Гипертауберовы алгебры, определенные гомоморфизмом банаховой алгебры Текст научной статьи по специальности «Математика»

Вестник КРАУНЦ. Физ.-мат. науки. 2019. Т. 26. № 1. C. 17-27. ISSN 2079-6641

DOI: 10.26117/2079-6641-2019-26-1-17-27

MSC 43A22, 16W20

HYPER-TAUBERIAN ALGEBRAS DEFINED BY A BANACH ALGEBRA HOMOMORPHISM

A. Ebadian1, A. Jabbari2

1 Department of Mathematics, Urmia University, Iran

2 Young Researchers and Elite Club, Ardabil Branch, Islamic Azad University, Ardabil, Iran

E-mail: jabbarial@yahoo.com, ali.jabbari@iauardabil.ac.ir

Let A and B be Banach algebras and T : B —> A be a continuous homomorphism. We consider left multipliers from A xTB into its the first dual i.e., A* x B* and we show that A xTB is a hyper-Tauberian algebra if and only if A and B are hyper-Tauberian algebras.

Keywords: Local operator, hyper-Tauberian algebra, Tauberian algebra

© Ebadian A., Jabbari A., 2019

УДК 519.53

ГИПЕРТАУБЕРОВЫ АЛГЕБРЫ, ОПРЕДЕЛЕННЫЕ ГОМОМОРФИЗМОМ БАНАХОВОЙ АЛГЕБРЫ

А. Ебадиан1, A. Жаббари2

1 Математический факультет, Университет Урмия, Иран и Клуб молодых исследователей и элит, Ардебильский филиал

2 Исламский университет Азад, Ардебиль, Иран E-mail: jabbarial@yahoo.com, ali.jabbari@iauardabil.ac.ir

Пусть A и B - банаховы алгебры, а T : B —> A - непрерывный гомоморфизм. Мы рассматриваем левые мультипликаторы из A xTB в его первое двойственное, т.е. A* x B*, и показываем, что A xTB является гипертауберовой алгеброй тогда и только тогда, когда A и B являются гипертауберовыми алгебрами.

Ключевые слова: локальный оператор, гипертауберова алгебра, тауберова алгебра.

© Ебадиан А., Жаббари A., 2019

Introduction

The notion of Hyper-Tauberian algebras is introduced by Samei [20]. These algebras are commutative Banach algebras that consist of all Tauberian algebras. Idea of definition of hyper-Tauberian algebras is related to the local derivation that this notion was introduced by Kadison [13].

Let (A, || ■ ||A) be a Banach algebra and (X, || ■ ||X) be a Banach space such that X is an A-bimodule. If the module actions maps i.e., A xX —>X and X x A —>X are continuous (in norm), then we say that X is a Banach A-bimodule. Now, let X be a Banach Abimodule, then one can see that the first dual of X i.e., X* is a Banach A-bimodule with the following module actions:

(x, a ■ f) = (x ■ a, f) and (x, f ■ a) = (a ■ x, f),

for every a e A, x e X and f e X*. A derivation from A into a Banach A-bimodule X is a linear map D : A —> X such that

D(ab) = a ■ D(b) + D(a) ■ b,

for every a,b e A. The set of all derivations from A into X is denoted by Z!(A,X); which is a linear subspace of B(A,X), the space of all bounded linear maps from A into X. For a fixed x e X, set Dx : A —> X, a m a ■ x — x ■ a. Derivations of this form are called inner derivations, and an inner derivation Dx is implemented by x. The set of all inner derivations from A into X is a linear subspace N!(A,X) of Z!(A,X). We denote the first cohomology group of a Banach algebra A with coefficients in a Banach A-bimodule X by H!(A,X), where it is equal to Z!(A,X)/N1 (A,X).

Let A be a Banach algebra, and let X be a Banach A-bimodule. An operator D: A —> X is called a local derivation if, for every a e A, there is a derivation Da : A —> X such that D(a) = Da(a). Kadison proved that every bounded local derivation from a von Neumann algebra A into a dual Banach A-bimodule X belongs to Z1 (A,X) and Johnson proved the same result to a C*-algebra A and Banach A-bimodule X [12].

