Derivations on the module extension Banach algebras
UDC 517.986 We correct some results presented in [M. Eshaghi Gordji, F. Habibian, A. Rejali,  Ideal amenability of module extension Banach algebras, Int. J. Contemp. Math. Sci.,  2, No. 5, 213–219 (2007)] and, using the obtained consequences, we find necessary and sufficien...
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| author | Teymouri , A. Bodaghi , A. Ebrahimi Bagha, D. Teymouri , A. Bodaghi, A. Ebrahimi Bagha, D. |
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We correct some results presented in [M. Eshaghi Gordji, F. Habibian, A. Rejali,  Ideal amenability of module extension Banach algebras, Int. J. Contemp. Math. Sci.,  2, No. 5, 213–219 (2007)] and, using the obtained consequences, we find necessary and sufficient conditions for the module extension $\mathcal A\oplus X$ to be $(\mathcal I\oplus Y)$-weakly amenable, where $\mathcal I$ is a closed ideal of the Banach algebra $\mathcal A$ and $Y$ is a closed $\mathcal A$-submodule of the Banach $\mathcal A$-bimodule $X.$ We apply this result to the module extension $\mathcal A\oplus(X_1\dotplus X_2),$ where $X_1,$ $X_2$ are two Banach $\mathcal A$-bimodules. |
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DOI: 10.37863/umzh.v73i4.240
UDC 517.986
A. Teymouri (Dep. Math., Central Tehran Branch, Islamic Azad Univ., Tehran, Iran),
A. Bodaghi (Dep. Math., Garmsar Branch, Islamic Azad Univ., Garmsar, Iran),
D. Ebrahimi Bagha (Dep. Math., Central Tehran Branch, Islamic Azad Univ., Tehran, Iran)
DERIVATIONS ON THE MODULE EXTENSION BANACH ALGEBRAS
ДИФЕРЕНЦIЮВАННЯ НА БАНАХОВИХ АЛГЕБРАХ РОЗШИРЕННЯ МОДУЛЯ
We correct some results presented in [M. Eshaghi Gordji, F. Habibian, A. Rejali, Ideal amenability of module extension
Banach algebras, Int. J. Contemp. Math. Sci., 2, № 5, 213 – 219 (2007)] and, using the obtained consequences, we find
necessary and sufficient conditions for the module extension \scrA \oplus X to be (\scrI \oplus Y )-weakly amenable, where \scrI is a closed
ideal of the Banach algebra \scrA and Y is a closed \scrA -submodule of the Banach \scrA -bimodule X. We apply this result to the
module extension \scrA \oplus (X1 \dotplus X2), where X1, X2 are two Banach \scrA -bimodules.
Виправлено деякi результати роботи [M. Eshaghi Gordji, F. Habibian, A. Rejali, Ideal amenability of module extension
Banach algebras, Int. J. Contemp. Math. Sci., 2, № 5, 213 – 219 (2007)] та за допомогою отриманих наслiдкiв знайдено
необхiднi та достатнi умови того, що розширення модуля \scrA \oplus X буде (\scrI \oplus Y )-слабко аменабельним, де \scrI —
замкнений iдеал банахової алгебри \scrA , а Y — замкнений \scrA -субмодуль банахового \scrA -бiмодуля X. Цi результати
застосовано до розширення модуля \scrA \oplus (X1 \dotplus X2), де X1, X2 — банаховi \scrA -бiмодулi.
1. Introduction. Let \scrA be a Banach algebra and X be a Banach \scrA -bimodule. Then X\ast is a Banach
\scrA -bimodule with module actions
\langle a \cdot x\ast , x\rangle = \langle x\ast , x \cdot a\rangle , \langle x\ast \cdot a, x\rangle = \langle x\ast , a \cdot x\rangle , a \in A, x \in X, x\ast \in X\ast .
A derivation from a Banach algebra \scrA into a Banach \scrA -bimodule X is a bounded linear mapping
D : \scrA - \rightarrow X such that D(ab) = D(a) \cdot b+ a \cdot D(b) for every a, b \in A. A derivation D : \scrA - \rightarrow X
is called inner if there exists x \in X such that D(a) = a \cdot x - x \cdot a = \delta x(a) for a \in \scrA . A Banach
algebra \scrA is called amenable if every bounded derivation D : \scrA - \rightarrow X\ast is inner for every Banach
\scrA -bimodule X, i.e., H1(\scrA , X\ast ) = \{ 0\} , where H1(\scrA , X\ast ) is the first cohomology group from \scrA
with coefficients in X\ast . This definition was introduced by B. E. Johnson in [10]. In addition, a
Banach algebra \scrA is weakly amenable if H1(\scrA ,\scrA \ast ) = \{ 0\} . Bade, Curtis and Dales [1] introduced
the notion of weak amenability for the first time for Banach algebras. They considered this concept
only for commutative Banach algebras. Next, Johnson defined the weak amenability for arbitrary
Banach algebras and showed that, for a locally compact group G, L1(G) is always weakly amenable
[11]. In [7], Gorgi and Yazdanpanah introduced and studied the concept of ideal amenability for a
Banach algebra. In fact, a Banach algebra \scrA is called \scrI -weakly amenable if H1(\scrA , \scrI \ast ) = \{ 0\} for a
closed two-sided ideal \scrI of \scrA , and ideally amenable if it \scrI -weakly amenable for every closed two-
sided ideal \scrI of \scrA . Weak amenability, and ideal amenability of module extension Banach algebras
are investigated in [17] and [6], respectively. Furthermore, ideal amenability of the (projective) tensor
product of Banach algebras is studied in [13]. An alternative notion of ideal amenability, namely,
quotient ideal amenability for Banach algebras was introduced and investigated by the authors in
[15]. Indeed, a Banach algebra \scrA is said to be quotient ideally amenable if all derivations from \scrA
into its annihilators of all ideals are inner. For the ideal Connes-amenability of dual Banach algebras,
we refer to [12].
c\bigcirc A. TEYMOURI, A. BODAGHI, D. EBRAHIMI BAGHA, 2021
566 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
DERIVATIONS ON THE MODULE EXTENSION BANACH ALGEBRAS 567
The main motivation for this work is taken from [4 – 6] and [17]. In this paper, for a Banach
algebra \scrA , we study the (\scrI \oplus \scrI )-weakly amenability of module extension \scrA \oplus \scrI , where \scrI is a closed
ideal of \scrA . We apply this result to show that \scrB (\scrH ) \oplus \scrK (\scrH ) is (\scrK (\scrH ) \oplus \scrK (\scrH ))-weakly amenable,
where \scrB (\scrH ) and \scrK (\scrH ) are bounded linear and compact operators on the infinite dimensional
Hilbert space \scrH , respectively. Finally, we prove that under what conditions the module extension
\scrA \oplus (X1 \.+X2) can be \scrJ -weakly amenable, where X1, X2 are two Banach \scrA -bimodules and \scrJ is
a closed ideal of \scrA \oplus (X1 \.+X2) .
