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# Banach algebras and automatic continuity

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... Higher derivations were introduced by Hasse and Schmidt [3], and algebraists sometimes call them Hasse-Schmidt derivations. For an account on higher derivations the reader is referred to the book [2]. Mirzavaziri [6] proved that there exists a one-to-one correspondence between higher derivations and the family of sequences of derivations on torsion free algebras. ...
... Proof. Let D = {d n } ∞ n=0 be an orthogonally higher ring derivation on A. Since for any n ∈ N, d n is an orthogonally additive mapping on the inner product space A, by Corollary 10 of [12], it follows that d n is of the form d n (x) = a n (∥x∥ 2 ...
... for all x, y ∈ A. Dividing the above equation by 2 4k and letting k → ∞, we get a n (∥xy∥ 2 ...
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In this paper, we prove that every orthogonally higher ring derivation is a higher ring derivation. Also we find the general solution of the pexider orthogonally higher ring derivations f n (x + y) = g n (x) + h n (y), x, y = 0, f n (xy) = i+j=n g i (x)h j (y). Then we prove that for any approximate pexider orthogonally higher ring derivation under some control functions φ(x, y) and ψ(x, y), there exists a unique higher ring derivation D = {d n } ∞ n=0 , near {f n } ∞ n=0 , {g n } ∞ n=0 and {h n } ∞ n=0 estimated by φ and ψ.
... It is well-known that on the second dual space A * * of a Banach algebra A there are two multiplications, called the first and second Arens products which make A * * into a Banach algebra [7]. If these products coincide on A * * , then A is said to be Arens regular. ...
... If these products coincide on A * * , then A is said to be Arens regular. It is shown [7] that every C * -algebra A is Arens regular. ...
... where {a i } i∈I and {x i } j∈I are nets in A and X that converge in the w * -topology, to Φ and u, respectively. One may refer to the monograph of Dales [7] for a full account of Arens product and w * -continuity of the above structures. Theorem 2.3. ...
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In this paper‎, ‎we prove that every continuous Jordan multiplier T from a‎ C*-algebra or group algebra A into a Banach A-bimodule X is exact a multiplier‎. We also characterize a continuous linear map on C*-algebras and standard operator algebra‎s ‎through the action on zero products‎. ‎Additionally‎, ‎we show that every continuous local multiplier T from a Banach algebra A with the property $(\mathbb{B})$ into a Banach A-bimodule X is a multiplier‎. 2010 Mathematics Subject Classification. Primary 47B47, 47B49 Secondary 15A86, 46H25.
... Recall that a bounded approximate identity for A is a bounded net {e λ } λ∈I in A such that e λ a −→ a and ae λ −→ a for every a ∈ A. It is known that the group algebra L 1 (G) for a locally compact group G and every C ⋆ -algebra has a bounded approximate identity bounded by one, see [7]. ...
... Consequently, T (ab) = T (a)b for all a, b ∈ A and hence T is a left multiplier. □ It is well-known that on the second dual space A * * of a Banach algebra A there are two multiplications, called the first and second Arens products which make A * * into a Banach algebra [7]. If these products coincide on A * , then A is said to be Arens regular. ...
... If these products coincide on A * , then A is said to be Arens regular. It is shown [7] that every C ⋆ -algebra A is Arens regular. ...
Article
Let A be a Banach ⋆-algebra, X be a Banach ⋆-A-bimodule and T : A −→ X be a continuous linear map. In this paper, by using orthogonality conditions on A, we characterize the map T on certain Banach algebra including C ⋆-algebras, group algebras, standard operator algebras and Banach algebras that is generated by idempotents. We also characterize a continuous linear map from zero Jordan product determined Banach algebra A into a Banach A-bimodule X, and give some applications of this result.
... It is well-known that the Gelfand transform is one-to-one if and only if A is semisimple. For more on the theory of commutative Banach algebras see, for example, [6,7,13]. ...
