• Title, Summary, Keyword: Hilbert $C^*$-module

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GENERALIZED JENSEN'S EQUATIONS IN A HILBERT MODULE

  • An, Jong Su;Lee, Jung Rye;Park, Choonkil
    • Korean Journal of Mathematics
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    • v.15 no.2
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    • pp.135-148
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    • 2007
  • We prove the stability of generalized Jensen's equations in a Hilbert module over a unital $C^*$-algebra. This is applied to show the stability of a projection, a unitary operator, a self-adjoint operator, a normal operator, and an invertible operator in a Hilbert module over a unital $C^*$-algebra.

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LINEAR MAPPINGS IN BANACH MODULES OVER A UNITAL C*-ALGEBRA

  • Lee, Jung Rye;Mo, Kap-Jong;Park, Choonkil
    • Journal of the Chungcheong Mathematical Society
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    • v.24 no.2
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    • pp.221-238
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    • 2011
  • We prove the Hyers-Ulam stability of generalized Jensen's equations in Banach modules over a unital $C^{\ast}$-algebra. It is applied to show the stability of generalized Jensen's equations in a Hilbert module over a unital $C^{\ast}$-algebra. Moreover, we prove the stability of linear operators in a Hilbert module over a unital $C^{\ast}$-algebra.

THE GENERALIZED INVERSES A(1,2)T,S OF THE ADJOINTABLE OPERATORS ON THE HILBERT C^*-MODULES

  • Xu, Qingxiang;Zhang, Xiaobo
    • Journal of the Korean Mathematical Society
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    • v.47 no.2
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    • pp.363-372
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    • 2010
  • In this paper, we introduce and study the generalized inverse $A^{(1,2)}_{T,S}$ with the prescribed range T and null space S of an adjointable operator A from one Hilbert $C^*$-module to another, and get some analogous results known for finite matrices over the complex field or associated rings, and the Hilbert space operators.

BESSEL MULTIPLIERS AND APPROXIMATE DUALS IN HILBERT C -MODULES

  • Azandaryani, Morteza Mirzaee
    • Journal of the Korean Mathematical Society
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    • v.54 no.4
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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.

ON FRAMES FOR COUNTABLY GENERATED HILBERT MODULES OVER LOCALLY C*-ALGEBRAS

  • Alizadeh, Leila;Hassani, Mahmoud
    • Communications of the Korean Mathematical Society
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    • v.33 no.2
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    • pp.527-533
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    • 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.

A GENERALIZATION OF STONE'S THEOREM IN HILBERT $C^*$-MODULES

  • Amyari, Maryam;Chakoshi, Mahnaz
    • 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.

INDEX AND STABLE RANK OF C*-ALGEBRAS

  • Kim, Sang Og
    • 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.

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RECONSTRUCTION THEOREM FOR STATIONARY MONOTONE QUANTUM MARKOV PROCESSES

  • Heo, Jae-Seong;Belavkin, Viacheslav P.;Ji, Un Cig
    • 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.