2. Module extension Banach algebra. We start this section with an example of Banach
semigroup algebras which is \scrI -weakly amenable.
Let S be a non-empty set. Consider l1(S) =
\Bigl\{
f \in \BbbC S :
\sum
s\in S
\bigm| \bigm| f(s)\bigm| \bigm| < \infty
\Bigr\}
with the norm
\| \cdot \| 1 given by \| f\| 1 =
\sum
s\in S
| f(s)| for f \in l1(S). We write \delta s for the characteristic function of
\{ s\} when s \in S. Suppose that S is a semigroup. We define the convolution of two elements f and
g of l1(S) by
(f \ast g)(s) =
\sum
uv=s
f(u)g(v), s \in S,
where
\sum
uv=s
f(u)g(v) = 0, when there are no elements u, v \in S with uv = s. Then\bigl(
l1(S), \ast , \| \cdot \| 1
\bigr)
becomes a Banach algebra that is called the semigroup algebra of S. Clearly,
l1(S) is commutative if and only if S is Abelian. Moreover, the dual space of l1(S) is l\infty (S), with
the duality
\langle f, \lambda \rangle =
\sum
s\in S
f(s)\lambda (s), f \in l1(S), \lambda \in l\infty (S).
Let S be a semigroup and E(S) = \{ e \in S : e2 = e\} be the set of idempotents in S. We note that if
\scrI is an ideal in S, then l1(\scrI ) is a closed ideal in l1(S).
Example 2.1. Let \BbbN be the commutative semigroup of positive integers. Consider (\BbbN ,\vee ) with
maximum operation m \vee n = \mathrm{m}\mathrm{a}\mathrm{x}\{ m,n\} . Obviously, each element of \BbbN is an idempotent. It is
easily verified that all ideals of (\BbbN ,\vee ) are exactly the sets \scrI n = \{ m \in \BbbN : m \geq n\} , and so l1(\scrI n)
are ideals of l1(\BbbN ). Indeed, for any element f =
\sum
r\in \scrI n
\alpha r\delta r and g =
\sum
s\in \BbbN
\beta s\delta s, we have
f \ast g =
\Biggl( \sum
r\in \scrI n
\alpha r\delta r
\Biggr) \Biggl( \sum
s\in \BbbN
\beta s\delta s
\Biggr)
=
\sum
r\vee s=t\in \scrI n
(\alpha r\beta s)\delta t \in l1(\scrI n).
Similarly, g \ast f \in l1(\scrI n). Since E(\BbbN ) = \BbbN and \BbbN is a commutative semigroup with maximum
operation, by [3] (Proposition 10.5), l1(\BbbN ) is weakly amenable, and thus l1(\BbbN ) is l1(\scrI n)-weakly
amenable by [1] (Theorem 1.5).
Let \scrA and X be a Banach algebra and a \scrA -bimodule, respectively. Consider \scrA \oplus X as a Banach
space with the following norm:
\| (a, x)\| = \| a\| + \| x\| , a \in \scrA , x \in x.
Then \scrA \oplus X is a Banach algebra with product
(a1, x1) \cdot (a2, x2) = (a1a2, x1 \cdot a2 + a1 \cdot x2).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
568 A. TEYMOURI, A. BODAGHI, D. EBRAHIMI BAGHA
\scrA \oplus X is called a module extension Banach algebra. Since (\scrA \oplus X)\ast = (0+X)\bot \.+(\scrA \oplus 0)\bot , where
\.+ denotes the direct \scrA -bimodule l\infty -sum, and (0\oplus X)\bot (resp., (\scrA \oplus 0)\bot ) is isometrically isomorphic
to \scrA \ast (resp., X\ast ) as \scrA -bimodule. For convenience, we simply identify the corresponding terms and
write
(\scrA \oplus X)\ast = \scrA \ast \.+X\ast .
Suppose that X and Y are \scrA -bimodules. Recall that an \scrA -module morphism from X to Y is a
bounded linear map T : X - \rightarrow Y such that
T (a \cdot x) = a \cdot T (x), T (x \cdot a) = T (x) \cdot a, a \in \scrA , x \in X.
Let \scrA and X be as the above. It is shown (without proof) in [5] that \scrJ is a closed ideal in
\scrA \oplus X if and only if there exist a closed ideal \scrI of \scrA and a closed \scrA -submodule Y of X such
that \scrJ = \scrI \oplus Y and (\scrI \cdot X) \cup (X \cdot \scrI ) \subseteq Y . We mention that one side of this result is not valid in
general. In the next lemma we correct it and indicate the proof completely.
Lemma 2.1. Let \scrA be a Banach algebra and X be a Banach \scrA -bimodule.
(i) If \scrI is a closed ideal of \scrA and Y is a closed \scrA -submodule of X such that (\scrI \cdot X)\cup (X \cdot \scrI ) \subseteq
\subseteq Y, then \scrJ = \scrI \oplus Y is a closed ideal of \scrA \oplus X .
(ii) Let \scrJ is a closed ideal in \scrA \oplus X and
\scrI = \{ a \in \scrA | (a, x) \in \scrJ for some x \in X\} ,
Y = \{ x \in X| (a, x) \in \scrJ for some a \in \scrA \} .
Then \scrI is an ideal of \scrA and Y is \scrA -submodule of X. Moreover, if \scrA has a approximate identity
and left action \scrA over X is zero, then \scrJ = \scrI \oplus Y.
Proof. (i) Consider (i, y) \in \scrJ = \scrI \oplus Y and (a, x) \in \scrA \oplus X. By assumption, we have
(i, y) \cdot (a, x) = (ia, i \cdot x+y \cdot a) \in (\scrI \oplus Y ), and thus (i, y) \cdot (a, x) \subseteq \scrJ . Similarly, (a, x) \cdot (i, y) \subseteq \scrJ .
Since \scrI and Y are closed, \scrJ is a closed ideal of \scrA \oplus X.
(ii) Let b \in \scrI and a \in \scrA . Then there exists x \in X such that (b, x) \in \scrJ . We obtain
(a, 0) \cdot (b, x) = (ab, a \cdot x) \in \scrJ ,
(b, x) \cdot (a, 0) = (ba, x \cdot a) \in \scrJ .
