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Arens regularity of module actions
Abstract
We study the Arens regularity of module actions of Banach left or right modules over Banach algebras. We prove that if A has a brai (blai), then the right (left) module action of A on A* is Arens regular if and only if A is reflexive. We find that Arens regularity is implied by the factorization of A* or A** when A is a left or a right ideal in A**. The Arens regularity and strong irregularity of A are related to those of the module actions of A on the nth dual A(n) of A. Banach algebras A for which Z(A**) = A but A ⊆ Zt(A* *) are found (here Z(A**) and Zt(A**) are the topological centres of A** with respect to the first and second Arens product, respectively). This also gives examples of Banach algebras such that A ⊆ Z(A**) ⊆ A**. Finally, the triangular Banach algebras T are used to find Banach algebras having the following properties: (i) T*T = TT* but Z(T**) ≠ Zt(T**); (ii) Z(T**) = Zt(T* *) and T*T = T* but TT* ≠ T*; (iii) Z(T**) = T but T is not weakly sequentially complete. The results (ii) and (iii) are new examples answering questions asked by Lau and Ülger.
... Many of the results of this paper are obtained by this theorem. In section 2, we discuss some conditions under which a Banach algebra A will be a (left or right) ideal in A * * and then we improve some results of [5]. In section 3, by using Theorem 2.1, we continue the studies of [8] about the second adjoint of derivations. ...
... π ℓ (A, X) = X). Some relationships between the factorization property and Arens regularity are stated in [2] and [5]. Proposition 3.3 and Theorem 3.1 from [5] are of these cases which together with Corollary 3.1 provide conditions for the Arens regularity of A. (1) If A * factors A on the right, then A is Arens regular. ...
... Some relationships between the factorization property and Arens regularity are stated in [2] and [5]. Proposition 3.3 and Theorem 3.1 from [5] are of these cases which together with Corollary 3.1 provide conditions for the Arens regularity of A. (1) If A * factors A on the right, then A is Arens regular. ...
In this paper, the relations between the topological centers of bounded bilinear mappings and some of their higher rank adjoints are investigated. Particularly, for a Banach algebra A, some results about the Banach A−modules and Arens regularity and strong Arens irregularity of module actions will be obtained.
... , 0, 0) in (6) where · denotes the Gaussian notation. We have an approximate oddness condition (6), one leads to ...
... and the functional inequality (6) for which the function ϕ : ...
... We refer the readers to [6,7] for details. We define a mapping ϕ f : f (a), a). ...
In this paper, we investigate homomorphisms from unital C*-algebras to unital Banach algebras and derivations from unital C*-algebras to Banach A-modules related to a Cauchy-Jensen functional inequality.
... We also say that A is (Arens) regular, left strongly irregular or right strongly irregular if the multiplication π of A enjoys the corresponding property. The subject of regularity of bounded bilinear mappings and Banach module actions have been investigated in [3], [6], [7] and [9]. In [7], Eshaghi Gordji and Fillali gave several significant results related to the topological centers of Banach module actions. ...
... The subject of regularity of bounded bilinear mappings and Banach module actions have been investigated in [3], [6], [7] and [9]. In [7], Eshaghi Gordji and Fillali gave several significant results related to the topological centers of Banach module actions. In [9], the authors have obtained a criterion for the regularity of f , from which they gave several results related to the regularity of Banach module actions with some ...
... We show that π r * 1 and π * 2 are permanently left strongly irregular ; see Theorem 2.2 below. This result improves some results of [6], [9] and [7]. For instance, it covers [ ...
We study the topological centers of some specific ad-joints of a Banach module action. Then, we investigate the Arens regularity and strong irregularity of these actions.
... Hence, (10) and (11) imply that φ is an A T -bimodule. For all a ∈ A T and x ∈ A we have ...
... It is easy to check that π * * * l (a * * , b * * ) = a * * □T * * (b * * ) and π * * * r (b * * , a * * ) = T * * (b * * )□a * * for all a * * , b * * ∈ A * * . The maps π l and π r are called Arens regular, if π * * * l = π t * * * t l and π * * * r = π t * * * t r , respectively for more details see [2,3,10]. As a result of the above Theorem, we have the following: (i) If A is Arens regular, then the left and right module actions π l and π r are Arens regular. ...
In this paper, we investigate a Banach algebra AT, where A is a Banach algebra and T is a left (right) multiplier on A. We study some concepts on AT such as n-weak amenability, cyclic amenability, biflatness, biprojectivity and Arens regularity. For the group algebra L1(G) of an infinite compact group G, it is shown that there is a multiplier T such that L1(G)T has not a bounded approximate identity. For ?1(S), where S is a regular semigroup with a finite number of idempotents, we show that there is a multiplier T such that Arens regularity of ?1(S)T implies that S is compact.
... Many functional analytic aspects concerning a triangular Banach algebra, such as the higher duals, Arens regularity, determination of certain cohomology groups, and the weak amenability, were extensively studied by several authors; see [7][8][9]. Their results were extended for certain class of module extension Banach algebras in a series of works; see, for example, [3,15,16,19] and references therein. We follow this stream and study the same aspects for a Morita context Banach algebra G = A M N B , in which A and B are Banach algebras, M and N are Banach (A, B) and (B, A)-bimodules, respectively, together with two (Morita) context maps M × N −→ A and N × M −→ B that are compatible with the module operations in the sense that G becomes a Banach algebra when it is equipped with the 1 -norm and 2 × 2 matrix-like operations. ...
... Taking N = 0 in Theorem 3.1, we arrive at the following result concerning the bidual of a triangular Banach algebra, which was already studied by Forrest and Marcoux [8] and Eshaghi Gordji and Filali [9]. ...
