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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 ...

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. ...

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. ...

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. ...

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. ...

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. ...

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- ...

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? ...

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. ...

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 * * }. ...

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 ...

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. ...

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]. ...

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}. ...

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 ...

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). ...

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} , ...

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]. ...

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]. ...

... 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. ...

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]. ...

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. ...

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). ...

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]. ...

A class of commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein-type inequality was introduced by Takahasi and Hatori. We generalize this property for the commutative Fréchet algebra (Ꮽ, p ℓ) ℓ∈ގ. Moreover, we verify and generalize some of the main results in the class of Banach algebras, for the Fréchet case. We prove that all Fréchet C *-algebras and also uniform Fréchet algebras are BSE algebras. Also, we show that C ∞ [0, 1] is not a Fréchet BSE algebra.

We study bounded bilinear maps on a C*-algebra A having product property at c ∈ A. This leads us to the question of when a C*-algebra is determined by products at c. In the first part of our paper, we investigate this question for compact C*-algebras, and in the second part, we deal with von Neumann algebras having non-trivial atomic part. Our results are applicable to descriptions of homomorphism-like and derivation-like maps at a fixed point on such algebras.

It is shown that there exists a normal uniform algebra, on a compact metrizable space, that fails to be strongly regular at some peak point. This answers a 31-year-old question of Joel Feinstein. Our example is R(K) for a certain compact planar set K. Furthermore, it has a totally ordered one-parameter family of closed primary ideals whose hull is a peak point. General results regarding lifting ideals under Cole root extensions are established. These results are applied to obtain a normal uniform algebra, on a compact metrizable space, with every point a peak point but again having a totally ordered one-parameter family of closed primary ideals.

This article serves four purposes: (1) A complete characterization of semigroup ideals in \({\mathbb {Z}}_{+}^{2}\) is given; (2) The concept of standard ideals is defined in most general set up; (3) Unlike the algebra \({\mathfrak {F}}({\mathbb {Z}}_{+};{\mathbb {C}})\), there is a non-standard ideal in the algebra \({\mathfrak {F}}({\mathbb {Z}}_{+}^{2};{\mathbb {C}})\); and (4) This is a first step in the direction of studying standard closed ideals in \(\ell ^1({{\mathbb {Z}}}_+^2, \, \omega )\). It is also proved that, under a certain condition on \(f\in {\mathfrak {F}}({\mathbb {Z}}_{+}^{2};{\mathbb {C}})\), the ideal \(I_{f}=f *{\mathfrak {F}}({\mathbb {Z}}_{+}^{2};{\mathbb {C}})\) is always a standard ideal. Though the proofs are elementary, the results will give more clarity about standard ideals in the formal power series algebras.

For a connected Lie group G it was shown by Lee, Ludwig, Samei and Spronk that its Fourier algebra A(G) is weakly amenable only if G is abelian. We extend this result to general connected locally compact groups, extending an approach developed in special cases by Choi and Ghandehari.

This article is devoted to the study of derivations in group algebras. Ideals of inner and quasi-inner derivations are constructed, which makes it possible to study the algebras of outer and quasi-outer derivations. We establish a relationship between derivations and characters on the groupoid of the adjoint action. Moreover, we study a connection between the structure of algebras and the number of ends of the conjugacy diagram and, as a consequence, the number of ends of the original group. The results obtained make it possible in the final analysis to give estimates for the dimension of the algebras of outer and quasi-outer derivations using only the combinatorial properties of the group. It is also possible to obtain information about the Hochschild one-cohomology.

Weak and cyclic amenability of certain function
algebras

This paper characterize two bi-linear maps bi-derivations and quasi-multipliers on the module extension Banach algebra $A\oplus_1 X$, where $A$ is a Banach algebra and $X$ is a Banach $A$-module. Under some conditions, it is shown that if every bi-derivation on $A\oplus_1 A$ is inner, then the quotient group of bounded bi-derivations and inner bi-derivations, is equal to space of quasi-multipliers of $A$. Moreover, it is proved that $\mathrm{QM}(A \oplus_1 A)=\mathrm{QM}(A)\oplus (\mathrm{QM}(A)+\mathrm{QM}(A)')$, where $\mathrm{QM}(A)'=\{m\in \mathrm{QM}(A):m(0,a)=m(a,0)=0\}$.