The above relations imply that ab, ba \in \scrI , which means \scrI is an ideal of \scrA . Similarly, one can
show that Y is a \scrA -submodule of X. Now, suppose that (a\alpha ) is an approximate identity for \scrA . By
Cohen’s factorization theorem (a\alpha ) is an approximate identity for X. It is obvious that \scrJ \subseteq \scrI \oplus Y.
Let y \in Y and a \in \scrI . In this case, there exist y0 \in X and a0 \in \scrA such that (a, y0), (a0, y) \in \scrJ .
We have \bigm\| \bigm\| (a\alpha , 0) \cdot (a, y0) - (a, 0)
\bigm\| \bigm\| =
\bigm\| \bigm\| (a\alpha a, a\alpha \cdot y0) - (a, 0)
\bigm\| \bigm\| =
\bigm\| \bigm\| a\alpha a - a
\bigm\| \bigm\| \rightarrow 0.
Hence, (a, 0) \in \scrJ . Similarly, (a0, 0) \in \scrJ , and so (a, y) = (a, 0)+(a0, y) - (a0, 0) \in \scrJ . Therefore,
\scrI \oplus Y \subseteq \scrJ .
Lemma 2.1 is proved.
Let \scrA be a Banach algebra, X be a Banach \scrA -bimodule, and \scrI \oplus Y be a closed ideal of \scrA \oplus X.
In view of [17] and that
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
DERIVATIONS ON THE MODULE EXTENSION BANACH ALGEBRAS 569
(\scrI \cdot X) \cup (X \cdot \scrI ) \subseteq Y (2.1)
the module actions are successively defined as follows:
first, for x \in X, F \in Y \ast , define x \cdot F, F \cdot x \in \scrI \ast by
\langle x \cdot F, i\rangle = \langle F, i \cdot x\rangle , \langle F \cdot x, i\rangle = \langle F, x \cdot i\rangle , i \in \scrI ; (2.2)
for a \in \scrA and u \in \scrI \ast , define a \cdot u, u \cdot a \in \scrI \ast via
\langle a \cdot u, i\rangle = \langle u, ia\rangle , \langle u \cdot a, i\rangle = \langle u, ai\rangle , i \in \scrI ; (2.3)
also, for a \in \scrA and F \in Y \ast , define a \cdot F, F \cdot a \in Y \ast through
\langle a \cdot F, y\rangle = \langle F, y \cdot a\rangle , \langle F \cdot a, y\rangle = \langle F, a \cdot y\rangle , y \in Y. (2.4)
Throughout this paper, we assume that \scrI is a closed ideal of Banach algebra \scrA and Y is a closed
\scrA -submodule of Banach \scrA -bimodule X such that condition (2.1) holds unless otherwise stated
explicitly.
Lemma 2.2. Suppose that X is a Banach \scrA -bimodule and \scrI \oplus Y is a closed ideal of \scrA \oplus X.
Then (\scrA \oplus X)-bimodule actions on \scrI \ast \.+Y \ast are given by the following formulas:
(a, x) \cdot (u, F ) = (a \cdot u+ x \cdot F, a \cdot F ), (2.5)
(u, F ) \cdot (a, x) = (u \cdot a+ F \cdot x, F \cdot a). (2.6)
Proof. For (i, y) \in (\scrI \oplus Y ), (u, F ) \in (\scrI \oplus Y )\ast , by using relations (2.2), (2.3) and (2.4), we
have
\langle (a, x) \cdot (u, F ), (i, y)\rangle = \langle (u, F ), (i, y) \cdot (a, x)\rangle =
= \langle (u, F ), (ia, i \cdot x+ y \cdot a)\rangle = \langle u, ia\rangle + \langle F, i \cdot x+ y \cdot a\rangle =
= \langle u, ia\rangle + \langle F, i \cdot x\rangle + \langle F, y \cdot a\rangle = \langle a \cdot u, i\rangle + \langle x \cdot F, i\rangle + \langle a \cdot F, y\rangle =
= \langle a \cdot u+ x \cdot F, i\rangle + \langle a \cdot F, y\rangle = \langle (a \cdot u+ x \cdot F, a \cdot F ), (i, y)\rangle .
Hence, equality (2.5) holds. The accuracy of relation (2.6) can be obtained similarly.
Lemma 2.2 is proved.
We wish to find necessary and sufficient conditions for a module extension Banach algebra
\scrA \oplus X to be \scrI \oplus Y -weakly amenable. However, to achieve our purposes in this paper, we need three
upcoming lemmas which were presented as Lemmas 2.1 and 2.3 of [5]. We include them without
proof.
Lemma 2.3. Suppose that \Gamma : X - \rightarrow \scrI \ast is a continuous \scrA -bimodule morphism. Then \=\Gamma :
\scrA \oplus X - \rightarrow (\scrI \oplus Y )\ast defined by \=\Gamma ((a, x)) = (\Gamma (x), 0) is a continuous derivation. The derivation \=\Gamma
is inner if and only if there exists F \in Y \ast such that a \cdot F - F \cdot a = 0 and \Gamma (x) = x \cdot F - F \cdot x for
a \in \scrA and x \in X.
Lemma 2.4. Suppose that D : \scrA - \rightarrow \scrI \ast is a continuous derivation. Then \=D : \scrA \oplus X - \rightarrow
- \rightarrow (\scrI \oplus Y )\ast defined by \=D((a, x)) = (D(a), 0) is also a continuous derivation, D is inner if and
only if \=D is inner.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
570 A. TEYMOURI, A. BODAGHI, D. EBRAHIMI BAGHA
Lemma 2.5. Suppose that T : X - \rightarrow Y \ast is a continuous \scrA -bimodule morphism, satisfying
x \cdot T (y) + T (x) \cdot y = 0 for all x, y \in X. Then \=T : \scrA \oplus X - \rightarrow (\scrI \oplus Y )\ast defined by \=T ((a, x)) =
= (0, T (x)) is a continuous derivation, \=T is inner if and only if T = 0.
In light of the above lemmas, we correct the proof of Theorem 2.4 from [5], and so we bring its
proof for the sake of completeness.
Theorem 2.1. Suppose that X is a Banach \scrA -bimodule and \scrI \oplus Y is a closed ideal of \scrA \oplus X.
Then the module extension Banach algebra \scrA \oplus X is (\scrI \oplus Y )-weakly amenable if and only if the
following conditions hold:
(i) H1(\scrA , \scrI \ast ) = \{ 0\} ;
(ii) the only continuous derivation D : \scrA - \rightarrow Y \ast for which there is a continuous operator K :
X - \rightarrow \scrI \ast such that K(a \cdot x) = D(a) \cdot x+ a \cdot K(x) and K(x \cdot a) = x \cdot D(a) +K(x) \cdot a (a \in \scrA )
are the inner derivations;
(iii) for every continuous \scrA -bimodule morphism \Gamma : X - \rightarrow \scrI \ast , there exists F \in Y \ast such that
a \cdot F - F \cdot a = 0 for a \in \scrA and \Gamma (x) = x \cdot F - F \cdot x for x \in X ;
(iv) the only continuous \scrA -bimodule morphism T : X - \rightarrow Y \ast for which x \cdot T (y)+T (x) \cdot y = 0
(x, y \in X) in \scrI \ast is T = 0.