The concept of amenability for Banach algebras was introduced by Johnson [10]. A Banach algebra A is called amenable if H1 (A, X*) = {0} for any A-bimodule X. A Banach algebra A is called weakly amenable if H 1(A,A*) = {0} i.e., every continuous derivation from A into A* is inner. The concept of weak amenability was first introduced by Bade, Curtis and Dales in [3] for commutative Banach algebras, and was extended to the noncommutative case by Johnson, see [11].

Let A and B be Banach algebras such that A is a Banach B-bimodule with compatible actions and appropriate norm. The semidirect product of these Banach algebras is defined on A x B as follows:

(a, b) (a', b') = (aa' + a ■ b' + b ■ a', bb')

for (a,b), (a',b') e A x B. By the above defined product on A x B, it becomes a Banach algebra with the i1 -norm that we denote it by A x B. Moreover, if A and B are commutative such that A is a symmetric Banach B-bimodule (i.e., a■ b = b■ a for every a e A and b e B), then A x B becomes a commutative Banach algebra.

Let A and B be Banach algebras and 6 e a(B), where a(B) is the space of all continuous homomorphisms from B onto C. Lau studied the Banach algebra A xg B in [15], with the norm ||(a,b)|| = ||a||A + ||b||B and with the following product:

(a, b) (a', b') = (aa' + 6 (b')a + 6 (b)a', bb'), (1)

for all (a,b), (a',b') e A xeB. Amenability and weak forms of amenability of A xeB are studied in [7, 16]. Let T : B —^ A be an algebra homomorphism, and A be a commutative Banach algebra. Following [5], we equip the Cartesian product space A x B with the following multiplication:

(a, b)(a', b') = (aa' + T (b)a' + T (b')a, bb'),

for all (a,b), (a',b') e A x B. By the above product, A x B becomes a Banach algebra; we denote it by A xTB. If A and B are Banach algebra and ||T|| < 1, then A xTB is a Banach algebra with the following norm

||(a, b)|| = ||a||A + 11 b |b ,

for (a,b) e A xTB. Arens regularity and various notions of amenability of this new Banach algebra considered in [5]. With a slight difference in definition of the multiplication xT from that given by Bhatt and Dabhi [5], we consider A xTB with the following multiplication

(a,b)(a',b') = (aa' + T(b)a' + aT(b'),bb'), (a,a' e A, b.b'e B). (2)

Note that if A is a commutative Banach algebra, then these multiplications coincide. Let A be a unital Banach algebra with unit eA, d e a(B), and define T0 : B —> A by To(b) = d (b)eA (b e B). Then A xTo B coincides with the product (1). The Banach algebra A xTB with the above multiplication that is a splitting of Banach algebra extension of Banach algebra B by A has been studied by many authors such as [1, 2, 6, 9, 17]. Splitting of Banach algebra extensions has important roles in studying of Banach algebras and they are good tools for giving counter examples for some concepts related to Banach algebras, for example see [4, 8, 23].

In this paper, we consider the Banach algebra A xTB with the multiplication (2). We show that A xTB is a Tauberian algebra if and only if A and B are Tauberian algebraa (Section 2) and in Section 3, we characterize left multipliers from A x T B into its the first dual. Finally, we show that if A and B are hyper-Tauberian then A xTB is a hyper-Tauberian and vice versa.

Tauberian algebra

In this section, we study on some basic properties of the Banach algebra A x T B. We identify (A xTB)* with A* x B* in the natural way

((a, b), (f, g)> = (a, f > + (b, g>

for all (a,b) e A xTB and (f,g) e A* x B*. Then by easy calculations, we have the following actions

(a, b) ■ (f, g) = (a ■ f + T(b) ■ f, T* (a ■ f) + b ■ g),

and

(f,g) ■ (a,b) = (f ■ a + f ■ T(b),T*(f ■ a) + g■ b).

for all (a,b) e A xrB and (f,g) e (A xtB)*.