... If A is a Banach function algebra on X, for every x ∈ X, the evaluation homomorphism ǫ x : A → C, f → f (x), is a character of A, and the mapping X → M(A), x → ǫ x , imbeds X onto a compact subset of M(A). When this mapping is surjective we call A natural and M(A) = X; [7,Definition 4.1.3]. For example, C (X) is a natural uniform algebra. ...
... Note that every semisimple commutative unital Banach algebra A can be seen, through its Gelfand transform, as a natural Banach function algebra on X = M(A). See, for example, [7,Chap. 4] and [13, Chap. ...
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We consider several notions of regularity, including strong regularity, bounded relative units, and Ditkin's condition, in the setting of vector-valued function algebras. Given a commutative Banach algebra $A$ and a compact space $X$, let $\mathcal{A}$ be a Banach $A$-valued function algebra on $X$ and let $\mathfrak{A}$ be the subalgebra of $\mathcal{A}$ consisting of scalar-valued functions. This paper is about the connection between regularity conditions of the algebra $\mathcal{A}$ and the associated algebras $\mathfrak{A}$ and $A$. That $\mathcal{A}$ inherits a certain regularity condition $P$ to $\mathfrak{A}$ and $A$ is the easy part of the problem. We investigate the converse and show that, under certain conditions, $\mathcal{A}$ receives $P$ form $\mathfrak{A}$ and $A$. The results apply to tensor products of commutative Banach algebras as they are included in the class of vector-valued function algebras.
... References [1], [13] and [14] will be our standard sources for undefined concepts related to Banach spaces, and [4] for notions related to Banach algebras. We use X ≈ Y to indicate linearly isomorphic Banach spaces, and A ∼ = B for isomorphic Banach algebras (that is, there is a Banach algebra isomorphism θ : A → B). ...
... 2. Closed subideals of L(X) which are isomorphic as Banach algebras Let X and Y be Banach spaces. It is a classical result of Eidelheit [8] (see also [4,Theorem 2.5.7]) that if θ : L(X) → L(Y ) is a Banach algebra isomorphism, then there is a linear isomorphism U ∈ L(X, Y ) such that θ(S) = U SU −1 for all S ∈ L(X). Chernoff [3, Corollary 3.2] established the following extension: Suppose that A ⊂ L(X) and B ⊂ L(Y ) are subalgebras such that the bounded finite rank operators F(X) ⊂ A and F(Y ) ⊂ B. If θ : A → B is a bijective algebra homomorphism, then there is a linear isomorphism U ∈ L(X, Y ) such that θ(S) = U SU −1 for all S ∈ A. As a consequence, if I, J ⊂ L(X) are closed ideals for which there is a Banach algebra isomorphism θ : I → J , then I = J (cf. also Remarks 2.5.(ii)). ...
... We leave the details to the interested reader. 4. Ideal of L(Z p ) generated by the closed K(Z p )-subideal I A We briefly return to the setting of Section 2 and the family F of non-trivial closed K(Z p )subideals. ...
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We exhibit a Banach space $Z$ failing the approximation property, for which there is an uncountable family $\mathcal F$ of closed subideals contained in the Banach algebra $\mathcal K(Z)$ of the compact operators on $Z$, such that the subideals in $\mathcal F$ are mutually isomorphic as Banach algebras. This contrasts with the behaviour of closed ideals of the algebras $\mathcal L(X)$ of bounded operators on $X$, where closed ideals $\mathcal I \neq \mathcal J$ are never isomorphic as Banach algebras. We also construct families of non-trivial closed subideals in the strictly singular operators $\mathcal S(X)$ for the classical spaces $X = L^p$ with $p \neq 2$, where the subideals are not pairwise isomorphic.
... It is well-known that the Gelfand transform is one-to-one if and only if A is semisimple. For more on the theory of commutative Banach algebras see, for example, [6,7,13]. ...
... If A is a Banach function algebra on X, for every x ∈ X, the evaluation homomorphism ǫ x : A → C, f → f (x), is a character of A, and the mapping X → M(A), x → ǫ x , imbeds X onto a compact subset of M(A). When this mapping is surjective we call A natural and M(A) = X; [7,Definition 4.1.3]. For example, C (X) is a natural uniform algebra. ...