Let (X, d) be a metric space and E be a unital commutative Banach algebra. In this paper, we prove that the regularity of unital commutative Banach algebra E is a necessary and sufficient condition for regularity of Lip d (X, E), where (X, d) is a compact metric space. Moreover, we show that the vector-valued Lipschitz algebra Lip d (X, E) is regular, where (X, d) is any metric space and E is a certain unital semisimple commutative *-Banach algebra. Furthermore, we study the regularity of some vector-valued function algebras. Mathematics Subject Classification. Primary: 46J10, 46E40, 16E65; Secondary: 46J05.

Let ?A and ?B be two homomorphisms on Banach algebras A and B, respectively. In this paper, we study ?-amenability, ?-weak amenability, ?-biflatness, and ?-biprojectivity of triangular Banach algebras of the form T?A,?B, where ? = ?A ? ?B.

Introduction
Let , and be Banach spaces and be a bilinear mapping. In 1951 Arens found two extension for as and from into . The mapping is the unique extension of such that from into is continuous for every , but the mapping is not in general continuous from into unless . Thus for all the mapping is continuous if and only if is Arens regular. Regarding as a Banach , the operation extends to and defined on . These extensions are known, respectively, as the first (left) and the second (right) Arens products, and with each of them, the second dual space becomes a Banach algebra.
Material and methods
The constructions of the two Arens multiplications in lead us to definition of topological centers for with respect to both Arens multiplications. The topological centers of Banach algebras, module actions and applications of them were introduced and discussed in some manuscripts. It is known that the multiplication map of every non-reflexive, -algebra is Arens regular. In this paper, we extend some problems from Banach algebras to the general criterion on module actions and bilinear mapping with some applications in group algebras.
Results and discussion
We will investigate on the Arens regularity of bounded bilinear mappings and we show that a bounded bilinear mapping is Arens regular if and only if the linear map with is weakly compact, so we prove a theorem that establish the relationships between Arens regularity and weakly compactness properties for any bounded bilinear mappings. We also study on the Arens regularity and weakly compact property of bounded bilinear mapping and we have analogous results to that of Dalse, lger and Arikan. For Banach algebras, we establish the relationships between Arens regularity and reflexivity.
Conclusion
The following conclusions were drawn from this research.
if and only if the bilinear mapping is Arens regular.
A bounded bilinear mapping is Arens regular if and only if the linear map with is weakly compact.
if and only if the bilinear mapping is Arens regular.
Assume that has approximate identity. Then is Arens regular if and only if is reflexive../files/site1/files/62/9Abstract.pdf

Let X be a compact Hausdorff space, E be a normed space, A(X,E) be a regular Banach function algebra on X , and A(X,E) be a subspace of C(X,E) . In this paper, first we introduce the notion of localness of an additive map S:A(X,E) → C(X,E) with respect to additive maps T1,...,Tn: A(X) → C(X) and then we characterize the general form of such maps for a certain class of subspaces A(X,E) of C(X,E) having A(X)-module structure. ./files/site1/files/71/9.pdf

In this paper, for a locally compact group and a fixed number , we give some sufficient conditions for the set to be spaceable in . Also, by some special Segal algebras which recently have been introduced, we find spaceable subsets of the Fourier algebra . Finally, we give some necessary and sufficient conditions for a locally compact group to be compact or discrete.

We prove that the faithful and uniqueness of norm properties are stable in different product algebras such as direct-sum product algebra, convolution product algebra, and module product algebra. Further, we exhibit that these properties are not stable in null product algebra, and also give a common sufficient condition in terms of algebra norm for the co-dimension of $\mathcal{A}^2 = \text{span} \{ ab : a,b \in \mathcal{A}\}$ to be finite in $\mathcal{A}$ and $\mathcal{A}^{2} = \mathcal{A} \ ( \text{when } \overline{\mathcal{A}^2} = \mathcal{A})$.

