New York Journal of Mathematics

New York J. Math. 15 (2009) 199–209.

On the structure of derivations on certain nonamenable nuclear Banach algebras

Ana L. Barrenechea and Carlos C. Peña

Abstract. We develop the structure of general bounded derivations on the algebra of 𝔛^{∗}-nuclear operators on an infinite-dimensional Banach space 𝔛 that admits a shrinking basis.

Contents

1. Introduction 199 2. An isomorphism theorem 201 3. Shrinking basis and tensor products 202 4. Some structure theorems 203 References 208

1. Introduction

Let (𝔛, 𝔜, ⟨∘, ∘⟩) be a dual pair of Banach spaces, i.e., a pair of Banach spaces (𝔛, 𝔜) with a nondegenerate, bounded, bilinear map ⟨∘, ∘⟩ : 𝔛 × 𝔜 →ℂ. By N_{𝔜} (𝔛) we denote the class of so-called 𝔜-nuclear operators T ∈ B (𝔛) which can be written as T x = ∑_{n=1}^{∞} ⟨x, y_{n}⟩ x_{n} if x ∈ 𝔛, with {x_{n}}_{n=1}^{∞} ⊆ 𝔛, {y_{n}}_{n=1}^{∞} ⊆ 𝔜 and ∑_{n=1}^{∞} ∥x_{n}∥ ∥y_{n}∥ < ∞. The infimum of these series taking over all such representations of T furnish a norm ∥T ∥ of T so that (N_{𝔜} (𝔛) , ∥∘∥) becomes a Banach space. By the universal property of the tensor product there is a unique linear function τ : 𝔛 ⊗ 𝔜 →N_{𝔜} (𝔛) so that τ (x ⊗ y) = x ⊙ y if x ∈ 𝔛, y ∈ 𝔜, with (x ⊙ y) (z) = ⟨z, y⟩ x if

199^{ISSN 1076-9803/09}

Received May 22, 2007; revised March 26, 2009.

Mathematics Subject Classification. 46H20 46H25.

Key words and phrases. Amenable, superamenable, biprojective and biflat Banach algebras. Bounded approximate identities. Multipliers of a Banach space on a fixed basis.

Injective and projective tensor product of Banach spaces.

x, z ∈ 𝔛, y ∈ 𝔜. Since 𝔛 ⊗ 𝔜 is dense in the projective tensor product then τ admits a unique extension Λ to 𝔛ˆ⊗𝔜. If

N (𝔛, 𝔜) = {𝔮 ∈ 𝔛ˆ⊗𝔜 : 𝔮 = ∑_{n=1}^{∞} x_{n} ⊗ y_{n}, ∑_{n=1}^{∞} ∥x_{n}∥ ∥y_{n}∥ < ∞}

then N (𝔛, 𝔜) is a linear subspace of 𝔛ˆ⊗𝔜. Given 𝔮 ∈ N (𝔛, 𝔜) it is readily seen that

∥𝔮∥_{π} = inf {∑_{n=1}^{∞} ∥x_{n}∥ ∥y_{n}∥ : 𝔮 = ∑_{n=1}^{∞} x_{n} ⊗ y_{n}}

and that N (𝔛, 𝔜) is closed in 𝔛ˆ⊗𝔜. As 𝔛 ⊗ 𝔜 ⊆N (𝔛, 𝔜) then N (𝔛, 𝔜) = 𝔛ˆ⊗𝔜 and Λ becomes surjective. Indeed, by the same reasoning it is easy to see that c_{0} (𝔛)⊗l^{1} (𝔜) ≈ 𝔛ˆ⊗𝔜, where ≈ denotes an isomorphism of Banach spaces. Now, by the open mapping theorem we deduce that N_{𝔜} (𝔛) ≈ 𝔛ˆ⊗𝔜/ ker Λ.

The projective tensor product has a Banach algebra structure given by the multiplication (x_{1} ⊗ y_{1}) (x_{2} ⊗ y_{2}) = ⟨x_{2}, y_{1}⟩ (x_{1} ⊗ y_{2}) if x_{1}, x_{2} ∈ 𝔛, y_{1}, y_{2} ∈ 𝔜. So, 𝔛ˆ⊗𝔜 becomes biprojective and hence biflat. Thus 𝔛ˆ⊗𝔜 is amenable if and only if it has a bounded approximate identity (cf. [8], Theorem 2.21). In the case of Banach pairings this is indeed the case, so that 𝔛ˆ⊗𝔜 is amenable if and only if dim (𝔛) < ∞ (cf. [6]). The amenability and superamenability of 𝔛ˆ⊗𝔜 impose finite dimensionality of 𝔛 and 𝔜. Moreover,

Theorem 1 (cf. [10], Theorem 4.3.5, p. 98). For a dual Banach pair

(𝔛, 𝔜, ⟨∘, ∘⟩) ,

the following assertions are equivalent:

(a) 𝔛ˆ⊗𝔜 is superamenable.