... Note that every semisimple commutative unital Banach algebra A can be seen, through its Gelfand transform, as a natural Banach function algebra on X = M(A). See, for example, [7,Chap. 4] and [13, Chap. ...
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We consider several notions of regularity, including strong regularity, bounded relative units, and Ditkin’s condition, in the setting of vector-valued function algebras. Given a commutative Banach algebra A and a compact space X, let A be a Banach A-valued function algebra on X and let A be the subalgebra of A consisting of scalar-valued functions. This paper is about the connection between regularity conditions of the algebra A and the associated algebras A and A. That A inherits a certain regularity condition P to A and A is the easy part of the problem. We investigate the converse and show that, under certain conditions, A receives P form A and A. The results apply to tensor products of commutative Banach algebras as they are included in the class of vector-valued function algebras.
... H 1 (A, J * ) = {0}. We call A ideally amenable if A is J -weakly amenable Communicated 1 for every closed ideal J of A [6]. It was shown that ideal amenability is different from amenability and weak amenability. ...
... For n ∈ N, the elements of ker δ n and im δ n−1 are the continuous n-cocycles and the continuous n-coboundaries, respectively; these linear spaces are denoted by Z n (A, E) = ker δ n and N n (A, E) = im δ n−1 [1]. ...
... Let A be a Banach algebra with a unit e and X be in A-mod. Then X is unital, if e · x = x for all x ∈ X [1]. We have similar definitions for X in mod-A and A-mod- ...
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In this paper, we study the notion of ideal amenability for ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^1$$\end{document}-Munn Banach algebras. In addition to the new results, our investigation leads to the completion and presentation of a different proof for some previous studies. Finally, we apply these results for Brandt and Rees semigroup algebras. In particular, we show some sufficient conditions for ideal amenability of ℓ1(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^1(G)$$\end{document}.
... Definition 1 ( [Dal01], Definition 1.8.1). Let A be an algebra over C, and M be an A-bimodule. ...
... This problem is stated in [Dal01] as follows: Let G be a locally compact group. Is every derivation from L 1 (G) to M(G) inner? ...
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The derivation problem is a familiar one concerning group algebras, particularly $L_1(G)$ and von Neumann algebras. In this paper, we study the Banach bimodule $\ell_p(G)$, which is generated by the $\ell_p$ norm over a specific class of groups with well-organized conjugacy classes. For this case, we will demonstrate that all $\ell_p(G)$ derivations are inner.
... Results on automatic continuity of linear maps defined on Banach algebras comprise a fruitful area of research intensively developed during the last sixty years. The reader is referred to [2,5,6,9,27] for a deep and extensive study on this subject. Let us recall some basic definitions and set the notations which are used in what follows. ...
... Let A be an algebra. A nonzero linear functional ϕ on A is called a character if ϕ(ab) = ϕ(a)ϕ(b) for all a, b ∈ A. The set of all characters on A is denoted by Φ A and is called the character space of A. According to [5,Proposition 1.3.37], the kernel of ϕ, ker ϕ, is a maximal ideal of A for every ϕ ∈ Φ A . Recall that an algebra (or ring) A is called prime if for a, b ∈ A, aAb = {0} implies that a = 0 or b = 0, and is semiprime if for a ∈ A, aAa = {0} implies that a = 0. ...
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Let $\mathcal{A}$ and $\mathcal{B}$ be two algebras and let $n$ be a positive integer. A linear mapping $D:\mathcal{A} \rightarrow \mathcal{B}$ is called a \emph{strongly generalized derivation of order $n$} if there exist families of linear mappings $\{E_k:\mathcal{A} \rightarrow \mathcal{B}\}_{k = 1}^{n}$, $\{F_k:\mathcal{A} \rightarrow \mathcal{B}\}_{k = 1}^{n}$, $\{G_k:\mathcal{A} \rightarrow \mathcal{B}\}_{k = 1}^{n}$ and $\{H_k:\mathcal{A} \rightarrow \mathcal{B}\}_{k = 1}^{n}$ which satisfy $D(ab) = \sum_{k = 1}^{n}\left[E_k(a) F_k(b) + G_k(a)H_k(b)\right]$ for all $a, b \in \mathcal{A}$. The purpose of this article is to study the automatic continuity of such derivations on Banach algebras and $C^{\ast}$-algebras.
