Commutative Semisimple Banach Algebra Research Articles

BSE-Properties of Vector-Valued Group Algebras

Let \(\mathcal {A}\) be a semisimple commutative Banach algebra with identity of norm one and G be a locally compact abelian group. In this paper, we study the BSE-Property for \(L^{1}(G,\mathcal {A})\) and show that \(L^{1}(G,\mathcal {A})\) is a BSE algebra if and only if \(\mathcal {A}\) is so.

Spectral Extension Property of Perturbed Triple Product on Semisimple Commutative Banach Algebras

Under a perturbed triple product defined on three semisimple commutative Banach algebras with the influence of two homomorphisms defined on two of them and that carry certain characteristics, we proved that the spectral extension property is stable.

On invariant and natural projections

We solve three open problems raised in Lau and Ülger, Trans. Amer. Math. Soc. 366 (2014), no. 8, 4151–4171 regarding invariant and natural projections. More precisely, we show that for a commutative C ∗ $C^*$ -algebra A ${\mathcal {A}}$ , every natural projection on A ∗ ${\mathcal {A}}^*$ is invariant under the canonical action of A ${\mathcal {A}}$ if and only if A ${\mathcal {A}}$ is scattered. We exhibit further classes of Banach algebras, comprising important examples from abstract harmonic analysis, for which the two types of projections differ, by establishing links of the latter property to almost periodicity, Arens regularity, and amenability. Moreover, for any commutative semisimple Banach algebra A ${\mathcal {A}}$ with a bounded approximate identity (BAI), we characterize the w ∗ $w^*$ -continuous natural projections on A ∗ ${\mathcal {A}}^*$ as the adjoints of idempotent multipliers on A ${\mathcal {A}}$ . We also prove that C $\mathbb {C}$ is the only (non-zero) commutative semisimple Banach algebra A ${\mathcal {A}}$ with a BAI such that every w ∗ $w^*$ -continuous projection on A ∗ ${\mathcal {A}}^*$ is natural.

Joint Continuity of Bi-multiplicative Functionals

For Banach algebras $mathcal{A}$ and $mathcal{B}$, we show that if $mathfrak{A}=mathcal{A}times mathcal{B}$ is unital, then each bi-multiplicative mapping from $mathfrak{A}$ into a semisimple commutative Banach algebra $mathcal{D}$ is jointly continuous. This conclusion generalizes a famous result due to$check{text{S}}$ilov, concerning the automatic continuity of homomorphisms between Banach algebras. We also prove that every $n$-bi-multiplicative functionals on $mathfrak{A}$ is continuous if and only if it is continuous for the case $n=2$.

Certain properties of Jordan homomorphisms, n-Jordan homomorphisms and n-homomorphisms on rings and Banach algebras

We investigate under what conditions n-Jordan homomorphisms between rings are n-homomorphism, or homomorphism; and under what conditions, n-Jordan homomorphisms are continuous. One of the main goals in this work is to show that every n-Jordan homomorphism f : A → B, from a unital ring A into a ring B with characteristic greater than n, is a multiple of a Jordan homomorphism and hence, it is an n-homomorphism if every Jordan homomorphism from A into B is a homomorphism. In particular, if B is an integral domain whose characteristic is greater than n, then every n-Jordan homomorphism f : A → B is an n-homomorphism. Along with some other results, we show that if A and B are unital rings such that the characteristic of B is greater than n, then every unital n-Jordan homomorphism f : A → B is a Jordan homomorphism and hence, it is an m-Jordan homomorphism for any positive integer m ≥ 2. We also investigate the automatic continuity of n-Jordan homomorphisms from a unital Banach algebra either into a semisimple commutative Banach algebra, onto a semisimple Banach algebra, or into a strongly semisimple Banach algebra whenever the n-Jordan homomorphism has dense range.

About the joint spectrum of vector-valued functions

We generalize the results of Abtahi and Farhangi about the joint spectrum and A-valued spectrum of a vector-valued map from the class of unital commutative semisimple Banach algebras to the case of unital topological algebras.

Automatic continuity of almost 3-homomorphisms and almost 3-Jordan homomorphisms

For a Banach algebra A and a semisimple commutative Banach algebra B, we prove that every 3-homomorphism $$\varphi :A\longrightarrow B$$ is automatically continuous, and we generalize this result for almost 3-homomorphism. We also investigate the automatic continuity of almost 3-Jordan homomorphisms and show that if A is a commutative Banach algebra, then every almost 3-Jordan homomorphism $$\varphi :A\longrightarrow \mathbb {C}$$ is automatically continuous.

Characterization of n-Jordan Homomorphisms and Automatic Continuity of 3-Jordan Homomorphisms on Banach Algebras

We show that every Banach algebra $${\mathcal {A}}$$ is 3-Jordan functionally continuous. More especially, every 3-Jordan homomorphism from a Banach algebra $${\mathcal {A}}$$ into a commutative semisimple Banach algebra $${\mathcal {B}}$$ is automatically continuous. We also prove the same result for n-Jordan homomorphism ($$n\geqslant 4$$) with the additional hypothesis that the Banach algebra $${\mathcal {A}}$$ is unital. A characterization of n-Jordan homomorphism for the special case $$n\in \lbrace 2,3,4\rbrace $$ is also given.

