Commutative Semisimple Banach Algebra Research Articles

Strong Ditkin sets in the spectrum of a commutative Banach algebra and the Fourier algebra

We extend the notion of a strong Ditkin set in the dual group for the $${L^1}$$ -algebra of a locally compact abelian group as well as a large number of results for such sets to the setting of a general regular and semisimple commutative Banach algebra and its spectrum. In particular, we study various stability and inheritance properties. Moreover, we present some applications to Fourier algebras of locally compact groups and an example of a compact, infinite double coset hypergroup for which every closed subset is a strong Ditkin set for its Fourier algebra.

On Uniform Topological Algebras

Abstract The uniform norm on a uniform normed Q-algebra is the only uniform Q-algebra norm on it. The uniform norm on a regular uniform normed Q-algebra with unit is the only uniform norm on it. Let A be a uniform topological algebra whose spectrum M(A) is equicontinuous, then A is a uniform normed algebra. Let A be a regular semisimple commutative Banach algebra, then every algebra norm on A is a Q-algebra norm on A.

Weak spectral synthesis in commutative Banach algebras. III

Let A be a regular and semisimple commutative Banach algebra with structure space Δ(A). Continuing the investigations of [7] and [9], we establish various results on weak spectral sets in Δ(A). To each closed subset of Δ(A) we associate a descending sequence of subsets of Δ(A) which proves to be a powerful tool in the study of weak spectral synthesis. Applications concern injection type properties, unions of weak spectral sets and projective tensor products. A number of interesting examples is discussed: algebras of m-times continuously differentiable functions and of Lipschitz functions, and L1(RN).

Peripherally multiplicative operators on unital commutative Banach algebras

Let $T: A \longrightarrow B$ be a surjective operator between two unital semisimple commutative Banach algebras $A$ and $B$ with $T1=1$. We show that if $T$ satisfies the peripheral multiplicativity condition $\sigma_\pi(Tf.Tg) = \sigma_\pi (f.g)$ for all $f$ and $g$ in $A$, where $\sigma_\pi(f)$ shows the peripheral spectrum of $f$, then $T$ is a composition operator in modulus on the $\check{S}$ilov boundary of $A$ in the sense that $|f(x)|=|Tf(\tau(x))|,$ for each $f\in A$ and $x\in \partial(A)$ where $\tau: \partial (A) \longrightarrow \partial (B)$ is a homeomorphism between $\check{S}$ilov boundaries of $A$ and $B$.

THE MULTIPLIER ALGEBRA AND BSE PROPERTY OF THE DIRECT SUM OF BANACH ALGEBRAS

AbstractThe notion of BSE algebras was introduced and first studied by Takahasi and Hatori and later studied by Kaniuth and Ülger. This notion depends strongly on the multiplier algebra $M( \mathcal{A} )$ of a commutative Banach algebra $ \mathcal{A} $. In this paper we first present a characterisation of the multiplier algebra of the direct sum of two commutative semisimple Banach algebras. Then as an application we show that $ \mathcal{A} \oplus \mathcal{B} $ is a BSE algebra if and only if $ \mathcal{A} $ and $ \mathcal{B} $ are BSE. We also prove that if the algebra $ \mathcal{A} \hspace{0.167em} {\mathop{\times }\nolimits}_{\theta } \hspace{0.167em} \mathcal{B} $ with $\theta $-Lau product is a BSE algebra and $ \mathcal{B} $ is unital then $ \mathcal{B} $ is a BSE algebra. We present some examples which show that the BSE property of $ \mathcal{A} \hspace{0.167em} {\mathop{\times }\nolimits}_{\theta } \hspace{0.167em} \mathcal{B} $ does not imply the BSE property of $ \mathcal{A} $, even in the case where $ \mathcal{B} $ is unital.

Solvability of the H ∞ Algebraic Riccati Equation in Banach Algebras

Let R be a commutative complex unital semisimple Banach algebra with the involution {Mathematical expression} . Sufficient conditions are given for the existence of a stabilizing solution to the H∞ Riccati equation when the matricial data has entries from R. Applications to spatially distributed systems are discussed. © 2011 Springer Basel AG.

Superstability of approximate d'Alembert harmonic functions

Abstract In this article, we study the superstability problem for the complex-valued functional equation f ( x + y + z ) + f ( x + y - z ) + f ( y + z - x ) + f ( z + x - y ) = 4 f ( x ) f ( y ) f ( z ) on an abelian group and on a commutative semisimple Banach algebra. As a result, we obtain application to harmonic functions satisfying the equation approximately.

Algebraic properties of isometries between groups of invertible elements in Banach algebras

We prove that an isometry T between open subgroups of the invertible groups of unital Banach algebras A and B is extended to a real-linear isometry up to translation between these Banach algebras. While a unital isometry between unital semisimple commutative Banach algebras need not be multiplicative, we prove in this paper that if A is commutative and A or B are semisimple, then ( T ( e A ) ) − 1 T is extended to an isometric real algebra isomorphism from A onto B. In particular, A − 1 is isometric as a metric space to B − 1 if and only if they are isometrically isomorphic to each other as metrizable groups if and only if A is isometrically isomorphic to B as a real Banach algebra; it is compared by the example of Żelazko concerning on non-isomorphic Banach algebras with the homeomorphically isomorphic invertible groups. Isometries between open subgroups of the invertible groups of unital closed standard operator algebras on Banach spaces are investigated and their general forms are given.

