• Communications of the Korean Mathematical Society
• /
• v.33 no.2
• /
• pp.527-533
• /
• 2018

Let $\mathcal{X}$ be a countably generated Hilbert module over a locally $C^*$-algebra $\mathcal{A}$ in multiplier module M($\mathcal{X}$) of $\mathcal{X}$. We propose the necessary and sufficient condition such that a sequence $\{h_n:n{{\in}}\mathbb{N}\}$ in M($\mathcal{X}$) is a standard frame of multipliers in $\mathcal{X}$. We also show that if T in $b(L_{\mathcal{A}}(\mathcal{X}))$, the space of bounded maps in set of all adjointable maps on $\mathcal{X}$, is surjective and $\{h_n:n{{\in}}\mathbb{N}\}$ is a standard frame of multipliers in $\mathcal{X}$, then $\{T{\circ}h_n:n{\in}\mathbb{N}}$ is a standard frame of multipliers in $\mathcal{X}$, too.

• Communications of the Korean Mathematical Society
• /
• v.27 no.1
• /
• pp.149-158
• /
• 2012

In this paper, we prove the superstability and the generalized Hyers-Ulam stability of Jordan *-homomorphisms between unital $C^*$-algebras associated with the following functional equation$$f(\frac{-x+y}{3})+f(\frac{x-3z}{c})+f(\frac{3x-y+3z}{3})=f(x)$$. Morever, we investigate Jordan *-homomorphisms between unital $C^*$-algebras associated with the following functional inequality $${\parallel}f(\frac{-x+y}{3})+f(\frac{x-3z}{3})+f(\frac{3x-y+3z}{3}){\parallel}\leq{\parallel}f(x)\parallel$$.

• Honam Mathematical Journal
• /
• v.23 no.1
• /
• pp.51-57
• /
• 2001

Let $\mathcal{B}(H)$ and $\mathcal{B}(K)$ denote the algebras of all bounded linear operators on Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, respectively. We show that if ${\phi}:\mathcal{B}(H){\rightarrow}\mathcal{B}(K)$ is an additive mapping satisfying ${\phi}({\mid}A{\mid}^2)={\mid}{\phi}(A){\mid}^2$ for every $A{\in}\mathcal{B}(H)$, then there exists a mapping ${\psi}$ defined by ${\psi}(A)={\phi}(I){\phi}(A)$, ${\forall}A{\in}\mathcal{B}(H)$ such that ${\psi}$ is the sum of $two^*$-homomorphisms one of which C-linear and the othere C-antilinear. We will also study some conditions implying the injective and rank-preserving of ${\psi}$.

• Communications of the Korean Mathematical Society
• /
• v.35 no.2
• /
• pp.455-468
• /
• 2020

An element in a ring R with identity is called invo-clean if it is the sum of an idempotent and an involution and R is called invoclean if every element of R is invo-clean. Let C(R) be the center of a ring R and g(x) be a fixed polynomial in C(R)[x]. We introduce the new notion of g(x)-invo clean. R is called g(x)-invo if every element in R is a sum of an involution and a root of g(x). In this paper, we investigate many properties and examples of g(x)-invo clean rings. Moreover, we characterize invo-clean as g(x)-invo clean rings where g(x) = (x-a)(x-b), a, b ∈ C(R) and b - a ∈ Inv(R). Finally, some classes of g(x)-invo clean rings are discussed.

• Journal of the Korean Mathematical Society
• /
• v.58 no.3
• /
• pp.609-631
• /
• 2021

Let C[0, T] denote a generalized analogue of Wiener space, the space of real-valued continuous functions on the interval [0, T]. Define $Z_{\vec{e},n}$ : C[0, T] → ℝⁿ⁺¹ by $$Z_{\vec{e},n}(x)=\(x(0),\;{\int}_0^T\;e_1(t)dx(t),{\cdots},\;{\int}_0^T\;e_n(t)dx(t)\)$$, where e₁,…, e_n are of bounded variations on [0, T]. In this paper we derive a simple evaluation formula for Radon-Nikodym derivatives similar to the conditional expectations of functions on C[0, T] with the conditioning function $Z_{\vec{e},n}$ which has an initial weight and a kind of drift. As applications of the formula, we evaluate the Radon-Nikodym derivatives of various functions on C[0, T] which are of interested in Feynman integration theory and quantum mechanics. This work generalizes and simplifies the existing results, that is, the simple formulas with the conditioning functions related to the partitions of time interval [0, T].

