we give a new proof and a non-commutative version) to show that the Schatten space S_{p} is a Q-algebra with the Schur product for 1 ≤ p ≤ 2.

1. Introduction and notation

In [10], Grothendieck stated what he called “the fundamental theorem of the metric theory of tensor products”, to be known later as Grothendieck’s theorem or Grothendieck’s inequality. In [14], Lindenstrauss and Pełczyński gave a detailed proof of this result and stated some of its consequences and equivalent formulations, making use of the theory of ℒ_{p} -spaces and using also the p-summing operators recently introduced in [18]. Since then, several equivalent formulations, extensions and different proofs have been obtained, together with innumerable applications. One of these extensions is the ‘non-commutative’ version given by Pisier (the interested reader can consult [13] or [22] for two recent applications). [8], [9] and [21] provide excellent expositions on this and related topics.

Grothendieck’s theorem can be seen as a matrix inequality associated to certain bilinear operators. In the late 70’s, several authors investigated multilinear extensions of Grothendieck’s matrix inequality (see, for instance, [3], [5], [24] and the references therein). Here, in Theorem 2.2, we give an elementary proof of one of these extensions (other proofs can be seen in [5] or [24]). In Corollary 2.3 we apply it to obtain that the Schatten space S_{p}, with 1 ≤ p ≤ 2 is a Q-algebra with the Schur product, answering (at least partially) an old problem that goes back to the work of Varopoulos in the 70’s. We refer to [16] for the history of the problem and the solution in the case 2 ≤ p ≤ 4. Let us just say here that the case 4 ≤ p ≤ ∞ is still open. Motivated by this application, we give a non-commutative extension of Theorem 2.2 in the line of Pisier’s result.

The notation and terminology used along the paper are standard in Banach space theory, as for instance in [8] or [9]. These books are also our main references for basic facts, definitions and unexplained notation. However, before going any further, we shall establish some terminology. All along this paper all the operators are supposed to be continuous. Given X, Y Banach spaces, ℒ (X, Y) will denote the Banach space of linear (and continuous) operators between them. X^{∗} will be the dual of X and B_{X} its unit ball. For a finite sequence (x_{i})_{i=1}^{m} ⊂ X and 1 ≤ p < ∞, we will write ∥(x_{i})_{i=1}^{m}∥_{p}^{ω} to denote

2000 Mathematics Subject Classification: Primary 47H60; Secondary 46J99, 46L05, 47L25.

Partially supported by BMF 2001-1284.

sup{(∑_{i=1}^{m} |x^{∗}(x_{i})|^{p})^{1/p} : x^{∗} ∈ B_{X^{∗}}}.

A linear operator u: X ⟶ Y is said to be absolutely summing if there exists a constant K > 0 such that ∑_{i=1}^{m} ∥u(x_{i})∥ ≤ K∥(x_{i})_{i=1}^{m}∥_{1}^{ω} for each finite sequence (x_{i})_{i=1}^{m} ⊂ X. π_{1}(u) will be the least of such constants K. We will use the fact that u: X ⟶ Y is absolutely summing if and only if there exists a constant K > 0 such that, for every m ∈ N, ∥ id_{l_{1}^{m}} ⊗u: l_{1}^{m} ⊗_{ε} X ⟶ l_{1}^{m} ⊗_{π} Y ∥ ≤ K, where ε and π denote the injective and projective tensor norms respectively [8, Proposition 11.1]. If so, π_{1}(u) is again the least of such constants. Grothendieck’s theorem tells us that there exists a constant K > 0 such that for every m ∈ N and for every u: l_{1}^{m} ⟶ l_{2}^{m}, we have that π_{1}(u) ≤ K∥u∥. The Grothendieck constant K_{G} is the least of them.

More generally, given s, r_{1}, …, r_{n} ∈ [1, ∞) such that

\frac{1}{s} ≤ \frac{1}{r_{1}} + ⋯ + \frac{1}{r_{n}} ,

we say that a multilinear operator T: X_{1} × ⋯ × X_{n} ⟶ Y is (s; r_{1}, …, r_{n})- summing if there exists a constant K > 0 such that

(∑_{i=1}^{m} ∥T (x_{i}^{1}, …, x_{i}^{n})∥^{s})^{1/s} ≤ K ∏_{j=1}^{n} ∥(x_{i}^{j})_{i=1}^{m}∥_{r_{j}}^{ω}

for every choice of sequences (x_{i}^{j})_{i=1}^{m} ⊂ X_{j}. Again ∥T ∥_{(s;r_{1},...,r_{n})} will denote the least of such constants.