Lemma 1. [9, Theorem 2.2] Let A and B be Banach algebras, and let T : B —> A be an algebra homomorphism with norm at most 1. Let F1 = {(9, 9 o T) : 9 e a (A)} and F2 = {(0, ¥) : ¥ e a(B)}. Then

(i) if a (A) = 0, then F = 0.

(ii) a (A xrB)= F U F2.

(iii) F1 and F2 are closed in a (A xTB).

We recall the following definitions and notions from [20]. For a Banach algebra A and a Banach A-bimodule X the annihilator of A in X and the annihilator of X in A are the following sets

AnnX(A) = {x e X : x■ a = 0 = a■ x, for all a e A},

and

AnnA(X) = {a e A : x■ a = 0 = a ■ x, for all x e X}.

For a commutative, semisimple and regular Banach algebra A the hull of a closed ideal I in A denoted by h(I). The hull of I is the following set

{t e a (A): a(t) = 0 for all a e I}.

For any element x e X, AnnA(x) is a closed ideal in A and the hull of AnnA(x) is called the support of x in a (A), denoted by suppA x or supp x. Let E c a (A), we consider the following sets

I(E) = {a e A : a|E = 0}, I0(E) = {a e A : a has a compact disjoint from E},

and

J(E) = {a e I(E): supp a is compact}.

The subset E of a (A) is called a set of synthesis for A if there is a unique closed ideal in A whose hull is E. We denote the set of all elements in A with the compact support by Ac. The Banach algebra A is called Tauberian algebra if Ac is dense in A [19]. Ideals of A xTB are investigated in [9, Proposition 2.4] and we write it as follows:

Lemma 2. Let A and B be Banach algebras and T : B —> A be a homomorphism with ||T|| < 1. Then ideals of A xTB are one of the following form

(i) A.

(ii) I xTB, where I is a closed ideal of A and T(B) c I.

(iii) I xT J, where I is a closed ideal of A and J is a closed ideal of B such that T(J) c I.

By Lemma , any subset E of a (A xTB) is a subset of F1 or F2. In other word, E = {(9, $ o T) : for some $ e a (A)} or E = {(0, $ o T) : for some $ e a (A)} or E = {(0, : for some ^ e a(B)}.

Theorem 1. Let A and B be commutative, semisimple, regular Banach algebras and T : B —^ A be a homomorphism with ||T|| < 1. Then A xTB is a Tauberian algebra if and only if A and B are Tauberian algebras.

Proof. Let A xTB be a Tauberian algebra. For every a e A, (a,0) e A xTB. Then there is a net

(aa,ba) c (A xtB)c = AcUBcU{(a,b): a e Ac, b e Bc},

such that (aa,bp) —> (a,0). This follows that aa —> a and consequently, Ac = A. Similarly, one can show that B is a Tauberian algebra.

Let A and B be Tauberian algebras and let (a,b) e A xTB. Then there are nets (aa)ae/ C Ac and (bp)peJ c Bc such that aa —> a and bp —^ b. We define an indexing directed set r = / x nae/J equipped with the product ordering, and for each (a,f) e r, we define cY = ca,f(a) = ca,p. Then by the Theorem on iterated limits [14],

lim cY = lim lim ca p. yer ae/peJ a,p

Now, set cY = ca,p = (aa,bp) e (A xTB)c. By the above arguments, we conclude that cY —^ (a,b). Thus, A xTB is a Tauberian algebra. □

Left multipliers from A xTB into (A xTB)*

Let A be a Banach algebra and X be a left (right) Banach A-module. A linear mapping T : A —> X is called a left (right) multiplier if T(ab) = a ■ T(b) (T(ab) = T(a) ■ b). In this section we characterize left multipliers from A xTB into (A xTB)* and a reason for investigating of left multipliers related to the next section.

Theorem 2. Let A and B be Banach algebras and T : A —^ B be a homomorphism with ||T|| < 1. If F : A xTB —^ A* x B* is a left multiplier, then

(i) there are coordinate maps F1 and F2 such that F = (F1, F2) and F1 and F2 are left multipliers on A and B, respectively.