Proof. Denote by \Delta 1 the projection from (\scrI \oplus Y )\ast onto \scrI \ast with kernel Y \ast . Let \Delta 2 be the projec-
tion \mathrm{i}\mathrm{d} - \Delta 1 : (\scrI \oplus Y )\ast - \rightarrow Y \ast and let \tau 1 : A - \rightarrow (\scrA \oplus X) and \tau 2 : X - \rightarrow (\scrA \oplus X) be the inclusion
mappings (i.e., \tau 1(a) = (a, 0) and \tau 2(x) = (0, x)). Then \Delta 1,\Delta 2 are \scrA -bimodule morphisms and
\tau 1, \tau 2 are algebra homomorphisms. We now proceed to prove the sufficiency. Suppose that condi-
tions (i) – (iv) hold. Assume that D : \scrA \oplus X - \rightarrow (\scrI \oplus Y )\ast is a continuous derivation. Then D \circ \tau 1 :
\scrA - \rightarrow (\scrI \oplus Y )\ast is a continuous derivation. This implies that \Delta 1\circ D\circ \tau 1 : \scrA - \rightarrow \scrI \ast and \Delta 2\circ D\circ \tau 1 :
\scrA - \rightarrow Y \ast are continuous derivations. By condition (i), \Delta 1 \circ D \circ \tau 1 is inner. It follows from
Lemma 2.4 that \Delta 1 \circ D \circ \tau 1 : \scrA \oplus X - \rightarrow (\scrI \oplus Y )\ast defined by
\Delta 1 \circ D \circ \tau 1((a, x)) =
\bigl(
\Delta 1 \circ D \circ \tau 1(a), 0
\bigr)
, (a, x) \in (\scrA \oplus X),
is also inner.
Claim 1: \Delta 2 \circ D \circ \tau 2 : X - \rightarrow Y \ast is trivial.
Let T = \Delta 2 \circ D \circ \tau 2. By condition (iv), it suffices to show that T is an \scrA -bimodule morphism
satisfying x \cdot T (y) + T (x) \cdot y = 0, x, y \in X. We have
0 = D(0, 0) = D
\bigl(
(0, x) \cdot (0, y)
\bigr)
= D
\bigl(
(0, x)
\bigr)
\cdot (0, y) + (0, x) \cdot D((0, y)) =
=
\bigl(
0,\Delta 2 \circ D \circ \tau 2(x)
\bigr)
\cdot (0, y) + (0, x) \cdot
\bigl(
0,\Delta 2 \circ D \circ \tau 2(y)
\bigr)
=
=
\bigl(
[\Delta 2 \circ D \circ \tau 2(x)]y, 0
\bigr)
+
\bigl(
x[\Delta 2 \circ D \circ \tau 2(y)], 0
\bigr)
.
Thus, x \cdot T (y) + T (x) \cdot y = 0. On the other hand,\bigl(
0, T (a \cdot x)
\bigr)
= \Delta 2 \circ D
\bigl(
(0, a \cdot x)
\bigr)
= \Delta 2 \circ D
\bigl(
(a, 0) \cdot (0, x)
\bigr)
=
= \Delta 2
\bigl(
D((a, 0)) \cdot (0, x) + (a, 0) \cdot D(0, x)
\bigr)
=
= \Delta 2
\bigl(
(a, 0) \cdot D(0, x)
\bigr)
= \Delta 2
\bigl(
aD \circ \tau 2(x)
\bigr)
= a \cdot T (x).
Similarly, T (x \cdot a) = T (x) \cdot a and so T is an \scrA -bimodule morphism. This proves the claim 1.
Let K = \Delta 1 \circ D \circ \tau 2 : X - \rightarrow \scrI \ast and D1 = \Delta 2 \circ D \circ \tau 1 : \scrA - \rightarrow Y \ast .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
DERIVATIONS ON THE MODULE EXTENSION BANACH ALGEBRAS 571
Claim 2: K(a \cdot x) = D1(a) \cdot x+ a \cdot K(x) and K(x \cdot a) = x \cdot D1(a) +K(x) \cdot a for a \in A and
x \in X. We have
(K(a \cdot x), 0) = D
\bigl(
(0, a \cdot x)
\bigr)
= D
\bigl(
(a, 0) \cdot (0, x)
\bigr)
=
= D((a, 0)) \cdot (0, x) + (a, 0) \cdot D
\bigl(
(0, x)
\bigr)
=
= (0,\Delta 2 \circ D \circ \tau 1(a)) \cdot (0, x) + (a, 0) \cdot (\Delta 1 \circ D \circ \tau 2(x), 0) =
=
\bigl(
[\Delta 2 \circ D \circ \tau 1(a)]x, 0
\bigr)
+
\bigl(
a[\Delta 1 \circ D \circ \tau 2(x)], 0
\bigr)
=
= (D1(a) \cdot x, 0) + (a \cdot K(x), 0).
Similarly, for every a \in \scrA and x \in X, we obtain (0,K(x\cdot a)) = (0, x\cdot D1(a)+K(x)\cdot a), and, hence,
by condition (ii), D1 = \Delta 2\circ D\circ \tau 1 is inner. Now, suppose that F \in Y \ast satisfies D1(a) = a\cdot F - F \cdot a
for a \in \scrA . Let K1 : X - \rightarrow \scrI \ast be defined by K1(x) = x \cdot F - F \cdot x for x \in X. Then K - K1 :
X - \rightarrow \scrI \ast is a continuous \scrA -bimodule morphism. In fact, from claim 2, for every a \in \scrA and
x \in X, we get
(K - K1)(a \cdot x) = K(a \cdot x) - K1(a \cdot x) =
= (D1(a) \cdot x+ a \cdot K(x)) - ((a \cdot x) \cdot F - F \cdot (a \cdot x)) =
= (a \cdot F - F \cdot a) \cdot x+ a \cdot K(x) - (a \cdot x \cdot F - F \cdot a \cdot x) =
= a(F \cdot x - x \cdot F ) + a \cdot K(x) = a \cdot (K - K1)(x).
