New York Journal of Mathematics
New York J. Math. 18 (2012) 657–665.
On dual-valued operators on Banach
algebras
Marı́a J. Aleandro and Carlos C. Peña
Abstract. Let U be a regular Banach algebra and let D : U → U ∗ be
a bounded linear operator, where U ∗ is the topological dual space of U.
We seek conditions under which the transpose of D becomes a bounded
derivation on U ∗∗ . We focus our attention on the class D (U) of bounded
derivations D : U → U ∗ so that ha, D(a)i = 0 for all a ∈ U . We consider
this matter in the setting of Beurling algebras on the additive group of
integers. We show that U is a weakly amenable Banach algebra if and
only if D (U) 6= {0}.
Contents
1. Introduction
2. Transposes and bounded derivations between U and U ∗
3. An application to Beurling algebras on the group (Z, +)
References
657
658
662
665
1. Introduction
Throughout this article U will be a Banach algebra. By and ♦ we will
denote the first and second Arens products on U ∗∗ (cf. [1]). The Banach
algebra U is said to be regular when these products coincide, in which case
we will simply write = ♦ = •. If U is regular it is readily seen that U ∗
becomes a Banach U ∗∗ -bimodule. As usual, B (U,U ∗ ) will denote the space
of bounded operators between U and U ∗ and Z 1 (U ∗∗ , U ∗ ) will be the space
of bounded derivations between U ∗∗ and U ∗ . As is well known, when endowed with the uniform norm B (U,U ∗ ) and Z 1 (U ∗∗ , U ∗ ) are Banach spaces.
By D (U) we will denote the class of D-derivations consisting of bounded
derivations D : U → U ∗ such that ha, D(a)i = 0 if a ∈ U. Clearly any inner derivation from U into U ∗ is a D-derivation. For problems related to
these special classes of derivations, their characterization and examples in
the context of Banach algebras of continuous functions or projective Banach algebras, we recommend [3]. In Proposition 1 we will characterize
Received May 17, 2011, and in revised form on August 22, 2012.
2010 Mathematics Subject Classification. 46H35, 47D30.
Key words and phrases. Arens products, amenable and weakly amenable Banach algebras, dual Banach algebras, Beurling algebras.
ISSN 1076-9803/2012
657
658
M. J. ALEANDRO AND C. C. PEÑA
those operators D ∈ B (U,U ∗ ) whose dual belongs to Z 1 (U ∗∗ , U ∗ ) under the
hypothesis that U is a regular Banach algebra. Further, the corresponding
problem if D ∈ Z 1 (U,U ∗ ) will be considered in Proposition 2. In Theorem
6 we will provide conditions under which D ∈ D (U) if D∗ ∈ Z 1 (U ∗∗ , U ∗ ) .
In Proposition 7 it will be shown that any D ∈ D (U) is (w, w) continuous.
This matter and examples in the setting of Beurling algebras on Z will be
considered in Theorem 8. For further information and background on the
subject of this paper, we recommend [11], §1.4, p. 46. In addition, important articles concerning the regularity of Banach algebras are [8], [12] and
[13]. Conditions under which the second transpose of a U ∗ -valued bounded
derivation on U becomes a bounded derivation on U ∗∗ endowed with the
first Arens product were investigated in [7] and [2].
2. Transposes and bounded derivations between U and U ∗
Proposition 1. If U is a regular Banach algebra and if D ∈ B (U,U ∗ ), then
the following assertions are equivalent:
(i) D∗ ∈ Z 1 (U ∗∗ , U ∗ ).
(ii) If a ∈ U and if Φ, Ψ ∈ U ∗∗ , then
haD∗ (Φ) , Ψi = hΨD(a) − D∗ (Ψ) a, Φi .
(iii) If a ∈ U and if Φ, Ψ ∈ U ∗∗ , then
hD∗ (Ψ) a, Φi = hD(a)Φ − aD∗ (Φ) , Ψi .
Proof. (i)⇒(ii). Let Φ, Ψ ∈ U ∗∗ and a ∈ U. Then
hΨD(a), Φi = hD(a), Φ • Ψi
= ha, D∗ (Φ • Ψ)i
= ha, D∗ (Φ)Ψ + ΦD∗ (Ψ)i
= haD∗ (Φ), Ψi + hD∗ (Ψ)a, Φi .
(ii)⇒(iii). Given Φ, Ψ ∈ U ∗∗ , a ∈ U, it will suffice to see that
(1)
hΨD(a), Φi − haD∗ (Φ) , Ψi = hD(a)Φ − aD∗ (Φ) , Ψi .
But (1) is an immediate consequence of the regularity of U.