... As is well-known [2], the second dual A * * of A endowed with the first Arens multiplication is a Banach algebra. The basic properties of this multiplication are as follows. ...
... For F fixed in A * * , the mapping E → EF is weak * -weak * continuous. For E fixed in A * * , the mapping F → EF is general not continuous unless E is in A (for more information see [2]). Whence the topological center of A * * with respect to this multiplication is defined as follows Z t (A * * ) = {E ∈ A * * ; F → EF is weak * -weak * continuous on A * * }. ...
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Let A be a Banach algebra with a bounded approximate identity bounded by 1. Two new topologies ?so and ?wo are introduced on A. We study these topologies and compare them with each other and with the norm topology. The properties of ?so and ?wo are then studied further and we pay attention to the group algebra L1(G) of a locally compact group G. Various necessary and sufficient conditions are found for a locally compact group G to be finite.
... For unexplained details from Banach algebra theory we refer the reader to [3,27] and from Orlicz space theory to [12-16, 18, 20, 26]. ...
... That ∞ is biflat follows from [28] (cf. [3,Theorem 2.9.65]) since ∞ is a commutative C * -algebra, therefore amenable by [9,Lemma 7.10]. As for the other case observe that 1 is even biprojective-see [ Recall that an approximate identity in a Banach algebra A is a net (e α ) α∈ ⊂ A such that ...
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We study Orlicz sequence algebras and their properties. In particular, we fully characterize biflat and biprojective Orlicz sequence algebras as well as weakly amenable and approximately (semi-)amenable Orlicz sequence algebras. As a consequence, we show the existence of a wide class of sequence algebras that behave differently—in terms of the amenability properties—from any of the algebras ℓp,1⩽p⩽∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell _{p}, 1\leqslant p\leqslant \infty$\end{document}.
... This in particular guarantees that the corresponding multiplier algebras have good properties and various approaches to multipliers coincide. We refer to [21], see also [22] and references therein, for more details. Following [22,Section 2], for a Banach algebra A we define the multiplier algebra as ...
... are nets that are convergent to a and b respectively, in the weak˚topology on B 2 induced by B 1 . The first Arens product is given by a˝b :" B 1 -lim α B 1 -lim β a α b β , and B is called Arens regular if the two products coincide, see [21] for more details. Our decision to use the second Arens product is arbitrary, and in any case we will be primarily interested in products av P B where a P B and v P B 2 , so that the first and second Arens products always agree. ...
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We initiate a study of (real or complex) Banach algebras associated to twisted \'etale groupoids $(\mathcal{G},\mathcal{L})$ and to twisted inverse semigroup actions, which provides a general unifying framework for numerous recent papers on $L^p$-operator algebras and the fully developed theory of groupoid $C^*$-algebras. Our main result is a version of the disintegration theorem that gives a bijective correspondence between representations of the twisted groupoid Banach algebra $F(\mathcal{G},\mathcal{L})$ and covariant representations of an inverse semigroup action. This powerful tool allows us to study $F(\mathcal{G},\mathcal{L})$ in terms of representations of the Banach algebra $C_0(X)$ and a (twisted) inverse semigroup $S$ of partial isometries subject to some relations. This works best when the groupoid is ample or when the target Banach algebra $B$ is a dual Banach algebra, for example when $B=B(E)$ for a reflexive Banach space $E$. When $E=L^p(\mu)$ for $p\in (1,\infty)\setminus \{2\}$ these relations force the partial isometries to be spatial, which explains why Phillips and others define $L^p$-analogues of Cuntz or graph algebras in terms of spatial partial isometries - the groupoid model forces that. We introduce and analyze in more detail $L^p$-operator algebras $F^{p}(\mathcal{G},\mathcal{L})$, $p\in[1,\infty]$, which are universal for representations of $F(\mathcal{G},\mathcal{L})$ on $L^p$-spaces, and which generalize algebras $F^{p}(\mathcal{G})$ introduced recently by Gardella and Lupini. As more concrete examples we discuss Banach (and $L^p$-operator algebras) associated to twisted partial group actions, twisted Renault-Deaconu groupoids and directed graphs. We also introduce tight inverse semigroup Banach algebras that cover all ample groupoid Banach algebras.