(b) 𝔛ˆ⊗𝔜 is amenable.

(c) 𝔛ˆ⊗𝔜 has a bounded approximate identity.

(d) 𝔛ˆ⊗𝔜 has a bounded left approximate identity.

(e) N_{𝔜} (𝔛) has a bounded left approximate identity.

(f) dim (𝔛) = dim (𝔜) < ∞.

Throughout this article by 𝔛 we will denote an infinite dimensional Banach space having a shrinking basis (see Section 3). The sets of all bounded finite rank operators on 𝔛 and of all bounded derivations on a Banach algebra 𝔘 will be denoted as 𝔉 (𝔛) and D (𝔘) respectively. As usual, the closure in B (𝔛) of 𝔉 (𝔛), the set of approximable operators on 𝔛, will be denoted as A (𝔛). We will be concerned about structure theorems of derivations on N_{𝔛^{∗}} (𝔛). For previous related studies on which the underlying space 𝔛 consists of nuclear or Hilbert–Schmidt operators on a separable Hilbert space the reader can see [1] or [2]. In Section 2 we will assume that 𝔛 satisfies the approximation property in the sense of A. Grothendieck, i.e., there is a net {S_{a}}_{a∈A} in F (𝔛) such that lim_{a∈A} S_{a} = Id_{𝔛} uniformly on compact subsets of 𝔛 (cf. [7], p. 165). Then there is an isometric isomorphism of Banach algebras between N_{𝔛^{∗}} (𝔛) and 𝔛ˆ⊗𝔛^{∗} (cf. [10], Theorem C.1.5). This will allow us to transfer our investigation to the frame of the projective Banach algebra 𝔛ˆ⊗𝔛^{∗} (see Remark 5). By Theorem 1 the determination of structure theorems of derivations on 𝔛ˆ⊗𝔛^{∗} has its own interest when 𝔛 is infinite-dimensional. To this end we will focus on the case when 𝔛 admits a shrinking basis. So, in Section 3 we will consider an infinite-dimensional Banach space 𝔛 endowed with a shrinking basis and we will show briefly how the basis may be used to construct a basis for 𝔛ˆ⊗𝔛^{∗}. Our main results rely on Theorem 7 in Section 4. In this theorem we will develop the precise structure of general bounded derivations on 𝔛ˆ⊗𝔛^{∗}. Later, we will introduce the notion of Hadamard derivations and we will prove in Theorem 10 that they constitute a Banach complementary subspace of D(𝔛ˆ⊗𝔛^{∗}). Finally, in Proposition 12 and Theorem 13 we will investigate how bounded operators on 𝔛 induce bounded derivations on 𝔛ˆ⊗𝔛^{∗}.

2. An isomorphism theorem

Proposition 2. Let τ : 𝔛ˆ⊗𝔛^{∗} → N_{𝔛^{∗}} (𝔛) be the unique bounded operator so that τ(x⊗x^{∗}) = x⊙x^{∗} for all basic tensors. If 𝔛 satisfies the approximation property then τ is a Banach algebra isometric isomorphism.

Proof. Let u ∈ 𝔛ˆ⊗𝔛^{∗} −{0}, say u = ∑_{n=1}^{∞} x_{n} ⊗x_{n}^{∗} with ∑_{n=1}^{∞} ∥x_{n}∥ ∥x_{n}^{∗}∥ < ∞. Indeed, we can assume that x_{n} → 0 and ∑_{n=1}^{∞} ∥x_{n}^{∗}∥ < ∞. Since {x_{n}}_{n=1}^{∞} ∪ {0} is compact and 𝔛 has the approximation property, we may find S ∈ F (𝔛) so that

∥Sx_{n} − x_{n}∥ < \frac{∥u∥_{π}}{2 ∑_{n=1}^{∞} ∥x_{n}^{∗}∥}

for all n ∈ ℕ. If v = ∑_{n=1}^{∞} Sx_{n} ⊗ x_{n}^{∗} then

∥u − v∥_{π} ≤ ∑_{n=1}^{∞} ∥Sx_{n} − x_{n}∥ ∥x_{n}^{∗}∥ < ∥u∥_{π} /2,

and so v ≠ 0. Let S = ∑_{m=1}^{p} y_{m} ⊙ y_{m}^{∗} with y_{m} ∈ 𝔛, y_{m}^{∗} ∈ 𝔛^{∗}. Since

v = ∑_{n=1}^{∞} (∑_{m=1}^{p} ⟨x_{n}, y_{m}^{∗}⟩ y_{m}) ⊗ x_{n}^{∗} = ∑_{m=1}^{p} y_{m} ⊗ ∑_{n=1}^{∞} ⟨x_{n}, y_{m}^{∗}⟩ x_{n}^{∗} ≠ 0

there is y^{∗} ∈ 𝔛^{∗} so that ∑_{n=1}^{∞} ⟨x_{n}, y^{∗}⟩ x_{n}^{∗} ≠ 0 in 𝔛^{∗}. Consequently, there exists y^{∗∗} ∈ 𝔛^{∗∗} so that

