PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 86(100) (2009), 107–114 DOI: 10.2298/PIM0900107B

ON BOUNDED DUAL-VALUED DERIVATIONS ON CERTAIN BANACH ALGEBRAS

A. L. Barrenechea and C. C. Peña

Communicated by Stevan Pilipović

Abstract. We consider the class 𝔇(U) of bounded derivations U →^{d} U^{∗} defined on a Banach algebra U with values in its dual space U^{∗} so that ⟨x, d(x)⟩ = 0 for all x ∈ U. The existence of such derivations is shown, but lacking the simplest structure of an inner one. We characterize the elements of 𝔇(U) if span(U^{2}) is dense in U or if U is a unitary dual Banach algebra.

1. Introduction

Throughout this article let U be a complex Banach algebra endowed with a norm ∥ ∘ ∥. Let U^{♯} be the algebra U plus an adjoined unit element e with the usual Banach algebra structure. As usual, by U^{∗} and (U^{♯})^{∗} we will denote the dual spaces of U and U^{♯} respectively. Let j : U ↪ U^{♯} and p : U^{♯} → U be the natural injection and the corresponding projection of U into U^{♯} and of U^{♯} onto U respectively. Then U^{♯} = ℂ·e ⨁ j(U), i.e., any element η ∈ U^{♯} can be written in a unique way as η = ae + j(x), with x ∈ U and a ∈ ℂ, and its U^{♯}-norm is given as ∥η∥_{U^{♯}} = |a| + ∥x∥. Indeed, since j is an isometric homomorphism then j(U) becomes a closed ideal of U^{♯}. Further, p∘j = Id_{U} while j ∘p is the linear projection of U^{♯} onto j(U). Thus, let e^{∗} ∈ (U^{♯})^{∗} be defined as ⟨η, e^{∗}⟩ ≜ a if η = ae + j(x) in U^{♯}. Then

(U^{♯})^{∗} = ℂ · e^{∗} ⨁ rank(p^{∗}),

where p^{∗} : U^{∗} → (U^{♯})^{∗} is the adjoint operator of p. It is well known that U^{∗} admits a Banach U-bimodule structure if for x, y ∈ U and x^{∗} ∈ U^{∗} we write

⟨y, xx^{∗}⟩ ≜ ⟨yx, x^{∗}⟩ and ⟨y, x^{∗}x⟩ ≜ ⟨xy, x^{∗}⟩.

2000 Mathematics Subject Classification: 46H35, 47D30.

Key words and phrases: Dual Banach algebras, approximation property, dual Banach pairs, nuclear operators, shrinking basis and associated sequence of coefficient functionals.

The U^{♯}-bimodule structure on (U^{♯})^{∗} is given as
(ae + j(x))(be^{∗} + p^{∗}(x^{∗})) ≜ (ab + ⟨x, x^{∗}⟩)e^{∗} + p^{∗}(ax^{∗} + xx^{∗}),
(be^{∗} + p^{∗}(x^{∗}))(ae + j(x)) ≜ (ab + ⟨x, x^{∗}⟩)e^{∗} + p^{∗}(ax^{∗} + x^{∗}x),

where a, b ∈ ℂ, x ∈ U, x^{∗} ∈ U^{∗}.

Given a Banach U-bimodule 𝔛 let Z^{1}(U, 𝔛) be the Banach space of bounded derivations d : U → 𝔛, i.e., those d ∈ B(U, 𝔛) that satisfy the Leibnitz rule d(xy) = d(x)y + xd(y) for all x, y ∈ U. A bounded derivation d is said to be inner if there is an element ϕ ∈ 𝔛 so that d(x) = xϕ−ϕx if x ∈ U. In that case we write d = ad_{ϕ} and the class of inner derivations from U into 𝔛 is denoted as N^{1}(U, 𝔛). The quotient H^{1}(U, 𝔛) ≜ Z^{1}(U, 𝔛)/N^{1}(U, 𝔛) defines the first Hochschild cohomology group of U with coefficients in 𝔛. Kamowitz lay the functional analytic overtones required to adapt the theory of Banach algebras to the Hochschild algebraic setting (cf. [6]; see also [4]). The theory of amenable Banach algebras was greatly influenced by Johnson’s memoire in 1972 (cf. [5]). A Banach algebra U is called amenable if H^{1}(U, 𝔛^{∗}) = {0} for any Banach U-bimodule 𝔛. A Banach algebra U it is called weakly amenable if H^{1}(U, U^{∗}) = {0}. This last notion generalizes that introduced by Bade, Curtis and Dales in [2].

