# Complete Normed Algebras

## Chapters (7)

Throughout this book the symbol F will be used to denote a field that is either the real field ℝ or the complex field ℂ.

In this section A denotes an algebra over F and we consider the purely algebraic theory of irreducible left A-modules.

In this section A will denote an algebra. Definitions and results will usually be stated only for left ideals; with the obvious changes, they apply to right ideals.

Recall from § 12 that a Banach star algebra is a complex Banach algebra A with an involution * satisfying, for all x, y∈A,α∈ℂ,
(i)
(x+y)*=x* + y*
(ii)
(αx)*=α* x*
(iii)
x**=x
(iv)
(xy)* =y* x*.

Let X, Y,Z be normed linear spaces over the same field F. A mapping ø: X × Y→Z is said to be bilinear if
(i)
for each y∈ Y, the mapping x→ø(x,y) is linear
(ii)
for each x∈ X, the mapping y→ø(x,y) is linear.

A will denote a complex Banach algebra with unit. As usual, a complex polynomial in one variable is said to be monic if the coefficient of the term of highest degree is 1. We denote by Pn
the set of all complex monic polynomials of degree n.

... In particular, the spectrum σ(R; M op (d p )) of an operator R ∈ M op (d p ) coincides with its spectrum σ(R; L (d p )) as an element of L (d p ). For the definition of the spectrum of an element in a unital Banach algebra we refer to [6], [15], for example. ...

... Theorem 2 on p. 98 of [6] implies that the maximal ideal space Φ of M (d 1 ) is homeomorphic with the spectrum σ(e 1 ; M (d 1 )) of the generator e 1 . Since M (d 1 ) is isometric to M op (d 1 ) we know from Proposition 6.1 that σ(e 1 ; M (d 1 )) = σ(T e 1 ; M op (d 1 )) = σ(S; M op (d 1 )) = σ(S; L (d 1 )) = D. ...

... In particular, the maximal ideal space of A (S, d p ) is homeomorphic to D (cf. [6,II Theorem 19.2]) and so, for any polynomial f , we have that σ(f (S); A (S, d p )) = σ(f (S); M op (d p )) = f (D). Every T ∈ A (S, d p ) ⊆ M op (d p ) is of the form T = T b for some unique element b ∈ M (d p ). ...

We investigate convolution operators in the sequence spaces $d_p$, for $1\le p<\infty$. These spaces, for $p>1$, arise as dual spaces of the \ces sequence spaces $ces_p$ thoroughly investigated by G.~Bennett. A detailed study is also made of the algebra of those sequences which convolve $d_p$ into $d_p$. It turns out that such multiplier spaces exhibit features which are very different to the classical multiplier spaces of $\ell^p$.

... Remark A. 4 The multiplication · : A × A → A is continuous with respect to the quasi-norm topology on A and left/right continuous. The proof goes exactly as in the Banach case. ...

... Following the pattern of Section 24 in [4], one can infer that the representation theory for quasi-Banach algebras goes exactly the same as for Banach algebras, since the main ingredients are the algebraic properties, the closedness criteria and the continuity of the representations. We then restate the same Lemmata 8.7, 8.8 and 8.9 in [24] in our setting as follows. ...

We generalize the results for Banach algebras of pseudodifferential operators obtained by Gröchenig and Rzeszotnik (Ann Inst Fourier 58:2279–2314, 2008) to quasi-algebras of Fourier integral operators. Namely, we introduce quasi-Banach algebras of symbol classes for Fourier integral operators that we call generalized metaplectic operators, including pseudodifferential operators. This terminology stems from the pioneering work on Wiener algebras of Fourier integral operators (Cordero et al. in J Math Pures Appl 99:219–233, 2013), which we generalize to our framework. This theory finds applications in the study of evolution equations such as the Cauchy problem for the Schrödinger equation with bounded perturbations, cf. (Cordero, Giacchi and Rodino in Wigner analysis of operators. Part II: Schrödinger equations, arXiv:2208.00505).

... Proof. Let {d n } be a spectrally bounded Jordan strong higher derivation associated with the corresponding numerical sequence {M n } and let P be a primitive ideal of A. It follows from part (ii) of Proposition 24-12 of [1] that, A has an irreducible representation π : A → L(X ) on a Banach space X with kernel P, where L(X ) is the algebra of all linear mappings on X . It follows from Lemma 2.2 that, d 1 (P) ⊆ P. Now, we are going to show that d 2 (P) ⊆ P. According to the proof of Lemma 1 of [16], we have d n k (a n ) = n j=1 a i,j =k, i=1,2,...,n d a n,1 (d a n−1,1 (...(d a 2,1 (d a 1,1 (a)))...)) ...