Similarly, K - K1 is a right \scrA -bimodule morphism. From the condition (iii), there is a G \in Y \ast such
that a \cdot G - G \cdot a = 0 for a \in \scrA and (K - K1)(x) = x \cdot G - G \cdot x for all x \in X. By Lemma 2.3,
we see that
K - K1 : \scrA \oplus X - \rightarrow (\scrI \oplus Y )\ast ,
(a, x) \mapsto \rightarrow (K - K1(x), 0)
is an inner derivation. Using claim 1, we arrive at
D((a, x)) =
\bigl(
\Delta 1 \circ D \circ \tau 1(a) +K(x), D1(a)
\bigr)
=
= \Delta 1 \circ D \circ \tau 1((a, x)) + (K - K1)((a, x)) +
\bigl(
K1(x), D1(a)
\bigr)
.
Since \bigl(
K1(x), D1(a)
\bigr)
= (x \cdot F - F \cdot x, a \cdot F - F \cdot a) =
= (a, x) \cdot (0, F ) - (0, F ) \cdot (a, x)
for a \in \scrA and x \in X, it gives an inner derivation from \scrA \oplus X into (\scrI \oplus Y )\ast . Hence, as a sum of
three inner derivation, D is inner. Therefore, \scrA \oplus X is (\scrI \oplus Y )-weakly amenable.
Now, we prove the necessity. Suppose that \scrA \oplus X is (\scrI \oplus Y )-weakly amenable. Let D :
\scrA - \rightarrow \scrI \ast be a continuous derivation with the property given in condition (ii), we define \=D :
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572 A. TEYMOURI, A. BODAGHI, D. EBRAHIMI BAGHA
\scrA \oplus X - \rightarrow (\scrI \oplus Y )\ast via
\=D((a, x)) := (K(x), D(a)), (a, x) \in (\scrA \oplus X).
Then \=D is a continuous derivation which is inner. Therefore, there exists (u, F ) \in (\scrI \oplus Y )\ast such
that
\=D((a, x)) = (a, x) \cdot (u, F ) - (u, F ) \cdot (a, x).
Once more, for some u \in \scrI \ast , we have
\bigl(
K(x), D(a)
\bigr)
= (x \cdot F - F \cdot x, a \cdot F - F \cdot a), and thus
D(a) = a \cdot F - F \cdot a. This means that D is inner, and condition (ii) holds. Moreover, conditions (i)
and (iv) hold by Lemmas 2.4 and 2.5, respectively. Furthermore, condition (iii) holds by Lemma 2.3.
Theorem 2.1 is proved.
Here, we make some comments on condition (iii) in Theorem 2.1 as follows:
Remark 2.1. The condition (iii) in Theorem 2.1 is equivalent to:
(iii)\prime there is no non-zero continuous \scrA -bimodule morphism \Gamma : X - \rightarrow \scrI \ast .
To prove this, suppose that (iii) holds. Taking F = 0 \in Y \ast , we see that condition (iii) holds.
Conversely, assume that condition (iii) holds and \Gamma : X - \rightarrow \scrI \ast is a continuous \scrA -bimodule mor-
phism. Then there is an F \in Y \ast with a \cdot F - F \cdot a = 0 for all a \in \scrA and \Gamma (x) = x \cdot F - F \cdot x for
all x \in X. Hence,
\langle \Gamma (x), a\rangle = \langle x \cdot F - F \cdot x, a\rangle = \langle F \cdot a - a \cdot F, x\rangle = 0, a \in \scrA , x \in X.
Therefore, \Gamma (x) = 0 for all x \in X. This shows that \Gamma = 0.
Proposition 2.1. If condition (iv) of Theorem 2.1 holds and \scrA has a bounded approximate
identity in \scrI , then \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(\scrI \cdot X + X \cdot \scrI ) is dense in Y and there is no non-zero continuous \scrA -
bimodule morphism T : X - \rightarrow Y \ast satisfying \langle T (y), x\rangle + \langle T (x), y\rangle = 0 for all x, y \in Y.
Proof. Suppose the assertion is false. Take 0 \not = f \in Y \ast such that f(\scrI \cdot X + X \cdot \scrI ) = 0. Let
F \in X\ast be a Hahn – Banach extension of f on X. Define T : X - \rightarrow Y \ast by T (x) := \langle F, x\rangle f. By
assumption, \scrA has a bounded approximate identity in \scrI , say (a\alpha ). Then, for y \in Y and a \in \scrA , we
have
T (a \cdot x) = \langle F, a \cdot x\rangle f = \langle x \cdot F, a\rangle f =
\bigl\langle
x \cdot F, \mathrm{l}\mathrm{i}\mathrm{m}
\alpha
(a\alpha a)
\bigr\rangle
f = \mathrm{l}\mathrm{i}\mathrm{m}
\alpha
\langle x \cdot F, a\alpha a\rangle f =
= \mathrm{l}\mathrm{i}\mathrm{m}
\alpha
\langle F, a\alpha a \cdot x\rangle f = 0.
Moreover,
\langle a \cdot T (x), y\rangle = \langle a \cdot \langle F, x\rangle f, y\rangle = \langle F, x\rangle \langle f, y \cdot a\rangle = \langle F, x\rangle
\bigl\langle
f, y \cdot \mathrm{l}\mathrm{i}\mathrm{m}
\alpha
(aa\alpha )
\bigr\rangle
=
= \mathrm{l}\mathrm{i}\mathrm{m}
\alpha
\langle F, x\rangle \langle f, y \cdot aa\alpha \rangle = 0.
In the above relation, we have used the fact that \scrI \cdot X+X \cdot \scrI \subseteq Y. Similarly, T (x) \cdot a = 0, and thus
T is a non-zero continuous \scrA -bimodule morphism and \scrI \cdot T (x) = T (x) \cdot \scrI = \{ 0\} . In other words,
for all x, y \in X, i \in \scrI , we get
\langle i \cdot T (x), y\rangle = \langle i\langle F, x\rangle f, y\rangle = \langle F, x\rangle \langle f, y \cdot i\rangle = 0.
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DERIVATIONS ON THE MODULE EXTENSION BANACH ALGEBRAS 573
The equality T (x) \cdot \scrI = \{ 0\} can be shown similarly. Since T (x) \subset (\scrI \cdot X)\bot
\bigcap
(X \cdot \scrI )\bot , one can
check that x \cdot T (y) = T (x) \cdot y = 0 in \scrI \ast for all x, y \in X. This leads to a contradiction with the
condition (iv) of Theorem 2.1. We now assume an \scrA -bimodule morphism T : X - \rightarrow Y \ast satisfies
\langle T (y), x\rangle + \langle T (x), y\rangle = 0 for x, y \in Y. Then, for any i \in \scrI ,
\langle x \cdot T (y) + T (x) \cdot y, i\rangle = \langle T (y), i \cdot x\rangle + \langle T (i \cdot x), y\rangle = 0.
