• Bulletin of the Korean Mathematical Society
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• v.22 no.2
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• pp.89-93
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• 1985

One of the most important problems in automatic continuity theory is to solve the question of continuity of an algebra homomorphism from a Banach algebra into a semisimple Banach algebra with dense range. Many results on this subject are obtained imposing some conditions on the domains or the ranges of homomorphisms. For most recent results and references in automatic continuity theory one may refer to [1], [4] and [5]. In this note we study some properties of homomorphisms from $C^{*}$-algebras into Banach algebras. It is shown that the range of an isomorphism from a $C^{*}$-algebra into a Banach algebra contains no non zero element of the radical of B. Using this result we show that the same holds for a continuous homomorphism, hence a Banach algebra which is the image of a $C^{*}$-algebra under a continuous homomorphism is necessarily semisimple. Thus if there is a homomorphism from a $C^{*}$-algebra onto a non-semisimple Banach algebra it must be discontinuous. Also it follows that every non zero homomorphism from a $C^{*}$-algebra into a radical algebra is discontinuous. Then we make a brief observation on the behavior of quasinilpotent element of noncommutative $C^{*}$-algebras in relation with continuous homomorphisms.momorphisms.

• Bulletin of the Korean Mathematical Society
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• v.30 no.1
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• pp.109-115
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• 1993

In 1940 Eidelheit showed that every homomorphism of a Banach algebra onto the Banach algebra B(X) of all bounded linear operators on a Banach space X is continuous. At about the same time, Gelfand proved that every homomorphism of a commutative Banach algebra into a commutative semi-simple Banach algebra is continuous. In [7] Johnson proved that every homomorphism of a Banach algebra onto non-commutative semi-simple Banach algebra is continuous, and this is still the most important result of this type. In this paper we are concerned with continuity of derivations on commutative Banach algebras and of homomorphisms into commutative Banach algebras. Throughout this paper we suppose that A is a commutative Banach algebra. R will denote the redical of A.

• Communications of the Korean Mathematical Society
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• v.14 no.2
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• pp.365-371
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• 1999

In this paper, we prove that the range of homomorphism from a C\ulcorner-algebra A into a commutative Banach algebra B whose radical is nil contains no non-zero element of the radical of B. Using this result we show that there is no non-zero homomorphism from a C\ulcorner-algebra into a commutative radical nil Banach algebra.

• Honam Mathematical Journal
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• v.26 no.1
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• pp.61-75
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• 2004

It is shown that every almost linear mapping $h:{\mathcal{A}}{\rightarrow}{\mathcal{B}}$ of a unital Poisson Banach algebra ${\mathcal{A}}$ to a unital Poisson Banach algebra ${\mathcal{B}}$ is a Poisson algebra homomorphism when h(xy) = h(x)h(y) holds for all $x,y{\in}\;{\mathcal{A}}$, and that every almost linear almost multiplicative mapping $h:{\mathcal{A}}{\rightarrow}{\mathcal{B}}$ is a Poisson algebra homomorphism when h(qx) = qh(x) for all $x\;{\in}\;{\mathcal{A}}$. Here the number q is in the functional equation given in the almost linear almost multiplicative mapping. We prove that every almost Poisson bracket $B:{\mathcal{A}}\;{\times}\;{\mathcal{A}}\;{\rightarrow}\;{\mathcal{A}}$ on a Banach algebra ${\mathcal{A}}$ is a Poisson bracket when B(qx, z) = B(x, qz) = qB(x, z) for all $x,z{\in}\;{\mathcal{A}}$. Here the number q is in the functional equation given in the almost Poisson bracket.

• 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].].

• Korean Journal of Mathematics
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• v.22 no.1
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• pp.91-121
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• 2014

Let X and Y be vector spaces. It is shown that a mapping $f:X{\rightarrow}Y$ satisfies the functional equation ${\ddag}$ $$2df(\frac{x_1+{\sum}_{j=2}^{2d}(-1)^jx_j}{2d})-2df(\frac{x_1+{\sum}_{j=2}^{2d}(-1)^{j+1}x_j}{2d})=2\sum_{j=2}^{2d}(-1)^jf(x_j)$$ if and only if the mapping $f:X{\rightarrow}Y$ is additive, and prove the Cauchy-Rassias stability of the functional equation (${\ddag}$) in Banach modules over a unital $C^*$-algebra, and in Poisson Banach modules over a unital Poisson $C^*$-algebra. Let $\mathcal{A}$ and $\mathcal{B}$ be unital $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras. As an application, we show that every almost homomorphism $h:\mathcal{A}{\rightarrow}\mathcal{B}$ of $\mathcal{A}$ into $\mathcal{B}$ is a homomorphism when $h(d^nuy)=h(d^nu)h(y)$ or $h(d^nu{\circ}y)=h(d^nu){\circ}h(y)$ for all unitaries $u{\in}\mathcal{A}$, all $y{\in}\mathcal{A}$, and n = 0, 1, 2, ${\cdots}$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras, and of Lie $JC^*$-algebra derivations in Lie $JC^*$-algebras.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.165-170
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• 2018

In this paper, we show that every unital n-Jordan homomorphism ${\varphi}$ from a Banach algebra ${\mathcal{A}}$ onto a semisimple commutative Banach algebra B is continuous.

• Bulletin of the Korean Mathematical Society
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• v.20 no.2
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• pp.71-74
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• 1983

The problems of the continuity of homomorphisms between Banach algebras have been studied widely for the last two decades to obtain various fruitful results, yet it is far from characterizing the calss of Banach algebras for which each homomorphism from a member of the class into a Banach algebra is conitnuous. For commutative Banach algebras A and B a simple proof shows that every homomorphism .theta. from A into B is continuous provided that B is semi-simple, however, with a non semi-simple Banach algebra B examples of discontinuous homomorphisms from C(K) into B have been constructed by Dales [6] and Esterle [7]. For non commutative Banach algebras the problems of automatic continuity of homomorphisms seem to be much more difficult. Many positive results and open questions related to this subject may be found in [1], [3], [5] and [8], in particular most recent development can be found in the Lecture Note which contains [1]. It is well-known that a$^{*}$-isomorphism from a $C^{*}$-algebra into another $C^{*}$-algebra is an isometry, and an isomorphism of a Banach algebra into a $C^{*}$-algebra with self-adjoint range is continuous. But a$^{*}$-isomorphism from a $C^{*}$-algebra into an involutive Banach algebra is norm increasing [9], and one can not expect each of such isomorphisms to be continuous. In this note we discuss an isomorphism from a commutative $C^{*}$-algebra into a commutative Banach algebra with dense range via separating space. It is shown that such an isomorphism .theta. : A.rarw.B is conitnuous and maps A onto B is B is semi-simple, discontinuous if B is not semi-simple.

• Communications of the Korean Mathematical Society
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• v.26 no.2
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• pp.253-259
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• 2011

Let A and B be Banach algebras with unit. Here we prove that an approximate algebra homomorphism f : A ${\rightarrow}$ B, in the sense of Rassias, is an algebra homomorphism.

• Korean Journal of Mathematics
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• v.19 no.1
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• pp.33-52
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• 2011

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.