... whence d 2 (a) + P ∈ Q( A P ) for each a ∈ P. Let a ∈ P and assume that d 2 (a) / ∈ P. Note that the normed division algebra D = {T ∈ L(X ) : bT (x) = T (bx), b ∈ A, x ∈ X } is CI (see p.128 of [1]). If π(d 2 (a)) ∈ CI, then bπ(d 2 (a))(x) = π(d 2 (a))(bx), and hence bd 2 (a)x = d 2 (a)bx for all b ∈ A, x ∈ X . ...

In this article, we prove Sinclair's Theorem for spectrally bounded Jordan higher derivations on Banach algebras. Furthermore, it is proved that the image of a higher derivation , under certain conditions, is contained in the Jacobson radical.

... One can in passing compare this result with the Gelfand-Naimark Theorem on nonunital commutative Banach algebras (see [3,Theorem 5] ). Let's describe in a few lines the space X * that allows this representation to work. ...

We illustrate the interplay between multiplicative structure and ordered structure on Banach spaces of real-valued functions via various versions of the Stone–Weierstrass Approximation Theorem.

... 2 of 14 is a bilinear map. For the basic properties concerning the tensor product of linear spaces, we refer the reader to [2]. Definition 1.1. ...

Given a fuzzy normed space $ \left( X,N \right) $, we will introduce the notion of fuzzy near best approximation within a relative distance $ \rho \geq 0 $. Some basic properties are characterized and also many examples for illustration are presented.

... The spaces of all bounded derivations and all inner derivations from A to X are denoted by Z 1 (A, X) and N 1 (A, X), respectively. Then the first (topological) cohomology group of A with coefficients in X is the quotient space Z 1 (A, X)/N 1 (A, X) and denoted by H 1 (A, X) (for reviewing these concepts one may see a standard text such as [1]). Much studies have been devoted to the calculation of cohomology groups H 1 (A, X) and the higher dimensions cohomology groups H n (A, X). ...

A class of (2m-1)-weakly amenable Banach algebras

... [5] Let A be a real or complex Banach algebra and p(t) = n k=0 b k t k a polynomial in the real variable t with coefficients in A. If for an infinite set of real values of t, p(t) ∈ M, where M is a closed linear subspace of A, then every b k lies in M. ...

Let $ \mathcal{A} $ be a Banach algebra and $ n > 1 $, a fixed integer. The main objective of this paper is to talk about the commutativity of Banach algebras via its projections. Precisely, we prove that if $ \mathcal{A} $ is a prime Banach algebra admitting a continuous projection $ \mathcal{P} $ with image in $ \mathcal{Z}(\mathcal{A}) $ such that $ \mathcal{P}(a^n) = a^n\; \text{for all} \; a \in \mathcal{G} $, the nonvoid open subset of $ \mathcal{A} $, then $ \mathcal{A} $ is commutative and $ \mathcal{P} $ is the identity mapping on $ \mathcal{A} $. Apart from proving some other results, as an application we prove that, a normed algebra is commutative iff the interior of its center is non-empty. Furthermore, we provide some examples to show that the assumed restrictions cannot be relaxed. Finally, we conclude our paper with a direction for further research.

... JB-algebras. Our general reference for JB-algebras are the books [9] and [5]. ...

... Note that σ(a) is a non-empty compact subset of C, and we have (see [1]): ...

In this paper, we study the properties of the S-spectrum of right linear bounded operators on a right quaternionic Banach space. We prove some relations between the S-spectrum and most of its important parts; the approximate S-spectrum, the compression S-spectrum and the surjective S-spectrum. Among other results, we provide some properties of duality and orthogonality on a right quaternionic Banach space.

... The term of regular Banach algebra is introduced byŠilove in [33] and generalized by Arenz in [6]. Some authors (see [11] and [28], for example) use terminology "complete regular" in place ofŠilov's "regular" to avoid the confusion with different concepts of regularity which was studied by J. von Neumann; see [35]. ...

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.

... We will identify B with its canonical image in B * * . For basic definitions and properties, the reader is referred to [2]. ...