Let λ > 1. A Banach space X is said to be an ℒ_{∞,λ} -space if, for every finitedimensional subspace E ⊂ X there exists another finite-dimensional subspace F, with E ⊂ F ⊂ X and such that there exists an isomorphism v: F ⟶ l_{∞}^{dim F} with ∥v∥ ∥v^{−1}∥ < λ. As basic examples we have that, for every compact Hausdorff space K and for every measure space (Ω, Σ, μ), C(K) and L_{∞}(μ) are ℒ_{∞,λ} -spaces for every λ > 1.

The trace class is a Q-algebra 289

A Banach algebra is said to be a uniform algebra if it is isometrically isomorphic (as Banach algebra) to a closed subalgebra of C(K) for some compact Hausdorff space K. By a Q-algebra we mean a commutative Banach algebra that is isomorphic (as Banach algebra) to C/I, where C is a uniform algebra and I a closed ideal of C. It is well known since the 60’s that every Q-algebra is an operator algebra, that is, isomorphic to a closed subalgebra of ℒ (H, H) for a Hilbert space H.

S_{∞} will be the Banach space of all compact operators on l_{2} (with the operator norm). The Schatten space S_{p}, with 1 ≤ p < ∞, will be the Banach space of all compact operators on l_{2} such that tr |u|^{p} < ∞, equipped with the norm ∥u∥_{S_{p}} = (tr |u|^{p})^{1/p}.

Finally, we have to define the Schur product in S_{p}. Using the canonical basis, any compact operator on l_{2} can be viewed as an infinite matrix (a_{k,h})_{k,h≥1} . The Schur product ∗ of two such matrices is just the pointwise product, that is

(a_{k,h})_{k,h≥1} ∗ (b_{k,h})_{k,h≥1} = (a_{k,h}b_{k,h})_{k,h≥1}.

2. The results

We will need the following

Lemma 2.1. Let n, m ∈ N and let u_{i} ∈ ℒ (l_{1}^{m}, l_{2}^{m}) with ∥u_{i}∥ ≤ 1, 1 ≤ i ≤ n. We write i: l_{1}^{m} ↪ l_{2}^{m} for the formal inclusion and we write P_{n}: l_{2}^{m}×⋯^{n}×l_{2}^{m} ⟶ l_{1}^{m} for the product operator P_{n}((x_{i}^{1})_{i=1}^{m}, …, (x_{i}^{n})_{i=1}^{m}) = (x_{i}^{1} ⋯ x_{i}^{n})_{i=1}^{m} (and also for its associate linear operator P_{n}: l_{2}^{m} ⊗_{π} ⋯ ⊗_{π} l_{2}^{m} ⟶ l_{1}^{m}). If we define v_{n} with the following diagram

\xymatrix{l_{1}^{m} ⊗_{ε} ⋯ ⊗_{ε} l_{1}^{m} \ar[d]_{u_{1}⊗···⊗u_{n}} \ar[r]^{v_{n}} & l_{2}^{m} \\ l_{2}^{m} ⊗_{π} ⋯ ⊗_{π} l_{2}^{m} \ar[r]_{P_{n}} & l_{1}^{m} \ar[u]_{i}}

Then π_{1}(v_{n}) ≤ K_{G}^{n}.

Proof. We are going to prove it by induction. The case n = 1 is obvious. For the general case, it is easy to see that the following diagram commutes

\xymatrix{(l_{1}^{m} ⊗_{ε} ⋯ ⊗_{ε} l_{1}^{m}) ⊗_{ε} l_{1}^{m} \ar[d]_{v_{n}⊗id} \ar[r]^{v_{n+1}} & l_{2}^{m} \\ l_{2}^{m} ⊗_{π} l_{1}^{m} \ar[d]_{id⊗u_{n+1}} \\ l_{2}^{m} ⊗_{π} l_{2}^{m} \ar[r]_{P_{2}} & l_{1}^{m} \ar[uu]_{i}}

By the induction hypothesis, π_{1}(v_{n}) ≤ K_{G}^{n}. Therefore ∥v_{n} ⊗ id∥ ≤ K_{G}^{n} and, thanks to Grothendieck’s theorem, we obtain that

π_{1}(v_{n+1}) ≤ ∥v_{n} ⊗ id∥ ∥id ⊗ u_{n+1}∥ ∥P_{2}∥π_{1}(i) ≤ K_{G}^{n+1}.