0 ≠ ⟨∑_{n=1}^{∞} ⟨x_{n}, y^{∗}⟩ x_{n}^{∗}, y^{∗∗}⟩ = ∑_{n=1}^{∞} ⟨x_{n}, y^{∗}⟩ ⟨x_{n}^{∗}, y^{∗∗}⟩ = (y^{∗} ⊗ y^{∗∗}) (u) ,

i.e., u ≠ 0 and the canonical map ι : 𝔛ˆ⊗𝔛^{∗} ↪ 𝔛 ⊗^{∨} 𝔛^{∗} is injective. But, if w = ∑_{j=1}^{q} z_{j} ⊗ z_{j}^{∗} in 𝔛 ⊗ 𝔛^{∗} then

∥τ (w)∥ = ∥τ (w)^{∗∗}∥

= sup_{∥x^{∗∗}∥_{𝔛^{∗∗}}=1} ∥x^{∗∗} ∘ τ (w)^{∗}∥

= sup_{∥x^{∗∗}∥_{𝔛^{∗∗}}=1} sup_{∥x^{∗}∥_{𝔛^{∗}=1}} |⟨x^{∗}, x^{∗∗} ∘ τ (w)^{∗}⟩|

= sup_{∥x^{∗∗}∥_{𝔛^{∗∗}}=1} sup_{∥x^{∗}∥_{𝔛^{∗}=1}} |⟨x^{∗} ∘ τ (w) , x^{∗∗}⟩|

= sup_{∥x^{∗∗}∥_{𝔛^{∗∗}}=1} sup_{∥x^{∗}∥_{𝔛^{∗}=1}} |∑_{j=1}^{q} ⟨z_{j}, x^{∗}⟩ ⟨z_{j}^{∗}, x^{∗∗}⟩|

= sup_{∥x^{∗∗}∥_{𝔛^{∗∗}}=1} sup_{∥x^{∗}∥_{𝔛^{∗}=1}} |(x^{∗} ⊗ x^{∗∗}) (w)| = ∥w∥_{ϵ} ,

i.e., τ |_{𝔛⊗𝔛}^{∗}: 𝔛 ⊗ 𝔛^{∗} ↪ F (𝔛) is an isometry. Therefore τ extends to an

isometric isomorphism τ̃ between 𝔛 ⊗^{∨} 𝔛^{∗} and A (𝔛). Certainly, τ = τ̃ ∘ ι becomes isometric. Indeed, we already know that τ is onto and by definition of the nuclear norm it is an isometric isomorphism. Finally, since

τ ((x_{1} ⊗ x_{1}^{∗}) (x_{2} ⊗ x_{2}^{∗})) = ⟨x_{2}, x_{1}^{∗}⟩ τ (x_{1} ⊗ x_{2}^{∗})
= ⟨x_{2}, x_{1}^{∗}⟩ x_{1} ⊙ x_{2}^{∗}
= (x_{1} ⊙ x_{1}^{∗}) ∘ (x_{2} ⊙ x_{2}^{∗})
= τ (x_{1} ⊗ x_{1}^{∗}) ∘ τ (x_{2} ⊗ x_{2}^{∗})

if x_{1}, x_{2} ∈ 𝔛, x_{1}^{∗}, x_{2}^{∗} ∈ 𝔛^{∗} the assertion follows. □

3. Shrinking basis and tensor products

If {x_{n}}_{n=1}^{∞} is a basis of 𝔛 there is {x_{n}^{∗}}_{n=1}^{∞} ⊆ 𝔛^{∗} so that ⟨x_{n}, x_{m}^{∗}⟩ = δ_{n,m} with n, m ∈ ℕ, i.e., {x_{n}^{∗}}_{n=1}^{∞} is the associated sequence of coefficient functionals (a.s.c.f.) of {x_{n}}_{n=1}^{∞} (see [12], Theorem 3.1, p. 20). Certainly, {x_{n}^{∗}}_{n=1}^{∞} need not be a basis of 𝔛^{∗} since 𝔛^{∗} may be nonseparable and so it may have no basis at all. However, {x_{n}^{∗}}_{n=1}^{∞} is a basis if 𝔛 is a reflexive Banach space (cf. [9]). In the sequel, we will assume that {x_{n}}_{n=1}^{∞} is a shrinking basis. This means, by definition, that the a.s.c.f., {x_{n}^{∗}}_{n=1}^{∞}, is a basis for 𝔛^{∗} (cf. [3], [5]).

