Book

# Complete Normed Algebras

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## 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 ). ...
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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. ...
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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 . ...
Article
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. ...
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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. ...
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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). ...
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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. ...
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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]): ...
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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]. ...
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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]. ...
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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 . ...
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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]. ...
Preprint
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]. ...
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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.
... 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. ...
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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. ...
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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. ...
Article
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. ...
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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] ...
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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.
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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.
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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.
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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}.
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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.
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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.
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