(iii)⇒(i). If a ∈ U and Φ, Ψ ∈ U ∗∗ we have
ha, D∗ (Φ • Ψ)i = hD(a), Φ • Ψi
= hD(a)Φ, Ψi
= hD∗ (Ψ) a, Φi + haD∗ (Φ) , Ψi
= ha, ΦD∗ (Ψ) + D∗ (Φ) Ψi .
Since a is arbitrary the claim holds.
Proposition 2. Let U be a regular Banach algebra and let kU ∗ : U ∗ ,→ U ∗∗∗
be the natural embedding of U ∗ into U ∗∗∗ . Given D ∈ Z 1 (U,U ∗ ), the
following assertions are equivalent:
ON DUAL-VALUED OPERATORS ON BANACH ALGEBRAS
659
(i) D∗ ∈ Z 1 (U ∗∗ , U ∗ ).
(ii) If a ∈ U and if Φ ∈ U ∗∗ , then kU ∗ (aD∗ (Φ)) + aD∗∗ (Φ) = 0.
(iii) If a ∈ U and if Φ ∈ U ∗∗ , then D∗∗ (aΦ) + kU ∗ (D∗ (aΦ)) = 0.
Proof. (i)⇒(ii). Let D∗ ∈ Z 1 (U ∗∗ , U ∗ ) , a ∈ U. Given Φ, Ψ ∈ U ∗∗ , consider
bounded nets {bi }i∈I , {cj }j∈J in U such that Φ = w∗ -limi∈I kU (bi ) and
Ψ = w∗ -limj∈J kU (cj ), where kU : U ,→ U ∗∗ denotes the usual isometric
embedding of U into its second dual space U ∗∗ by means of evaluations.
Hence
hD∗ (Ψ) a, Φi = lim hbi , D∗ (Ψ) ai = lim hD (abi ) , Ψi = lim lim hcj , D (abi )i .
i∈I
i∈I
i∈I j∈J
Further,
(2)
hΨD(a) − D∗ (Ψ) a, Φi = hD(a), Φ • Ψi − ha, ΦD∗ (Ψ)i
= lim lim (hbi cj , D(a)i − hcj , D (abi )i)
i∈I j∈J
= − lim lim hcj , aD(bi )i
i∈I j∈J
= − lim haD (bi ) , Ψi
i∈I
= − hD∗ (Ψa) , Φi
and the conclusion follows from Proposition 1 and (2).
(ii)⇒(iii). If a ∈ U and Φ, Ψ ∈ U ∗∗ we write
(3) hD∗ (Ψ) a, Φi = hD∗ (Ψa) + ΨD(a), Φi = hΨD(a), Φi − haD∗ (Φ) , Ψi .
Moreover, hΨD(a), Φi = hD(a)Φ, Ψi because U is regular. Hence, by (3) we
obtain
hD∗ (Ψ) a, Φi = hD(a)Φ − aD∗ (Φ) , Ψi = − hD∗ (aΦ) , Ψi .
(iii)⇒(i). If a ∈ U and Φ, Ψ ∈ U ∗∗ we write
ha, D∗ (Φ • Ψ)i = hD(a)Φ, Ψi
= haD∗ (Φ) − D∗ (aΦ) , Ψi
= haD∗ (Φ) , Ψi + hD∗ (Ψ) a, Φi
= ha, D∗ (Φ) Ψ + ΦD∗ (Ψ)i .
Corollary 3. Let U be a regular Banach algebra. Given D ∈ Z 1 (U, U ∗ )
such that D∗ ∈ Z 1 (U ∗∗ , U ∗ ), then
UD∗∗ (U ∗∗ ) ∪ D∗∗ (U ∗∗ ) U ,→ U ∗ .
Theorem 4 (cf. [3, Theorem 2.1]). Let U be a general Banach algebra such
that U 2 is dense in U, where
U 2 = span {xy : x, y ∈ U} .
Then for D ∈ Z 1 (U, U ∗ ), the following assertions are equivalent:
(i) D ∈ D (U).
660
M. J. ALEANDRO AND C. C. PEÑA
(ii) hx, D(y)i + hy, D(x)i = 0 for all x, y ∈ U.
(iii) D∗ ◦ kU ∈ Z 1 (U, U ∗ ) .
(iv) D + D∗ ◦ kU = 0U ,U ∗ .
Corollary 5. Let U be a general Banach algebra such that U 2 is dense in
U. If D ∈ Z 1 (U,U ∗ ), then D ∈ D (U) if and only if for all a, b, c ∈ U the
following identity
hab, D(c)i + hca, D(b)i + hbc, D(a)i = 0
(4)
holds.
