• Bulletin of the Korean Mathematical Society
• /
• v.58 no.5
• /
• pp.1193-1208
• /
• 2021

Let 𝓐 be a von Neumann algebra with no central summands of type I₁. In this article, we study Lie n-derivation on von Neumann algebra and prove that every additive Lie n-derivation on a von Neumann algebra has standard form at zero product as well as at projection product.

• Communications of the Korean Mathematical Society
• /
• v.31 no.1
• /
• pp.101-113
• /
• 2016

In this article, we considered the stability of the following (${\alpha}$, ${\beta}$, ${\gamma}$)-derivation $${\alpha}D[x,y]={\beta}[D(x),y]+{\gamma}[x,D(y)]$$ and homomorphisms associated to the quadratic type functional equation $$f(kx+y)+f(kx+{\sigma}(y))=2kg(x)+2g(y),\;x,y{\in}A$$, where ${\sigma}$ is an involution of the Lie $C^*$-algebra A and k is a fixed positive integer. The Hyers-Ulam stability on unbounded domains is also studied. Applications of the results for the asymptotic behavior of the generalized quadratic functional equation are provided.

• Honam Mathematical Journal
• /
• v.38 no.2
• /
• pp.269-277
• /
• 2016

In a paper of S. H. Choi [2], the author studied the derivations of a restricted Weyl Type non-associative algebra, and determined a 1-dimensional vector space of derivations. We describe all the derivations of this algebra and prove that they form a 3-dimensional Lie algebra.

• Honam Mathematical Journal
• /
• v.27 no.4
• /
• pp.583-593
• /
• 2005

The Weyl-type non-associative algebra ${\overline{WN_{g_n,m,s_r}}$ and its subalgebra ${\overline{WN_{n,m,s_r}}$ are defined and studied in the papers [2], [3], [9], [11], [12]. We find the derivation group $Der_{non}({\overline{WN_{1,0,0_{[2]}}})$ the non-associative simple algebra ${\overline{WN_{1,0,0_{[2]}}}$.

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