Motivated by the elaborate works of Forrest, Marcoux, and Zhang [Trans. Amer. Math. Soc. 354 (2002), 1435–1452 and 4131–4151] on determining the first cohomology group and studying n-weak amenability of triangular and module extension Banach algebras, we investigate the same notions for a Morita context Banach algebra G=AMNB, where A and B are Banach algebras, M and N are Banach (A,B) and (B,A)-bimodules, respectively. We describe the nth-dual G(n) of G and characterize the structure of derivations from G to G(n) for studying the first cohomology group H1G,G(n) and characterizing the n-weak amenability of G. Our study provides some improvements of certain known results on the triangular Banach algebras. The results are then applied to the full matrix Banach algebras Mk(A). Some examples illustrating the results are also included, and several questions are also left undecided.
... A Banach algebra A is said to be Arens regular, if its product π(a, b) = ab considered as a bilinear mapping π : A × A −→ A is Arens regular. For a discussion of Arens regularity for bounded bilinear maps and Banach algebras, see [2], [4], [5], [7], [8] and [9]. For example, every C * -algebra is Arens regular, see [3]. ...
... The maps m and g are Arens regular, so by comparing equations (2-4), (2)(3)(4)(5) and (2-6), we conclude that f i * * * * i = f j * * * * j = f r * * * * r . Now by theorem 2.1 proof follows. ...
Let $f:X\times Y\times Z\longrightarrow W $ be a bounded tri-linear map on normed spaces. We say that $f$ is close-to-regular when $f^{t****s}=f^{s****t}$ and $f$ is Aron-Berener regular when all natural extensions are equal. In this manuscript, we have some results on the Aron-Berner regular maps. We investigate the relation between Arens regularity of bounded bilinear maps and Aron-Berner regularity of bounded tri-linear maps. We also give a simple criterion for the Aron-Berner regularity of tri-linear maps.
... Let X, Y and Z be normed spaces and let m : X × Y → Z be a bounded bilinear mapping. Arens in [1] offers two natural extensions m * * * and m t * * * t of m from X * * × Y * * into Z * * that he called m is Arens regular whenever m * * * = m t * * * t , for more information see [9,10,14]. Let A be a Banach algebra, regarding A as a Banach A-bimodule, the operation π : A × A −→ A extends to π * * * and π t * * * t defined on A * * × A * * . These extensions are known, respectively, as the first (left) and the second (right) Arens products, and with each of them, the second dual space A * * becomes a Banach algebra. ...
... In this way, we write Z ℓ B * * (A * * ) = Z(π ℓ ), Z r A * * (B * * ) = Z(π t ℓ ) and Z r B * * (A * * ) = Z(π t r ), where π ℓ : A × B → B and π r : B × A → B are the left and right module actions of A on B, for more information related to the Arens regularity of module actions on Banach algebras, see [2,4,9,10]. ...
In this paper, we study approximate identity properties, some propositions from Baker, Dales, Lau in general situations and we establish some relationships between the topological centers of module actions and factorization properties with some results in group algebras. We consider under which sufficient and necessary conditions the Banach algebra $A\widehat{\otimes}B$ is Arens regular.
... The left topological center of m ∈ Bil(X × Y, Z) is Z (m) = {Φ ∈ X * * ; m * * * (Φ, · ) : Y * * → Z * * is w * -w * continuous} and the right topological center of m is Z r (m) = Z (m t ) (see [2,3,6]). It is not hard to see that Z (m) = {Φ ∈ X * * ; m * * * (Φ, Γ) = m t * * * t (Φ, Γ) for all Γ ∈ Y * * }, and therefore Z r (m) = {Γ ∈ Y * * ; m * * * (Φ, Γ) = m t * * * t (Φ, Γ) for all Φ ∈ X * * }. ...
... Similarly, if Z r (m) = Y, then m is right strongly Arens irregular. There exist bilinear operators which are left strongly Arens irregular but not right strongly Arens irregular (and vice versa), see [3]. ...
We explore the relation between Arens regularity of a bilinear operator and the weak compactness of the related linear operators. Since every bilinear operator has natural factorization through the projective tensor product a special attention is given to Arens regularity of the tensor operator. We consider topological centers of a bilinear operator and we present a few results related to bilinear operators which can be approximated by linear operators.
... On the other hand, motivated by Wedderburn's principal theorem [1], splitting of Banach algebra extensions has been a major question in the theory of Banach algebras and several researchers have used the splitting of Banach algebra extensions as a tool for the study of Banach algebras. For example, module extensions as generalizations of Banach algebra extensions were introduced and first studied by Gourdeau [13] were used to show that amenability of A (2) , the second dual space of A, implies amenability of A; Filali and Eshaghi Gordji [10] used triangular Banach algebras to answer some questions asked by Lau andÜlger [17]; Zhang [22] used module extensions to construct an example of a weakly amenable Banach algebra which is not 3-weakly amenable. For some other applications of splitting of Banach algebra extensions we refer the reader to the references [2,3,4,11,21]. ...
... Let us recall from [10] that the topological centres of the left and right module actions of A (2) on A (2) are as follows: ...
Let ${\mathcal A}$ and ${\frak A}$ be Banach algebras such that ${\mathcal A}$ is a Banach ${\frak A}$-bimodule with compatible actions. We define the product ${\cal A}\rtimes{\frak A}$, which is a strongly splitting Banach algebra extension of ${\frak A}$ by $\cal A$. After characterization of the multiplier algebra, topological centre, (maximal) ideals and spectrum of ${\cal A}\rtimes{\frak A}$, we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of ${\cal A}\rtimes{\frak A}$ in relation to that of the algebras $\cal A$, $\frak A$ and the action of $\frak A$ on $\cal A$. We also compute the first cohomology group $H^1{(}{\cal A}\rtimes{\frak A},({\cal A}\rtimes{\frak A})^{(n)}{)}$ for all $n\in {\Bbb N}\cup\{0\}$ as well as the first-order cyclic cohomology group $H_\lambda^1{(}{\cal A}\rtimes{\frak A},({\cal A}\rtimes{\frak A})^{(1)}{)}$, where $({\cal A}\rtimes{\frak A})^{(n)}$ is the n-th dual space of ${\cal A}\rtimes{\frak A}$ when $n\in{\Bbb N}$ and ${\cal A}\rtimes{\frak A}$ itself when $n=0$. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and $n$-weak amenability of ${\cal A}\rtimes{\frak A}$.