Proof. (⇒) For a, b, c ∈ U and D ∈ D (U)
hab, D(c)i + hca, D(b)i + hbc, D(a)i = hab, D(c)i + hca, D(b)i − ha, D(bc)i
= 0.
(⇐) If a, b ∈ U let {bn } and {cn } be sequences in U such that b =
limn→∞ (bn cn ) , then
ha, D(b)i + hb, D(a)i = lim {ha, D(bn cn )i + hbn cn , D(a)i}
n→∞
= lim {ha, D(bn )cn + bn D(cn )i + hbn cn , D(a)i}
n→∞
= lim {hcn a, D(bn )i + habn , D(cn )i + hbn cn , D(a)i}
n→∞
= 0.
Theorem 6. Let U be a regular Banach algebra, and let D ∈ Z 1 (U,U ∗ ) .
(i) If U 2 is dense in U and D∗ ∈ Z 1 (U ∗∗ , U ∗ ) then D ∈ D (U) .
(ii) Suppose D ∈ D (U) has the property that
(5)
lim lim hcj , aD(bi )i = lim lim hcj , aD(bi )i
i∈I j∈J
j∈J i∈I
for every pair of bounded sequences in U, {bi }i∈I , {cj }j∈J , and every
a ∈ U for which both iterated limits exist. Then D∗ ∈ Z 1 (U ∗∗ , U ∗ ).
Proof. (i) By Proposition 2 if D∗ ∈ Z 1 (U ∗∗ , U ∗ ), the equality (4) holds for
all a, b, c ∈ U. Thus the conclusion follows from Corollary 5.
(ii) If a, b ∈ U, then aD∗∗ (kU (b)) = kU ∗ (aD(b)). So, by Theorem 4 we
get
0 = kU ∗ (aD(b)) − aD∗∗ (kU (b))
= kU ∗ (aD∗ (kU (−b))) + aD∗∗ (kU (−b)) .
If Φ ∈ U ∗∗ let {bi }i∈I be a bounded net in U such that Φ = w∗ -limi∈I kU (bi ).
Define ζ ∈ U ∗ by hc, ζi , hD∗ (kU (c) a) , Φi. Thus ζ = w∗ -limi∈I aD (bi ) and
kU ∗ (ζ) = aD∗∗ (Φ). For, let Ψ ∈ U ∗∗ such that Ψ = w∗ -limj∈J kU (cj ) in
U ∗∗ for some bounded net {cj }j∈J in U. So, by (5) we have
hΨ, aD∗∗ (Φ)i = lim lim hcj , aD(bi )i = lim lim hcj , aD(bi )i = hζ, Ψi .
i∈I j∈J
j∈J i∈I
ON DUAL-VALUED OPERATORS ON BANACH ALGEBRAS
661
Consequently,
hΨ, kU ∗ (aD∗ (Φ)) + aD∗∗ (Φ)i = hΨ, kU ∗ (aD∗ (Φ) + ζ)i
= haD∗ (Φ) + ζ, Ψi
= lim hcj , aD∗ (Φ) + ζi
j∈J
= lim [hD(cj a), Φi + hζ, kU (cj )i]
j∈J
= lim lim [hbi , D (cj a)i + haD(bi ), kU (cj )i]
j∈J i∈I
= lim lim hcj , a (D∗ (kU (bi )) + D (bi ))i
j∈J i∈I
= 0.
Since Ψ was arbitrary, kU ∗ (aD∗ (Φ)) + aD∗∗ (Φ) = 0 and the conclusion
follows from Proposition 2.
Proposition 7. If D ∈ D (U) then D∗ is (w, w)-continuous.
Proof. If D ∈ D, let {Φi }i∈I be a net in U ∗∗ such that w-limi∈I D∗ (Φi ) 6=
0U ∗ . There exists Θ ∈ U ∗∗ and a subnet {Φi }i∈I1 of {Φi }i∈I such that
|hD∗ (Φi ), Θi| ≥ 1 if i ∈ I1 .
Let {aj }j∈J be a bounded net in U such that
Θ = w∗ − lim kU (aj ) .
j∈J
Since {kU ∗ (D (aj ))}j∈J is a bounded net in U ∗∗∗ by the Banach–Alaoglu
theorem there is a subnet {aj }j∈J1 such that the limit w∗ -limj∈J1 kU ∗ (D (aj ))
defines an element M in U ∗∗∗ . As D∗∗ ∈ (w∗ , w∗ ),
D∗∗ (Θ) = w∗ − lim D∗∗ (kU (aj )).
j∈J1
In particular, by Theorem 4 we deduce that D∗∗ ◦ kU = kU ∗ ◦ D. Hence, if
i ∈ I1 we obtain
1 ≤ |hD∗ (Φi ), Θi|
= |hΦi , D∗∗ (Θ)i|
= lim |hΦi , D∗∗ (kU (aj ))i|
j∈J1
= lim |hΦi , kU ∗ (D (aj ))i|
j∈J1
= |hΦi , M i| ,
i.e., w-limi∈I Φi 6= 0U ∗∗ .