Characterization of mixed n-Jordan and pseudo n-Jordan homomorphisms

In this article, a new notion of n-Jordan homomorphism namely the mixed n-Jordan homomorphism is introduced. It is proved that how a mixed (n + 1)-Jordan homomorphism can be a mixed n-Jordan homomorphism and vice versa. By means of some examples, it is shown that the mixed n-Jordan homomorphisms are different from the n-Jordan homomorphisms and the pseudo n-Jordan homomorphisms. As a consequence, it shown that every mixed Jordan homomorphism from Banach algebra A into commutative semisimple Banach algebra B is automatically continuous. Under some mild conditions, every unital pseudo 3-Jordan homomorphism is a homomorphism.

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Some ergodic properties of multipliers on commutative Banach algebras

A commutative semisimple regular Banach algebra $A$ with the Gelfand space $ \Sigma _{A}$ is called a Ditkin algebra if each point of $\Sigma _{A}\cup \left\{ \infty \right\} $ is a set of synthesis for $A$. Generalizing the Choquet-Deny theorem, it is shown that if $T$ is a multiplier of a Ditkin algebra $A,$ then $\left\{ \varphi \in A^{\ast }:T^{\ast }\varphi =\varphi \right\} $ is finite dimensional if and only if \textnormal{card}$\mathcal{F}_{T}$ is finite, where $\mathcal{F}_{T}=\left\{ \gamma \in \Sigma _{A}:\widehat{T}\left( \gamma \right) =1\right\} $ and $ \widehat{T}$ is the Helgason-Wang representation of $T.$

BSE-Property for Some Certain Segal and Banach Algebras

For a commutative semi-simple Banach algebra A which is an ideal in its second dual, we give a necessary and sufficient condition for an essential abstract Segal algebra in A to be a BSE-algebra. We show that a large class of abstract Segal algebras in the Fourier algebra A(G) of a locally compact group G are BSE-algebra if and only if they have bounded weak approximate identities. In addition, in the case that G is discrete, we show that $$A_{\mathrm{cb}}(G)$$ is a BSE-algebra if and only if G is weakly amenable. We study the BSE-property of some certain Segal algebras implemented by local functions that were recently introduced by J. Inoue and S.-E. Takahasi. Finally, we give a similar construction for the group algebra implemented by a measurable and sub-multiplicative function.

A characterization of Jordan and 5-Jordan homomorphisms between Banach algebras

We prove that each surjective Jordan homomorphism from a Banach algebra [Formula: see text] onto a semiprime commutative Banach algebra [Formula: see text] is a homomorphism, and each 5-Jordan homomorphism from a unital Banach algebra [Formula: see text] into a semisimple commutative Banach algebra [Formula: see text] is a 5-homomorphism.

Mean Ergodic Theorems for Multipliers on Banach Algebras

Let A be a complex commutative semisimple Banach algebra. In this paper, we study some ergodic properties of Cesaro bounded multipliers on A. The results are linked to the sets of synthesis and the main applications are concerned with Fourier and Fourier-Stieltjes algebras on locally compact groups. We study also the structure of ideals associated with multipliers of A and A-invariant projections of the dual space of A. Some related problems are also discussed.

A Characterization of Bi-Jordan Homomorphisms on Banach Algebras

For Banach algebras A and B, we show that if U=A×B is unital and commutative, each bi-Jordan homomorphism from U into a semisimple commutative Banach algebra D is a bihomomorphism.

A Characterization of Multipliers of a Lau Algebra Constructed by Semisimple Commutative Banach Algebras

A necessary and sufficient condition for a Lau type binary operation defined by two mappings to be an algebra-operation is given in terms of multipliers. Also a characterization of multipliers of a Lau algebra constructed by semisimple commutative Banach algebras is given in terms of multipliers of original Banach algebras.

Some spectral properties of multipliers on commutative Banach algebras

In this paper, some results concerning spectral properties of multipliers on commutative semisimple Banach algebras are obtained. The study concentrates on multipliers on uniformly regular Banach algebras whose Helgason–Wang representation vanish at infinity. A spectral mapping theorem for convolution operators induced by representations of locally compact abelian groups is also given.

Multipliers of perturbed Cartesian products with an application to BSE-property

Given semisimple commutative Banach algebras \({\mathcal{A}}\) and \({\mathcal{B}}\) and a norm decreasing homomorphism \({\mathcal{T} : \mathcal{B} \rightarrow \mathcal{B}}\), we characterize the multipliers of the perturbed product Banach algebra \({\mathcal{A}\times_T \mathcal{B}}\). As an application it is shown that \({\mathcal{A}\times_T \mathcal{B}}\) has the Bochner–Schoenberg–Eberlein property if and only if both \({\mathcal{A}}\) and \({\mathcal{B}}\) have this property.

On a theorem of Reiter and spectral synthesis

Motivated by and extending a theorem of Reiter on sets of synthesis in \(\mathbb {R}^N\), we establish a general result for Fourier algebras of locally compact groups, even in the wider context of regular, semisimple and Tauberian commutative Banach algebras, which contains Reiter’s theorem as a special case and explains why it holds. In addition, we give a number of examples and several further results on weak spectral sets.

On the spectra of multipliers on commutative Banach algebras

In this paper, we present some results concerning spectral properties of multipliers on commutative semi-simple Banach algebras. The study concentrates on multipliers whose (local) spectra are in a sense natural. Particular emphasis is given to the determination of the (local) spectra of multipliers in connection with Beurling algebras, Dunford property (C) and Hille-Yosida space.