• 대한수학회보
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• 제45권1호
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• pp.157-167
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• 2008

A mapping f : $M{\rightarrow}N$ between Hilbert $C^*$-modules approximately preserves the inner product if $$\parallel<f(x),\;f(y)>-<x,y>\parallel\leq\varphi(x,y)$$ for an appropriate control function $\varphi(x,y)$ and all x, y $\in$ M. In this paper, we extend some results concerning the stability of the orthogonality equation to the framework of Hilbert $C^*$-modules on more general restricted domains. In particular, we investigate some asymptotic behavior and the Hyers-Ulam-Rassias stability of the orthogonality equation.

• 대한수학회보
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• 제24권1호
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• pp.52-52
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• 1987

• 대한수학회보
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• 제58권2호
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• pp.461-479
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• 2021

This paper includes a general version of Bessel multipliers in Hilbert C∗-modules. In fact, by combining analysis, an operator on the standard Hilbert C∗-module and synthesis, we reach so-called generalized Bessel multipliers. Because of their importance for applications, we are interested to determine cases when generalized multipliers are invertible. We investigate some necessary or sufficient conditions for the invertibility of such operators and also we look at which perturbation of parameters preserve the invertibility of them. Subsequently, our attention is on how to express the inverse of an invertible generalized frame multiplier as a multiplier. In fact, we show that for all frames, the inverse of any invertible frame multiplier with an invertible symbol can always be represented as a multiplier with an invertible symbol and appropriate dual frames of the given ones.

• 대한수학회지
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• 제54권4호
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• pp.1063-1079
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• 2017

Two standard Bessel sequences in a Hilbert $C^*$-module are approximately duals if the distance (with respect to the norm) between the identity operator on the Hilbert $C^*$-module and the operator constructed by the composition of the synthesis and analysis operators of these Bessel sequences is strictly less than one. In this paper, we introduce (a, m)-approximate duality using the distance between the identity operator and the operator defined by multiplying the Bessel multiplier with symbol m by an element a in the center of the $C^*$-algebra. We show that approximate duals are special cases of (a, m)-approximate duals and we generalize some of the important results obtained for approximate duals to (a, m)-approximate duals. Especially we study perturbations of (a, m)-approximate duals and (a, m)-approximate duals of modular Riesz bases.

• 대한수학회지
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• 제38권1호
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• pp.125-134
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• 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.

• 충청수학회지
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• 제15권1호
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• pp.53-60
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• 2002

In this paper, we prove the generalized Hyers-Ulam-Rassias stability of linear operators in Banach modules over a unital $C^*$-algebra associated with its unitary group.

• 호남수학학술지
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• 제28권4호
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• pp.533-541
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• 2006

let E be a Finsler modules over $C^*$-algebras. A with norm-map $\rho$ and L(E) set of all A-linear bonded operators on E. We show that the canonical homomorphism ${\phi}:L(E){\rightarrow}L(E_I)$ sending each operator T to its restriction $T|E_I$ is injective if and only if I is an essential ideal in the underlying $C^*$-algebra A. We also show that $T{\in}L(E)$ is a bounded below if and only if ${\mid}{\mid}x{\mid}{\mid}={\mid}{\mid}{\rho}{\prime}(x){\mid}{\mid}$ is complete, where ${\rho}{\prime}(x)={\rho}(Tx)$ for all $x{\in}E$. Also, we give a necessary and sufficient condition for the equivalence of the norms generated by the norm map.

• 대한수학회지
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• 제57권3호
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• pp.691-706
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• 2020

Necessary and sufficient conditions are provided under which the weighted Moore-Penrose inverse A^†_MN exists, where A is an adjointable operator between Hilbert C^⁎-modules, and the weights M and N are only self-adjoint and invertible. Relationship between weighted Moore-Penrose inverses A^†_MN is clarified when A is fixed, whereas M and N are variable. Perturbation analysis for the weighted Moore-Penrose inverse is also provided.

• Nonlinear Functional Analysis and Applications
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• 제27권2호
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• pp.249-259
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• 2022

Let 𝓐 be a unital C^*-algebra. Then it follows that $\sum\limits_{i=1}^{n}(a_ia^*_i)^{\frac{1}{2}}{\leq}\sqrt{n}\(\sum\limits_{i=1}^{n}a_ia^*_i\)^{\frac{1}{2}}$, ∀n ∈ ℕ, ∀a₁, …, a_n ∈ 𝓐. By modifications of arguments of Botelho-Andrade, Casazza, Cheng, and Tran given in 2019, for certain n-tuple x = (a₁, …, a_n) ∈ 𝓐ⁿ, we give a method to compute a positive element c_x in the C^*-algebra 𝓐 such that the equality $$\sum\limits_{i=1}^{n}(a_ia^*_i)^{\frac{1}{2}}=c_x\sqrt{n}\(\sum\limits_{i=1}^{n}a_ia^*_i\)^{\frac{1}{2}}$$ holds. We give an application for the integral of Kasparov. We also derive a formula for the exact constant for the continuous ℓ₁ - ℓ₂ inequality.

• 대한수학회지
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• 제50권1호
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• pp.61-80
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• 2013

In this paper, we study ${\alpha}$-completely positive maps between locally $C^*$-algebras. As a generalization of a completely positive map, an ${\alpha}$-completely positive map produces a Krein space with indefinite metric, which is useful for the study of massless or gauge fields. We construct a KSGNS type representation associated to an ${\alpha}$-completely positive map of a locally $C^*$-algebra on a Krein locally $C^*$-module. Using this construction, we establish the Radon-Nikod$\acute{y}$m type theorem for ${\alpha}$-completely positive maps on locally $C^*$-algebras. As an application, we study an extremal problem in the partially ordered cone of ${\alpha}$-completely positive maps on a locally $C^*$-algebra.