This show that x \cdot T (y) + T (x) \cdot y = 0 for all x, y \in Y, and therefore T = 0.
Proposition 2.1 is proved.
Let X0 be an \scrA -bimodule with trivial right module action, i.e., X0\scrA = \{ 0\} such that \scrA has a
bounde approximate identity on \scrI . Suppose that Y0 is a submodule of X0 and \scrI is a closed ideal of
\scrA . We, firstly, observe that conditions (iii) and (iv) in Theorem 2.1 are reduced, respectively, to:
(iii)\prime 0 for any continuous \scrA -bimodule morphism \Gamma : X0 - \rightarrow \scrI \ast there is F \in Y0
\ast such that
F \cdot a = 0 for a \in \scrA and \Gamma (x) = x \cdot F for x \in X0 ;
(iv)\prime 0 \scrI X0 is dense in Y0.
Indeed, the equivalence of (iii) and (iii)\prime 0 in this case is clear. Now, if (iv) holds for X = X0, then
Proposition 2.1 necessitates that span(\scrI \cdot X0) is dense in Y0, and so (iv)\prime 0 holds. Conversely, with
having the condition (iv)\prime 0, any \scrA -bimodule morphism T : X0 - \rightarrow Y \ast
0 is trivial, because the left
\scrA -module action on Y \ast
0 is trivial.
Suppose that \scrA has a bounded approximate identity on \scrI . From Proposition 1.5 in [10], condition
(ii) in Theorem 2.1 always holds for X = X0. This verifies the following consequence.
Theorem 2.2. Suppose that \scrA is \scrI -weakly amenable Banach algebra with a bounded appro-
ximate identity on \scrI . Then \scrA \oplus X0 is (\scrI \oplus Y )-weakly amenable if and only if \scrI X0 is dense in Y0.
By a similar way, Theorem 2.2 is valid when X0 is a \scrA -bimodule that left module action is
trivial.
The rest of this section we will be concerned with the two cases X = \scrI and X = \scrA \ast as Banach
\scrA -bimodules for which \scrI is a closed ideal of \scrA . Firstly, we note that if \scrA is not ideally amenable,
then there is an ideal \scrI 1 of \scrA such that H1(\scrA , \scrI 1\ast ) \not = \{ 0\} . It follows from Theorem 2.1 that, for
such \scrI , the module extension \scrA \oplus \scrI is not (\scrI 1 \oplus Y )-weakly amenable. For the ideal amenability of
\scrA \oplus \scrA \ast , we have the following result.
Proposition 2.2. For any Banach algebra \scrA , \scrA \oplus \scrA \ast is never ideally amenable.
Proof. Since the identity mapping from X = \scrA \ast onto \scrI \ast is a non-zero continuous \scrA -bimodule
morphism, Remark 2.1 implies that condition (iii) of Theorem 2.1 does not holds.
Proposition 2.2 is proved.
We now consider the case X = \scrI and \scrI \oplus \scrI be a closed ideal of \scrA \oplus \scrI . We show that under
which conditions H1(\scrA \oplus \scrI , \scrI \ast \.+\scrI \ast ) = \{ 0\} . In other words, we show when \scrA \oplus \scrI is (\scrI \oplus \scrI )-weakly
amenable. Note that for X = \scrI conditions (iii) and (iv) in Theorem 2.1 hold if and only if there
is no non-zero \scrA -bimodule morphism T from \scrI into \scrI \ast . Besides, we see in the case X = \scrI that
conditions (i) and (ii) in Theorem 2.1 are the same.
In light of Theorem 5.4 of [17], we have the upcoming result.
Theorem 2.3. Let \scrI be a closed ideal of a Banach algebra \scrA such that \scrA is ideally amenable
and \scrA has a bounded approximate identity on \scrI . Then \scrA \oplus \scrI is (\scrI \oplus \scrI )-weakly amenable if and
only if \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{ ij - ji; i, j \in \scrI \} is dense in \scrI .
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574 A. TEYMOURI, A. BODAGHI, D. EBRAHIMI BAGHA
Proof. If span \{ ij - ji; i, j \in \scrI \} is not dense in \scrI , then there is a non-zero linear functional f
in \scrI \ast such that \langle f, ij - ji\rangle = 0 for all i, j \in \scrI . This means that i \cdot f = f \cdot i for i \in \scrI . Define T :
\scrI - \rightarrow \scrI \ast via T (i) = i \cdot f = f \cdot i. By assumption, \scrA has a bounded approximate identity in \scrI , say
(a\alpha ). For every j \in \scrI and a \in \scrA , we have
T (a \cdot i) = (a \cdot i) \cdot f =
\bigl(
\mathrm{l}\mathrm{i}\mathrm{m}
\alpha
(aa\alpha ) \cdot i
\bigr)
\cdot f = \mathrm{l}\mathrm{i}\mathrm{m}
\alpha
(aa\alpha \cdot i) \cdot f = 0.
Moreover,
\langle a \cdot T (i), j\rangle = \langle i \cdot f, j \cdot a\rangle = \langle f, (j \cdot a) \cdot i\rangle = 0.
Similarly, T (i) \cdot a = 0, and hence T is a non-zero continuous \scrA -bimodule morphism and, clearly,
x \cdot T (y) + T (x) \cdot y = 0 on \scrI \ast for all x, y \in \scrI . Since \scrA is ideally amenable and \scrI has a bounded
approximate identity, \scrI is weakly amenable by [7] (Theorem 1.9). Now, Proposition 1.3 from [2]
implies that \scrI 2, the linear span of all product elements ij, i, j \in \scrI , is dense in \scrI , and so there
are i, j \in \scrI such that \langle f, ij\rangle \not = 0. This shows that T \not = 0. Therefore, in this case \scrA \oplus \scrI is not
(\scrI \oplus \scrI )-weakly amenable.
For the converse, assume that span\{ ij - ji; i, j \in \scrI \} is dense in \scrI . Then, for any given continuous
\scrA -bimodule morphism T : \scrI - \rightarrow \scrI \ast , we have T (a) = a \cdot f = f \cdot a, where f is weak\ast cluster point
of (T (a\alpha )). This means that f(ij - ji) = 0 for all i, j \in \scrI . It follows that f = 0 and hence T = 0.
Thus, the conditions (iii) and (iv) in Theorem 2.1 hold. The other two conditions hold automatically.
Therefore, \scrA \oplus \scrI is (\scrI \oplus \scrJ )-weakly amenable by Theorem 2.1.
Theorem 2.3 is proved.
From Theorem 2.3 we have immediately the next direct consequence.