Generalizations of spectrally multiplicative surjections between uniform algebras

Let $$ A $$ and ℬ be unital semisimple commutative Banach algebras. It is shown that if surjections S,T: $$ A $$ → ℬ with S(1)=T(1)= 1 and α ∈ ℂ \ {0} satisfy r(S(a)T(b) − α)= r(ab− α) for all a,b ∈ $$ A $$ , then S=T and S is a real algebra isomorphism, where r(a) is the spectral radius of a. Let I be a nonempty set, A and B be uniform algebras. Let ρ, τ: I → A and S,T: I → B be maps satisfying σ π (S(p)T(q)) ⊂ σ π (ρ(p) τ(q)) for all p,q ∈ I, where σ π (f) is the peripheral spectrum of f. Suppose that the ranges ρ(I), τ(I) ⊂ A and S(I),T(I) ⊂ B are closed under multiplication in a sense, and contain peaking functions “enough”. There exists a homeomorphism ϕ: Ch(B)→Ch(A) such that S(p)(y)= ρ(p)(ϕ(y)) and T(p)(y)= τ(p)(ϕ(y)) for every p ∈ I and y ∈ Ch(B), where Ch(A) is the Choquet boundary of A.

Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras

Abstract Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces M A and M B, respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: M B → M A and a closed and open subset K of M B such that $$ \widehat{T\left( a \right)}\left( y \right) = \left\{ \begin{gathered} \widehat{T\left( e \right)}\left( y \right)\hat a\left( {\phi \left( y \right)} \right) y \in K \hfill \\ \widehat{T\left( e \right)}\left( y \right)\overline {\hat a\left( {\phi \left( y \right)} \right)} y \in M_\mathcal{B} \backslash K \hfill \\ \end{gathered} \right. $$ for all a ∈ A, where e is unit element of A. If, in addition, $$ \widehat{T\left( e \right)} = 1 $$ and $$ \widehat{T\left( {ie} \right)} = i $$ on M B, then T is an algebra isomorphism.

Quasicompact endomorphisms of commutative semiprime Banach algebras

This paper is a continuation of our study of compact, power compact, Riesz, and quasicompact endomorphisms of commutative Banach algebras. Previously it has been shown that if $B$ is a unital commutative semisimple Banach algebra with connected character space, and $T$ is a unital endomorphism of $B$, then $T$ is quasicompact if and only if the operators $T^n$ converge in operator norm to a rank-one unital endomorphism of $B$. In this note the discussion is extended in two ways: we discuss endomorphisms of commutative Banach algebras which are semiprime and not necessarily semisimple; we also discuss commutative Banach algebras with character spaces which are not necessarily connected. In previous papers we have given examples of commutative semisimple Banach algebras $B$ and endomorphisms $T$ of $B$ showing that $T$ may be quasicompact but not Riesz, $T$ may be Riesz but not power compact, and $T$ may be power compact but not compact. In this note we give examples of commutative, semiprime Banach algebras, some radical and some semisimple, for which every quasicompact endomorphism is actually compact.

Sufficient conditions for the projective freeness of Banach algebras

Let R be a unital semi-simple commutative complex Banach algebra, and let M ( R ) denote its maximal ideal space, equipped with the Gelfand topology. Sufficient topological conditions are given on M ( R ) for R to be a projective free ring, that is, a ring in which every finitely generated projective R-module is free. Several examples are included, notably the Hardy algebra H ∞ ( X ) of bounded holomorphic functions on a Riemann surface of finite type, and also some algebras of stable transfer functions arising in control theory.

Approximately -Jordan Homomorphisms on Banach Algebras

Let , and let be two rings. An additive map is called -Jordan homomorphism if for all . In this paper, we establish the Hyers-Ulam-Rassias stability of -Jordan homomorphisms on Banach algebras. Also we show that (a) to each approximate 3-Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra there corresponds a unique 3-ring homomorphism near to , (b) to each approximate -Jordan homomorphism between two commutative Banach algebras there corresponds a unique -ring homomorphism near to for all .