Proposition 3 (cf. [11], [12]). Let 𝔛 be a Banach space, let {x_{n}}_{n=1}^{∞} be a shrinking basis of 𝔛 and let {x_{n}^{∗}}_{n=1}^{∞} be the a.s.c.f. Then the system of all basic tensor products x_{n} ⊗ x_{m}^{∗} is a basis of 𝔛ˆ⊗𝔛^{∗}, when arranged into a single sequence as follows: if m ∈ ℕ let n ∈ ℕ so that (n − 1)^{2} < m ≤ n^{2} and then write

z_{m} = x_{σ_{1}(m)} ⊗ x_{σ_{2}(m)},

with

σ(m) = \begin{cases}(m − (n − 1)^{2}, n) & if (n − 1)^{2} + 1 ≤ m ≤ (n − 1)^{2} + n, \\ (n, n^{2} − m + 1) & if (n − 1)^{2} + n ≤ m ≤ n^{2}. \end{cases}

Remark 4. In particular, σ : ℕ → ℕ × ℕ becomes a bijective function. Since 𝔛^{∗} ˆ⊗𝔛 ↪ (𝔛ˆ⊗𝔛^{∗})^{∗} we will also write z_{m}^{∗} = x_{σ_{1}(m)}^{∗} ⊗ x_{σ_{2}(m)}, m ∈ ℕ. Thus {z_{m}^{∗}}_{n=1}^{∞} becomes the a.s.c.f. of {z_{m}}_{n=1}^{∞}.

Remark 5. A Banach space 𝔛 with a shrinking basis {x_{n}}_{n=1}^{∞} satisfies the approximation property. For, it suffices to observe that if {x_{n}^{∗}}_{n=1}^{∞} is the corresponding a.s.c.f. the sequence {∑_{n=1}^{m} x_{n} ⊙ x_{n}^{∗}}_{m=1}^{∞} in F (𝔛) converges uniformly on compact subsets of 𝔛 to Id_{𝔛} (cf. [10], p. 255).

The proof of the following is straightforward.

Proposition 6. With the above notation, for n, m ∈ ℕ the following assertions hold:

σ_{1}^{−1}({n}) = {n^{2} − k + 1}_{1≤k≤n} ∪ {k^{2} + n}_{k≥n},
σ_{2}^{−1}({m}) = {(m − 1)^{2} + k}_{1≤k≤m} ∪ {k^{2} − m + 1}_{k>m}.

In particular

(1)  σ^{−1}(n, m) = \begin{cases}n^{2} − m + 1 & if 1 ≤ m ≤ n, \\ (m − 1)^{2} + n & if m > n. \end{cases}

4. Some structure theorems

Theorem 7. Let 𝔛 be an infinite-dimensional Banach space with a shrinking basis {x_{n}}_{n=1}^{∞}. Given δ ∈ D(𝔛ˆ⊗𝔛^{∗}) there are unique sequences {𝔥_{n}}_{n∈ℕ} and {𝔶_{u}^{v}}_{u,v∈ℕ} so that if u, v ∈ ℕ then

(2) δ (z_{σ^{−1}(u,v)}) = (𝔥_{u} − 𝔥_{v}) z_{σ^{−1}(u,v)} + ∑_{n=1}^{∞} (𝔶_{u}^{n} · z_{σ^{−1}(n,v)} − 𝔶_{n}^{v} · z_{σ^{−1}(u,n)}) .

In the sequel we will say that they are the 𝔥 and 𝔶 sequences of δ.

Proof. Let {𝔥_{u,v}^{n}}_{u,v,n∈ℕ} ⊆ ℂ so that δ (z_{σ^{−1}(u,v)}) = ∑_{n=1}^{∞} 𝔥_{u,v}^{n}z_{n} if u, v ∈ ℕ. If u, v, t ∈ ℕ then x_{u} ⊗ x_{v}^{∗} = (x_{u} ⊗ x_{t}^{∗}) (x_{t} ⊗ x_{v}^{∗}) and so

(3)  δ (x_{u} ⊗ x_{v}^{∗}) = ∑_{n=1}^{∞} 𝔥_{u,v}^{n} (x_{σ_{1}(n)} ⊗ x_{σ_{2}(n)}^{∗})

= δ (x_{u} ⊗ x_{t}^{∗}) (x_{t} ⊗ x_{v}^{∗}) + (x_{u} ⊗ x_{t}^{∗}) δ (x_{t} ⊗ x_{v}^{∗})