Now, we can give our elementary proof of the following multilinear Grothendieck’s inequality.

Theorem 2.2. For every m ∈ N, n ≥ 2, (a_{i_{1}···i_{n}})_{i_{j}=1}^{m} ⊂ K and x_{i_{1}}^{1}, …, x_{i_{n}}^{n} ∈ B_{l_{2}^{m}} we have

|∑_{i_{j}=1}^{m} a_{i_{1}···i_{n}} ∑_{k=1}^{m} x_{i_{1}}^{1}(k) ⋯ x_{i_{n}}^{n}(k)| ≤ K_{G}^{n−1} sup_{|t_{i_{j}} |≤1}|∑_{i_{j}=1}^{m} a_{i_{1}···i_{n}}t_{i_{1}} ⋯ t_{i_{n}} |.

Proof. Let us write z_{i_{n}} := ∑_{i_{1},...,i_{n−1}=1}^{m} a_{i_{1},...,i_{n}}e_{i_{1}} ⊗⋯ ⊗e_{i_{n−1}} ∈ l_{1}^{m}⊗_{ε} ⋯^{n−1}⊗_{ε}

l_{1}^{m}. It is easy to see that:

∥(z_{i_{n}})_{i_{n}=1}^{m}∥_{1}^{ω} = sup{|∑_{i_{j}=1}^{m} a_{i_{1}···i_{n}}t_{i_{1}} ⋯ t_{i_{n}} | : |t_{i_{j}}| ≤ 1}.

For every j ∈ {1, …, n − 1} we define the operator u_{j}: l_{1}^{m} ⟶ l_{2}^{m}, by u_{j}(e_{i_{j}}) = x_{i_{j}}^{j}, (1 ≤ i_{j} ≤ m). Since (e_{i_{j}}) are the extremal points of the ball of l_{1}^{m} and since ∥x_{i_{j}}^{j}∥ ≤ 1, we have that ∥u_{j}∥ ≤ 1. Applying Lemma 2.1 we get

∑_{i_{n}=1}^{m} ∥v_{n−1}(z_{i_{n}})∥ ≤ K_{G}^{n−1}∥(z_{i_{n}})_{i_{n}=1}^{m}∥_{1}^{ω}.

Finally,

\begin{aligned}∑_{i_{n}=1}^{m} ∥v_{n−1}(z_{i_{n}})∥ & ≥ | ∑_{i_{n}=1}^{m}⟨v_{n−1}(z_{i_{n}}), x_{i_{n}}^{n}⟩| \\ & = | ∑_{i_{1},...,i_{n}=1}^{m} a_{i_{1},...,i_{n}} ∑_{k=1}^{m} x_{i_{1}}^{1}(k) ⋯ x_{i_{n}}^{n}(k)|. \end{aligned}

Corollary 2.3. For any 1 ≤ p ≤ 2, the Schatten space with the Schur product (S_{p}, ∗) is a Q-algebra.

Proof. Davie’s criterion (see [7]) tells us that a commutative Banach algebra A is a Q-algebra if and only if there exists a positive constant K such that

∥ ∑_{i_{1},...,i_{n}=1}^{m} a_{i_{1},...,i_{n}}x_{i_{1}} ⋯ x_{i_{n}} ∥ ≤ K^{n} sup_{|t_{i_{j}} |≤1}| ∑_{i_{1},...,i_{n}=1}^{m} a_{i_{1}···i_{n}}t_{i_{1}} ⋯ t_{i_{n}} |,

for every sequence x_{1}, …, x_{m} ∈ A with ∥x_{i}∥ ≤ 1 and for every choice of a_{i_{1}···i_{n}} ∈ C. Therefore, it is enough to show that

∥ ∑_{i_{1},...,i_{n}=1}^{m} a_{i_{1}···i_{n}}x_{i_{1}}^{1} ∗ ⋯ ∗ x_{i_{n}}^{n} ∥_{S_{p}} ≤ K_{G}^{n−1} sup_{|t_{i_{j}} |≤1}| ∑_{i_{1},...,i_{n}=1}^{m} a_{i_{1}···i_{n}}t_{i_{1}} ⋯ t_{i_{n}} |,

for each choice of a_{i_{1}···i_{n}} ∈ C and x_{i_{1}}^{1}, …, x_{i_{n}}^{n} ∈ B_{S_{p}}.

