Homomorphisms on a class of commutative Banach algebras

Rocky Mountain Journal of Mathematics, 2016

Let X be a completely regular Hausdorff space. We denote by C(X, A) the algebra of all continuous functions on X with values in a complex commutative unital Banach algebra A. Let C b (X, A) be its subalgebra consisting of all bounded continuous functions and endowed with the uniform norm. In this paper, we give conditions equivalent to the density of a natural continuous image of X × M(A) in the maximal ideal space of C b (X, A).

2021

The unital commutative Banach algebra of all complex-valued continuous functions on a compact Hausdorff space K_{j} is denoted by C(K_{j}). A map $\ps i $ : C(K_{1}) \ri ght arrow C(K_{2}) is a unital homomorphism (a homomorphism which preserves identity) if and only if there is a

Bulletin of the Belgian Mathematical Society - Simon Stevin, 2005

Let X be a compact plane set. A(X) denotes the uniform algebra of all continuous complex-valued functions on X which are analytic on intX. For 0 < α ≤ 1, Lipschitz algebra of order α, Lip(X, α) is the algebra of all complex-valued functions f on X for which p α (f) = sup{ |f (x)−f (y)| |x−y| α : x, y ∈ X, x = y} < ∞. Let Lip A (X, α) = A(X) Lip(X, α), and Lip n (X, α) be the algebra of complex-valued functions on X whose derivatives up to order n are in Lip(X, α). Lip A (X, α) under the norm f = f X +p α (f), and Lip n (X, α) for a certain plane set X under the norm f = n k=0 f (k) X +pα(f (k)) k! are natural Banach function algebras, where f X = sup x∈X |f (x)|. In this note we study endomorphisms of algebras Lip A (X, α) and Lip n (X, α) and investigate necessary and sufficient conditions for which these endomorphisms to be compact. Finally, we determine the spectra of compact endomorphisms of these algebras.

International Journal of Nonlinear Analysis and Applications, 2016

For a Banach algebra $fA$, we introduce ~$c.c(fA)$, the set of all $phiin fA^*$ such that $theta_phi:fAto fA^*$ is a completely continuous operator, where $theta_phi$ is defined by $theta_phi(a)=acdotphi$~~ for all $ain fA$. We call $fA$, a completely continuous Banach algebra if $c.c(fA)=fA^*$. We give some examples of completely continuous Banach algebras and a sufficient condition for an open problem raised for the first time by J.E Gal`{e}, T.J. Ransford and M. C. White: Is there exist an infinite dimensional amenable Banach algebra whose underlying Banach space is reflexive? We prove that a reflexive, amenable, completely continuous Banach algebra with the approximation property is trivial.

2021

We study automatic injectivity of surjective algebra homomorphisms from $\mathscr{B}(X)$, the algebra of (bounded, linear) operators on $X$, to $\mathscr{B}(Y)$, where $X$ is one of the following \emph{long} sequence spaces: $c_0(\lambda)$, $\ell_{\infty}^c(\lambda)$, and $\ell_p(\lambda)$ ($1 \leqslant p &lt; \infty$) and $Y$ is arbitrary. \textit{En route} to the proof that these spaces do indeed enjoy such a property, we classify two-sided ideals of the algebra of operators of any of the aforementioned Banach spaces that are closed with respect to the `sequential strong operator topology&#39;.

We prove the Hyers-Ulam stability of partitioned functional equations in Banach modules over a unital C * -algebra. It is applied to show the stability of algebra homomorphisms in Banach algebras associated with partitioned functional equations in Banach algebras.

arXiv (Cornell University), 2018

We propose a unified approach to the study of isometries on algebras of vectorvalued Lipschitz maps and those of continuously differentiable maps by means of the notion of natural C(Y)-valuezations that take values in unital commutative C *-algebras. 2010 Mathematics Subject Classification. 46E40,46B04,46J10,46J15. Key words and phrases. isometries, vector-valued maps, admissible quadruples, vector-valued Lipschitz algebras, continuously differentiable maps. F L = F ∞(K) + L α (F), F ∈ Lip α (K, E) (resp. lip α (K, E)), which is often called the ℓ 1-norm or the sum norm. The norm, which is called the max norm, • M of Lip α (K, E) (resp. lip α (K, E)) is defined by F M = max{ F ∞ , L α (F)}, F ∈ Lip α (K, E) (resp. lip α (K, E)). Note that Lip α (K, E) (resp. lip α (K, E)) is a Banach space with respect to • L and • M respectively. If E is a unital (commutative) Banach algebra, then the norm • L is sub-multiplicative. Hence Lip α (K, E) (resp. lip α (K, E)) is a unital (commutative) Banach

Bull. Malays. Math. Sci. Soc.(2), 2007

A linear map ϕ : A → B between (Banach) algebras is called 3-homomorphism if ϕ(abc) = ϕ(a)ϕ(b)ϕ(c) for each a, b, c ∈ A. We investigate 3-homomorphisms on Banach algebras with bounded approximate identities and establish in two ways that every involution preserving homomorphism between C * -algebras is norm decreasing.

Journal of Fourier Analysis and Applications, 1996

Keywords and Phrases. Banach algebra, absolute convergence, synthesis Acknowledgements and Notes. The research of the first author was supported by the Israeli Ministry of Science and the Arts through the Ma'agara program for absorption of immigrant mathematicians at the Technion Israel Institute of Technology. The second author acknowledges the support of the Minerva Foundation in Germany through the Emmy Noether Mathematics Institute in Bar-Ilan University. The authors thank H. Feichtinger and G. Zimmermann for helpful discussions and valuable remarks. We are indebted to the referee for various remarks which definitely made the presentation better.

Advances in Mathematics, 2005

Let U and V be convex and balanced open subsets of the Banach spaces X and Y respectively. In this paper we study the following question: Given two Fréchet algebras of holomorphic functions of bounded type on U and V respectively that are algebra-isomorphic, can we deduce that X and Y (or X * and Y * ) are isomorphic? We prove that if X * or Y * has the approximation property and H wu (U ) and H wu (V ) are topologically algebra-isomorphic, then X * and Y * are isomorphic (the converse being true when U and V are the whole space). We get analogous results for H b (U ) and H b (V ), giving conditions under which an algebra-isomorphism between H b (X) and H b (Y ) is equivalent to an isomorphism between X * and Y * . We also obtain characterizations of different algebra-homomorphisms as composition operators, study the structure of the spectrum of the algebras under consideration and show the existence of homomorphisms on H b (X) with pathological behaviors. 22 and 23. Since T * is reflexive, if U ⊂ T * and V ⊂ Y are convex and balanced open sets and H b (U ) and H b (V ) are topologically algebra-isomorphic, then T * and Y are isomorphic. Moreover, H b (T * ) and H b (Y ) are isomorphic if and only if T * and Y are isomorphic.