Using the notion of a symmetric virtual diagonal for a Banach algebra, we prove that a Banach algebra is symmetrically amenable if its second dual is symmetrically amenable. We introduce symmetric operator amenability in the category of completely contractive Banach algebras as an operator algebra analogue of symmetric amenability of Banach algebras. We give some equivalent formulations of symmetric operator amenability of completely contractive Banach algebras and investigate some hereditary properties of symmetric operator amenable algebras. We show that amenability of locally compact groups is equivalent to symmetric operator amenability of its Fourier algebra. Finally, we discuss about Jordan derivation on symmetrically operator amenable algebras.

... In this section, we quote some basic lemmas that will be used as tools in subsequent section. The following lemma, due to Bonsall and Duncan [2], is crucial for developing the proof of our main results. Ä ÑÑ 2.1º Let X be a real or complex Banach algebra and P (t) = n k=0 b k t k a polynomial in the real variable t with coefficients in X . ...

Let X be a Banach algebra. In this article, on the one hand, we proved some results concerning the continuous projection from X to its center. On the other hand, we investigate the commutativity of X under specific conditions. Finally, we included some examples and applications to prove that various restrictions in the hypotheses of our theorems are necessary.

... 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]. ...

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.

... 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]. ...

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.

... x ∈ E}, then F = {x * : x ∈ F }, and hence Z is closed under involution. That is, (Z, µ F ) is a Banach algebra with an involution, which is the restriction of the involution on A. It should be noted that the restriction of f to (Z, µ F ) is continuous because of a comment between Definition 10 and Theorem 11, and because of Theorem 11 in Section 37 of [7]. □ Remark 2.5. ...

Some techniques, which were already used to derive automatic continuity results, are chosen and modified, and extended results, as well as generalized results, are obtained. A technique of using the open mapping theorem and a technique of using the Hahn Banach extension theorem are explained. Results in connection with measurable cardinals are also obtained. Results for multiplicative linear functionals, positive linear functionals, and uniqueness of topology are obtained. For example, sequential continuity of real multiplicative linear functionals on sequentially complete LMC algebras is obtained when Michael's open problem is concerned only with the boundedness of multiplicative linear functionals. The continuity of positive linear function-als on F-algebras with identity elements and involution is derived when these functionals are continuous on the set of all involution-symmetric elements. Possibilities of extending the concept of positive linear functionals are considered to derive results for the continuity of such functionals on topological groups and topological vector spaces with additional structures. The technique for the Carpenter's uniqueness theorem is modified to derive the boundedness of some homomorphisms. The entire article is oriented toward Michael's problem.

... The reader is referred to [3,6] and [21] for details on Banach algebras and to [4,25] for more information on dynamical systems. ...

Let σ be a linear *-endomorphism on a C *-algebra A so that σ(A) acts on a Hilbert space H which including K(H) and let {αt} t∈R be a σ-C *-dynamical system on A with the generator δ. In this paper, we demonstrate some conditions under which {αt} t∈R is implemented by a C0-groups of unitaries on H. In particular, we prove that for a rank-one projection p ∈ A, which is invariant under αt, there is a C0-group {ut} t∈R of unitaries in B(H) such that αt(a) = utσ(a)u * t. Furthermore, introducing the concepts of σ-inner endomorphism and σ-bijective map, we prove that each σ-bijective linear endomorphism on A is a σ-inner endomorphism, where σ ia idempotent. Finally, as an application, we characterize each so-called σ-C *-dynamical system on the concrete C *-algebra A := B(H) × B(H), where H is a separable Hilbert space and σ is the linear *-endomorphism σ(S, T) = (0, T) on A.

... In particular, the spectrum σ(R; M op (d p )) of an operator R ∈ M op (d p ) coincides with its spectrum σ(R; L (d p )) as an element of L (d p ). For the definition of the spectrum of an element in a unital Banach algebra we refer to [6], [15], for example. ...

We investigate convolution operators in the sequence spaces dp, for 1≤p<∞. These spaces, for p>1, arise as dual spaces of the Cesàro sequence spaces cesp thoroughly investigated by G. Bennett. A detailed study is also made of the algebra of those sequences which convolve dp into dp. It turns out that such multiplier spaces exhibit features which are very different to the classical multiplier spaces of ℓp.

... Following the pattern of Section 24 in [4], one can infer that the representation theory for quasi-Banach algebras goes exactly the same as for Banach algebras, since the main ingredients are the algebraic properties, the closedness criteria and the continuity of the representations. We then restate the same Lemmata 8.7, 8.8 and 8.9 in [24] in our setting as follows. ...