(ii) F2(aa', 0) = T* (a ■ F1 (a', 0)) and F1 (0, bb') = T(b) ■ F1 (0, b') for every a, a' e A and b, b' e B.

(iii) if AnnA(A*) = A or A is without of order, then F1(T(b), 0) = F1(0,b) for every b e B. Similarly, if AnnB(B*) = B or B is without of order, then T*(F1(a, 0)) = F2(a,0) for every a e A.

Proof. Let F = (F1,F2), where F1 and F2 are coordinate maps related to F and it is easy to check that they are linear and continuous. (i)-(ii) For every a,a' e A, we have

(F1(aa',0),F2(aa',0)) = F(aa',0) = F((a,0)(a',0)) = (a,0) ■ F(a',0)

= (a, 0) ■ (F1 (a', 0), F2(a', 0))

= (a ■ F1(a', 0), T *(a ■ F1(a', 0))). (3)

Therefore F1(aa', 0) = a ■ F1(a', 0) and F2(aa', 0) = T*(a ■ F1(a', 0)) for every a, a' e A. This means that F1 on A is a left multiplier. For every b, b' e B,

(F1 (0,bb'),F2(0,bb')) = F(0,bb') = F((b,0)(0,b')) = (0,b) ■ F(0,b')

= (0,b) ■ (F1 (0,b'),F2(0,b'))

= (T(b) ■ F1 (0,b'),b■ F2(0,b')). (4)

The above relations show that F1(0,bb') = T(b) ■ F1(0,b') and F2(0,bb') = b ■ F2(0,b') for every b, b' e B. This shows that F2 is a left multiplier on B.

(iii) By (1) and (2) we have

F ((a, ft) (a, ft')) = F (aa' + aT (b') + T (b)a', bb')

= (Fi (aa' + aT (b') + T (b)a', bb'), F2(aa' + aT (b') + T (b)a', bb')) = (Fi (aa', 0), 0) + (Fi (aT (b'), 0), 0) + (Fi (T (b) a', 0), 0) +(Fi(0, bb'), 0) + (0, F2(aa', 0)) + (0, F2(aT (b'), 0)) +(0, F2(T (b)a', 0)) + (0, F2(0, bb')) = (a ■ Fi (a', 0), 0) + (a ■ Fi (T(b'), 0), 0) + (T(b) ■ Fi (a', 0), 0) (T(b) ■ Fi (0, b'), 0) + (0, T* (a ■ Fi (a', 0))) +(0, T* (a ■ Fi (T(b'), 0))) + (0, T* (T(b) ■ Fi (a', 0))) +(0, b ■ F2 (0, b')) (5)

On the other hand,

(a,b) ■ F(a',b') = (a,b) ■ F((a',0) + (0,b')) = (a,b) ■ F(a',0) + (a,b) ■ F(0,b') = (a,b) ■ (Fi (a', 0),F2(a', 0)) + (a, b) ■ (Fi (0, b'),Fi(0,b')) = (a ■ Fi (a', 0) + T(b) ■ Fi (a', 0), T* (a ■ Fi (a', 0)) + b ■ Fi(a/, 0))

+(a ■ Fi (0, b') + T(b) ■ Fi (0, b'), T* (a ■ Fi (0, b')) + b ■ F2(0, b')) = (a ■ Fi (a', 0), 0) + (T(b) ■ Fi (a', 0), 0) + (0, T* (a ■ Fi (a', 0))) +(0, b ■ F2(a', 0)) + (a ■ Fi (0, b'), 0) + (T(b) ■ Fi (0, b'), 0) +(0, T* (a ■ Fi (0, b'))) + (0, b ■ F2(0, b')) (6)

The relations (5) and (6) imply that

a ■ Fi (T (b'), 0) = a ■ Fi(0, b')

T*(T(b) ■ Fi(a', 0)) = b ■ F2(a', 0).