= ∑_{n∈σ_{1}^{−1}({t})} 𝔥_{t,v}^{n} (x_{u} ⊗ x_{σ_{2}(n)}^{∗})

+ ∑_{n∈σ_{2}^{−1}({t})} 𝔥_{u,t}^{n} (x_{σ_{1}(n)} ⊗ x_{v}^{∗})

= ∑_{n=1}^{t} {𝔥_{u,t}^{(t−1)^{2}+n}z_{σ^{−1}(n,v)} + 𝔥_{t,v}^{t^{2}−n+1}z_{σ^{−1}(u,n)}}

+ ∑_{n=t}^{∞} {𝔥_{u,t}^{n^{2}−t+1}z_{σ^{−1}(n,v)} + 𝔥_{t,v}^{n^{2}+t}z_{σ^{−1}(u,n+1)}} .

Therefore by (3) we get 𝔥_{u,v}^{n} = 0 if n ∉ σ_{1}^{−1} ({u}) ∪ σ_{2}^{−1} ({v}) for all u, v, n ∈ ℕ. Now, by evaluating 𝔥_{u,v}^{σ^{−1}(u,v)} from (3) the following assertions hold:

(4)  t ≤ u < v ⇒ 𝔥_{u,v}^{(v−1)^{2}+u} = 𝔥_{t,v}^{(v−1)^{2}+t} + 𝔥_{u,t}^{u^{2}−t+1},
u < t < v ⇒ 𝔥_{u,v}^{(v−1)^{2}+u} = 𝔥_{t,v}^{(v−1)^{2}+t} + 𝔥_{u,t}^{(t−1)^{2}+u},
u < v ≤ t ⇒ 𝔥_{u,v}^{(v−1)^{2}+u} = 𝔥_{t,v}^{t^{2}−v+1} + 𝔥_{u,t}^{(t−1)^{2}+u}.

In particular, 𝔥_{n,n}^{σ^{−1}(n,n)} = 0 if n ∈ ℕ. Likewise,

(5)  t < u = v ⇒ 𝔥_{t,u}^{(u−1)^{2}+t} + 𝔥_{u,t}^{u^{2}−t+1} = 0,
t = u = v ⇒ 𝔥_{t,t}^{t^{2}−t+1} = 0,
u = v < t ⇒ 𝔥_{t,u}^{t^{2}−u+1} + 𝔥_{u,t}^{(t−1)^{2}+u} = 0.

Finally,

(6)  t ≤ v < u ⇒ 𝔥_{u,v}^{u^{2}−v+1} = 𝔥_{t,v}^{(v−1)^{2}+t} + 𝔥_{u,t}^{u^{2}−t+1},
v < t ≤ u ⇒ 𝔥_{u,v}^{u^{2}−v+1} = 𝔥_{t,v}^{t^{2}−v+1} + 𝔥_{u,t}^{u^{2}−t+1},
v < u < t ⇒ 𝔥_{u,v}^{u^{2}−v+1} = 𝔥_{t,v}^{t^{2}−v+1} + 𝔥_{u,t}^{(t−1)^{2}+u}.

Consequently, from (4), (5) and (6) we deduce that

(7)  𝔥_{u,v}^{σ^{−1}(u,v)} = 𝔥_{u,t}^{σ^{−1}(u,t)} + 𝔥_{t,v}^{σ^{−1}(t,v)} if u, v ∈ ℕ.

By (7) we can write

(8)  𝔥_{u,v}^{σ^{−1}(u,v)} = 𝔥_{u,1}^{σ^{−1}(u,1)} + 𝔥_{1,v}^{σ^{−1}(1,v)} = 𝔥_{u,1}^{σ^{−1}(u,1)} − 𝔥_{v,1}^{σ^{−1}(v,1)}.

Let us write

(9)  𝔥_{n} = 𝔥_{n,1}^{σ^{−1}(n,1)}, n ∈ ℕ.

Now, let n, u, v ∈ ℕ, n < u. By (3) we have

\begin{aligned}if n ≤ t & ⇒ & 𝔥_{u,v}^{σ^{−1}(n,v)} = 𝔥_{u,t}^{(t−1)^{2}+n}, \\ if t < n & ⇒ & 𝔥_{u,v}^{σ^{−1}(n,v)} = 𝔥_{u,t}^{n^{2}−t+1}, \end{aligned}

i.e., 𝔥_{u,v}^{σ^{−1}(n,v)} = 𝔥_{u,t}^{σ^{−1}(n,t)}. The same conclusion holds if u < n. Therefore we deduce the existence of doubly indexed sequences {𝔶_{u}^{p}}_{u,p∈ℕ}, {𝔷_{v}^{q}}_{v,q∈ℕ} so

that 𝔶_{u}^{p} = 𝔥_{u,n}^{σ^{−1}(p,n)} and 𝔷_{v}^{q} = 𝔥_{n,v}^{σ^{−1}(n,q)} if u, v, p, q, n ∈ ℕ. In particular, we already know that 𝔶_{n}^{n} = 𝔷_{n}^{n} = 0 if n ∈ ℕ. Thus