... By an A-valued uniform algebra on K we mean a closed subalgebra A of C (K, A) that contains the constant functions and separates points of K; see [1,12]. A comprehensive discussion on complex function algebras appears in [5,Chapter 4]. ...
... Thus φ = φ a on P N0 (K), a dense subspace of P N (K), whence φ = φ a on P N (K). Finally, the mappingK N → M(P N (K)), a → φ a , is an embedding of K onto M(P N (K)); see [5,Chapter 4]. ...
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Let $E$ be a Banach space and $A$ be a commutative Banach algebra with identity. Let ${P}(E, A)$ be the space of $A$-valued polynomials on $E$ generated by bounded linear operators (an $n$-homogenous polynomial in ${P}(E,A)$ is of the form $P=\sum_{i=1}^\infty T^n_i$, where $T_i:E\to A$ ($1\leq i <\infty$) are bounded linear operators and $\sum_{i=1}^\infty \|T_i\|^n < \infty$). For a compact set $K$ in $E$, we let ${P}(K, A)$ be the closure in $C(K,A)$ of the restrictions $P|_K$ of polynomials $P$ in ${P}(E,A)$. It is proved that ${P}(K, A)$ is an $A$-valued uniform algebra and that, under certain conditions, it is isometrically isomorphic to the injective tensor product $\mathcal{P}_N(K)\hat\otimes_\epsilon A$, where $\mathcal{P}_N(K)$ is the uniform algebra on $K$ generated by nuclear scalar-valued polynomials. The character space of ${P}(K, A)$ is then identified with $\hat{K}_N\times \mathfrak{M}(A)$, where $\hat K_N$ is the nuclear polynomially convex hull of $K$ in $E$, and $\mathfrak{M}(A)$ is the character space of $A$.
... First, we recall some standard notions, for further details, see [3] and [9]. Let A be a Banach algebra and E a Banach A-bimodule, a linear map ...
... Clearly, max{ ab , ba } ≤ a M b for all a, b ∈ A, so that a M ≤ a (a ∈ A). The annihilator ideal of A denoted by ann(A) is defined as (3) ann(A) = {a ∈ A : ab = ba = 0, b ∈ A}. ...
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Let X be a compact Hausdorff space, we show that for a norm irregular Banach algebra A with a bounded approximate identity, if A has an approximate diagonal which is bounded with respect to the multiplier norm on A⊗A, then C(X, A) has an approximate diagonal.
... for a neighborhood U 0 ∋ u in σ Z n (G, U ). Shrinking the initial U so that all u ′ · w u ′ in (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) belong to U 0 , we have our conclusion. ...
... The kernel of the addition map (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) can be identified with the intersection T 1 C ∩ T 1 G via ...
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Consider a compact group $G$ acting on a real or complex Banach Lie group $U$, by automorphisms in the relevant category, and leaving a central subgroup $K\le U$ invariant. We define the spaces ${}_KZ^n(G,U)$ of $K$-relative continuous cocycles as those maps $G^n\to U$ whose coboundary is a $K$-valued $(n+1)$-cocycle; this applies to possibly non-abelian $U$, in which case $n=1$. We show that the ${}_KZ^n(G,U)$ are analytic submanifolds of the spaces $C(G^n,U)$ of continuous maps $G^n\to U$ and that they decompose as disjoint unions of fiber bundles over manifolds of $K$-valued cocycles. Applications include: (a) the fact that $Z^n(G,U)\subset C(G^n,U)$ is an analytic submanifold and its orbits under the adjoint of the group of $U$-valued $(n-1)$-cochains are open; (b) hence the cohomology spaces $H^n(G,U)$ are discrete; (c) for unital $C^*$-algebras $A$ and $B$ with $A$ finite-dimensional the space of morphisms $A\to B$ is an analytic manifold and nearby morphisms are conjugate under the unitary group $U(B)$; (d) the same goes for $A$ and $B$ Banach, with $A$ finite-dimensional and semisimple; (e) and for spaces of projective representations of compact groups in arbitrary $C^*$ algebras (the last recovering a result of Martin's).