We generalize the results for Banach algebras of pseudodifferential operators obtained by Gr\"ochenig and Rzeszotnik in [24] to quasi-algebras of Fourier integral operators. Namely, we introduce quasi-Banach algebras of symbol classes for Fourier integral operators that we call generalized metaplectic operators, including pseudodifferential operators. This terminology stems from the pioneering work on Wiener algebras of Fourier integral operators [11], which we generalize to our framework. This theory finds applications in the study of evolution equations such as the Cauchy problem for the Schr\"odinger equation with bounded perturbations, cf. [7].

... One of the earliest results in this area is the following, which was obtained independently by Gleason [2], Kahane and Żelazko [5], and now known as the Gleason-Kahane-Żelazko theorem (see also [1] ...

Let A and B be two unital Banach algebras and 𝔘 = A × B. We prove that the bilinear mapping φ: 𝔘 → ℂ is a bi-Jordan homomorphism if and only if φ is unital, invertibility preserving and jointly continuous. Additionally, if A is commutative, then φ is a bi-homomorphism.

We establish some Lie–Trotter formulae for unital complex Jordan–Banach algebras, showing that for any elements a, b, c in a unital complex Jordan–Banach algebra \(\mathfrak {A}\) the identities
hold. These formulae are actually deduced from a more general result involving holomorphic functions with values in \(\mathfrak {A}\). These formulae are employed in the study of spectral-valued (non-necessarily linear) functionals \(f:\mathfrak {A}\rightarrow \mathbb {C}\) satisfying \(f(U_x (y))=U_{f(x)}f(y),\) for all \(x,y\in \mathfrak {A}\). We prove that for any such a functional f, there exists a unique continuous (Jordan-) multiplicative linear functional \(\psi :\mathfrak {A}\rightarrow \mathbb {C}\) such that \( f(x)=\psi (x),\) for every x in the connected component of the set of all invertible elements of \(\mathfrak {A}\) containing the unit element. If we additionally assume that \(\mathfrak {A}\) is a JB\(^*\)-algebra and f is continuous, then f is a linear multiplicative functional on \(\mathfrak {A}\). The new conclusions are appropriate Jordan versions of results by Maouche, Brits, Mabrouk, Schulz, and Touré.

Let A be a Banach algebra and let x∈A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in A$$\end{document} have the property that its spectrum does not separate 0 from infinity. It is well known that x has a logarithm, i.e., there exists y∈A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\in A$$\end{document} with x=ey\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=e^y$$\end{document}. We will use this statement to identify measures defined on a locally compact group to have logarithms. Also, we will show that the converse of the above statement is in general not true. Our results will be related to infinitely divisible probability measures.

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.

Let $(A, \|\cdot\|)$ be any normed algebra (not necessarily complete nor unital). Let $a \in A$ and let $V_A(a)$ denote the spatial numerical range of $a$ in $(A, \|\cdot\|)$. Let $A_e = A + {\mathbb C} 1$ be the unitization of $A$. If $A$ is faithful, then we get two norms on $A_e$; namely, the operator norm $\|\cdot\|_{op}$ and the $\ell^1$-norm $\|\cdot\|_1$. Let $A^{op} = (A, \|\cdot\|_{op})$, $A_e^{op} = (A_e, \|\cdot\|_{op})$, and $A_e^1 = (A_e, \|\cdot\|_1)$. We can calculate the spatial numerical range of $a$ in all these three normed algebras. Because the spatial numerical range highly depend on the identity as well as on the completeness and the regularity of the norm, they are different. In this paper, we study the relations among them. Most of the results proved in \cite{BoDu:71, BoDu:73} will become corollaries of our results. We shall also show that the completeness and regularity of the norm is not required in \cite[Theorem 2.3]{GaHu:89}.

Let \({{\mathcal {H}}}_1\) and \({{\mathcal {H}}}_2\) be two separable Hilbert spaces, \(K_1\in {{\mathcal {B}}}({{\mathcal {H}}}_1)\) and \(K_2\in {{\mathcal {B}}}({{\mathcal {H}}}_2)\). Based on some previous results about tensor product of frames, in this paper we generalize them to tensor product of K-frames. We provide equivalent conditions for that the tensor product of two \(K_1\)-frame and \(K_2\)-frame is a \(K_1\otimes K_2\)-frame. Moreover, we investigate whenever the tensor product of two Bessel sequences F and G in \({{\mathcal {H}}}_1\) and \({{\mathcal {H}}}_2\), respectively, is a \(K_1\otimes K_2\)-dual frame for \(F\otimes G\).