... Let X, Y and Z be normed spaces. Recall that a bounded bilinear map m from X × Y into Z factors if m is onto, see [8] for example. In the following we define a strong version of this concept. ...
... The following two corollaries of Theorem 3.2 extend some results of [8], in the sense that A does not necessarily have a BRAI. ...
Let m be a bounded bilinear mapping on Banach spaces. First we give a sufficient condition for strong irregularity of m and apply the result to extend some earlier results b Ulger. Next, we study in-variance of irregularity for m under extension of its domain from left or right. As a consequence, we improve some known results about Arens regularity. Finally, we investigate invariance of irregularity for direct sum of a family of Banach algebras.
... f is said to be strongly (Arens) irregular whenever f * * * and f r * * * r are equal only on X × Y * * and X * * × Y . Regularity and strong irregularity of bounded bilinear maps are investigated by many authors, for example see [6,8,10]. The interested reader may also refer to [4,5] for more information on the subject of Arens regularity. ...
... In this paper, we give a sufficient condition for strong irregularity of certain bounded bilinear maps (Theorem 4.1 infra), then we apply it to determine the topological centers of certain normed module actions. In particular, in Theorem 5.1 for the approximately unital normed A −modules (π 1 , X ) and (X , π 2 ) we show that π r * r some older results, but also provides a unified approach to give a simple direct proof for several known results from [2,3,6,7,8,10,11] concerning to the relation between Arens regularity and reflexivity. ...
We provide a sufficient condition for strong (Arens) irregularity of certain
bounded bilinear maps, which applies in particular to the adjoint of Banach
module actions. We then apply our result to improve several known results
concerning to the relation between Arens regularity of certain Banach module
actions and reflexivity.
... The constructions of the two Arens multiplications in A * * lead us to definition of topological centers for A * * with respect to both Arens multiplications. The topological centers of Banach algebras , module actions and applications of them were introduced and discussed in [1, 5, 9, 14, 17, 18, 25, 26]. In first section, we find some relationships between the topological centers of the second dual of Banach algebra A and module actions with some conclusions in group algebras. ...
... Lau and Losert in [17], for locally compact group G, show that the topological center of L 1 (G) * * is L 1 (G). Neufang in [21] and [22] has studied the topological centers of L 1 (G) * * and M (G) * * in spacial case, Eshaghi and Filali in [9] have studied these problems on module actions. In the following example for a compact group G, by using the proceeding theorem, we study the topological centers of L 1 ...
In this paper, first we study some Arens regularity properties of module actions. Let $B$ be a Banach $A-bimodule$ and let ${Z}^\ell_{B^{**}}(A^{**})$ and ${Z}^\ell_{A^{**}}(B^{**})$ be the topological centers of the left module action $\pi_\ell:~A\times B\rightarrow B$ and the right module action $\pi_r:~B\times A\rightarrow B$, respectively. We investigate some relationships between topological center of $A^{**}$, ${Z}_1({A^{**}})$ with respect to the first Arens product and topological centers of module actions ${Z}^\ell_{B^{**}}(A^{**})$ and ${Z}^\ell_{A^{**}}(B^{**})$. On the other hand, if $A$ has Mazure property and $B^{**}$ has the left $A^{**}-factorization$, then $Z^\ell_{A^{**}}(B^{**})=B$, and so for a locally compact non-compact group $G$ with compact covering number $card(G)$, we have $Z^\ell_{M(G)^{**}}{(L^1(G)^{**})}= {L^1(G)}$ and $Z^\ell_{L^1(G)^{**}}{(M(G)^{**})}= {M(G)}$. By using the Arens regularity of module actions, we study some cohomological groups properties of Banach algebra and we extend some propositions from Dales, Ghahramani, Gr{\o}nb{\ae}k and others into general situations and we investigate the relationships between some cohomological groups of Banach algebra $A$. We obtain some results in Connes-amenability of Banach algebras, and so for every compact group $G$, we conclude that $H^1_{w^*}(L^\infty(G)^*,L^\infty(G)^{**})=0$. Suppose that $G$ is an amenable locally compact group. Then there is a Banach $L^1(G)-bimodule$ such as $(L^\infty(G),.)$ such that $Z^1(L^1(G),L^\infty(G))=\{L_{f}:~f\in L^\infty(G)\}$ where for every $g\in L^1(G)$, we have $L_f(g)=f.g$.
... The constructions of the two Arens multiplications in A * * lead us to definition of topological centers for A * * with respect to both Arens multiplications. The topological centers of Banach algebras, module actions and applications of them were introduced and discussed in [3, 5, 6, 9, 15, 16, 17, 18, 19, 24, 25]. In this paper, we extend some problems from [3, 5, 6, 16, 22] to the general criterion on module actions with some applications in group algebras. ...
... respectively, clearly these two sets are subsets of A * . The extension of bilinear maps on normed space and the concept of regularity of bilinear maps were studied by [1, 2, 5, 6, 9]. We start by recalling these definitions as follows. ...
In this paper, we study the Arens regularity properties of module actions and we extend some proposition from Baker, Dales, Lau and others into general situations. For Banach $A-bimodule$ $B$, let $Z_1(A^{**})$, ${Z}^\ell_{B^{**}}(A^{**})$ and ${Z}^\ell_{A^{**}}(B^{**})$ be the topological centers of second dual of Banach algebra $A$, left module action $\pi_\ell:~A\times B\rightarrow B$ and right module action $\pi_r:~B\times A\rightarrow B$, respectively. We establish some relationships between them and factorization properties of $A^*$ and $B^*$. We search some necessary and sufficient conditions for factorization of $A^*$, $B$ and $B^*$ with some results in group algebras. We extend the definitions of the left and right multiplier for module actions.
