• Journal of the Chungcheong Mathematical Society
• /
• v.19 no.1
• /
• pp.9-18
• /
• 2006

In this paper, the completion of a quasi-normed algebra and the completion of a quasi-normed module over a Banach algebra are constructed.

• Kyungpook Mathematical Journal
• /
• v.46 no.3
• /
• pp.433-436
• /
• 2006

In this note we show that a representable difference algebra is equivalent to a commutative BCK-algebra.

• Research in Mathematical Education
• /
• v.8 no.2
• /
• pp.107-114
• /
• 2004

University teachers of linear algebra often feel annoyed and disarmed when faced with the inability of their students to cope with concepts that they consider to be very simple. Usually, they lay the blame on the impossibility for the students to use geometrical intuition or the lack of practice in basic logic and set theory. J.-L. Dorier [(2002): Teaching Linear Algebra at University. In: T. Li (Ed.), Proceedings of the International Congress of Mathematicians (Beijing: August 20-28, 2002), Vol. III: Invited Lectures (pp. 875-884). Beijing: Higher Education Press] mentioned that the situation could not be improved substantially with the teaching of Cartesian geometry or/and logic and set theory prior to the linear algebra. In East Asian countries, science-orientated mathematics curricula of the high schools consist of calculus with many other materials. To understand differential and integral calculus efficiently or for other reasons, students have to learn a lot of content (and concepts) in linear algebra, such as ordered pairs, n-tuple numbers, planar and spatial coordinates, vectors, polynomials, matrices, etc., from an early age. The content of linear algebra is spread out from grades 7 to 12. When the high school teachers teach the content of linear algebra, however, they do not concern much about the concepts of content. With small effort, teachers can help the students to build concepts of vocabularies and languages of linear algebra.

• Journal of Educational Research in Mathematics
• /
• v.17 no.3
• /
• pp.253-270
• /
• 2007

by A.C. Clairaut was written based on the historico-genetic principle such as his . In this paper, by analyzing his we can induce six principles that Clairaut adopted to teach algebra: necessity and curiosity as a motive of studying algebra, harmony of discovery and proof, complementarity of generalization and specialization, connection of knowledge to be learned with already known facts, semantic approaches to procedural knowledge of mathematics, reversible approach. These can be considered as strategies for teaching algebra accorded with beginner's mind. Some of them correspond with characteristics of , but the others are unique in the domain of algebra. And by comparing Clairaut's approaches with school algebra, we discuss about some mathematical subjects: setting equations in relation to problem situations, operations and signs of letters, rule of signs in multiplication, solving quadratic equations, and general relationship between roots and coefficients of equations.

• 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.

• Honam Mathematical Journal
• /
• v.32 no.3
• /
• pp.467-480
• /
• 2010

There are various papers on finding all the derivations of a non-associative algebra and an anti-symmetrized algebra (see [2], [3], [4], [5], [6], [10], [13], [15], [16]). We and all the derivations of the growing algebra WN($e^{{\pm}x_1x_2x_3}$, 0, 3)[1] with the set of all right annihilators $T_3$ = $\{id,\;\partial_1,\;\partial_2,\;\partial_3\}$ in the paper. The dimension of $Der_{non}$(WN($e^{{\pm}x_1x_2x_3}$, 0, 3)$_{[1]}$) of the algebra WN($e^{{\pm}x_1x_2x_3}$, 0, 3)$_{[1]}$ is one and every derivation of the algebra WN($e^{{\pm}x_1x_2x_3}$, 0, 3)$_{[1]}$ is outer. We show that there is a class P of purely outer algebras in this work.

• Journal of the Korean Mathematical Society
• /
• v.34 no.4
• /
• pp.859-869
• /
• 1997

It is shown that for $A_{k, m}$ a k-homogeneous $C^*$-algebra over $S^{2n - 1} \times S^1$ such that no non-trivial matrix algebra can be factored out of $A_{k, m}$ and $A_{k, m} \otimes M_l(C)$ has a non-trivial bundle structure for any positive integer l, we construct an $A_{k, m^-} C(S^{2n - 1} \times S^1) \otimes M_k(C)$-equivalence bimodule to show that every k-homogeneous $C^*$-algebra over $S^{2n - 1} \times S^1)$. Moreover, we prove that the tensor product of the k-homogeneous $C^*$-algebra $A_{k, m}$ with a UHF-algebra of type $p^\infty$ has the tribial bundle structure if and only if the set of prime factors of k is a subset of the set of prime factors of pp.

• Communications of the Korean Mathematical Society
• /
• v.31 no.2
• /
• pp.247-259
• /
• 2016

We first give the constructions of (Jordan) higher derivations on a trivial extension algebra and then we provide some sufficient conditions under which a Jordan higher derivation on a trivial extension algebra is a higher derivation. We then proceed to the trivial generalized matrix algebras as a special trivial extension algebra. As an application we characterize the construction of Jordan higher derivations on a triangular algebra. We also provide some illuminating examples of Jordan higher derivations on certain trivial extension algebras which are not higher derivations.

• Bulletin of the Korean Mathematical Society
• /
• v.54 no.2
• /
• pp.455-461
• /
• 2017

The Lie algebra analogue of Schur's result which is proved by Moneyhun in 1994, states that if L is a Lie algebra such that dimL/Z(L) = n, then $dimL_{(2)}={\frac{1}{2}}n(n-1)-s$ for some non-negative integer s. In the present paper, we determine the structure of central factor (for s = 0) and the factor Lie algebra $L/Z_2(L)$ (for all $s{\geq}0$) of a finite dimensional nilpotent Lie algebra L, with n-dimensional central factor. Furthermore, by using the concept of n-isoclinism, we discuss an upper bound for the dimension of $L/Z_n(L)$ in terms of $dimL_{(n+1)}$, when the factor Lie algebra $L/Z_n(L)$ is finitely generated and $n{\geq}1$.