(10) δ (x_{u} ⊗ x_{v}^{∗}) = 𝔥_{u,v}^{σ^{−1}(u,v)}z_{σ^{−1}(u,v)} + ∑_{n=1}^{∞} (𝔶_{u}^{n} · z_{σ^{−1}(n,v)} + 𝔷_{v}^{n} · z_{σ^{−1}(u,n)})

and

δ_{u,v} · δ (x_{u} ⊗ x_{v}^{∗}) = δ (x_{u} ⊗ x_{v}^{∗}) (x_{u} ⊗ x_{v}^{∗}) + (x_{u} ⊗ x_{v}^{∗}) δ (x_{u} ⊗ x_{v}^{∗}) .

Hence

δ_{u,v} · δ (x_{u} ⊗ x_{v}^{∗})

= δ_{u,v}[2𝔥_{u,v}^{σ^{−1}(u,v)}z_{σ^{−1}(u,v)} + ∑_{n=1}^{∞} (𝔶_{u}^{n} · z_{σ^{−1}(n,v)} + 𝔷_{v}^{n} · z_{σ^{−1}(u,n)})]

+ (𝔶_{u}^{v} + 𝔷_{v}^{u}) · (x_{u} ⊗ x_{v}^{∗}) ,

i.e., 𝔶_{u}^{v} + 𝔷_{v}^{u} = 0 if u ≠ v in ℕ. Finally, (2) follows from (8), (9) and (10). □

Remark 8. If n, m ∈ ℕ then

⟨δ (z_{m}) , z_{n}^{∗}⟩ = \begin{cases}𝔥_{σ_{1}(m)} − 𝔥_{σ_{2}(m)} & if m = n, \\ 𝔶_{σ_{1}(m)}^{σ_{1}(n)} & if σ_{1} (n) ≠ σ_{1} (m) and σ_{2} (n) = σ_{2} (m), \\ −𝔶_{σ_{2}(n)}^{σ_{2}(m)} & if σ_{1} (n) = σ_{1} (m) and σ_{2} (n) ≠ σ_{2} (m), \\ 0 & if σ_{1} (n) ≠ σ_{1} (m) and σ_{2} (n) ≠ σ_{2} (m) \end{cases}

and

δ (𝔮) = ∑_{n=1}^{∞} (∑_{m=1}^{∞} ⟨δ (z_{m}), z_{n}^{∗})⟩ ⟨𝔮, z_{m}^{∗}⟩) z_{n} if 𝔮 ∈ 𝔛ˆ⊗𝔛^{∗}.

Definition 9. With the notation of Theorem 7 and if X = {x_{n}}_{n=1}^{∞} is a fixed basis of a Banach space 𝔛, a derivation δ ∈ D(𝔛ˆ⊗𝔛^{∗}) is called an X - Hadamard derivation if its 𝔶-sequence is the null sequence. We will denote the set of all those derivations by D_{X} (𝔛ˆ⊗𝔛^{∗}).

Theorem 10. Let 𝔛 be a Banach space with a shrinking basis X = {x_{n}}_{n=1}^{∞} and let {z_{n}}_{n=1}^{∞} be the induced basis of 𝔛ˆ⊗𝔛^{∗} as in Proposition 3. Then (a) D_{X} (𝔛ˆ⊗𝔛^{∗}) is a Banach subspace of D(𝔛ˆ⊗𝔛^{∗}).

(b) D_{X} (𝔛ˆ⊗𝔛^{∗}) ↪ M(𝔛ˆ⊗𝔛^{∗}, {z_{n}}_{n=1}^{∞}), i.e., there is an isometric isomorphism from the space of X -Hadamard derivations into the multiplier Banach space of 𝔛ˆ⊗𝔛^{∗} on the basis {z_{n}}_{n=1}^{∞}.

(c) The Banach space D_{X} (𝔛ˆ⊗𝔛^{∗}) is complementable.

Proof. (a) Observe that

D_{X} (𝔛ˆ⊗𝔛^{∗}) = ⋂_{n=1}^{∞} {δ ∈ D(𝔛ˆ⊗𝔛^{∗}) : δ (z_{σ^{−1}(n,n)}) = 0} .

(b) By Theorem 7 given δ ∈ D_{X} (𝔛ˆ⊗𝔛^{∗}) we can write

(11)  δ (𝔮) = ∑_{n=1}^{∞} (𝔥_{σ_{1}(n)} − 𝔥_{σ_{2}(n)}) ⟨𝔮, z_{m}^{∗}⟩ · z_{n}, 𝔮 ∈ 𝔛ˆ⊗𝔛^{∗}.