Isometries between groups of invertible elements in Banach algebras

We show that if $T$ is an isometry (as metric spaces) from an open subgroup of the group of invertible elements in a unital semisimple commutative Banach algebra $A$ onto a open subgroup of the group of invertible elements in a unital Banach algebra $B$,

On surjective ring homomorphisms between semi-simple commutative Banach algebras

On surjective ring homomorphisms between semi-simple commutative Banach algebras

On left Jordan derivations of rings and Banach algebras

It is well known that there are no nonzero linear derivations on complex commutative semisimple Banach algebras. In this paper we prove the following extension of this result. Let A be a complex semisimple Banach algebra and let D : A → A be a linear mapping satisfying the relation D(x 2) = 2xD(x) for all x ∈ R. In this case D = 0. Throughout, R will represent an associative ring with center Z(R). A ring R is n-torsion free, where n > 1 is an integer, if nx = 0, x ∈ R implies x = 0. As usual the commutator xy − yx will be denoted by [x, y]. We shall use the commutator identities [xy, z] = [x, z] y + x [y, z] and [x, yz] = [x, y] z + y [x, z] for all x, y, z ∈ R. Recall that a ring R is prime if for a, b ∈ R, aRb = (0) implies that either a = 0 or b = 0, and is semiprime in case aRa = (0) implies that a = 0. An additive mapping D is called a derivation if D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R, and is called a Jordan derivation in case D(x 2) = D(x)x + xD(x) is fulfilled for all x ∈ R. Obviously, any derivation is a Jordan derivation. The converse is in general not true. Herstein ([8]) has proved that any Jordan derivation on a 2-torsion free prime ring is a derivation (see also [1]). Cusack ([5]) has generalized Herstein’s result to 2-torsion free semiprime rings (see [2] for an alternative proof). An additive mapping D : R → R is called a left derivation if D(xy) = yD(x) + xD(y) holds for all pairs x, y ∈ R and is called a left Jordan derivation (or Jordan left derivation) in case D(x 2) = 2xD(x) is fulfilled for all x ∈ R. In this paper by a Banach algebra we mean a Banach algebra over the complex field.

BANACH ALGEBRAS WITH LARGE GROUPS OF UNITARY ELEMENTS

We study unitary Banach algebras, as defined by M. L. Hansen and R. V. Kadison in 1996, as well as some related concepts like maximal or uniquely maximal Banach algebras. We show that a norm-unital Banach algebra is uniquely maximal if and only if it is unitary and has minimality of the equivalent norm. We prove that every unitary semisimple commutative complex Banach algebra has a conjugate-linear involution mapping each unitary element to its inverse, and that, endowed with such an involution, becomes a hermitian *-algebra. The possibility of removing the requirement of commutativity in the above statement is also considered. The paper concludes by translating to real algebras some results previously known in the complex case. In particular, we show that every maximal semisimple finite-dimensional real Banach algebra is isometrically isomorphic to a real C*-algebra.

On branching index of algebraic extension spectrum points

The paper studies the branching A-index and the topological branching index of points of the spectrum of Arens-Hoffman algebraic extension of a semisimple commutative Banach algebras A.

Homomorphisms of commutative Banach algebras and extensions to multiplier algebras with applications to Fourier algebras

Let A and B be semisimple commutative Banach algebras with bounded approximate identities. We investigate the problem of extending a homomorphism ' : A ! B to a homomorphism of the multiplier algebras M(A) and M(B) of A and B, respectively. Various sufficient conditions in terms ofB (or B and ') are given that allow the construction of such extensions. We exhibit a number of classes of Banach algebras to which these criteria apply. In addition, we prove a polar decomposition for homomorphisms from A into A with closed range. Our results are applied to Fourier algebras of locally compact groups. Introduction. Let A and B be semisimple commutative Banach alge- bras and suppose that A has a bounded approximate identity (e�)�. We study homomorphisms ϕ from A to B from various aspects. Let I' be the largest ideal of B for which (ϕ(e�))� serves as an approximate identity, and let Z' denote the zero set of I' in the Gelfand spectrum � (B) of B. In Section 1 we find criteria for Z' to be open in � (B) and I' to be comple- mented by a certain ideal J' (Theorems 1.4 and 1.5). The results are applied to the related question of when a homomorphism ϕ : A → B extends to a homomorphism, φ : M(A) → M(B), between the multiplier algebras M(A) and M(B). This extension problem is the main objective of the paper. When I' is complemented as above, we give in Theorem 2.1 (which is a basic result of the paper) an explicit construction of an extension φ : M(A) → M(B). Moreover, if in addition B is a BSE-algebra (named after Bochner-Schoenberg-Eberlein), then all homomorphisms from M(A)

Order convolution and vector-valued multipliers

Let I = (0,1) with the usual topology. For x, y 2 I, we define xy = max(x, y). Then I becomes a locally compact commutative topological semigroup. The Banach space L 1 (I) of all Lebesgue integrable functions on I becomes a commutative semisimple Banach algebra with order convolution as multiplication. A bounded linear operator T on L 1 (I) is called a multiplier of L 1 (I) if T(f ⋆ g) = f ⋆ T g for all f, g 2 L 1 (I). The space of multipliers of L 1 (I) was determined by Johnson and Lahr. Let X be a Banach space and L 1 (I, X) be the Banach space of all X-valued Bochner integrable functions on I. We show that L 1 (I, X) becomes an L 1 (I)-Banach module. Suppose X and Y are Banach spaces. A bounded linear operator T from L 1 (I, X) to L 1 (I, Y ) is called a multiplier if T(f ⋆ g) = f ⋆ T g for all f 2 L 1 (I) and g 2 L 1 (I, X). In this paper, we characterize the multipliers from L 1 (I, X) to L 1 (I, Y ).