Let E be a Banach space. For a topological space X, let \({{\cal C}_b}(X,E)\) be the space of all bounded continuous E-valued functions on X, and let \({{\cal C}_K}(X,E)\) be the subspace of \({{\cal C}_b}(X,E)\) consisting of all functions having a pre-compact image in E. We show that \({{\cal C}_K}(X,E)\) is isometrically isomorphic to the injective tensor product \({{\cal C}_b}(X){{\hat \otimes}_\varepsilon}E\), and that \({{\cal C}_b}(X,E) = {{\cal C}_b}(X){{\hat \otimes}_\varepsilon}E\) if and only if E is finite dimensional. Next, we consider the space Lip(X, E) of E-valued Lipschitz operators on a metric space (X, d) and its subspace LipK(X, E) of Lipschitz compact operators. Utilizing the results on \({{\cal C}_b}(X,E)\), we prove that LipK(X, E) is isometrically isomorphic to a tensor product \({\rm{Lip}}(X){{\hat \otimes}_\alpha}E\), and that \({\rm{Lip}}(X,E) = {\rm{Lip}}(X){{\hat \otimes}_\alpha}E\) if and only if E is finite dimensional. Finally, we consider the space D1(X, E) of continuously differentiable functions on a perfect compact plane set X and show that, under certain conditions, D1(X, E) is isometrically isomorphic to a tensor product \({D^1}(X){\hat \otimes _\beta}E\).

Let [Formula: see text] be a direct set, [Formula: see text] be a family of Banach algebras with bounded approximate identity (or unital) and [Formula: see text] be a set. We consider the Banach algebra [Formula: see text]. We show that this algebra has a bounded approximate identity (or is unital) if and only if [Formula: see text] is finite. We also characterize the left multipliers of these algebras and investigate their amenability of them. Moreover, we characterize the character spaces (Gelfand spaces) of these algebras in a special case.

We set up a general theory leading to a quantum Wasserstein distance of order 1 between channels in an operator algebraic framework. This gives a metric on the set of channels from one composite system to another, which is deeply connected to reductions of the channels. The additivity and stability properties of this metric are studied.

In this paper we show that, for an ideal J of a unital complex Banach algebra A, we have (i) under certain conditions the ? -quasi centralizer, the quasi centralizer, and the centralizer of J are all identical, and so they are subsets of the ? -quasi centralizer of J. (ii) If J is closed and a is a quasi-centralizer element of J, then DaJ, a restriction of the inner derivation of a to J is topologically nilpotent. (iii) For each complex number ? and each x in J we have, (? – a) x = 0 if and only if x (? – a) = 0.
في هذا البحث تم إثبات أنه إذا كان J مثالياً في جبربناخ الوحدوي العقدي فإن: (1) في حال تحقق شروط معينة تكون مجموعات شبه الممركز من نوع، وشبه الممركز، والممركز جميعها متساوية وبذلك تصبح هذه المجموعات جزئية من مجموعة شبه الممركز من نوع وذلك للمثالي J. (2) إذا كان J مغلق و a عنصر ممركزي لــ J فإن الاشتقاق الداخلي لــ a

We prove that every n-Jordan derivation in the sense of Herstein (Bull Am Math Soc 67:517–531, 1961, p. 528) on n!-torsion free unital commutative rings is a derivation. Furthermore, we prove that every continuous n-Jordan derivation on semiprime normed algebras is a derivation. The results of this paper improve and generalize the main results of Bridges and Bergen (Proc Am Math Soc 90:25–29, 1984), but under weaker assumptions. Some applications and examples of our results are also provided.

In this paper, we study additive properties of the generalized Drazin inverse in a Banach algebra. We first show that a + b ? Ad under the condition that a, b ? Ad, aba? = ?a?bab?a?, and then give some explicit expressions for the generalized Drazin inverse of the sum a+b under some weaker conditions than those used in the previous papers. Some known results are extended.

In this paper, we look at the question of when various ideals in the Fourier algebra $A(G)$ or its closures $A_M(G)$ and $A_{cb}(G)$ in, respectively, its multiplier and $cb$-multiplier algebra are Arens regular. We show that in each case, if a non-zero ideal is Arens regular, then the underlying group $G$ must be discrete. In addition, we show that if an ideal $I$ in $A(G)$ has a bounded approximate identity, then it is Arens regular if and only if it is finite-dimensional.