So, the sequence 𝔥_{δ} = {𝔥_{σ_{1}(n)} − 𝔥_{σ_{2}(n)}}_{n=1}^{∞} becomes a multiplier of 𝔛ˆ⊗𝔛^{∗}. Indeed, by (11) we have δ = M_{𝔥_{δ}}, where M is the usual isometric algebraic isomorphism (cf. [13]) of M(𝔛ˆ⊗𝔛^{∗}, {z_{n}}_{n=1}^{∞}) into B(𝔛ˆ⊗𝔛^{∗}) given by

M ({c_{n}}_{n=1}^{∞}) (𝔮) = ∑_{n=1}^{∞} c_{n} ⟨𝔮, z_{n}^{∗}⟩ · z_{n} if 𝔮 ∈ 𝔛ˆ⊗𝔛^{∗}.

(c) Let D_{X}^{⊥}(𝔛ˆ⊗𝔛^{∗}) be the set of bounded derivations on 𝔛ˆ⊗𝔛^{∗} with null 𝔥-sequences. Since

D_{X}^{⊥}(𝔛ˆ⊗𝔛^{∗}) = ⋂_{n=1}^{∞} {δ ∈ D(𝔛ˆ⊗𝔛^{∗}) : ⟨δ (z_{n^{2}}) , z_{n^{2}}^{∗}⟩ = 0}

we deduce that D_{X}^{⊥}(𝔛ˆ⊗𝔛^{∗}) is a Banach space and by Theorem 7 we have

D(𝔛ˆ⊗𝔛^{∗}) = D_{X} (𝔛ˆ⊗𝔛^{∗}) + D_{X}^{⊥}(𝔛ˆ⊗𝔛^{∗}).

Finally, it is immediate that D_{X} (𝔛ˆ⊗𝔛^{∗}) ∩ D_{X}^{⊥}(𝔛ˆ⊗𝔛^{∗}) = {0}. □

Proposition 11. Let 𝔛 be an infinite-dimensional Banach space with a shrinking basis {x_{n}}_{n=1}^{∞}. Given δ ∈ D_{X}^{⊥}(𝔛ˆ⊗𝔛^{∗}) there are unique subsets {y_{n}}_{n=1}^{∞} and {y_{n}^{∗}}_{n=1}^{∞} of 𝔛 and 𝔛^{∗} respectively so that

(12)  δ (z_{σ^{−1}(u,v)}) = y_{u} ⊗ x_{v}^{∗} − x_{u} ⊗ y_{v}^{∗} if u, v ∈ ℕ.

Proof. By Theorem 7 each series in (2) converges. Since

\overline{lim}_{p,q→∞} ∥∑_{n=p}^{p+q} 𝔶_{u}^{n} · x_{n}∥ = \frac{1}{∥x_{v}^{∗}∥}\overline{lim}_{p,q→∞} ∥(∑_{n=p}^{p+q} 𝔶_{u}^{n} · x_{n}) ⊗ x_{v}^{∗}∥_{π}

= \frac{1}{∥x_{v}^{∗}∥}\overline{lim}_{p,q→∞} ∥∑_{n=p}^{p+q} 𝔶_{u}^{n} · (x_{n} ⊗ x_{v}^{∗})∥_{π} = 0

the series ∑_{n=1}^{∞} 𝔶_{u}^{n}·x_{n} converges to an element y_{u} ∈ 𝔛 if u ∈ ℕ. Analogously, let y_{v}^{∗} = ∑_{n=1}^{∞} 𝔶_{n}^{v} · x_{n}^{∗}, v ∈ ℕ. Now (12) holds and our claim follows. □

The proof of the following is straightforward.

Proposition 12. Let 𝔛 be a Banach space, T ∈ B (𝔛). There exists a unique δ_{T} ∈ D(𝔛ˆ⊗𝔛^{∗}) so that

δ_{T} (x ⊗ x^{∗}) = T (x) ⊗ x^{∗} − x ⊗ T^{∗} (x^{∗}) if x ∈ 𝔛 and x^{∗} ∈ 𝔛^{∗}.

Theorem 13. Let 𝔛 be an infinite-dimensional Banach space with a shrinking basis {x_{n}}_{n=1}^{∞}. Let δ ∈ D(𝔛ˆ⊗𝔛^{∗}) whose associated 𝔥-sequence established in Theorem 7 is zero and let {y_{n}}_{n=1}^{∞}, {y_{n}^{∗}}_{n=1}^{∞} be the unique sequences determinated by δ in Proposition 11. Then, if

(13)  sup_{n∈ℕ} ∥∑_{i=1}^{n} y_{i} ⊗ x_{i}^{∗}∥_{π} < ∞

there is a unique T ∈ B (𝔛) so that δ = δ_{T}.