662
M. J. ALEANDRO AND C. C. PEÑA
3. An application to Beurling algebras on the group (Z, +)
Given a function w : Z → R+ let U , `1 (Z, w) be the space of complex
P
sequences {am }m∈Z such that kak1,w , m∈Z |am | w(m) is finite. With the
natural vector space operations U, k◦k1,w is a Banach space. Further, let
us suppose that w is a weight function, i.e., w (m + n) ≤ w (m) w (n) for all
m, n ∈ Z and w (0) = 1. Then, for a, b ∈ U the convolution product
(
)
X
a∗b,
am bn−m
m∈Z
n∈Z
is well defined and U becomes a Banach algebra. These algebras are called
Beurling algebras on the additive group Z (cf. [6], [9]). The topological dual
U ∗ consists of all functions λ : Z → C such that
kλk∞,w−1 , sup |λ (m)| w(m)−1 : m ∈ Z
is finite. Indeed, U is a dual Banach algebra
whose predual can be identified
−1 consisting of those sequences λ ∈
with the the
closed
subspace
c
Z,
w
0
`∞ Z, w−1 such that λw−1 vanishes at infinity. Since the additive group
of integers is discrete and countable there are weights w on Z such that
`1 (Z, w) is regular. Further, U is regular if
inf
i≤j
w (mi + nj )
=0
w (mi ) w (nj )
for all sequences of distinct elements of Z (see [5]). For instance, U is not
regular if w (m) = 1 or w(m) = exp (|m|), and it is regular if w(m) = 1 + |m|
for all m ∈ Z.
Theorem 8. Let D ∈ Z 1 (U,U ∗ ) .
(i) There is a unique complex sequence {λm }m∈Z such that
)
(
|λm+p−1 |
|m|
(6)
kDk = sup
sup
,
w (p)
m∈Z w (m) p∈Z
and if a ∈ U we have
(
(7)
D (a) =
)
X
m∈Z
mλm+p−1 am
.
p∈Z
(ii) If we write D0 (a) , {−ma−m }m∈Z for a ∈ U then D0 ∈ D(U) and
any other element of D(U) is a constant multiple of D0 .
(iii) D(U) 6= {0} if and only if U is a non-weakly amenable Banach algebra.
(iv) If D ∈ D (U) then D (U) ⊆ c0 Z, w−1 .
(v) If D ∈ D (U) then D∗ + D ◦ kc∗0 (Z,w−1 ) = 0`∞ (Z,w−1 )∗ ,`∞ (Z,w−1 ) .
(vi) If D ∈ D (U) then D ◦ kc∗0 (Z,w−1 ) = k`∗1 (Z,w) ◦ D∗∗ .
ON DUAL-VALUED OPERATORS ON BANACH ALGEBRAS
663
Proof. (i) If m ∈ Z, let em be the characteristic function of {m}considered
as an element of U and let D (em ) = {λm,p }p∈Z in `∞ Z, w−1 . Since D
satisfies the Leibnitz rule, the following identities λm+p,q = λm,p+q + λp,m+q
hold for all m, p, q ∈ Z. Let us write λm , λ1,m for m ∈ Z. It is readily seen
that λm,p = mλm+p−1 if m, p ∈ Z. Hence (7) holds
since for each p ∈ Z the
∗
∞
−1
linear form µ → hep , µi belongs to ` Z, w
. Now,
|λm,p |
1
em
= sup
sup
sup D w (m) w(m)
w(p)
−1
m∈Z
m∈Z
∞,w
p∈Z
|λm+p−1 |
|m|
sup
≤ kDk .
w (p)
m∈Z w(m) p∈Z
= sup
We can assume that D 6= 0. If 0 < t < kDk there exist m, p ∈ Z such that
|mλm+p−1 | /w(m)w(p) > t. Otherwise, we can choose u, v ∈ [U]1 such that
X
X
t < |hv, D(u)i| ≤
|vp |
|mλm+p−1 um | ≤ t kuk1,w kvk1,w ≤ t,
p∈Z
m∈Z
which is absurd. Thus (6) follows.
(ii) It is straightforward to see that D0 ∈ D(U). Moreover, with the above
notation let D ∈ D(U) and m, p ∈ Z. By Theorem 4(ii) we see that
0 = hem , D(ep )i + hep , D(em )i = (m + p) λm+p−1 .