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

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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 }.

We consider the class of all linear functionals $L$ on a unital commutative real algebra $A$ that can be represented as an integral w.r.t. to a Radon measure with compact support in the character space of $A$. Exploiting a recent generalization of the classical Nussbaum theorem, we establish a new characterization of this class of moment functionals solely in terms of a growth condition intrinsic to the given linear functional. To the best of our knowledge, our result is the first to exactly identify the compact support of the representing Radon measure. We also describe the compact support in terms of the largest Archimedean quadratic module on which $L$ is nonnegative and in terms of the smallest submultiplicative seminorm w.r.t. which $L$ is continuous. Moreover, we derive a formula for computing the measure of each singleton in the compact support, which in turn gives a necessary and sufficient condition for the support to be a finite set. Finally, some aspects related to our growth condition for topological algebras are also investigated.

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.

This modern introduction to operator theory on spaces with indefinite inner product discusses the geometry and the spectral theory of linear operators on these spaces, the deep interplay with complex analysis, and applications to interpolation problems. The text covers the key results from the last four decades in a readable way with full proofs provided throughout. Step by step, the reader is guided through the intricate geometry and topology of spaces with indefinite inner product, before progressing to a presentation of the geometry and spectral theory on these spaces. The author carefully highlights where difficulties arise and what tools are available to overcome them. With generous background material included in the appendices, this text is an excellent resource for graduate students as well as researchers in operator theory, functional analysis, and related areas.

In this paper, we investigate the commutativity of a Banach algebra [Formula: see text] provided with a continuous derivation satisfying algebraic identities involving nonvoid open subsets of [Formula: see text] Furthermore, we provide examples to show that various restrictions in the hypothesis of our theorems are not superfluous.

Let \({\mathcal {A}}\) and \({\mathcal {B}}\) be Banach algebras, and let \(\theta\) be a multiplicative linear functional on \(\mathcal {B}\). Then the \(\theta -\)Lau product \(\mathcal {A}\times _\theta \mathcal {B}\) is also a Banach algebra. We shall study the relation between the various spectra of elements of \(\mathcal {A}\times _\theta \mathcal {B}\) in relation with the spectra of elements of \(\mathcal {A}\) and \(\mathcal {B}\).

The stability problem of the functional equation
was evoked by Ulam in 1940.
In mathematical modeling of physical problems,
the deviations in measurements will result with
errors and deviations can be dealt with the stability
of equations.
Hence, the stability of equations is essential in
mathematical models.
In this paper, we prove the hyperstability of a cubic
functional equation on a restricted domain.
The method of the proof of the main theorem is
motivated by an idea used by Brzdek in 2013
And further by Piszczek, it is based on a fixed point
theorem for functional spaces obtained by Brzdek
[1-2].

The 4th International Conference on Research in Applied Mathematics and
Computer Science (ICRAMCS 2022) is aimed to bring researchers and
professionals to discuss recent developments in both applied mathematics
and computer science and to create a professional knowledge exchange
platform between mathematicians, computer science and other disciplines.
This conference is the result of international cooperation bringing together
African and European universities. It is a privileged place for meetings and
exchanges between young researchers and high-level African and
international decision makers in the fields of mathematics and applied
computing.
This conference has several major objectives, in particular:
▪ To bring together doctoral students and research professors in the
fields of applied sciences and new technologies.
▪ To consolidate the scientific cooperation between the university and
the socio-economic environment in the field of applied sciences.
▪ To allow young researchers to present and discuss their research
work before a panel of specialists and university professors.
▪ To contribute to the development of a database, which can help
decision makers to opt for a better management strategy.
The abstracts of these conference proceedings were presented at the 4th
International Conference on Research in Applied Mathematics and
Computer Science (ICRAMCS 2022). These conference proceedings include
abstracts that underwent a rigorous review by two or more reviewers.
These papers represent current important work in the field of Mathematics
& Computer Science and are elaborations of the ICRAMCS conference
reports.

In this paper, we prove that each n-Jordan homomorphism φ from Banach algebra A into a semisimple commutative Banach algebra B is automatically continuous. Some useful results about characterization of n-Jordan homomorphisms and interesting examples of them on Banach algebras are given as well.

We study the interpolation properties of weakly compact bilinear operators by the real method and also by the complex method. We also study the factorization property of weakly compact bilinear operators through reflexive Banach spaces.

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