• Journal of applied mathematics & informatics
• /
• v.11 no.1_2
• /
• pp.431-436
• /
• 2003

Given vectors x and y in a Hilbert space, an interpolating operator is a bounded operator T such that Tx = y. An interpolating operator for n vectors satisfies the equation $Tx_i\;:\;y_i,\;for\;i\;=\;1,\;2,\;{\cdots},\;n$. In this article, we obtained the following : $Let\;x\;=\;\{x_i\}\;and\;y=\{y_\}$ be two vectors in a separable complex Hilbert space H such that $x_i\;\neq\;0$ for all $i\;=\;1,\;2;\cdots$. Let L be a commutative subspace lattice on H. Then the following statements are equivalent. (1) $sup\;\{\frac{\$\mid${\sum_{k=1}}^l\;\alpha_{\kappa}E_{\kappa}y\$\mid$}{\$\mid${\sum_{k=1}}^l\;\alpha_{\kappa}E_{\kappa}x\$\mid$}\;:\;l\;\in\;\mathbb{N},\;\alpha_{\kappa}\;\in\;\mathbb{C}\;and\;E_{\kappa}\;\in\;L\}\;<\;\infty\;and\;$\mid$y_n\$\mid$x_n$\mid$^{-1}\;=\;1\;for\;all\;n\;=\;1,\;2,\;\cdots$. (2) There exists an operator A in AlgL such that Ax = y, A is a unitary operator and every E in L reduces, A, where AlgL is a tridiagonal algebra.

• Journal of the Korean Mathematical Society
• /
• v.33 no.3
• /
• pp.669-677
• /
• 1996

Let $M_n$ be the $C^*$-algebra of all $n \times n$ matrices over the complex field, and $P[M_m, M_n]$ the convex cone of all positive linear maps from $M_m$ into $M_n$ that is, the maps which send the set of positive semidefinite matrices in $M_m$ into the set of positive semi-definite matrices in $M_n$. The convex structures of $P[M_m, M_n]$ are highly complicated even in low dimensions, and several authors [CL, KK, LW, O, R, S, W]have considered the possibility of decomposition of $P[M_m, M_n] into subcones.

• Bulletin of the Korean Mathematical Society
• /
• v.26 no.2
• /
• pp.119-126
• /
• 1989

Throughout this paper, X will always denote a Banach space over the complex numbers C, and L(X) will denote the Banach algebra of all continuous linear operators on X. Operator will always mean continuous linear operator. An n-tuple of operators T$_{1}$,..,T$_{n}$ on X will be denoted by over ^ T=(T$_{1}$,..,T$_{n}$ ). Let L$^{n}$ (X) be the set of all n-tuples of operators on X. X' will denote the dual space of X, S(X) its unit sphere and .PI.(X) the subset of X*X' defined by .PI.(X)={(x,f).mem.X*X': ∥x∥=∥f∥=f(x)=1}.

• Honam Mathematical Journal
• /
• v.32 no.2
• /
• pp.217-226
• /
• 2010

Using the Bu$\c{s}$neag's model ([1, 2, 3]), the notion of pseudo-valuations (valuations) on a ${\mathbf{BCK/BCI}}$-algebra is introduced, and a pseudo-metric is induced by a pseudo-valuation on ${\mathbf{BCK/BCI}}$-algebras. Based on the notion of (pseudo) valuation, we show that the binary operation in ${\mathbf{BCK/BCI}}$-algebras is uniformly continuous.

• Journal of the Korean Mathematical Society
• /
• v.38 no.1
• /
• pp.125-134
• /
• 2001

Independent $\sigma$-algebras Α and Β on X, L$^2$(X, Α V Β), L$^2$(X x X, Α x Β), and the Hilbert space tensor product L$^2$(X,Α), (※Equations, See Full-text) L$^2$(X,Β), are isomorphic. In this note, we show that various Hilbert C(sup)*-algebra tensor products provide the analogous roles when independence is weakened to conditional independence.