... Baker, Lau and Pym in [19] proved that for Banach algebra A with bounded right approximate identity, (A * A) ⊥ is an ideal of right annihilators in A * * and A * * ∼ = (A * A) * ⊕ (A * A) ⊥ . Also in [9,5,11], the authors established the Arens regularity of module actions of Banach left or right modules over Banach algebras. They proved that if A has a bounded left approximate identity, then the right (left) module action of A on A * is Arens regular if and only if A is reexive. ...
In this paper, we establish some relationships between Left and right weakly completely continuous operators and topological centers of module actions and relationships between the factorization and the kinds of amenability. We define the locally topological center of the left and right module actions and investigate some of its properties. Also, we want to examine some conditions that under those the duality of a Banach algebra is strongly Connes-amenable. Finally, we generalize the concept of the weakly strongly connes amenable to even dual in higher orders.
... If we continue dualizing we shall reach (π * * * r * r 2 , X * * * , π * * * * 1 ) and (π r * * * * r 2 , X * * * , π r * * * r * 1 ) are the dual Banach (A * * , )-module and Banach (A * * , ♦)-module of (π * * * 1 , X * * , π * * * 2 ) and (π r * * * r 1 , X * * , π r * * * r 2 ), respectively (see [15]). In [8], Eshaghi Gordji and Fillali show that if a Banach algebra A has a bounded left (or right) approximate identity, then the left (or right) module action of A on A * is Arens regular if and only if A is reflexive. ...
Let X, Y, Z and W be normed spaces and f : X × Y × Z −→ W be a bounded tri-linear mapping. In this manuscript, we introduce the topological centers of bounded tri-linear mapping and we invistagate their properties. We study the relationships between weakly compactenss of bounded linear mappings and regularity of bounded tri-linear mappings. We extend some factorization property for bounded tri-linear mappings. We also establish the relations between regularity and factorization property of bounded tri-linear mappings.
... If we continue dualizing we shall reach (π * * * r * r 2 , X * * * , π * * * * 1 ) and (π r * * * * r 2 , X * * * , π r * * * r * 1 ) are the dual Banach (A * * , ) − module and Banach (A * * , ♦)−module of (π * * * 1 , X * * , π * * * 2 ) and (π r * * * r 1 , X * * , π r * * * r 2 ), respectively. In [8], Eshaghi Gordji and Fillali show that if a Banach algebra A has a bounded left (or right) approximate identity, then the left (or right) module action of A on A * is Arens regular if and only if A is reflexive. ...
Let $X,Y,Z$ and $W$ be normed spaces and $f:X\times Y\times Z\longrightarrow W $ be a bounded tri-linear mapping. In this Article, we define the topological centers for bounded tri-linear mapping and we invistagate thier properties. We study the relationships between weakly compactenss of bounded linear mappings and regularity of bounded tri-linear mappings. For both bounded tri-linear mappings $f$ and $g$, let $f$ factors through $g$, we present necessary and suficient condition such that the extensions of $f$ factors through extensions of $g$. Also we establish relations between regularity and factorization property of bounded tri-linear mappings.
... A triple (π 1 , X, π 2 ) is said to be a Banach A−module if (X, π 1 ) and (X, π 2 ) are left and right Banach A−modules, respectively, and π 1 (a, π 2 (x, b)) = π 2 (π 1 (a, x), b). (see, for instance, [10] and [7]) , d), b, c) = π 2 (D(a, b, c), d) + π 1 (a, D(d, b, c)), D(a, d, c)), D(a, b, d)). ...
Let $f:X\times Y\times Z\longrightarrow W $ be a bounded tri-linear map on normed spaces. We say that $f$ is close-to-regular when $f^{t****s}=f^{s****t}$ and we say that $f$ is completely regular when all natural extensions are equal. In this manuscript, we have some results on the close-to-regular maps and investigate the close-to-regularity of tri-linear maps. We investigate the relation between Arens regularity of bounded bilinear maps and close-to-regularity bounded tri-linear maps. We give a simple criterion for the completely regularity of tri-linear maps. We provide a necessary and sufficient condition such that the fourth adjoint $D^{****}$ of a tri-derivation is again a tri-derivation.
... If we continue dualizing we shall reach (π * * * r * r 2 , X * * * , π * * * * 1 ) and (π r * * * * r 2 , X * * * , π r * * * r * 1 ) are the dual Banach (A * * , □) − module and Banach (A * * , ♢)−module of (π * * * 1 , X * * , π * * * 2 ) and (π r * * * r 1 , X * * , π r * * * r 2 ), respectively. In [8], Eshaghi Gordji and Fillali show that if a Banach algebra A has a bounded left (or right) approximate identity, then the left (or right) module action of A on A * is Arens regular if and only if A is reflexive. ...
In this paper, we will study on some topologies induced by order convergences in a vector lattice. We will investigate the relationships of them.
... In [5] Eshaghi Gordji and Filali show that left module action of a Banach algebra on ( ) factors. Now let 2 be the right module action of on ( ) . ...
... Similarly, the projection map í µí¼ 2 : M ⊕ 1 A → A is defined by (í µí±¥, í µí±) í®í¸ → í µí±. For more information about Banach modules and module extensions, we refer the reader to [36,37]. ...
This is a survey presenting the most significant results concerning approximate (generalized) derivations, motivated by the notions of Ulam and Hyers-Ulam stability. Moreover, the hyperstability and superstability issues connected with derivations are discussed. In the section before the last one we highlight some recent outcomes on stability of conditions defining (generalized) Lie derivations.
... (i) If θ = 0, then [11]) Let S = A ⊕ X be the module extension Banach algebra corresponding A and X. ...
Let $A$ and $B$ be Banach algebras, $\theta: A\to B$ be a continuous Banach algebra homomorphism and $I$ be a closed ideal in $B$. Then the direct sum of $A$ and $I$ with respect to $\theta$, denoted $A\bowtie^{\theta}I$, with a special product becomes a Banach algebra which is called the amalgamated Banach algebra. In this paper, among other things, we compute the topological centre of $A\bowtie^{\theta}I$ in terms of that of $A$ and $I$. Using this, we provide a characterization of the Arens regularity of $A\bowtie^{\theta}I$. Then we determine the character space of $A\bowtie^{\theta}I$ in terms of that of $A$ and $I$. Moreover, we study the weak amenability of $A\bowtie^{\theta}I$.