Let G be a locally compact Abelian group, and w : G → (0, ∞) be a Borel measurable weighted function. In this paper, the algebraic and topological properties of group algebra are studied and assessed. We show that the weighted group algebra L 1 (G, w) is regular if and only if w is a nonquasianalytic weight function. Also L 1 (G, w) is Tauberian, for any Borel measurable weight function w on the group G.

Let $E$ be a Banach space, and $\mathcal B(E)$ the algebra of all bounded linear operators on $E$. The question of amenability of $\mathcal B(E)$ goes back to Johnson's seminal memoir \cite{johnson} from 1972. We present the first general criteria applying to very wide classes of Banach spaces, given in terms of the Banach space geometry of $E$, which imply that $\mathcal B(E)$ is non-amenable. We cover all spaces for which this is known so far (with the exception of one particular example), with much shorter proofs, such as $\ell_p$ for $p \in [1, \infty]$ and $c_0$, but also many new spaces: the numerous classes of spaces covered range from all $\mathcal{L}_p$-spaces for $p \in (1, \infty)$ to Lorentz sequence spaces and reflexive Orlicz sequence spaces, to the Schatten classes $S_p$ for $p \in [1,\infty]$, and to the James space $J$, the Schlumprecht space $S$, and the Tsirelson space $T$, among others. Our approach also highlights the geometric difference to the only space for which $\mathcal B(E)$ \emph{is} known to be amenable, the Argyros--Haydon space, which solved the famous scalar-plus-compact problem.

Let $G$ be a locally compact Abelian group, and $w: G\to (0, \infty)$ be a Borel measurable weighted function. In this paper, the algebraic and topological properties of group algebra are studied and assessed. We show that the weighted group algebra $L^{1}(G, w)$ is regular if and only if $w$ is a nonquasianalytic weight function. Also $L^{1}(G, w)$ is Tauberian, for any Borel measurable weight function $w$ on the group $G$.

The work is devoted to a survey of known results related to the study of derivationsin group algebras, bimodules and other algebraic structures, as well as to various generalizationsand variations of this problem. A review of results on derivations in L_1 (G) algebras, invon Neumann algebras, and in Banach bimodules is given. Algebraic problems are discussed,in particular, derivations in groups, (σ,τ)-derivations, and the Fox calculus. A review ofsome results related to the application to pseudodifferential operators and the constructionof the symbolic calculus is also given. In conclusion, some results related to the description ofderivations as characters on the groupoid of the adjoint action are described. Some applicationsare also described: to coding theory, the theory of ends of metric spaces, and rough geometry.

Let H be an ultraspherical hypergroup and let $A(H)$ be the Fourier algebra associated with $H.$ In this paper, we study the dual and the double dual of $A(H).$ We prove among other things that the subspace of all uniformly continuous functionals on $A(H)$ forms a $C^*$ -algebra. We also prove that the double dual $A(H)^{\ast \ast }$ is neither commutative nor semisimple with respect to the Arens product, unless the underlying hypergroup H is finite. Finally, we study the unit elements of $A(H)^{\ast \ast }.

Let \({\mathcal {A}}\) be an algebra. Let \(N({\mathcal {A}})\) be the set of all algebra norms on \({\mathcal {A}}\). There is a natural equivalence relation \(\sim \) on \(N({\mathcal {A}})\). Let \({\widetilde{N}}({\mathcal {A}})\) be the collection of equivalence classes in \((N({\mathcal {A}}), \sim )\). Then there is a natural partial order \(\le \) on \({\widetilde{N}}({\mathcal {A}})\). Using this partial relation, we study different types of algebra norms on \({\mathcal {A}}\). We also study uniqueness and existence of norms with some specific properties.

It is shown that there exists a compact planar set K such that the uniform algebra R(K) is nontrivial and strongly regular. This answers a 53-year-old question of Donald Wilken.