Hence λm,p = λm+p−1 = 0 if m + p 6= 0 while λm,−m = mλ−1 . Consequently
D (em ) = λ−1 me−m and D = λ−1 D0 .
(iii) Observe that U is not weakly amenable if and only if
(8)
|m|
< +∞
w(m)w(−m)
m∈Z
sup
(cf. [10], Corollary 4.8). Further, by (6),
(9)
|m|
m∈Z w(m)w(−m)
kD0 k = sup
and the conclusion now follows.
(iv) If a ∈ U and m ∈ Z by (9) we have
|−ma−m |
|m|
=
|a−m | w(−m) ≤ kD0 k |a−m | w(−m),
w(m)
w(m)w(−m)
i.e., limm→∞ (−ma−m ) /w(m) = 0.
(v) Let K be the subset of elements F ∈ `∞ (Z)∗ whose induced finitely
additive set function µF (E) , hχE , F i defined for all E ∈ P (Z) vanishes on
finite subsets of Z. Certainly
`∞ (Z)∗ = k`1 (Z) `1 (Z) ⊕ K
(cf. [4, Theorem 3.2]). Further, since Id`1 (Z,w) = kc∗0 (Z,w−1 ) ◦ k`1 (Z,w) then
h
i
∗
(10)
`∞ Z, w−1 = k`1 (Z,w) `1 (Z, w) ⊕ ker kc∗0 (Z,w−1 ) .
664
M. J. ALEANDRO AND C. C. PEÑA
Let Aw : `1 (Z) → `1 (Z, w) be the isometric isomorphism such that
Aw (x) , {x(m)/w (m)}m∈Z
if x ∈ `1 (Z) . Then
h
i
∗
A∗∗
(K)
=
ker
k
w
c0 (Z,w−1 ) .
For, let be given F ∈ K and λ ∈ c0 Z, w−1 . Then
D
E ∗
(12)
λ, kc∗0 (Z,w−1 ) (A∗∗
w (F )) = Aw kc0 (Z,w−1 ) (λ) , F
= {λ (m) /w(m)}m∈Z , F
Z
λ
dµF .
=
Z w
(11)
But {em }m∈Z can be considered as a Schauder basis of c0 Z, w−1 . Moreover, using (12) we can write
*
+
D
E
X
(13)
λ (m) em , kc∗0 (Z,w−1 ) (A∗∗
λ, kc∗0 (Z,w−1 ) (A∗∗
w (F )) =
w (F ))
m∈Z
=
X
D
E
λ (m) em , kc∗0 (Z,w−1 ) (A∗∗
(F
))
w
m∈Z
=
X
Z
λ (m)
m∈Z
Z
em
dµF
w
= 0.
Since λ was arbitrary then kc∗0 (Z,w−1 ) (A∗∗
w (F )) = 0`1 (Z,w) . On the other
h
i
∗∗
hand, given Φ ∈ ker kc∗0 (Z,w−1 ) we set F , A−1
(Φ) . If m ∈ Z, let χ∞
w
{m}
be the characteristic function of {m} considered as an element of `∞ (Z).
Given a ∈ `1 (Z, w) we see that
D
E D
∞ E
−1 ∗
χ∞
,
F
=
A
χ{m} , Φ
w
{m}
= w (m) kc0 (Z,w−1 ) (em ) , Φ
D
E
= w (m) em , kc∗0 (Z,w−1 ) (Φ)
= 0.
Therefore, F ∈ K and (8) holds. If Φ ∈ U ∗∗ , then by (10) and (11), there are
unique elements a ∈ U and F ∈ K such that Φ = kU (a) + A∗∗
w (F ) . Finally,
∗
it is easy to verify that a = kc0 (Z,w−1 ) (Φ) and given b ∈ U we have
hb, D0∗ (Φ)i = hb, −D0 (a)i + A∗∗
w (F ) , kc0 (Z,w−1 ) (D0 (b))
D
E
= b, − D0 ◦ kc∗0 (Z,w−1 ) (Φ) .
(vi) It suffices to apply Theorem 4 and (v).
ON DUAL-VALUED OPERATORS ON BANACH ALGEBRAS
665
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CONICET - UNCPBA. FCExactas, Dpto. de Matemáticas, NUCOMPA.
aleandro@exa.unicen.edu.ar
UNCPBA. FCExactas, Dpto. de Matemáticas, NUCOMPA.
ccpenia@exa.unicen.edu.ar
This paper is available via http://nyjm.albany.edu/j/2012/18-35.html.