... In [EF007, Theorem 2.4], the theorem was proved under the conditions that A is a right ideal in A * * and A * * A = A * * and no Arens regularity was assumed. However, it turned out later on that in this situation A must be Arens regular; see [EF07,Theorem 4.3]. ...
Let A be a Banach algebra, A*, A** and A*** be its first, second and third dual, respectively. Let R: A*** → A* be the restriction map, J: A* → A*** be the canonical injection and Λ: A*** → A*** be the composition of R and J: Let D: A→ A* be a continuous derivation and D〞: A** → A*** be its second transpose. We obtain a necessary and sufficient condition for Λ ο D〞: A** → (A**)* to be a derivation. We apply this to prove some results on weak amenability of second dual Banach algebras.
... It is known that the Arens regularity of module actions has been a major tool in the study of Banach algebras. For example, Arens regularity of module actions as a generalization of Arens regularity of Banach algebras were introduced and first studied by Filali and Eshaghi Gordji [6] and they used this notion to answer some questions regarding Arens regularity of Banach algebras raised by Lau andÜlger [16]. Also in [17], Arens regularity of module actions were considered by Mohammadzadeh and Vishki to investigate the conditions under which the second adjoint of a derivation into a dual Banach module is again a derivation, that extends the results of Dales, Rodriguez-Palacios and Velasco in [5] for a general derivation. ...
Let $\cal X$ be a $\cal G$-space; That is, $\cal G$ is a locally compact
group acting continuously from the left on a locally compact Hausdorff space
$\cal X$, for instances, a homogeneous space, dynamical system and some
semigroup compactification of $\cal G$ and etc. Let also, the Banach space
$LUC({\cal G})$ and $C_b({\cal X})$ are considered as usual and for a given $s$
in $\cal G$, we consider the left translation of a function $F\in C_b({\cal
X})$ by $l_sF(y)=F(s\cdot y)$ ($y\in {\cal X}$). Following Lau and Chu
\cite{chulau}, we use the notation $LUC({\cal X},{\cal G})$ to denote the
Banach space of all functions $F\in C_b({\cal X})$ for which the map $s \mapsto
l_sF$ from $\cal G$ into $C_b({\cal X})$ is continuous. In this paper, we study
some properties of the first dual space of $LUC({\cal X},{\cal G})$. In
particular, we introduce a left action of $LUC({\cal G})^*$ on $LUC({\cal
X},{\cal G})^*$ to make it a Banach left module and then we investigate the
Banach subspace ${{\frak{Z}}({\cal X},{\cal G})}$ of $LUC({\cal G})^*$, as the
topological centre related to this module action. Among other things, we
consider the faithfulness of this module action and we extend the main results
of Lau~\cite{lau} from locally compact groups to ${\cal G}$-spaces. Moreover,
apart from some characterization of the space ${{\frak{Z}}({\cal X},{\cal
G})}$, we apply our results to special $\cal G$ and $\cal X$.
... A Banach algebra A is Arens regular when Z A (A * * ) = A * * , or equivalently, Z t A (A * * ) = A * * . (see, [6]). Let A be a Banach algebra and X is a Banach A−bimodule. ...
... holds. The details of the constructions may be found in many places, including the book [2] and the articles [4,5,6,8]. Throughout this paper, the first (second) Arens multiplication is denoted by 2 (respectively ♦). Thise multiplications can be defined on M(X) * * by ...
Let M(X) be the space of all finite regular Borel measures on X. A general measure algebra is a subspace of M(X), which is an L-space and has a multiplication preserving the probability measures. Let L ⊆ M(X) be a general measure algebra on a locally compact space X. In this paper, we investigate the relation between Arens regularity of L and the topology of X. We find conditions under which the Arens regularity of L implies the compactness of X. We show that these conditions are necessary. We also present some examples in showing that the new conditions are different from Theorem 3.1 of [7].
... continuous for all x ∈ X (see [8] for the classical case). Note that the transpose f t of f is a continuous map which satisfies the relations (1) and (2). ...
We extend the notion of Arens regularity and module Arens regularity of Banach algebras to Arens regularity of module actions. We also investigate the more general notion of Arens regularity for bilinear maps. Finally, we find necessary and sufficient conditions for module Arens regularity of the semigroup algebra of an inverse semigroup.
... In [16], Ulger showed that the Arens regularity of a bounded bilinear map can be characterized by its weakly compactness or its reflexiveness and simplified proofs of some old results. For more recent results, the reader is referred to [6] and [14]. On the other hand, special attention has been focused on the bilinear maps arisen from Banach algebras. ...
In this paper, we study the topological centers of bilinear maps induced by unitary representations, giving a characterization when the center is minimal and also conditions which guarantee that the center is maximal. Various examples whose topological centers are maximal, minimal or neither will be given. We will also investigate the topological centers of sub-representations, direct sums and tensor products.
... If both equalities A * A = AA * = A * hold, then we say that A * factors on both sides. The extension of bilinear maps on normed space and the concept of regularity of bilinear maps were studied by [1, 2, 3, 6, 8, 14]. We start by recalling these definitions as follows. ...
In this paper, we will study the topological centers of $n-th$ dual of Banach $A-module$ and we extend some propositions from Lau and \"{U}lger into $n-th$ dual of Banach $A-modules$ where $n\geq 0$ is even number. Let $B$ be a Banach $A-bimodule$. By using some new conditions, we show that ${{Z}^\ell}_{A^{(n)}}(B^{(n)})=B^{(n)}$ and ${{Z}^\ell}_{B^{(n)}}(A^{(n)})=A^{(n)}$. We also have some conclusions in group algebras.