In fact, Davie’s criterion follows for the particular case in which x_{i}^{1} = ⋯ = x_{i}^{n} for every 1 ≤ i ≤ m.

So let us choose a_{i_{1}···i_{n}} ∈ C and x_{i_{1}}^{1}, …, x_{i_{n}}^{n} ∈ B_{S_{p}}. [9, Theorem 4.7(a)] assures that ∥x∥_{S_{p}} ≤ ∑_{k,h=1}^{∞}|x(k, h)| for each x = (x(k, h))_{k,h≥1} ∈ S_{p}. Therefore, we have that

\begin{aligned}∥ ∑_{i_{1},...,i_{n}=1}^{m} a_{i_{1}···i_{n}}x_{i_{1}}^{1} ∗ ⋯ ∗ x_{i_{n}}^{n} ∥_{S_{p}} & ≤ ∑_{k,h=1}^{∞}| ∑_{i_{1},...,i_{n}=1}^{m} a_{i_{1}···i_{n}}x_{i_{1}}^{1}(k, h) ⋯ x_{i_{n}}^{n}(k, h)| \\ & = ∑_{k,h=1}^{∞} ε_{k,h} ∑_{i_{1},...,i_{n}=1}^{m}a_{i_{1}···i_{n}}x_{i_{1}}^{1}(k, h) ⋯ x_{i_{n}}^{n}(k, h) \end{aligned}

for some |ε_{k,h}| = 1. Calling x̄_{i}^{1}(k, h) = ε_{k,h}x_{i}^{1}(k, h) (trivially |x̄_{i}^{1}(k, h)| = |x_{i}^{1}(k, h)| for every k, h) and using Theorem 2.2 for (a_{i_{1}···i_{n}})_{i_{j}=1}^{m} and x̄_{i_{1}}^{1} , x_{i_{2}}^{2}, …, x_{i_{n}}^{n} we obtain

\begin{matrix}∥ ∑_{i_{1},...,i_{n}=1}^{m} a_{i_{1}···i_{n}}x_{i_{1}}^{1} ∗ ⋯ ∗ x_{i_{n}}^{n} ∥_{S_{p}} \\ ≤ K_{G}^{n−1} sup_{|t_{i_{j}} |≤1}| ∑_{i_{1},...,i_{n}=1}^{m} a_{i_{1}···i_{n}}t_{i_{1}} ⋯ t_{i_{n}} | ∏_{j=1}^{n} sup_{i_{j}}( ∑_{k,h=1}^{∞} |x_{i_{j}}^{j}(k, h)|^{2})^{1/2}. \end{matrix}

Finally, for each 1 ≤ j ≤ n and each 1 ≤ i_{j} ≤ m, we have that

( ∑_{k,h=1}^{∞} |x_{i_{j}}^{j}(k, h)|^{2})^{1/2} = ∥x_{i_{j}}^{j}∥_{S_{2}} ≤ ∥x_{i_{j}}^{j}∥_{S_{p}} ≤ 1,

and we are done.

Remark 2.4. The previous corollary proves in fact that l_{p} ˆ⊗_{π}l_{q} and l_{p}(l_{q}) are also Q-algebras with the Schur product when 1 ≤ p, q ≤ 2.

In the next corollary, in a completely different direction, we point out how Theorem 2.2 can be seen as a result concerning absolutely summing multilinear operators. It improves previous results of Botelho, Meléndez and Tonge in [4] and [15] and it has been recently used by Pellegrino in [17].

Corollary 2.5. If X_{j} is an ℒ_{∞,λ_{j}} -space (1 ≤ j ≤ n), then every multilinear form T: X_{1} × ⋯ × X_{n} ⟶ K is (1; 2 …, 2)-summing and ∥T ∥_{(1;2,...,2)} ≤ K_{G}^{n−1} ∏_{j=1}^{n} λ_{j}∥T ∥ holds.

Proof. The proof is quite standard. Because of the local behavior of the (1; 2, …, 2)-summing multilinear operators, it is enough to prove that if T: l_{∞}^{m} × ⋯ × l_{∞}^{m} ⟶ K is a multilinear form, then for every (x_{r}^{j})_{r=1}^{m} ⊂ l_{∞}^{m} such that ∥(x_{r}^{j})_{r=1}^{m}∥_{2}^{ω} ≤ 1, the following holds

∑_{r=1}^{m} |T (x_{r}^{1}, …, x_{r}^{n})| ≤ K_{G}^{n−1}∥T ∥.