... By Theorem 3.1 in Chen and Zhang (2021), D is inner. Since C 0 ðG; 1=xÞ Ã ¼ MðG; xÞ, and from Remark 2.1 we know C 0 ðG; 1=xÞ is an essential Banach L 1 ðG; xÞ-module, the multiplier algebra MðL 1 ðG; xÞÞ is isometrically isomorphic to MðG; xÞ, so D satisfies the conditions of Theorem 2.9.53 in Dales (2000). ...
Article
Let G be a locally compact group and $$\omega$$ be a diagonally bounded weight function on G. In this paper, we investigate derivations from the Banach algebra $$L^{\infty }_{0}(G,1/\omega )^*$$ into $$L^1(G,\omega )$$ and show that these derivations are inner. In addition, we prove these derivations are zero maps when G is abelian and $$\omega$$ is symmetric.
... So, if a is quasi-invertible, there is a unique element b ∈ A such that a • b = b • a = 0, b is the quasi-inverse of a, and is denoted by a q . We write q − invA for the set of all quasi-invertible elements of A [3,4]. Clearly a • b = 0 if and only if (−a, 1)(−b, 1) = e in A # , and so invA # = {(−a, 1) : a ∈ q − invA} , ...
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It is known that even duals of a Banach algebra A with one of Arens products are Banach algebras, these products are natural multiplications extending the one on A. But the essence of A * is completely different. By defining new products, we investigate some algebraic and spectral properties of odd duals of A. We will show relations between these products and Arens products, weak or weak-star continuity, commutativity and unit elements of these algebras. We also determine the spectrum and multiplier algebra for A * , and we calculate the quasi-inverses, spectrum and spectral radius for elements of these kinds of algebras.
... The following result originates in the work of Eidelheit [13, Lemma 1 and Theorem 1], as already mentioned in the Introduction. It can for instance be found in [8,Theorem 5.1.14] or [42,Theorem,page 107]. ...
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We show that for each of the following Banach spaces~$X$, the quotient algebra $\mathscr{B}(X)/\mathscr{I}$ has a unique algebra norm for every closed ideal $\mathscr{I}$ of $\mathscr{B}(X)\colon$ - $X= \bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{c_0}$\quad and its dual,\quad $X= \bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{\ell_1}$, - $X= \bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{c_0}\oplus c_0(\Gamma)$\quad and its dual, \quad $X= \bigl(\bigoplus_{n\in\N}\ell_2^n\bigr)_{\ell_1}\oplus\ell_1(\Gamma)$,\quad for an uncountable cardinal number~$\Gamma$, - $X = C_0(K_{\mathcal{A}})$, the Banach space of continuous functions vanishing at infinity on the locally compact Mr\'{o}wka space~$K_{\mathcal{A}}$ induced by an uncountable, almost disjoint family~$\mathcal{A}$ of infinite subsets of~$\mathbb{N}$, constructed such that $C_0(K_{\mathcal{A}})$ admits "few operators". Equivalently, this result states that every homomorphism from~$\mathscr{B}(X)$ into a Banach algebra is continuous and has closed range. The key step in our proof is to show that the identity operator on a suitably chosen Banach space factors through every operator in $\mathscr{B}(X)\setminus\mathscr{I}$ with control over the norms of the operators used in the factorization. These quantitative factorization results may be of independent interest.
... This extends the usual convolution for integrable functions, defined by (f * g)(x) = G f (xy −1 )g(y)dy, in such a way that L 1 (G) is an ideal in M (G). The spaces L p (G) for 1 ⩽ p ⩽ ∞ and C(G) are L 1 (G)-modules (actually M (G)-modules) under convolution; see for instance Rudin [33, Section 1.1] and [12,Theorem 5.6.34]. ...