Let f : X × Y × Z −→ W be a bounded tri-linear map on normed spaces. We say that f is close-to-regular when f t * * * * s = f s * * * * t and we say that f is Aron-Berner regular when all natural extensions are equal. In this manuscript, we give a simple criterion for the close-to-regularity of tri-linear maps.
Let $B$ be a Banach $A-bimodule$. We introduce the weak topological centers of left module action and we show it by $\tilde{{Z}}^\ell_{B^{**}}(A^{**})$. For a compact group, we show that $L^1(G)=\tilde{Z}_{M(G)^{**}}^\ell(L^1(G)^{**})$ and on the other hand we have $\tilde{Z}_1^\ell{(c_0^{**})}\neq c_0^{**}$. Thus the weak topological centers are different with topological centers of left or right module actions. In this manuscript, we investigate the relationships between two concepts with some conclusions in Banach algebras. We also have some application of this new concept and topological centers of module actions in the cohomological properties of Banach algebras, spacial, in the weak amenability and $n$-weak amenability of Banach algebras.
In this paper, we study some cohomlogical properties of Banach algebras. For a Banach algebra $A$ and a Banach $A$-bimodule $B$, we investigate the vanishing of the first Hochschild cohomology groups $H^1(A^n,B^m)$ and $H_{w^*}^1(A^n,B^m)$, where $0\leq m,n\leq 3$. For amenable Banach algebra $A$, we show that there are Banach $A$-bimodules $C$, $D$ and elements $\mathfrak{a}, \mathfrak{b}\in A^{**}$ such that $$Z^1(A,C^*)=\{R_{D^{\prime\prime}(\mathfrak{a})}:~D\in Z^1(A,C^*)\}=\{L_{D^{\prime\prime}(\mathfrak{b})}:~D\in Z^1(A,D^*)\}.$$ where, for every $b\in B$, $L_{b}(a)=ba$ and $R_{b}(a)=a b,$ for every $a\in A$. Moreover, under a condition, we show that if the second transpose of a continuous derivation from the Banach algebra $A$ into $A^*$ i.e., a continuous linear map from $A^{**}$ into $A^{***}$, is a derivation, then $A$ is Arens regular. Finally, we show that if $A$ is a dual left strongly irregular Banach algebra such that its second dual is amenable, then $A$ is reflexive.
In this Article, we give a simple criterion for the regularity of a tri-linear mapping. We provide if $f:X\times Y\times Z\longrightarrow W $ is a bounded tri-linear mapping and $h:W\longrightarrow S$ is a bounded linear mapping, then $f$ is regular if and only if $hof$ is regular. We also shall give some necessary and sufficient conditions such that the fourth adjoint $D^{****}$ of a tri-derivation $D$ is again tri-derivation.
We associate a Banach algebra \(\mathbf A _\Phi \) to each bounded bilinear map \(\Phi : \mathrm X\times \mathrm Y\rightarrow \mathrm {Z}\) on Banach spaces, and we find out that this Banach algebra can be useful for many purposes in the theory of Banach algebras. For example, the construction of \(\mathbf A _\Phi \) enables us to provide many simple examples of Banach algebras with different topological centers, which are neither Arens regular nor either left or right strongly Arens irregular. It also gives examples of Banach algebras which are not n-weakly amenable for each natural number n. We also find out that the dual of \(\mathbf A _\Phi \) does not enjoy the factorization property of any level.
Let X, Y and Z be normed spaces. In this article we give a tool to investigate Arens regularity of a bounded bilinear map f : X Y Z. Also, under some assumptions on and , we give some new results to determine reflexivity of the spaces.
Associated with a locally compact group \(\mathcal G\) and a \(\mathcal G\)-space \(\mathcal X\) there is a Banach subspace \(LUC({\mathcal X},{\mathcal G})\) of \(C_b({\mathcal X})\), which has been introduced and studied by Chu and Lau (Math Z 268:649–673, 2011). In this paper, we study some properties of the first dual space of \(LUC({\mathcal X},{\mathcal G})\). In particular, we introduce a left action of \(LUC({\mathcal G})^*\) on \(LUC({\mathcal X},{\mathcal G})^*\) to make it a Banach left module and then we investigate the Banach subalgebra \({{\mathfrak {Z}}({\mathcal X},{\mathcal G})}\) of \(LUC({\mathcal G})^*\), as the topological centre related to this module action, which contains \(M({\mathcal G})\) as a closed subalgebra. Also, we show that the faithfulness of this module action is related to the properties of the action of \(\mathcal G\) on \(\mathcal X\) and we prove an analogue of the main result of Lau (Math Proc Cambridge Philos Soc 99:273–283, 1986) for \({\mathcal G}\)-spaces. Sufficient and/or necessary conditions for the equality \({{\mathfrak {Z}}({\mathcal X},{\mathcal G})}=M({\mathcal G})\) or \(LUC({\mathcal G})^*\) are given. Finally, we apply our results to some special cases of \(\mathcal G\) and \(\mathcal X\) for obtaining various examples whose topological centres \({{\mathfrak {Z}}({\mathcal X},{\mathcal G})}\) are \(M({\mathcal G})\), \(LUC({\mathcal G})^*\) or neither of them.
Let (Formula presented.) and (Formula presented.) be Banach algebras such that (Formula presented.) is a Banach (Formula presented.)-bimodule with compatible actions. We define the product (Formula presented.), which is a strongly splitting Banach algebra extension of (Formula presented.) by (Formula presented.). After characterization of the multiplier algebra, topological center, (maximal) ideals and spectrum of (Formula presented.), we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of (Formula presented.) in relation to that of the algebras (Formula presented.), (Formula presented.) and the action of (Formula presented.) on (Formula presented.). We also compute the first cohomology group (Formula presented.) for all (Formula presented.) as well as the first-order cyclic cohomology group (Formula presented.), where (Formula presented.) is the (Formula presented.)th dual space of (Formula presented.) when (Formula presented.) and (Formula presented.) itself when (Formula presented.). These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and (Formula presented.)-weak amenability of (Formula presented.). In this context, several open questions arise.