Proof. Write T (x_{u}) = y_{u}, u ∈ ℕ. Then

sup_{∥x∥_{𝔛}≤1} ∥∑_{i=1}^{n} y_{i} · ⟨x, x_{i}^{∗}⟩∥_{𝔛} = ∥∑_{i=1}^{n} y_{i} ⊙ x_{i}^{∗}∥_{B(𝔛)}

= ∥τ (∑_{i=1}^{n} y_{i} ⊗ x_{i}^{∗})∥_{B(𝔛)}

= ∥∑_{i=1}^{n} y_{i} ⊗ x_{i}^{∗}∥_{π}

≤ sup_{n∈ℕ} ∥∑_{i=1}^{n} y_{i} ⊗ x_{i}^{∗}∥_{π} < ∞

and by linearity T becomes a bounded linear operator on 𝔛 (cf. [4]). So, if x ∈ 𝔛 and v ∈ ℕ we have

⟨x, T^{∗} (x_{v}^{∗})⟩ = ⟨T (x), x_{v}^{∗}⟩

= ∑_{u=1}^{∞} ⟨x, x_{u}^{∗}⟩ ⟨T (x_{u}), x_{v}^{∗}⟩

= ∑_{u=1}^{∞} ⟨x, x_{u}^{∗}⟩ ⟨∑_{n=1}^{∞} 𝔶_{u}^{n} · x_{n}, x_{v}^{∗}⟩

= ∑_{u=1}^{∞} ⟨x, x_{u}^{∗}⟩ · 𝔶_{u}^{v} = ⟨x, y_{v}^{∗}⟩ ,

i.e., T^{∗} (x_{v}^{∗}) = y_{v}^{∗}. Thus by (12) T_{δ} (z_{n}) = δ (z_{n}) if n ∈ ℕ, i.e., T = T_{δ}. □

208 A. L. Barrenechea and C. C. Peña

Example 14. Assume that {𝔶_{n}^{m}}_{n,m∈ℕ} ∈ l^{1} (ℕ × ℕ) and that {x_{n}}_{n=1}^{∞} is a bounded shrinking basis of 𝔛, i.e., assume that

0 < inf_{n∈ℕ}∥x_{n}∥ ≜ ι < σ ≜ sup_{n∈ℕ}∥x_{n}∥ < ∞.

Then (13) holds. For, there exists M > 0 so that 1 ≤ ∥x_{n}∥ ∥x_{n}^{∗}∥ ≤ M if n ∈ ℕ (cf. [12], Theorem 3.1, p. 20). Thus,

∥∑_{i=1}^{n} y_{i} ⊗ x_{i}^{∗}∥_{π} ≤ \frac{M}{ι} ∑_{i=1}^{n} ∥y_{i}∥ ≤ \frac{M}{ι} ∑_{i=1}^{n} ∑_{j=1}^{∞}|𝔶_{i}^{j}| ∥x_{j}∥

≤ \frac{Mσ}{ι} ∑_{i=1}^{∞} ∑_{j=1}^{∞}|𝔶_{i}^{j}| ∀n ∈ ℕ.

Example 15. For 𝔮 ∈ 𝔛ˆ⊗𝔛^{∗} we have

δ_{x_{1}⊙x_{2}^{∗}} (𝔮) = ⟨𝔮, z_{4}^{∗}⟩ z_{1} − ⟨𝔮, z_{1}^{∗}⟩ z_{2}

+ ∑_{n=2}^{∞} [⟨𝔮, z_{(n−1)^{2}+2}^{∗}⟩ z_{(n−1)^{2}+1} − ⟨𝔮, z_{n^{2}}^{∗}⟩ z_{n^{2}−1}]

= ∑_{n=1}^{∞} [⟨𝔮, z_{σ^{−1}(2,n)}⟩ x_{1} ⊗ x_{n}^{∗} − ⟨𝔮, z_{σ^{−1}(n,1)}⟩ x_{n} ⊗ x_{2}^{∗}] .

With the notation of Theorem 7, since 𝔶_{2}^{1} = 1 then

δ_{x_{1}⊙x_{2}^{∗}} ∈ D(𝔛ˆ⊗𝔛^{∗}) − D_{X} (𝔛ˆ⊗𝔛^{∗}).

Indeed, it is easy to see that 𝔥_{n} = 0 if n ∈ ℕ.

Acknowledgements. The authors express our gratitude to the referees for their helpful suggestions and their advice for the writing of the final form of this article.

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UNCPBA — Departamento de Matemáticas — NUCOMPA-Argentina analucia@exa.unicen.edu.ar ccpenia@exa.unicen.edu.ar

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