• 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.

• 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].].

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.653-662
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• 2005

In this paper we shall give the relationships among $T_R,\;End_{R}(M),\;SEnd_{R}(M)\;and\;SAut_R(M)$ when M is a perfect R-module. If M and N are perfect modules, we get $SAut_{R}(M {\times}N){\cong}SAut_{R}(M){\times}SAut_R(N)$. Also we shall discuss that $_x(M)_H$ is a subgroup of $_x(M)$ if M is quasi-perfect and $_x(M)_H$ is a normal subgroup of $_x(M)$ if M is perfect.