For every x e B*, b e B and a' e A,

(7)

(x,b • F2(a',0)) = (x, T*(T(b) • Fi(a', 0))) = (T(x), T(b) • Fi(a',0)) = (T (x)T (b), Fi(a/, 0)) = (T (xb), Fi(a/, 0)) = (xb, T*(Fi(a/, 0))) = (x, b • T*(Fi(a/, 0))). (8)

Then we can write (7) as follows:

a ■ Fi(T (b'), 0) = a ■ Fi(0, b')

b ■ T * (Fi (a', 0)) = b ■ F2(a', 0).

(9)

Now if one of the assumptions in (iii) holds, we conclude the desire. □ We now consider the converse of the above Theorem as follows: Theorem 3. Let A and B be Banach algebras and T : A —> B be a homomorphism with ||T|| < i. // FA : A —^ A* and FB : B —^ B* are le/t multipliers, then F : A xTB —^ A* x B* de/ined as

F (a, b) = (Fa (a + T (b)), T* o Fa (a + T (b)) + FB(b)) (10)

for every (a, b) g A x T B, is a left multiplier.

Proof. For every (a,b), (a', b') g A xTB, we have

(a, b) ■ F (a', b') = (a, b) ■ (FA (a' + T(b')), T*o FA (a' + T(b')) + FB(b')) = (a ■ Fa (a' + T(b')) + T(b) ■ FA (a' + T(b')), b ■ FB(b')

+b ■ T* o Fa (a' + T(b')) + T* (a ■ Fa (a' + T(b')))). (11)

Also, for all a,a'g A and b, b',x g B, we have

(x, T* oFA(T(b)a' + T(bb'))) = (T(x),Fa(T(b)a' + T(bb')))

= (T (x), T (b) ■ Fa (a' + T (b')))

= (xb, T *(FA(a' + T(b'))))

= (x, b ■ T *(FA (a' + T(b')))). (12)

Then by (12) we have the following

F ((a, b) (a', b')) = (Fa (aa' + T(b)a' + aT(b') + T(bb')), T* o Fa (aa' + aT(b') ) +T* oFa(T(b)a' + T(bb')) + FB(bb')) = (a ■ Fa (a' + T (b')) + T (b) ■ Fa (a' + T (b')), b' ■ FB(b')

+b ■ T * o Fa (a' + T (b')) + T *(a ■ Fa (a' + T (b' )))). (13)

Then the relations (11) and (13) imply that F is a left multiplier. □

Hyper-Tauberian algebra

Let X and Y be left (right) Banach A-modules. An operator T : X —> Y is called local with respect to the left (right) A-module action if supp T(x) c supp x, for every x e X .If A is a Tauberian algebra and X is a left (right) Banach A-module, then a bounded operator T : A —> X is called local if supp T(a) c supp a, for every a e Ac [20, Proposition 2].

In this section we assume that all Banach algebras are commutative, semisimple and regular. The Banach algebra A is a said to be a hyper-Tauberian algebra if every bounded local operator T : A —> A* is a multiplier. Hyper-Tauberian algebras are defined by Samei in [20]. He proved that every hyper-Tauberian algebra is Tauberian algebra and is weakly amenable [20, Theorem 5]. In light of Lemma , we have the following Lemma.

Lemma 3. Let A and B be Banach algebras and T : B —> A be a homomorphism with ||T|| < 1. Then

(i) supp (a, 0) = {t e a (A): a(t) = 0}.

(ii) supp (0,b) = {s e a(B): b(s) = 0}.

(iii) supp (a, b) = {(t, s) e a (A xTB) : a(t) + b(s) = 0}.

Theorem 4. Let A and B be Banach algebras and T : B —> A be a homomorphism with ||T|| < 1. Then A xTB is a hyper-Tauberian algebra if and only if A and B are hyper-Tauberian algebras.

Proof. Let A and B be hyper-Tauberian algebras. Since AxALB = B, so AxATB is a hyper-Tauberian algebra. Then by [20, Theorem 9], A xTB is hyper-Tauberian.

