• Bulletin of the Korean Mathematical Society
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• v.30 no.2
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• pp.187-191
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• 1993

In this note we prove the following result: Let A be a complex Banach *-algebra with continuous involution and let B be an $A^{*}$-algebra./T(A) = B. Then T is continuous (Theorem 2). From above theorem some others results of special interest and some well-known results follow. (Corollaries 3,4,5,6 and 7). We close this note with some generalizations and some remarks (Theorems 8.9.10 and question). Throughout this note we consider only complex algebras. Let A and B be complex algebras. A linear mapping T from A into B is called jordan homomorphism if T( $x^{1}$) = (Tx)$^{2}$ for all x in A. A linear mapping T : A .rarw. B is called spectrally-contractive mapping if .rho.(Tx).leq..rho.(x) for all x in A, where .rho.(x) denotes spectral radius of element x. Any homomorphism algebra is a spectrally-contractive mapping. If A and B are *-algebras, then a homomorphism T : A.rarw.B is called *-homomorphism if (Th)$^{*}$=Th for all self-adjoint element h in A. Recall that a Banach *-algebras is a complex Banach algebra with an involution *. An $A^{*}$-algebra A is a Banach *-algebra having anauxiliary norm vertical bar . vertical bar which satisfies $B^{*}$-condition vertical bar $x^{*}$x vertical bar = vertical bar x vertical ba $r^{2}$(x in A). A Banach *-algebra whose norm is an algebra $B^{*}$-norm is called $B^{*}$-algebra. The *-semi-simple Banach *-algebras and the semi-simple hermitian Banach *-algebras are $A^{*}$-algebras. Also, $A^{*}$-algebras include $B^{*}$-algebras ( $C^{*}$-algebras). Recall that a semi-prime algebra is an algebra without nilpotents two-sided ideals non-zero. The class of semi-prime algebras includes the class of semi-prime algebras and the class of prime algebras. For all concepts and basic facts about Banach algebras we refer to [2] and [8].].er to [2] and [8].].

• Bulletin of the Korean Mathematical Society
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• v.38 no.4
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• pp.709-718
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• 2001

In this paper we show that if D and G are continuous linear Jordan derivations on a Banach algebra A satisfying [D(x), x]x - x[G(x),x] $\epsilon$ rad(A)for all $\epsilon$ A, then both D and G map A into rad(A).

• Bulletin of the Korean Mathematical Society
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• v.37 no.3
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• pp.429-435
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• 2000

Our main goal is to show that if there exist Jordan derivations D, E and G on a noncommutative 2-torsion free prime ring R such that$(G^2(x)+E(x))D(x)=0\ or\ D(x)(G^2(x)+E(x))=0\ for\ all\ x\inR$, then we have D=o or E=0, G=0.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.451-459
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• 2019

Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be two operator ${\ast}$-rings such that ${\mathcal{A}}$ is prime. In this paper, we show that if the map ${\Phi}:{\mathcal{A}}{\rightarrow}{\mathcal{B}}$ is bijective and preserves Jordan or ${\ast}$-Jordan triple product, then it is additive. Moreover, if ${\Phi}$ preserves Jordan triple product, we prove the multiplicativity or anti-multiplicativity of ${\Phi}$. Finally, we show that if ${\mathcal{A}}$ and ${\mathcal{B}}$ are two prime operator ${\ast}$-algebras, ${\Psi}:{\mathcal{A}}{\rightarrow}{\mathcal{B}}$ is bijective and preserves ${\ast}$-Jordan triple product, then ${\Psi}$ is a ${\mathbb{C}}$-linear or conjugate ${\mathbb{C}}$-linear ${\ast}$-isomorphism.

• Communications of the Korean Mathematical Society
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• v.28 no.3
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• pp.535-558
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• 2013

The purpose of this paper is to prove that the noncommutative version of the Singer-Wermer Conjecture is affirmative under certain conditions. Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D:A{\rightarrow}A$ such that $D(x)^3[D(x),x]{\in}rad(A)$ for all $x{\in}A$. In this case, we show that $D(A){\subseteq}rad(A)$.

• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.31-35
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• 2020

With the motivation to extend the Zelasko's theorem on commutative algebras, it was shown in [2] that if n ∈ {3, 4} is fixed, A, B are commutative algebras and h : A → B is an n-Jordan homomorphism, then h is an n-ring homomorphism. In this paper, we extend this result for all n ≥ 3.

• Journal of the Chungcheong Mathematical Society
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• v.5 no.1
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• pp.75-85
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• 1992

Let R be an associative ring and let G be a finite group of Jordan automorphisms of R. Let $R^G$ be the set of elements in R fixed by all $g{\in}G$. In this paper we will study the relationship between the Levitzki radical of $R^G$ and R as that a Jordan ring. We also show that if R is a P.I. algebra, then the algebraicity of $R^G$ implies the algebraicity of R.

• Korean Journal of Mathematics
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• v.31 no.4
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• pp.495-504
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• 2023

This article deals with non-degenerate linear transformations on Euclidean Jordan algebras. First, we study non-degenerate for cone invariant, copositive, Lyapunov-like, and relaxation transformations. Further, we study that the non-degenerate is invariant under principal pivotal transformations and algebraic automorphisms.

• Communications of the Korean Mathematical Society
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• v.13 no.3
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• pp.489-499
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• 1998

Let A be the non-commutative Banach algebra with identity satisfying certain conditions. We show that if D is a derivation on A, then D(A) is contained in the radical of A.

• Journal of applied mathematics & informatics
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• v.6 no.2
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• pp.611-630
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• 1999

In this paper we show that the Derivation D(A) on the non-commutative Banach algebra A with identity satisfying certain conditions is contained in the radical of A and will show some examples satisfying such properties.