In [3] it was proved that a non-unital abelian Banach algebra U is weakly amenable if and only if U^{♯} is weakly amenable but the general case still remains open. Our goal in this article is to seek relationships between derivations on a non-abelian non-unital Banach algebra U with values in U^{∗} and derivations in U^{♯} with values in (U^{♯})^{∗}. Our investigation naturally bring us to introduce the notion of 𝔇-derivations on U in Definition 2.1. Althought any element of N^{1}(U, U^{∗}) is a 𝔇-derivation on U sometimes there exist non-inner 𝔇-derivations as we will see in the examples 2.2 and 2.3. In Proposition 2.1 we consider certain Banach projective tensor products all of whose derivations are 𝔇-derivations. In Theorem 2.1 we characterize 𝔇-derivations on U on Banach algebras U so that U^{2} is dense in U, where U^{2} = span{xy : x, y ∈ U}. The relationship between 𝔇-derivations on U and their extensions to the unitization U^{♯} are studied in Proposition 2.2. In this context, inner 𝔇-derivations on U are characterized in Corollary 2.1. Finally, in Proposition 2.3 we characterize 𝔇-derivations on dual Banach algebras.

2. On 𝔇-derivations

Definition 2.1. A derivation d ∈ Z^{1}(U, U^{∗}) is called a 𝔇-derivation on U if ⟨x, d(x)⟩ = 0 for all x ∈ U. Let 𝔇(U) be the set of 𝔇-derivations on U.

Example 2.1. All inner U^{∗}-valued derivations on U are 𝔇-derivations on U.

Example 2.2. Let U ≜ C_{0}^{(1)}[a, b] be the commutative Banach algebra of functions x : [a, b] → ℂ with continuous derivative ẋ so that x(a) = x(b) endowed with the norm ∥x∥ ≜ ∥x∥_{∞} + ∥ẋ∥_{∞} . Then we define d : U → U^{∗} as

⟨y, d(x)⟩ = ∫_{a}^{b}y dx if x, y ∈ U.

The above Riemann–Stieltjes integral is well defined, d becomes clearly a ℂ-linear functional and

|⟨y, d(x)⟩| ⩽ ∫_{a}^{b}|y| d|x| ⩽ ∥y∥_{∞} |ẋ∥_{∞} ⩽ ∥y∥ ∥x∥,

i.e., d ∈ B(U, U^{∗}) and ∥d(x)∥ ⩽ ∥x∥ for all x ∈ U. Indeed,

⟨y, d(x_{1}x_{2})⟩= ∫_{a}^{b}y \frac{d}{dt} (x_{1}x_{2}) dt

= ∫_{a}^{b}y( \frac{dx_{1}}{dt} x_{2} + x_{1} \frac{dx_{2}}{dt} ) dt

= ∫_{a}^{b}yx_{2}dx_{1} + ∫_{a}^{b}yx_{1}dx_{2}

= ⟨yx_{2}, d(x_{1})⟩ + ⟨yx_{1}, d(x_{2})⟩

= ⟨y, d(x_{1})x_{2} + x_{1}d(x_{2})⟩,

i.e., d ∈ Z^{1}(U, U^{∗}). Certainly, it is a nonzero 𝔇-derivation since for x ∈ U we see that

⟨x, d(x)⟩ = ∫_{a}^{b}x\frac{dx}{dt} dt = \frac{x^{2}}{2} |_{a}^{b} = 0.

Further, d is not inner because U is abelian and so N^{1}(U, U^{∗}) = {0}.

Remark 2.1. Given a dual Banach pair (𝔛, 𝔜, ⟨∘, ∘⟩) by the universal property characteristic of general tensor products there is a unique operation on 𝔛 ⊗ 𝔜 so that

(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.