Corollary 2.1. For any commutative Banach algebra \scrA which has a bounded approximate iden-
tity in \scrI , \scrA \oplus \scrI is not (\scrI \oplus \scrI )-weakly amenable.
Consider the algebra \scrA (X) of approximable operators on a Banach space X. It is well-known
that \scrA (X) is the closure in \scrB (X) of ideal of continuous finite-rank operators on X, where \scrB (X)
denotes the Banach algebra of all bounded linear operator on X. We also denote the algebra of
compact operator on a Banach space X by \scrK (X).
Example 2.2. (i) For a Banach space X, it is shown in [8] that \scrA (X) = \scrK (X) if \scrA (X) are
amenable. Consider lp = lp(\BbbN ), 1 < p < \infty , which is a reflexive Banach space. By [10], \scrK (lp)
is amenable for 1 < p < \infty . It is also proved in [8] (Theorem 6.9) that \scrA (X) is amenable for
X = lp \oplus lq (1 < p, q < \infty ) if and only if either p = q or one of p or q is 2. Therefore,
\scrA (l2) = \scrK (l2) is amenable. Thus, \scrA (l2) as a closed ideal is \scrB (l2) and has a bounded approximate
identity. Since l2 is a Hilbert space, by [16] \scrB (l2) is a C\ast -algebra, and thus it is ideally amenable [7]
(Corollary 2.2). In fact, H1(\scrB (l2),\scrA (l2)\ast ) = 0. It is shown in [14] that \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{ ab - ba, a, b \in \scrA (l2)\}
is dense in \scrA (l2). Therefore, it follows from Theorem 2.3 that \scrB (l2) \oplus \scrA (l2) is (\scrA (l2) \oplus \scrA (l2))-
weakly amenable.
(ii) Let \scrB (\scrH ) bounded linear operators on the infinite dimensional Hilbert space \scrH . It is well-
known that \scrB (\scrH ) has exactly two non-zero closed ideals \scrK (\scrH ) and \scrB (\scrH ). According to a classical
result due to Halmos, every element in \scrB (\scrH ) can be written as a sum of two commutators ([9] (The-
orem 8) and [14] (Theorem 1)). On the other hand, \scrB (\scrH ) and \scrK (\scrH ) have an identity and as
C\ast -algebras which are ideally amenable by [7] (Corollary 2.2). Now, Theorem 2.3 implies that
\scrB (\scrH )\oplus \scrK (\scrH ) is (\scrK (\scrH )\oplus \scrK (\scrH ))-weakly amenable.
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DERIVATIONS ON THE MODULE EXTENSION BANACH ALGEBRAS 575
3. Derivations on \bfscrA \oplus (\bfitX 1 \.+\bfitX 2). Suppose that X1 and X2 are two Banach \scrA -bimodules.
We denote by X1 \.+X2 the direct module sum of X1 and X2, i.e., the l1 direct sum of X1 and X2
with the module actions given by
a \cdot (x1, x2) = (a \cdot x1, a \cdot x2), (x1, x2) \cdot a = (x1 \cdot a, x2 \cdot a), a \in \scrA , x1 \in X1, x2 \in X2.
For this module actions we have the following equality:
(x1, x2) \cdot (f\ast
1 , f
\ast
2 ) = x1 \cdot f\ast
1 + x2 \cdot f\ast
2 , (x1, x2) \in X1 \.+X2, (f\ast
1 , f
\ast
2 ) \in (X1 \.+X2)
\ast .
In this section, we investigate the \scrJ -ideal amenability for Banach algebra \scrA \oplus (X1 \.+X2). In
analogy with Lemma 2.1, we have the next lemma for \scrA \oplus (X1 \.+X2). Since the proof is similar, is
omitted.
Lemma 3.1. Let \scrA be a Banach algebra and X1, X2 be two Banach \scrA -bimodules. If \scrI is a
closed ideal of \scrA , and Y1, Y2 are closed \scrA -submodules of X1, X2, respectively, then \scrI \oplus (Y1\dotplus Y2) is
a closed ideal of \scrA \oplus (X1\dotplus X2) provided that (\scrI \cdot X1)
\bigcup
(X1 \cdot \scrI ) \subseteq Y1 and (\scrI \cdot X2)
\bigcup
(X2 \cdot \scrI ) \subseteq Y2.
The idea of proof of the next result is taken from [17] (Lemma 7.1), but we bring its proof for
the sake of completeness.
Theorem 3.1. Suppose that \scrA \oplus X1 is (\scrI \oplus Y1)-weakly amenable and \scrA \oplus X2 is (\scrI \oplus Y1)-weakly
amenable. Then the following are equivalent:
(i) \scrA \oplus (X1 \dotplus X2) is (\scrI \oplus (Y1 \dotplus Y2))-weakly amenable;
(ii) there is no non-zero continuous \scrA -bimodule morphism \Gamma : Y1 - \rightarrow Y \ast
2 ;
(iii) there is no non-zero continuous \scrA -bimodule morphism \Lambda : Y2 - \rightarrow Y \ast
1 .
Proof. (i) \Rightarrow (ii). Assume that \scrA \oplus (X1 \dotplus X2) is (\scrI \oplus (Y1 \dotplus Y2))-weakly amenable. Let \Gamma :
Y1 - \rightarrow Y \ast
2 be a continuous \scrA -bimodule morphism and T : Y1 \.+Y2 - \rightarrow (Y1 \.+Y2)
\ast be the continuous
\scrA -bimodule morphism defined through
T ((y1, y2)) = ( - \Gamma \ast (y2),\Gamma (y1)), (y1, y2) \in (Y1 \.+Y2).
For each (y1, y2), (z1, z2) \in (Y1 \.+Y2) and i \in \scrI , we get\bigl\langle
(y1, y2) \cdot T
\bigl(
(z1, z2)
\bigr)
+ T
\bigl(
(y1, y2)
\bigr)
\cdot (z1, z2), i
\bigr\rangle
=
=
\bigl\langle
- y1 \cdot \Gamma \ast (z2) + y2 \cdot \Gamma (z1), i
\bigr\rangle
+
\bigl\langle
- \Gamma \ast (y2) \cdot z1 + \Gamma (y1) \cdot z2, i
\bigr\rangle
=
=
\bigl\langle
- \Gamma (y1) \cdot z2 + y2 \cdot \Gamma (z1), i
\bigr\rangle
+
\bigl\langle
- y2 \cdot \Gamma (z1) + \Gamma (y1) \cdot z2, i
\bigr\rangle
= 0.
Thus, (y1, y2) \cdot T
\bigl(
(z1, z2)
\bigr)
+T
\bigl(
(y1, y2)
\bigr)
\cdot (z1, z2) = 0. It follows from condition (iv) of Theorem 2.1
that T = 0. This implies that \Gamma = 0.