Let A xT B be hyper-Tauberian. First, we show that A is hyper-Tauberian. Let F : A —> A* be a bounded local operator. Consider the projection map nA : A xTB —> A defined by nA(a,b) = a + T(b), for all (a,b) e A xTB. Then nA*oFonA : A xTB —^ A* x B* is a bounded local operator. Because by Lemma 3, we have

supp nAoF o nA(a,b) = supp nJ*oF(a + T(b)) c supp nA(a + T(b))

= supp(a + T (b), 0) = {t e a (A) : (a + T(b))(t) = 0} = {(t, t o T) e a (A xtB) : (a + T (b))(t) = 0} c supp (a, b).

This means that F o nA is local and therefore it is a multiplier. Then

n*oFonA((a,b)(a',b')) = (a,b) ■ n*oFonA((a',b'))

= (a,b) ■ n*oF(a' + T(b')) = (a,b) ■ (F(a' + T(b')),0) = (a ■ F(a' + T(b')) + T(b) ■ F(a' + T(b')), T* (a ■ F(a' + T(b')))) = (a ■ F(a') + a ■ F(T(b')) + T(b) ■ F(a')

+T(b) ■ F(T(b')), T*(a ■ F(a' + T(b')))). (14)

On the other hand

nA*o F o nA((a, b)(a', b')) = nA*o F o nA(aa' + T(b)a' + aT(b'), bb')

= n*o F (aa' + T (b)a' + aT (b') + T (bb'))

= (F (aa' + T (b) a' + aT (b') + T (bb')), 0)

= (F(aa')+ F(T(b)a') + F(aT(b')) + F(T(bb'))),0). (15)

By taking b = b' = 0 and using relations (14) and (15), we conclude that F is a multiplier. This shows that A is hyper-Tauberian.

Finally, we prove that B is hyper-Tauberian. Define F : A xT B axtb = B by F(a,b) = (a,b)+ A, for every (a,b) eAxTB. Clearly, F is a bounded and onto homomorphism. Since A xTB is hyper-Tauberian algebra, by [20, Theorem 12], B is hyper-Tauberian. □

Amenability of A xTB studied in [6]. Authors in [6, 18] proved that weak amenability of A xTB implies weak amenability of A and B, but converse is not true in general. Samei in [20] proved that every hyper-Tauberian algebra is weakly amenable. By this fact and above Theorem we have the following result.

Corollary. Let A and B be hyper-Tauberian algebras and T: B —> A be a homomorphism with ||T|| < i. Then A xTB is weakly amenable i/ and only i/ A and B are weakly amenable.

Example. Let G and H be locally compact abelian groups and A(G) and A(H) be Fourier algebras on them. Define T : A(G) —> A(H) by T(a)(h) = a(t(h)) for every h e H, where t : H —> G is continuous. According to [22], if T is an isometry, then t is of the form t(h) = g0(h) for every g e G and 0 : H —> G is a group homomorphism. Thus, A(G) xTA(H) is a hyper-Tauberian algebra, by [20, Proposition 18] and Theorem

Acknowledgement

The authors sincerely thank the anonymous reviewer for his/her careful reading and constructive comments to improve the quality of the first draft of this paper.

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[14] Kelly J.L., General topology, American Book, Van Nostrand, Reinhold, 1969.

[15] Lau A. T-M., "Analysis on a class of Banach algebras with application to harmonic analysis on locally compact groups and semigroups", Fund. Math., 118 (1983), 161-175 https://doi.org/10.4064/fm-118-3-161-175.

[16] Monfared M.S., "On certain products of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups", Studia Math., 2007, № 178(3), 277294 https://doi.org/10.4064/sm178-3-4.

[17] Nemati M., Javanshiri H., "Some homological and cohomological notions on T-Lau product of Banach algebras", Banach J. Math. Anal., 2015, №9(2), 183-195 https://doi.org/10.15352/bjma/09-2-13.

[18] Ramezanpour M., "Weak amenability of the Lau product of Banach algebras defined by a Banach algebra morphism", Bull. Korean Math. Soc., 2017, №54(6), 1991-1999. https://doi.org/10.4134/BKMS.b160690.

[19] Rickart C.E., General theory o/ Banach algebra, Van Nostrand, Princeton, 1960.

