Ring Homomorphisms on Real Banach Algebras
نویسنده
چکیده
Let B be a strictly real commutative real Banach algebra with the carrier space Φ B. If A is a commutative real Banach algebra, then we give a representation of a ring homomorphism ρ : A → B, which needs not be linear nor continuous. If A is a commutative complex Banach algebra, then ρ(A) is contained in the radical of B. 1. Introduction and results. Ring homomorphisms are mappings between two rings that preserve both addition and multiplication. In particular, we are concerned with ring homomorphisms between two commutative Banach algebras. If R is the real number field, then the zero map and the identity are typical examples of ring homomorphisms on R. Furthermore, the converse is true: if ρ is a nonzero ring homomorphism on R, then ρ(t) = t for every t ∈ R. For if ρ is nonzero, then ρ(1) = ρ(1) 2 implies ρ(1) = 1, and hence ρ preserves
منابع مشابه
$n$-Jordan homomorphisms on C-algebras
Let $nin mathbb{N}$. An additive map $h:Ato B$ between algebras $A$ and $B$ is called $n$-Jordan homomorphism if $h(a^n)=(h(a))^n$ for all $ain A$. We show that every $n$-Jordan homomorphism between commutative Banach algebras is a $n$-ring homomorphism when $n < 8$. For these cases, every involutive $n$-Jordan homomorphism between commutative C-algebras is norm continuous.
متن کاملAlgebras defined by homomorphisms
Let $mathcal{R}$ be a commutative ring with identity, let $A$ and $B$ be two $mathcal{R}$-algebras and $varphi:Blongrightarrow A$ be an $mathcal{R}$-additive algebra homomorphism. We introduce a new algebra $Atimes_varphi B$, and give some basic properties of this algebra. Generalized $2$-cocycle derivations on $Atimes_varphi B$ are studied. Accordingly, $Atimes_varphi B$ is considered from th...
متن کاملFuzzy Stability of Ring Homomorphisms and Ring Derivations on Fuzzy Banach Algebras
In this paper, we establish the Hyers–Ulam–Rassias stability of ring homomorphisms and ring derivations on fuzzy Banach algebras.
متن کاملAutomatic continuity of surjective $n$-homomorphisms on Banach algebras
In this paper, we show that every surjective $n$-homomorphism ($n$-anti-homomorphism) from a Banach algebra $A$ into a semisimple Banach algebra $B$ is continuous.
متن کامل