Let us define T_{i_{1},...,i_{n}} = T (e_{i_{1}}, …, e_{i_{n}}) and

h_{r} = ∑_{i_{1},...,i_{n}=1}^{m} T_{i_{1}···i_{n}}x_{r}^{1}(i_{1}) ⋯ x_{r}^{n}(i_{n}), θ_{r} = \{ \begin{matrix}\frac{|h_{r}|}{h_{r}}, & h_{r} ≠ 0, \\ 0, & h_{r} = 0. \end{matrix}

If y_{i_{1}}^{1}(r) = θ_{r}x_{r}^{1}(i_{1}) and y_{i_{j}}^{j}(r) = x_{r}^{j}(i_{j}) for j ≥ 2, we have that

(∑_{r=1}^{m} |y_{i_{j}}^{j}(r)|^{2})^{1/2} ≤ (∑_{r=1}^{m} |x_{r}^{j}(i_{j})|^{2})^{1/2} ≤ ∥(x_{r}^{j})_{r=1}^{m}∥_{2}^{ω} ≤ 1.

Therefore y_{i_{j}}^{j} ∈ B_{l_{2}^{m}} and so, by Theorem 2.2,

| ∑_{i_{1},...,i_{n}=1}^{m} T_{i_{1}···i_{n}} ∑_{r=1}^{m} y_{i_{1}}^{1}(r) ⋯ y_{i_{n}}^{n}(r)| ≤ K_{G}^{n−1}∥T ∥.

But

\begin{aligned}| ∑_{i_{1},...,i_{n}=1}^{m} T_{i_{1}···i_{n}} ∑_{r=1}^{m} y_{i_{1}}^{1}(r) ⋯ y_{i_{n}}^{n}(r)| & = |∑_{r=1}^{m} θ_{r} ∑_{i_{1},...,i_{n}=1}^{m} T_{i_{1}···i_{n}}x_{r}^{1}(i_{1}) ⋯ x_{r}^{n}(i_{n})| \\ & = ∑_{r=1}^{m} |T (x_{r}^{1}, …, x_{r}^{n})| \end{aligned}

and we are done.

Remark 2.6. In [6] the authors use the fact that every n-linear form T: c_{0} × ⋯ × c_{0} ⟶ K is (1; 1, …, 1)-summing to obtain quite easily certain bounds for polynomials and multilinear forms on c_{0} , which had been obtained previously and with different techniques in [1], [2] and [25]. The above result could help to obtain better bounds along the same lines.

Finally, we are going to extend Theorem 2.2 to the non-commutative setting. To do that, we are going to use a tricky induction argument of [24] and several deep results of Haagerup, Pisier and Tomczak-Jaegermann. Though it is not the standard in the theory of C^{∗} -algebras, following [21] we will denote |x|^{2} = \frac{1}{2}(xx^{∗} + x^{∗}x) when x is an element of a C^{∗} -algebra.

The starting point is the following non-commutative version of Grothendieck’s theorem:

Theorem 2.7 ([21, (9.3)]). There exists a universal constant C such that, if A_{1} , A_{2} are C^{∗} -algebras and T: A_{1} × A_{2} ⟶ C is a bilinear form, we have that

∑_{i=1}^{m}|T (x_{i}^{1}, x_{i}^{2})| ≤ C∥T ∥∥∑_{i=1}^{m} |x_{i}^{1}|^{2}∥^{1/2}∥∑_{i=1}^{m} |x_{i}^{2}|^{2}∥^{1/2}.

Remark 2.8. This result was proved for the first time in [20], in the presence of some approximation condition. The general case is due to Haagerup [11]. The formulation given in [20] and [11] is slightly different, but equivalent (as it is shown in [12, Remark 2.10(b)]).

To generalize it to the n-linear case, we need a pair of results. The first one is a consequence of [21, Theorem 4.1] and [23]. The other can be obtained easily from [21, Theorem 9.4].

Theorem 2.9. Let A be a C^{∗} -algebra and Y a finite-dimensional Banach space. For every linear operator u: A ⟶ Y there exists a Hilbert space H and operators v: X ⟶ H and w: H ⟶ Y such that u = wv and

∥w∥ ∥v∥ ≤ (4\sqrt{e} C_{2}(Y))^{3/2}∥u∥.

