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

• Honam Mathematical Journal
• /
• v.30 no.1
• /
• pp.21-32
• /
• 2008

We introduce some notions and results which have been discussed, and we define the notion of fuzzy normal B-algebra and obtain the structure of quotient B-algebra via fuzzy normal B-algebra. Moreover, we generalize a fundamental theorem of B-homomorphism for B-algebras via fuzzy normal B-algebras.

• Honam Mathematical Journal
• /
• v.31 no.3
• /
• pp.407-419
• /
• 2009

The simple non-associative algebra $N(e^{A_S},q,n,t)_k$ and its simple sub-algebras are defined in the papers [1], [3], [4], [5], [6], [12]. We define the non-associative algebra $\overline{WN_{(g_n,\mathfrak{U}),m,s_B}}$ and its antisymmetrized algebra $\overline{WN_{(g_n,\mathfrak{U}),m,s_B}}$. We also prove that the algebras are simple in this work. There are various papers on finding all the derivations of an associative algebra, a Lie algebra, and a non-associative algebra (see [3], [5], [6], [9], [12], [14], [15]). We also find all the derivations $Der_{anti}(WN(e^{{\pm}x^r},0,2)_B^-)$ of te antisymmetrized algebra $WN(e^{{\pm}x^r}0,2)_B^-$ and every derivation of the algebra is outer in this paper.

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

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

• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.2
• /
• pp.323-333
• /
• 2010

The biproduct bialgebra has been generalized to generalized biproduct bialgebra $B{\times}^L_H\;D$ in [5]. Let (D, B) be an admissible pair and let D be a bialgebra. We show that if generalized biproduct bialgebra $B{\times}^L_H\;D$ is a Hopf algebra with antipode s, then D is a Hopf algebra and the identity $id_B$ has an inverse in the convolution algebra $Hom_k$(B, B). We show that if D is a Hopf algebra with antipode $s_D$ and $s_B$ in $Hom_k$(B, B) is an inverse of $id_B$ then $B{\times}^L_H\;D$ is a Hopf algebra with antipode s described by $s(b{\times}^L_H\;d)={\Sigma}(1_B{\times}^L_H\;s_D(b_{-1}{\cdot}d))(s_B(b_0){\times}^L_H\;1_D)$. We show that the mapping system $B{\leftrightarrows}^{{\Pi}_B}_{j_B}\;B{\times}^L_H\;D{\rightleftarrows}^{{\pi}_D}_{i_D}\;D$ (where $j_B$ and $i_D$ are the canonical inclusions, ${\Pi}_B$ and ${\pi}_D$ are the canonical coalgebra projections) characterizes $B{\times}^L_H\;D$. These generalize the corresponding results in [6].

• Journal of the Chungcheong Mathematical Society
• /
• v.26 no.1
• /
• pp.91-103
• /
• 2013

Let (D,B) be an admissible pair. Then recall that $B\;{\times}^L_HD^{{\rightarrow}{\pi}_D}_{{\leftarrow}i_D}\;D$ are bialgebra maps satisfying ${\pi}_D{\circ}i_D=I$. We have solved a converse in case D is a Hopf algebra. Let D be a Hopf algebra with antipode $S_D$ and be a left H-comodule algebra and a left H-module coalgebra over a field $k$. Let A be a bialgebra over $k$. Suppose $A^{{\rightarrow}{\pi}}_{{\leftarrow}i}D$ are bialgebra maps satisfying ${\pi}{\circ}i=I_D$. Set ${\Pi}=I_D*(i{\circ}s_D{\circ}{\pi}),B=\Pi(A)$ and $j:B{\rightarrow}A$ be the inclusion. Suppose that ${\Pi}$ is an algebra map. We show that (D,B) is an admissible pair and $B^{\leftarrow{\Pi}}_{\rightarrow{j}}A^{\rightarrow{\pi}}_{\leftarrow{i}}D$ is an admissible mapping system and that the generalized biproduct bialgebra $B{\times}^L_HD$ is isomorphic to A as bialgebras.

• Communications of the Korean Mathematical Society
• /
• v.31 no.2
• /
• pp.209-216
• /
• 2016

In this paper, we introduce the notion of symmetric bi-derivations of a B-algebra and investigate some related properties. We study the notion of symmetric bi-derivations of a 0-commutative B-algebra and state some related properties.

• The Pure and Applied Mathematics
• /
• v.30 no.4
• /
• pp.407-416
• /
• 2023

In this paper, we introduce the concept of C^*-algebra-valued quasi b-metric space and prove some existence and uniqueness theorems. Furthermore, we prove the Hyers-Ulam stability results for fixed point problems via C^*-algebra-valued extended quasi b-metric space.

• Kyungpook Mathematical Journal
• /
• v.48 no.4
• /
• pp.529-543
• /
• 2008

It is shown that every almost linear mapping h : $A{\rightarrow}B$ of a unital PoissonC*-algebra A to a unital Poisson C*-algebra B is a Poisson C*-algebra homomorph when $h(2^nuy)\;=\;h(2^nu)h(y)$ or $h(3^nuy)\;=\;h(3^nu)h(y)$ for all $y\;\in\;A$, all unitary elements $u\;\in\;A$ and n = 0, 1, 2,$\codts$, and that every almost linear almost multiplicative mapping h : $A{\rightarrow}B$ is a Poisson C*-algebra homomorphism when h(2x) = 2h(x) or h(3x) = 3h(x for all $x\;\in\;A$. Here the numbers 2, 3 depend on the functional equations given in the almost linear mappings or in the almost linear almost multiplicative mappings. We prove the Cauchy-Rassias stability of Poisson C*-algebra homomorphisms in unital Poisson C*-algebras, and of homomorphisms in Poisson Banach modules over a unital Poisson C*-algebra.