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... Given a compact Hausdorff space K and a (unital) uniform algebra A on K, the Choquet boundary and Shilov boundary for A will be denoted by χA and ∂A, respectively. We refer the reader to [12,30] for background on uniform algebras and related notions. ...
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Inspired by the recent work of Cascales et al., we introduce a generalized concept of ACK structure on Banach spaces. Using this property, which we call by the quasi-ACK structure, we are able to extend known universal properties on range spaces concerning the density of norm attaining operators. We provide sufficient conditions for quasi-ACK structure of spaces and results on the stability of quasi-ACK structure. As a consequence, we present new examples satisfying (Lindenstrauss) property B$$^k$$, which have not been known previously. We also prove that property B$$^k$$ is stable under injective tensor products in certain cases. Moreover, ACK structure of some Banach spaces of vector-valued holomorphic functions is also discussed, leading to new examples of universal BPB range spaces for certain operator ideals.
... Sanjani Monfared [16] extended this product to arbitrary Banach algebras A and B. The θ−Lau products are significance and utility. Because, the θ−Lau product is a strongly splitting Banach algebra extension of B by A; for the study of extensions of Banach algebras see [3,7]. Also, many properties are not shared by arbitrary strongly splitting extensions, while the θ−Lau products exhibit them; see [16]. ...
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In this paper, we study Jordan derivation-like maps on the $\theta-$Lau products of algebras. We characterize them and prove that under certain condition any Jordan derivation-like maps on the $\theta-$Lau products is a derivation-like map. Moreover, we investigate the concept of centralizing for Jordan derivation-like maps on the $\theta-$Lau products of algebras.
... If A is a projective (respectively, flat) as a Banach A-bimodule, then A is named biprojective (respectively, biflat). The Banach algebra A is amenable if and only if A biflat and it has bounded approximate identity; see [5]. In a special case, BM(G, H) has an identity, so BM(G, H) is biflat if and only if it is amenable. ...
... If A is a projective (respectively, flat) as a Banach A-bimodule, then A is named biprojective (respectively, biflat). The Banach algebra A is amenable if and only if A biflat and it has bounded approximate identity; see [5]. In a special case, BM(G, H) has an identity, so BM(G, H) is biflat if and only if it is amenable. ...
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Let $G$ and $H$ be locally compact groups. $BM(G, H)$ denoted the Banach algebra of bounded bilinear forms on $C_{0}(G)\times C_{0}(H)$.In this paper, the homological properties of Bimeasure algebras are investigated. It is found and approved that the Bimeasure algebras $BM(G, H)$ is amenable if and only if $G$ and $H$ are discrete. The correlation between the weak amenability of $BM(G, H)$ and $M(G\times H)$ is assessed. It is found and approved that the biprojectivity of the bimeasure algebra $BM(G, H)$ is equivalent to the finiteness of $G$ and $H$. Furthermore, we show that the bimeasure group algebra $BM_{a}(G, H)$ is a BSE algebra. It will be concluded that $BM(G, H)$ is a BSE- algebra if and only if $G$ and $H$ are discrete groups.
... Therefore (5) implies that ω x ∈ LU C(S). Let m be a linear functional in LU C(S) * which satisfies (ii). ...
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We investigate the bounded derivations and bounded crossed homomorphisms from S into X* (the first dual of X). We show that the innerness of these bounded derivations implies that S is inner amenable. We prove that every left (right) crossed homomorphism on a semigroup is principal if and only if it is left (right) amenable. Finally, we show that every bounded left (right) crossed homomorphism from S into M(X), the Banach space of all Borel measures on X, is principal. In the locally compact group case, this is an answer for the derivation problem on locally compact groups.
... for all x, y in A. For the properties of algebras of type F or B 0 the reader is referred to [7], [10], [11], [15] and [16]. ...
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Multilinear mappings and Banach algebras
• K. B. Laursen