Theorem 2.10. Let A be a C^{∗} -algebra, H a Hilbert space and u: A ⟶ H a linear operator. Then

∑_{i=1}^{m} ∥u(x_{i})∥^{2} ≤ 4∥u∥^{2}∥∑_{i=1}^{m} |x_{i}|^{2}∥,

for each finite sequence (x_{i})_{i=1}^{m} ⊂ A.

So let us prove our result:

Theorem 2.11. If A_{1}, …, A_{n} are C^{∗} -algebras, every n-linear form T: A_{1} × ⋯ × A_{n} ⟶ C satisfies

∑_{i=1}^{m} |T (x_{i}^{1}, …, x_{i}^{n})| ≤ C^{n−1}∥T ∥∥∑_{i=1}^{m} |x_{i}^{1}|^{2}∥^{1/2} ⋯ ∥∑_{i=1}^{m} |x_{i}^{n}|^{2}∥^{1/2}

where C is an universal constant.

Proof. We are going to reason by induction. The case n = 2 is Theorem 2.7. To prove the general case we can suppose ∥T ∥ ≤ 1 and

∥∑_{i=1}^{m} |x_{i}^{j}|^{2}∥^{1/2} ≤ 1 for every 1 ≤ j ≤ n.

We consider the following operators:

\begin{aligned}u & : l_{2}^{m} ⟶ A_{1}, \\ v & : A_{1} ⟶ ℒ^{n−1}(A_{2}, …, A_{n}), \\ w & : ℒ^{n−1}(A_{2}, …, A_{n}) ⟶ l_{1}^{m}, \\ i & : l_{1}^{m} ↪ l_{2}^{m}, \end{aligned}  \begin{aligned}u(e_{i}) & = x_{i}^{1}, \\ v(x^{1}) & = T (x^{1}, · , …, ·), \\ w(S) & = (S(x_{i}^{2}, …, x_{i}^{n}))_{i=1}^{m}, \\ i(e_{i}) & = e_{i}. \end{aligned}

Clearly ∥v∥ ≤ 1 and by the induction hypothesis ∥w∥ ≤ C^{n−2} . Now, by Theorem 2.9 there exists a Hilbert space H and operators ṽ: A_{1} ⟶ H, w̃: H ⟶ l_{1}^{m} such that w̃ṽ = wv and ∥w̃∥ ≤ C^{n−2} , ∥ṽ∥ ≤ C′, where C′ is a universal constant.

We can now consider ṽu: l_{2}^{m} ⟶ H. By Theorem 2.10,

∑_{i=1}^{m} ∥ṽu(e_{i})∥^{2} = ∑_{i=1}^{m} ∥ṽ(x_{i}^{1})∥ ≤ 4∥ṽ∥^{2}∥∑_{i=1}^{m} |x_{i}^{1}|^{2}∥.

Then, by [9, Corollary 4.8], the Hilbert–Schmidt norm of ṽu satisfies ∥ṽu∥_{S_{2}} ≤ 4C′. But, by Grothendieck’s theorem and [9, Theorem 4.10], it is also true that the Hilbert–Schmidt norm of iw̃: H ⟶ l_{2}^{m} satisfies ∥iw̃∥_{S_{2}} ≤ K_{G}C^{n−2} .

Therefore, by [19, Theorem 15.5.9], the composition iwvu: l_{2}^{m} ⟶ l_{2}^{m} satisfies that ∥iwvu∥_{S_{1}} ≤ 4K_{G}C′C^{n−2} and so ∥iwvu∥_{S_{1}} ≤ C^{n−1} if we choose C greater than 4K_{G}C′ and big enough to satisfy the case n = 2.

Using [19, Theorem 15.5.7], we can conclude that

∑_{i=1}^{m} |T (x_{i}^{1}, …, x_{i}^{n})| = ∑_{i=1}^{m}|(iwvu(e_{i}) | e_{i})| ≤ ∥iwvu∥_{S_{1}} ≤ C^{n−1}

where ( · | ·) denotes the scalar product in l_{2}^{m}.

Acknowledgement. This paper was written while the author was visiting Kent State University to which thanks are acknowledged. The author would also like to thank F. Bombal, A. Tonge and I. Villanueva for many helpful conversations.

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Received 18 January 2005