In this paper, we extend some propositions of Banach algebras into module actions and establish the relationships between topological centers of module actions. We introduce some new concepts as Lw*w-property and Rw*w-property for Banach modules and obtain some conclusions in the topological center of module actions and Arens regularity of Banach algebras.
We utilize the notion of module extension to reduce the problem of stability of derivations to that of ring homomorphisms studied by R. Badora in the context of Banach bimodules over Banach algebras.
Some new results concerning Arens regularity of Banach algebras are presented. n-weak amenability of module extensions of Banach algebras and w∗-continuous derivations on Banach algebras are studied.
Assume that A, B are Banach algebras and that m: A×B → B, m ′: A × A → B are bounded bilinear mappings. We study the relationships between Arens regularity of m, m ′ and the Banach algebras A, B. For a Banach A-bimodule B, we show that B factors with respect to A if and only if B ** is unital as an A ** -module. Let Ze′′ (B **) = B ** where e ′′ is a mixed unit of A ** Then B * factors on both sides with respect to A if and only if B ** has a unit as A ** -module.
In this paper, we extend some problems of Arens regularity and factorizations properties of Banach algebras for general structures and we establish the relationships between topological centers and factorization properties of left module actions with some conclusions in the Arens regularity of Banach algebras. To a Banach algebra A, we extend the definition of *-involution algebra to a Banach A - bimodule B with some results in the factorizations properties of B*. We have some applications in group algebras.
La dualidad de espacios normados es central para el abordaje de cuestiones geométricas y topológicas de los mismos. En el contexto de espacios de Banach la completitud permite avanzar mucho más, siendo especialmente rico este escenario si se tratare con álgebras de Banach. El análisis funcional moderno es difícil de abordar por la diversidad de enfoques y la complejidad de las construcciones, no obstante ser las mismas en general concurrentes y guiadas por una suerte de derrotero intrínseco. Pareciera haber capítulos, cada uno con valía e interés propios, con límites a veces difusos. Hay también especialistas que tienen el mérito y la capacidad de integrarlos, complementarlos y enriquecerlos a veces de manera magistral.
En este trabajo, que procuramos en lo posible sea autocontenido, nos hemos de concentrar en cuestiones ligadas a la regularidad de álgebras de Banach. Se trata de un aspecto teórico de carácter algebraico, con fuertes implicancias que permiten una mejor comprensión de la estructura de las correspondientes álgebras.
Let A be a unital C*-algebras. B be a Bausch algebra and let X be a Banach A-module. By using fixed pint methods, we prove that: i) Every almost linear mapping h : A -> B which satisfies h(2(n)uy) = h(2(n)u)h(y) for all U is an element of A(+), all y is an element of A, and all n = 0,1, 2, ..., is a homomorphism. ii) Every almost linear continuous mapping d : A -> X is a derivation when d(2(n)uy) = d(2(n)u)y + 2(n)ud(y) holds for all u is an element of A(+), all Y is an element of A, and all n = 0, 1, 2,....
Let H be a Hilbert space. we show that the following statements are equivalent: (a) B(H) is finite dimension, (b) every left Banach module action l: B(H)×H → H, is Arens regular (c) every bilinear map f: B(H)*→ B(H) is Arens regular. Indeed we show that a Banach space X is reflexive if and only if every bilinear map f: X* × X → X* is Arens regular.
In this article, for Banach left and right module actions, we will
extend some propositions from Lau and $\ddot{U}lger$ into general
situations and we establish the relationships between topological
centers of module actions. We also introduce the new concepts as
$Lw^*w$-property and $Rw^*w$-property for Banach $A-bimodule$ $B$ and we
investigate the relations between them and topological center of module
actions. We have some applications in dual groups.
We study Arens regularity of the left and right module actions of on , where is the nth dual space of a Banach algebra , and then
investigate (quotient) Arens regularity of as a module extension
of Banach algebras.
It is known that a Banach algebra inherits amenability from its second Banach dual A��. No example is yet known whether this fails if one considers the weak amenability instead, but the property is known to hold for the group algebra L1(G), the Fourier algebra A(G) when G is amenable, the Banach algebras A which are left ideals in A��, the dual Banach algebras, and the Banach algebras A which are Arens regular and have every derivation from A into Aweakly compact. In this paper, we extend this class of algebras to the Banach algebras for which WAP(A) � A�, the Banach algebras A which are right ideals in A�� and satisfy AA�� = A��, and to the Figa-Talamanca-Herz algebra Ap(G) for G amenable. We also provide a short proof of the interesting recent criterion on when the second adjoint of a derivation is again a derivation.
Let $A$ be a Banach algebra and $A^{**}$ be the second dual of it. We show that by some new conditions, $A$ is weakly amenable whenever $A^{**}$ is weakly amenable. We will study this problem under generalization, that is, if $(n+2)-th$ dual of $A$, $A^{(n+2)}$, is $T-S-$weakly amenable, then $A^{(n)}$ is $T-S-$weakly amenable where $T$ and $S$ are continuous linear mappings from $A^{(n)}$ into $A^{(n)}$.
For Banach left and right module actions, we extend some propositions from Lau and $\ddot{U}lger$ into general situations and we establish the relationships between topological centers of module actions. We also introduce the new concepts as $Lw^*w$-property and $Rw^*w$-property for Banach $A-bimodule$ $B$ and we obtain some conclusions in the topological center of module actions and Arens regularity of Banach algebras. we also study some factorization properties of left module actions and we find some relations of them and topological centers of module actions. For Banach algebra $A$, we extend the definition of $\ast-involution$ algebra into Banach $A-bimodule$ $B$ with some results in the factorizations of $B^*$. We have some applications in group algebras.