(ii) \Rightarrow (iii). Suppose that \Lambda : Y2 - \rightarrow Y \ast
1 is a continuous \scrA -bimodule morphism. It is easily
verified that the mapping \Gamma : Y1 - \rightarrow Y \ast
2 defined by \Gamma = \Lambda \ast | Y1 is a continuous \scrA -bimodule mor-
phism. Consequently, \Gamma = 0. This implies that \Lambda \ast = 0. Since \Lambda \ast is weak\ast -weak\ast continuous and
Y1 is weak\ast dense in Y \ast \ast
1 , we have \Lambda = 0.
(iii) \Rightarrow (ii). The proof is similar to the preceding implication.
(ii) + (iii) \Rightarrow (i). Since \scrA \oplus X1 is (\scrI \oplus Y1)-weakly amenable and \scrA \oplus X2 is (\scrI \oplus Y1)-weakly
amenable, conditions (i) – (iii) of Theorem 2.1 hold automatically for X = X1 \.+X2 and Y = Y1 \.+Y2.
For condition (iv), assume that T : X - \rightarrow Y \ast is a continuous \scrA -bimodule morphism fulfilling
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576 A. TEYMOURI, A. BODAGHI, D. EBRAHIMI BAGHA
(x1, x2) \cdot T
\bigl(
(y1, y2)
\bigr)
+ T
\bigl(
(x1, x2)
\bigr)
\cdot (y1, y2) = 0,
\bigl(
x1, x2), (y1, y2) \in X.
Let Pi : Y \ast - \rightarrow Y \ast
i be the natural projections, \iota : Y - \rightarrow X and \tau i : Yi - \rightarrow Y be the natural
embedding for i = 1, 2. Choosing x2 = y2 = 0 and x1 = y1 = 0, we arrive at
x1 \cdot P1 \circ T \circ \iota \circ \tau 1(y1) + P1 \circ T \circ \iota \circ \tau 1(x1) \cdot y1 = 0,
x2 \cdot P2 \circ T \circ \iota \circ \tau 2(y2) + P2 \circ T \circ \iota \circ \tau 2(x2) \cdot y2 = 0
for all xi, yi \in Yi and i = 1, 2. Applying condition (iv) of Theorem 2.1 to the \scrA \oplus Xi is (\scrI \oplus Yi)-
weakly amenable, for i = 1, 2, we have Pi \circ T \circ \iota \circ \tau i = T | Yi = 0. Moreover, the parts (ii) and (iii)
imply that P1 \circ T \circ \iota \circ \tau 2 : Y2 - \rightarrow Y \ast
1 and P2 \circ T \circ \iota \circ \tau 1 : Y1 - \rightarrow Y \ast
2 are trivial. These show that
T = 0, that is, the condition (iv) of Theorem 2.1 holds.
Theorem 3.1 is proved.
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Received 22.07.18
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| resource_txt_mv | umjimathkievua/e8/b34d067c6811f16fa370d98dd04b77e8.pdf |
| spelling | umjimathkievua-article-2402025-03-31T08:48:15Z Derivations on the module extension Banach algebras Derivations on the module extension Banach algebras Teymouri , A. Bodaghi , A. Ebrahimi Bagha, D. Teymouri , A. Bodaghi, A. Ebrahimi Bagha, D. Amenability Banach algebra Derivation Module extension Amenability Banach algebra Derivation Module extension UDC 517.986 We correct some results presented in [M. Eshaghi Gordji, F. Habibian, A. Rejali,&nbsp; Ideal amenability of module extension Banach algebras, Int. J. Contemp. Math. Sci.,&nbsp; 2, No. 5, 213–219 (2007)] and, using the obtained consequences, we find necessary and sufficient conditions for the module extension $\mathcal A\oplus X$ to be $(\mathcal I\oplus Y)$-weakly amenable, where $\mathcal I$ is a closed ideal of the Banach algebra $\mathcal A$ and $Y$ is a closed $\mathcal A$-submodule of the Banach $\mathcal A$-bimodule $X.$ We apply this result to the module extension $\mathcal A\oplus(X_1\dotplus X_2),$ where $X_1,$ $X_2$ are two Banach $\mathcal A$-bimodules. УДК 517.986 Диференцiювання на банахових алгебрах розширення модуля Виправлено деякі результати роботи [M. Eshaghi Gordji, F. Habibian, A. Rejali,&nbsp; Ideal amenability of module extension Banach algebras, Int. J. Contemp. Math. Sci.,&nbsp; 2, No. 5, 213–219 (2007)] та за допомогою отриманих наслідків знайдено необхідні та достатні умови того, що розширення модуля $\mathcal A\oplus X$ буде $(\mathcal I\oplus Y)$-слабко аменабельним, де $\mathcal I$ — замкнений ідеал банахової алгебри $\mathcal A,$ а $Y$ — замкнений $\mathcal A$-субмодуль банахового $\mathcal A$-бімодуля $X.$ Ці результати застосовано до розширення модуля $\mathcal A\oplus(X_1\dotplus X_2),$ де $X_1,$ $X_2$ — банахові $\mathcal A$-бімодулі. Institute of Mathematics, NAS of Ukraine 2021-04-21 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/240 10.37863/umzh.v73i4.240 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 4 (2021); 566 - 576 Український математичний журнал; Том 73 № 4 (2021); 566 - 576 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/240/9010 Copyright (c) 2021 Abasalt Bodaghi, Akram Teymur, Davood Ebrahimi Bagha |
| spellingShingle | Teymouri , A. Bodaghi , A. Ebrahimi Bagha, D. Teymouri , A. Bodaghi, A. Ebrahimi Bagha, D. Derivations on the module extension Banach algebras |
| title | Derivations on the module extension Banach algebras |
| title_alt | Derivations on the module extension Banach algebras |
| title_full | Derivations on the module extension Banach algebras |
| title_fullStr | Derivations on the module extension Banach algebras |
| title_full_unstemmed | Derivations on the module extension Banach algebras |
| title_short | Derivations on the module extension Banach algebras |
| title_sort | derivations on the module extension banach algebras |
| topic_facet | Amenability Banach algebra Derivation Module extension Amenability Banach algebra Derivation Module extension |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/240 |
| work_keys_str_mv | AT teymouria derivationsonthemoduleextensionbanachalgebras AT bodaghia derivationsonthemoduleextensionbanachalgebras AT ebrahimibaghad derivationsonthemoduleextensionbanachalgebras AT teymouria derivationsonthemoduleextensionbanachalgebras AT bodaghia derivationsonthemoduleextensionbanachalgebras AT ebrahimibaghad derivationsonthemoduleextensionbanachalgebras |