Then 𝔛 ⊗ 𝔜 becomes an algebra. Further, if for u ∈ 𝔛 ⊗ 𝔜 we write

∥u∥_{π} = inf{ ∑_{j=1}^{n} ∥x_{j} ∥ ∥y_{j} ∥ : u = ∑_{j=1}^{n} x_{j} ⊗ y_{j} }

then (𝔛⊗𝔜, ∥∘∥_{π}) becomes a normed algebra. The completion of this algebra is the well known projective Banach tensor algebra 𝔛 ˆ⊗ 𝔜 (cf. [8, B.2.2, p. 250]). Then, 𝔛 ˆ⊗ 𝔜 is amenable if and only if dim(𝔛) = dim(𝔜) < ∞ (cf. [8, Th. 4.3.5, p. 98]). So, if 𝔛 is an infinite dimensional Banach space the determination of structure theorems of bounded derivations on 𝔛 ˆ⊗ 𝔛^{∗} has its own interest. Moreover, several Banach operator algebras can be represented as certain tensor products of the above type. For instance, if the Banach space 𝔛 has the approximation property, then N_{𝔛^{∗}} (𝔛) ≈ 𝔛 ˆ⊗ 𝔛^{∗}, where ≈ denotes an isometric isomorphism and N_{𝔛^{∗}} (𝔛) is the Banach space of 𝔛^{∗}-nuclear operators on 𝔛 (cf. [8, Th. C.1.5, p. 256]). In this setting the authors recently researched on structure theorems and properties of derivations on some non-amenable nuclear Banach algebras (see [1]).

Proposition 2.1. Let 𝔛 is an infinite dimensional Banach space endowed with and shrinking basis {x_{n}}_{n=1}^{∞} and an associated sequence of coefficient functionals {x_{n}^{∗}}_{n=1}^{∞} and let U ≜ 𝔛 ˆ⊗ 𝔛^{∗}. Then 𝔇(U) ⊇ Z^{1}(U, U^{∗}).

Proof. Let d ∈ Z^{1}(U, U^{∗}). The system of basic tensor products z_{n,m} ≜ x_{n} ⊗ x_{m}^{∗} can be arranged into a basis {z_{n,m} }_{n,m=1}^{∞} of 𝔛 ˆ⊗ 𝔛^{∗} (the reader can see [9], or else [10, Th. 18.1, p. 172]). Given p, q, r, s, t ∈ ℕ we have

(2.1)  ⟨z_{p,q}, d(z_{r,t})⟩= ⟨z_{p,q}, d(z_{r,s} ) · z_{s,t} + z_{r,s} · d(z_{s,t})⟩
= ⟨z_{s,t} · z_{p,q}, d(z_{r,s} )⟩ + ⟨z_{p,q} · z_{r,s} , d(z_{s,t})⟩
= δ_{p,t}⟨z_{s,q} , d(z_{r,s} )⟩ + δ_{q,r} ⟨z_{p,s} , d(z_{s,t})⟩,

where δ denotes the usual Kronecker’s symbol. By (2.1), ⟨z_{p,q}, d(z_{r,t})⟩ = 0 if p ≠ t and q ≠ r. Using (2.1) we also get

(2.2)  ⟨z_{p,q}, d(z_{q,p})⟩ = ⟨z_{s,q} , d(z_{q,s})⟩ + ⟨z_{p,s} , d(z_{s,p})⟩

if p, q, s ∈ ℕ. By (2.2) we see that

(2.3)  ⟨z_{p,p}, d(z_{p,p})⟩ = 0 if p ∈ ℕ.

On the other hand, by (2.1) we obtain

(2.4)  ⟨z_{p,q}, d(z_{p,q})⟩ = 0 if p, q ∈ ℕ, p ≠ q.

Now, let F be a finite subset of ℕ × ℕ , {λ_{(n,m)} }_{(n,m)∈F} ⊆ ℂ and let

u = ∑_{(n,m)∈F} λ_{(n,m)} z_{n,m} .

By (2.3) and (2.4) we see that

(2.5)  ⟨u, d(u)⟩ = ∑_{(n,m),(p,q)∈F} λ_{(n,m)} λ_{(p,q)}[⟨z_{n,m} , d(z_{p,q})⟩ + ⟨z_{p,q}, d(z_{n,m} )⟩].