Some of the results on automatic continuity of intertwining operators and homomorphisms that were obtained between 1960 and 1973 are here collected together to provide a detailed discussion of the subject. The book will be appreciated by graduate students of functional analysis who already have a good foundation in this and in the theory of Banach algebras.

In this paper we investigate the weighted convolution algebras l p (ω n ), where 1≦ p <∞ and {ω n } is a sequence of positive weights satisfying the following conditions. If p = 1 we require ω 0 = 1, and ω t+s ≦ω t ω s .Then

Let P be the algebra of polynomials in one inderminate x over the complex field C. Suppose ∥ · ∥ is a norm on P such that the coefficient functionals c j : ∑α i x ¹ → α j ( j = 0,1,2,…) are all continuous with respect to ∥·∥, and Let K ⊂ C be the set of characters on P which are ∥·∥-continuous. then K is compact, C\ K is connected, and 0∈ K . K. Let A be the completion of P with respect to ∥·∥. Then A is a singly generated Banach algebra, with space of characters (homeomorphic with) K . The functionals c j have unique extensions to bounded linear functionals on A , and the map a →∑ C i ( a ) x ⁱ ( a ∈ A ) is a homomorphism from A onto an algebra of formal power series with coefficients in C. We say that A is an algebra of power series if this homomorphism is one-to-one, that is if a ∈ A and a ≠O imply c j ( a )≠ 0 for some j .

We prove that a commutative unital Banach algebra which is a valuation ring must reduce to the field of complex numbers, which implies that every homomorphism from l∞ onto a Banach algebra is continuous. We show also that if bϵ [b Rad B]− for some nonnilpotent element b of the radical of a commutative Banach algebra B, then the set of all primes of B cannot form a chain, and we deduce from this result that every homomorphism from (K) onto a Banach algebra is continuous.

Haar Measure and Convolution The Dual Group and the Fourier Transform Fourier-Stieltjes Transforms Positive-Definite Functions The Inversion Theorem The Plancherel Theorem The Pontryagin Duality Theorem The Bohr Compactification A Characterization of B(Γ)

This paper surveys the known results on automatic continuity of positive functionals on topological *-algebras and then shows how two theorems on Banach *-algebras extend to complete metrizable topological *-algebras. The two theorems concerned are Loy's theorem on separable Banach *-algebras A with centre Z such that AZ is of countable codimension and Varopoulos' result on Banach *-algebras with bounded approximate identity. Both theorems have the conclusion that all positive functionals on such algebras are continuous. The extension of the second theorem requires the algebra to be locally convex and the approximate identity to be ‘uniformly bounded’. Neither extension requires the algebra to be LMC. This means that the proof of the first theorem is quite different from the corresponding Banach algebra result (which used spectral theory). The proof of the second is closer to the previously known LMC version, but actually neater by being more general. It is also shown that the well-known estimate of |f(a*ba)| for a positive functional f on a Banach *-algebra may be obtained without the usual use of spectral theory. The paper concludes with a list of open questions.

Let M(G) denote the convolution algebra of finite regular Borel measures on a locally compact abelian group G, and let Δ denote the maximal ideal space of M(G). It is well-known that on certain subsets of Δ the Gelfand transforms μ^ of members μ of M(G) behave like holomorphic functions. The simplest way to exhibit this is to use Taylor’s description of Δ as the semigroup of all continuous semicharacters of a compact semigroup S - the structure semigroup of M(G) (see [10]). If f ∈ Δ (= S^) and f(s) ^ 0 for all s ∈ S, then ⨍ ∈ A for Re(z) ≧ 0. Thus, provided ⨍z≠⨍, there is an analytic disc around ⨍ in the sense that μ(⨍z) is holomorphic on Re(z)≧0 for all μ ∈ M(G). Using this fact, Taylor (loc. cit.) has shown that if ⨍ is a strong boundary point of M(G), then |⨍|z = |⨍|.

Multilinear mappings and Banach algebras

- K. B. Laursen