• Korean Journal of Mathematics
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• v.7 no.1
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• pp.71-77
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• 1999

We show that if the stable rank of $B^{\alpha}$ is one, then the stable rank of B is less than or equal to the order of G for any action of a finite group G. Also we give a short proof to the known fact that if the action of a finite group on a $C^*$-algebra B is saturated then the canonical conditional expectation from B to $B^{\alpha}$ is of index-finite type and the crossed product $C^*$-algebra is isomorphic to the algebra of compact operators on the Hilbert $B^{\alpha}$-module B.

• Bulletin of the Korean Mathematical Society
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• v.49 no.1
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• pp.63-74
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• 2012

Based on the Hilbert $C^*$-module structure we study the reconstruction theorem for stationary monotone quantum Markov processes from quantum dynamical semigroups. We prove that the quantum stochastic monotone process constructed from a covariant quantum dynamical semigroup is again covariant in the strong sense.

• East Asian mathematical journal
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• v.7 no.1
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• pp.53-59
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• 1991

• The Pure and Applied Mathematics
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• v.18 no.1
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• pp.31-39
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• 2011

Stone's theorem states that "A bounded linear operator A is infinitesimal generator of a $C_0$-group of unitary operators on a Hilbert space H if and only if iA is self adjoint". In this paper we establish a generalization of Stone's theorem in the framework of Hilbert $C^*$-modules.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.4
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• pp.367-373
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• 2007

For a unital *-subalgebra of the space $\mathcal{L}^a(X)$ of all adjointable maps on a Hilbert $\mathcal{B}$-module X with a $C^*$-algebra $\mathcal{B}$, we study unitary operator (in such algebra)-valued multiplier ${\sigma}$ on a normal, generating subsemigroup S of a group G with its extension to G. A dilation of a projective isometric ${\sigma}$-representation of S is established as a projective unitary ${\rho}$-representation of G for a suitable unitary operator (in some algebra)-valued multiplier ${\rho}$ associated with the multiplier ${\sigma}$ which is explicitly constructed.

• Korean Journal of Mathematics
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• v.27 no.3
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• pp.657-660
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• 2019

We give necessary and sufficient condition for the Greenberg's generalized conjecture conjecture on certain imaginary quadratic fields.

• Bulletin of the Korean Mathematical Society
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• v.45 no.1
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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.

• Bulletin of the Korean Mathematical Society
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• v.58 no.2
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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.

• Journal of the Chungcheong Mathematical Society
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• v.15 no.1
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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.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.201-210
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• 2004

We examine a k-configuration X in P$^2$ or P$^3$ whose minimal free resolution has a non-cancelable Betti number in the last free module. We also find partial answers to the question: which Artinian O-sequences are level or not?