As we already observed, those summands in (2.5) so that n ≠ q and m ≠ p are zero. By symmetry, it suffices to consider n = q and then

⟨z_{n,m} , d(z_{p,n})⟩ + ⟨z_{p,n} , d(z_{n,m} )⟩= ⟨z_{n,n} , z_{n,m} · d(z_{p,n}) + d(z_{n,m} ) · z_{p,n} ⟩
= ⟨z_{n,n} , d(z_{n,n} )⟩
= 0.

Consequently ⟨u, d(u)⟩ = 0. Finally, the result holds since u → ⟨u, d(u)⟩ is continuous on U and 𝔛 ⊗ 𝔛^{∗} is dense in U. □

Example 2.3. Let U ≜ l^{p} ˆ⊗ l^{q}, with 1 < p, q < ∞, 1/p + 1/q = 1. If x ∈ l^{p},

x^{∗} ∈ l^{q} let

\underline{d}_{x,x^{∗}} : l^{p} × l^{q} → ℂ, \underline{d}_{x,x^{∗}} (y, y^{∗}) ≜ ⟨x, y^{∗}⟩− ⟨y, x^{∗}⟩.

Then \underline{d}_{x,x^{∗}} ∈ B^{2}(l^{p}, l^{q}, ℂ), i.e., \underline{d}_{x,x^{∗}} is a bounded bilinear form between l^{p} × l^{q} and ℂ. By the universal characteristic property of the projective tensor product of Banach spaces there is a unique \overline{d}_{x,x^{∗}} ∈ U^{∗} so that ∥\overline{d}_{x,x^{∗}} ∥ = ∥\underline{d}_{x,x^{∗}} ∥ and ⟨y ⊗ y^{∗}, \overline{d}_{x,x^{∗}} ⟩ = \underline{d}_{x,x^{∗}} (y, y^{∗}) if y ∈ l^{p}, y^{∗} ∈ l^{q}. The following map is then induced

\overline{d}: l^{p} × l^{q} → U^{∗}, \overline{d}(x, x^{∗}) ≜ \overline{d}_{x,x^{∗}} .

It is readily seen that \overline{d} ∈ B^{2}(l^{p}, l^{q}, U^{∗}) and so there is a unique d ∈ B(U, U^{∗}) so that ∥d∥ = ∥\overline{d}∥ and d(x ⊗ x^{∗}) = \overline{d}(x, x^{∗}) if x ∈ l^{p}, x^{∗} ∈ l^{q}. Consequently, the following identity

⟨y ⊗ y^{∗}, d(x ⊗ x^{∗})⟩ = ⟨x, y^{∗}⟩ − ⟨y, x^{∗}⟩

holds if x, y ∈ l^{p} and x^{∗}, y^{∗} ∈ l^{q}. It is straightforward to see that d ∈ Z^{1}(U, U^{∗}) and hence it is a 𝔇-derivation. Let us see that d ∉ N^{1}(U, U^{∗}). For, let us assume that d is inner, say d = ad_{T} for some T ∈ U^{∗}. Let us consider the usual basis {x_{n}}_{n=1}^{∞} of l^{p}, x_{n} = {δ_{n,m} }_{m=1}^{∞} if n ∈ ℕ. So, {x_{n}}_{n=1}^{∞} is obviously a shrinking basis and its associated sequence of coefficient functionals are x_{n}^{∗} = {δ_{n,m} }_{m=1}^{∞} if n ∈ ℕ. With the notation of Proposition 2.1, since ⟨z_{n,m} , d(z_{p,q})⟩ = δ_{m,p} − δ_{n,q} for all n, m, p, q ∈ ℕ we deduce that T (z_{n,m} ) = 1 if n, m ∈ ℕ and n ≠ m. However, let us write

u ≜ \frac{1}{ζ(q)^{1/q}} ∑_{n=1}^{∞} \frac{1}{n} · z_{1,1+n} ,

where ζ denotes the Riemann zeta function. Then u ∈ U is well defined,

∥u∥_{π} = lim_{N→∞} ∥ \frac{1}{ζ(q)^{1/q}} ∑_{n=1}^{N} \frac{1}{n} · z_{1,1+n} ∥_{π}

= \frac{1}{ζ(q)^{1/q}} lim_{N→∞} ∥x_{1} ⊗ ∑_{n=1}^{N} \frac{1}{n}x_{n+1}^{∗} ∥_{π}

= \frac{1}{ζ(q)^{1/q}} lim_{N→∞} ∥ ∑_{n=1}^{N} \frac{1}{n}x_{n+1}^{∗} ∥_{l^{q}}

= 1,

and as

T (u) = \frac{1}{ζ(q)^{1/q}} ∑_{n=1}^{∞} \frac{1}{n} = ∞

then T can not be bounded.