[20] Samei E. Hyper-Tauberian algebras and weak amenability of Figa-Talamanca-Herz algebras, J. Func. Anal, 2006, №231(1), 195-220 https://doi.org/10.1016/j-.jfa.2005.05.005.

[21] Samei E., "Local properties of the Hochschild cohomology of C*-algebras", J. Aust. Math. Soc, 2008, №84, 117-130. https://doi.org/10.1017/S1446788708000049.

[22] Walter M. E., "W*-algebras and nonabelian harmonic analysis", J. Func. Anal., 1972, № 11, 17-38 https://doi.org/10.1016/0022-1236(72)90077-8.

[23] Zhang Y., "Weak amenability of module extensions of Banach algebras Trans.", Amer. Math. Soc, 354(10) (2002), 4131-4151 https://doi.org/10.1090/S0002-9947-02-03039-8.

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[11] Johoson B.E. Weak amenability of group algebras // Bull. Lond. Math. Soc. 1991. no. 23(3). pp. 281-284. https://doi.org/10.1112/blms/23.3.281

[12] Johnson B.E. Local derivations on C*-algebras are derivations // Trans. Amer. Math. Soc. 2001. vol. 353. no. 1. pp. 313-325. https://doi.org/10.1090/S0002-9947-00-02688-X

[13] Kadison R.V. Local derivation // J. Algebra. 1990. vol. 130. no. 2. pp. 494-509. https://doi.org/10.1016/0021-8693(90)90095-6

[14] Kelly J. L. General topology. Van Nostrand, Reinhold: American Book, 1969.

[15] Lau A. T-M. Analysis on a class of Banach algebras with application to harmonic analysis on locally compact groups and semigroups // Fund. Math. 1983. no. 118. pp. 161-175. https://doi.org/10.4064/fm-118-3-161-175

[16] Monfared M.S. On certain products of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups // Studia Math. 2007. no. 178(3). pp. 277-294. https://doi.org/10.4064/sm178-3-4

[17] Nemati M., Javanshiri H. Some homological and cohomological notions on T-Lau product of Banach algebras // Banach J. Math. Anal. 2015. no. 9(2). pp. 183-195. https://doi.org/10.15352/bjma/09-2-13

[18] Ramezanpour M. Weak amenability of the Lau product of Banach algebras defined by a Banach algebra morphism // Bull. Korean Math. Soc. 2017. no. 54(6). pp. 1991-1999. https://doi.org/10.4134/BKMS.b160690

[19] Rickart C. E. General theory of Banach algebra. Van Nostrand, Princeton, 1960.

[20] Samei E. Hyper-Tauberian algebras and weak amenability of Figa-Talamanca-Herz algebras //J. Func. Anal. 2006. no. 231(1). pp. 195-220. https://doi.org/10.1016/j-.jfa.2005.05.005

[21] Samei E. Local properties of the Hochschild cohomology of C*-algebras // J. Aust. Math. Soc. 2008. no. 84. pp. 117-130. https://doi.org/10.1017/S1446788708000049

[22] Walter M. E. W*-algebras and nonabelian harmonic analysis // J. Func. Anal. 1972. no. 11. pp. 17-38. https://doi.org/10.1016/0022-1236(72)90077-8

[23] Zhang Y.Weak amenability of module extensions of Banach algebras Trans. // Amer. Math. Soc. 2002. no. 354(10). pp. 4131-4151. https://doi.org/10.1090/S0002-9947-02-03039-8

Для цитирования: Ebadian A., Jabbari A. Hyper-Tauberian algebras defined by a Banach algebra homomorphism // Вестник КРАУНЦ. Физ.-мат. науки. 2019. Т. 26. № 1. C. 17-27. DOI: 10.26117/2079-6641-2019-26-1-17-27

For citation: Ebadian A., Jabbari A. Hyper-Tauberian algebras defined by a Banach algebra homomorphism, Vestnik KRAUNC. Fiz.-mat. nauki. 2019, 26: 1, 17-27. DOI: 10.26117/20796641-2019-26-1-17-27

Поступила в редакцию / Original article submitted: 01.03.2019