In this paper, we will study some Arens regularity properties of module actions. Let $B$ be a Banach $A-bimodule$ and let ${Z}^\ell_{B^{**}}(A^{**})$ and ${Z}^\ell_{A^{**}}(B^{**})$ be the topological centers of the left module action $\pi_\ell:~A\times B\rightarrow B$ and the right module action $\pi_r:~B\times A\rightarrow B$, respectively. In this paper, we will extend some problems from topological center of second dual of Banach algebra $A$, $Z_1(A^{**})$, into spaces ${Z}^\ell_{B^{**}}(A^{**})$ and ${Z}^\ell_{A^{**}}(B^{**})$. We investigate some relationships between ${Z}_1({A^{**}})$ and topological centers of module actions. For an unital Banach $A-module$ $B$ we show that ${Z}^\ell_{A^{**}}(B^{**}){Z}_1({A^{**}})={Z}^\ell_{A^{**}}(B^{**})$ and as results in group algebras, for locally compact group $G$, we have ${Z}^\ell_{{L^1(G)}^{**}}(M(G)^{**})M(G)={Z}^\ell_{{L^1(G)}^{**}}(M(G)^{**})$ and ${Z}^\ell_{M(G)^{**}}({L^1(G)}^{**})M(G)={Z}^\ell_{M(G)^{**}}({L^1(G)}^{**})$. For Banach $A-bimodule$ $B$, if we assume that $B^*B^{**}\subseteq A^*$, then $~B^{**}{Z}_1(A^{**})\subseteq {Z}^\ell_{A^{**}}(B^{**})$ and moreover if $B$ is an unital as Banach $A-module$, then we conclude that $B^{**}{Z}_1({A^{**}})={Z}^\ell_{A^{**}}(B^{**})$. Let ${Z}^\ell_{A^{**}}(B^{**})A\subseteq B$ and suppose that $B$ is $WSC$, so we conclude that ${Z}^\ell_{A^{**}}(B^{**})=B$. If $\overline{B^{*}A}\neq B^*$ and $ B^{**}$ has a left unit $A^{**}-module$, then $Z^\ell_{B^{**}}(A^{**})\neq A^{**}$. We will also establish some relationships of Arens regularity of Banach algebras $A$, $B$ and Arens regularity of projective tensor product $A\hat{\otimes}B$.
Let A and B be unital Banach algebras, and let M be a Banach A, B-module. Then T = [0ABM] becomes a triangular Banach algebra when equipped with the Banach space norm ∥ [0abm] ∥ = ∥ a ∥ A + ∥ m ∥ M + ∥ b ∥ B. A Banach algebra T is said to be n-weakly amenable if all derivations from T into its nth dual space T(n) are inner. In this paper we investigate Arens regularity and n-weak amenability of a triangular Banach algebra T in relation to that of the algebras A, B and their action on the module M.
Let A be a Banach algebra, and let D : A → A*be a continuous derivation, where A*is the topological dual space of A. The paper discusses the situation when the second transpose D**:A**→ (A**)*is also a derivation in the case where A" has the first Arens product.
Let A be a Banach algebra with a bounded,approximate identity. Let Z1 and eZ2 be, respectively, the topological centers of the algebras A and (AA). In this paper, for weakly sequentially complete Banach algebras, in particular for the group and Fourier algebras L,(G )a nd A ( G), we study the sets Z1, eZ2, the relations between them and with several other subspaces of A,or A.
Let A be a Banach algebra with a bounded approximate identity and let Z1(A )a nd Z 2 ( A) be the left and right topological centers of A. It is shown that i) AA = AA is not sucient for Z1(A )= Z 2 ( A); ii) the inclusion ^ AZ1(A) ^ A is not sucient for Z2(A)^ A ^ A; iii) Z1(A )= Z 2 ( A )= ^ Ais not sucient for A to be weakly sequentially complete. These results answer three questions of Anthony To-Ming Lau and Ali ¨ Ulger.
Synopsis
We give a survey of the current state of knowledge on the Arens second dual of a Banach algebra, including some simplified proofs of known results, some new results, some open problems and a full bibliography of the subject.
Let X and Y be two Banach spaces. We show that a bounded bilinear form m: X × Y → C is Arens regular iff it is weakly compact. This result permits us to find very short proofs of some known results as well as some new results. Some of them are: Any C*-algebra, the disk algebra and the Hardy class H∞ are Arens regular under every possible product. We also characterize the Arens regularity of certain bilinear mappings.
A short proof is given that if E is a super-reflexive Banach space, then B(E), the Banach algebra of operators on E with composition product, is Arens regular. Some remarks are made on necessary conditions on E for B(E) to be Arens regular.
Let X be a compact totally ordered space made into a semigroup by the multiplication xy =max{ x , y }. Suppose that there is a continuous regular Borel measure μ on X with supp μ= X . Then the space L ¹ (μ) of μ-integrable functions becomes a Banach algebra when provided with convolution as multiplication. The second dual L ¹ (μ)** therefore has two Arens multiplications, each making it a Banach algebra. We shall always consider L ¹ (μ)** to have the first of these: if F , G ∈ L ¹ (μ)** and F = w *−lim i ϕ i , G = w *−lim j ψ j , where (ϕ i ), (ψ j ) are bounded nets in L ¹ (μ), then
formula here
Let
G \cal G
be a locally compact group. Consider the Banach algebra
L1(G)** L_{1}(\cal G)^{**}
, equipped with the first Arens multiplication, as well as the
algebra LUC
(G)* (\cal G)^*
, the dual of the space of bounded left uniformly
continuous functions on
G \cal G
, whose product extends the convolution in
the measure algebra M
(G) (\cal G)
. We present (for the most interesting case
of a non-compact group) completely different - in particular,
direct - proofs and even obtain
sharpened versions of
the results, first proved by Lau-Losert in [9] and Lau in
[8], that the topological centres of the latter algebras
precisely are
L1(G) L_{1}(\cal G)
and M
(G) (\cal G)
, respectively. The special interest of
our new approach lies in the fact that it shows a fairly general pattern
of solving the topological centre problem for various kinds of Banach
algebras; in particular, it avoids the use of any measure theoretical
techniques. At the same time, deriving both results in perfect
parallelity, our method reveals the nature of their close relation.