Theorem 2.1. Let U be a Banach algebra so that U^{2} is dense in U. Let us denote k_{U} : U ↪ U^{∗∗} to the usual isometric embedding of U into its second dual space U^{∗∗} by means of evaluations. Given d ∈ Z^{1}(U, U^{∗}) the following assertions are equivalent:

(i) d ∈ 𝔇(U).

(ii) ⟨x, d(y)⟩ + ⟨y, d(x)⟩ = 0 for all x, y ∈ U.

(iii) d^{∗} ∘ k_{U} ∈ Z^{1}(U, U^{∗}).

(iv) d + d^{∗} ∘ k_{U} = 0.

Proof. (i) ⇒ (ii) Given x, y ∈ U we have

0 = ⟨x + y, d(x + y)⟩ = ⟨x, d(y)⟩ + ⟨y, d(x)⟩.

112 BARRENECHEA AND PEÑA

(ii) ⇒ (iii) If x, y, z ∈ U we have

⟨z, (d^{∗} ∘ k_{U})(xy)⟩= ⟨(d(z), k_{U}(xy))⟩

= ⟨xy, d(z)⟩ = ⟨x, yd(z)⟩ = ⟨x, d(yz) − d(y)z⟩ = ⟨x, d(yz)⟩ − ⟨zx, d(y)⟩ = ⟨d(yz), k_{U}(x)⟩ + ⟨y, d(zx)⟩ = ⟨yz, d^{∗}(k_{U}(x))⟩ + ⟨d(zx), k_{U}(y)⟩ = ⟨z, d^{∗}(k_{U}(x))y⟩ + ⟨z, xd^{∗}(k_{U}(y))⟩ = ⟨z, (d^{∗} ∘ k_{U})(x)y + x(d^{∗} ∘ k_{U})(y)⟩.

(iii) ⇒ (iv) For x, y, z ∈ U we have

⟨xy, d(z)⟩= ⟨d(z), k_{U}(xy)⟩

= ⟨z, (d^{∗} ∘ k_{U})(xy)⟩ = ⟨z, (d^{∗} ∘ k_{U})(x)y + x(d^{∗} ∘ k_{U})(y)⟩ = ⟨yz, (d^{∗} ∘ k_{U})(x)⟩ + ⟨zx, (d^{∗} ∘ k_{U})(y)⟩ = ⟨x, d(y)z + yd(z)⟩ + ⟨y, d(z)x + zd(x)⟩ = ⟨zx, d(y)⟩ + 2⟨xy, d(z)⟩ + ⟨yz, d(x)⟩.

Therefore,

⟨z, d(xy)⟩ = −⟨xy, d(z)⟩ = −⟨d(z), k_{U}(xy)⟩ = −⟨z, (d^{∗} ∘ k_{U})(xy)⟩, i.e., (d + d^{∗} ∘ k_{U})(xy) = 0 if x, y ∈ U. Since U^{2} is dense in U the claim follows.

(iv) ⇒ (i) If x ∈ U then

0 = ⟨x, d(x) + (d^{∗} ∘ k_{U})(x)⟩ = 2⟨x, d(x)⟩.  □

Proposition 2.2. Let U be a Banach algebra and let d ∈ 𝔇(U). There is a unique d^{♯} ∈ 𝔇 (U^{♯}) so that the following diagram commutes

U −−−−→^{d} U^{∗}
j ↓  ↓^{p^{∗}}
U^{♯} −−−−→^{d^{♯}} t(U^{♯})^{∗}.

Proof. Consider d^{♯} ≜ p^{∗} ∘ d ∘ p. Thus, d^{♯} ∈ B(U^{♯}, (U^{♯})^{∗}) and d^{♯} ∘ j = p^{∗} ∘ d. If η, μ ∈ U^{♯} we get

(2.6)  ⟨η, d^{♯}(η)⟩ = ⟨p(η), d(p(η))⟩ = 0

and if η = ae + j(x), μ = be + j(y) for uniquely determined a, b ∈ ℂ and x, y ∈ U then

(2.7) d^{♯}(η)μ + ηd^{♯}(μ) = p^{∗}(d(x))(be + j(y)) + (ae + j(x))p^{∗}(d(y))
= (⟨y, d(x)⟩ + ⟨x, d(y)⟩)e^{∗} + p^{∗}(ad(y) + bd(x) + d(xy)
= ad^{♯}(y) + bd^{♯}(x) + p^{∗}(d(xy))
= p^{∗}(d(ay + bx + xy))
= d^{♯}(ημ).

Thus, by (2.6) and (2.7) we conclude that d^{♯} ∈ 𝔇(U^{♯}). As we already observed, j ∘ p projects U^{♯} onto j(U). Since j(U) is complemented in U^{♯} by ℂ·e then d^{♯} is uniquely determined. □

Corollary 2.1. A 𝔇-derivation on U is inner if and only if its associated derivation d^{♯} : U^{♯} → (U^{♯})^{∗} by Proposition 2.2 is inner.

Proof. Let x^{∗} ∈ U^{∗}, a ∈ ℂ. Hence, it is easy to see that (ad_{x^{∗}} )^{♯} = ad_{p^{∗} (x^{∗})} and if d^{♯} = ad_{ae^{∗} +p^{∗} (x^{∗})} , then d = ad_{x^{∗}} . □

Remark 2.2. Let us consider a dual Banach algebra U, i.e., U ≈ (U_{∗})^{∗}, where U_{∗} is a closed submodule of U^{∗}. Although U_{∗} need not be unique, we will assume that U is realized as the dual space of a fixed closed submodule U_{∗} of U^{∗}. It is known that a dual Banach algebra has a unit if and only if it has a bounded approximate identity (see [7, Prop. 1.2]).

Proposition 2.3. Let U be a dual Banach algebra with unit and let d ∈ Z^{1}(U, U^{∗}) so that d(U) ⊆ k_{U_{∗}}(U_{∗}). Then d ∈ 𝔇(U) if and only if d^{∗} + d ∘ k_{U_{∗}}^{∗} = 0.

Proof. (⇒) Given x ∈ U let x_{∗} ∈ U_{∗} be the unique element so that d(x) = k_{U_{∗}}(x_{∗}). If x^{∗∗} ∈ U^{∗∗} by Theorem 2.1(iv) we have

⟨x, (d ∘ k_{U_{∗}}^{∗})(x^{∗∗})⟩= ⟨d(k_{U_{∗}}^{∗}(x^{∗∗})), k_{U}(x)⟩
= ⟨k_{U_{∗}}^{∗}(x^{∗∗}), (d^{∗} ∘ k_{U})(x)⟩
= −⟨k_{U_{∗}}^{∗}(x^{∗∗}), d(x)⟩
= −⟨x_{∗}, k_{U_{∗}}^{∗}(x^{∗∗})⟩
= −⟨k_{U_{∗}}(x_{∗}), x^{∗∗}⟩
= −⟨d(x), x^{∗∗}⟩
= −⟨x, d^{∗}(x^{∗∗})⟩.
(⇐) If x, y ∈ U we obtain
⟨y, (d^{∗} ∘ k_{U})(x)⟩ = −⟨y, (d ∘ k_{U_{∗}}^{∗} ∘ k_{U})(x)⟩ = −⟨y, d(x)⟩,

i.e., d + d^{∗} ∘ k_{U} = 0 and our claim follows. □

Acknowledgement. The authors express their recognition to the reviewer for his care reading of this manuscript and his insight to improve their results.

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NUCOMPA – Departamento de Matemáticas (Received 10 09 2008) Universidad Nacional del Centro (Revides 20 07 2009) Tandil Argentina

   analucia@exa.unicen.edu.ar
    ccpenia@exa.unicen.edu.ar