• Journal of the Korean Mathematical Society
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• v.46 no.3
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• pp.657-673
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• 2009

The fixed point algebra $C^*(E)^{\gamma}$ of a gauge action $\gamma$ on a graph $C^*$-algebra $C^*(E)$ and its AF subalgebras $C^*(E)^{\gamma}_{\upsilon}$ associated to each vertex v do play an important role for the study of dynamical properties of $C^*(E)$. In this paper, we consider the stability of $C^*(E)^{\gamma}$ (an AF algebra is either stable or equipped with a (nonzero bounded) trace). It is known that $C^*(E)^{\gamma}$ is stably isomorphic to a graph $C^*$-algebra $C^*(E_{\mathbb{Z}}\;{\times}\;E)$ which we observe being stable. We first give an explicit isomorphism from $C^*(E)^{\gamma}$ to a full hereditary $C^*$-subalgebra of $C^*(E_{\mathbb{N}}\;{\times}\;E)({\subset}\;C^*(E_{\mathbb{Z}}\;{\times}\;E))$ and then show that $C^*(E_{\mathbb{N}}\;{\times}\;E)$ is stable whenever $C^*(E)^{\gamma}$ is so. Thus $C^*(E)^{\gamma}$ cannot be stable if $C^*(E_{\mathbb{N}}\;{\times}\;E)$ admits a trace. It is shown that this is the case if the vertex matrix of E has an eigenvector with an eigenvalue $\lambda$ > 1. The AF algebras $C^*(E)^{\gamma}_{\upsilon}$ are shown to be nonstable whenever E is irreducible. Several examples are discussed.

• 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.

• Communications of the Korean Mathematical Society
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• v.17 no.3
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• pp.475-485
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• 2002

We estimate the stable rank, connected stable rank and general stable rank of the multiplier algebras of $C^{＊}$-algebras under some conditions and prove that the ranks of them are infinite. Moreover, we show that for any $\sigma$-unital subhomogeneous $C^{＊}$-algebra, its stable rank is equal to that of its multiplier algebra.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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• pp.813-823
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• 2024

In this paper, we prove that every 2-local derivation on several classes of C^*-algebras, such as unital properly infinite, type I or residually finite-dimensional C^*-algebras, is a derivation. We show that the following statements are equivalent: (1) every 2-local derivation on a C^*-algebra is a derivation, (2) every 2-local derivation on a unital primitive antiliminal and no properly infinite C^*-algebra is a derivation. We also show that every 2-local derivation on a group C^*-algebra C^*(𝔽) or a unital simple infinite-dimensional quasidiagonal C^*-algebra, which is stable finite antiliminal C^*-algebra, is a derivation.

• Kyungpook Mathematical Journal
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• v.47 no.2
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• pp.203-219
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• 2007

We estimate the stable rank and connected stable rank of group $C^*$-algebra of certain disconnected solvable Lie groups such as semi-direct products of connected solvable Lie groups by the integers.

• Journal of the Korean Mathematical Society
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• v.40 no.5
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• pp.789-830
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• 2003

By forming completions of families of noncommutative polynomials, we define a notion of noncommutative continuous function and locally bounded Borel function that give a noncommutative analogue of the functional calculus for elements of commutative $C^{*}$-algebras and von Neumann algebras. These notions give a precise meaning to $C^{*}$-algebras defined by generator and relations and we show how they relate to many parts of operator and operator algebra theory.

• Bulletin of the Korean Mathematical Society
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• v.47 no.5
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• pp.997-1000
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• 2010

We prove that the reduced free product of $k\;{\times}\;k$ matrix algebras over abelian $C^*$-algebras is not the minimal tensor product of reduced free products of $k\;{\times}\;k$ matrix algebras over abelian $C^*$-algebras. It is shown that the reduced group $C^*$-algebra associated with a group having the property T of Kazhdan is not isomorphic to a reduced free product of abelian $C^*$-algebras or the minimal tensor product of such reduced free products. The infinite tensor product of reduced free products of abelian $C^*$-algebras is not isomorphic to the tensor product of a nuclear $C^*$-algebra and a reduced free product of abelian $C^*$-algebra. We discuss the freeness of free product $II_1$-factors and solidity of free product $II_1$-factors weaker than that of Ozawa. We show that the freeness in a free product is related to the existence of Cartan subalgebras in free product $II_1$-factors. Finally, we give a free product factor which is not solid in the weak sense.

• Bulletin of the Korean Mathematical Society
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• v.37 no.2
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• pp.249-254
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• 2000

We give an easy proof of the fact that every noncommutative torus $A_{\omega}$ is stably isomorphic to the noncommutative torus $C(\widehat{S\omega}){\;}\bigotimes{\;}A_p$ which hasa trivial bundle structure. It is well known that stable isomorphism of two separable $C^{*}-algebras$ is equibalent to the existence of eqivalence bimodule between the two stably isomorphic $C^{*}-algebras{\;}A_{\omega}$ and $C(\widehat{S\omega}){\;}\bigotimes{\;}A_p$.

• Journal of the Korean Society of Industry Convergence
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• v.6 no.3
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• pp.171-176
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• 2003

The poly(triazine bissulfide) synthesized from 6-dibutylamino-1,3,5-triazine-2,4-dithiol with bis(4-chloro-3-nitrophenyl) sulfone in the presence of the phase transfer catalyst, the maximum algebra viscosity (0.57 dL/g) is stable at reaction temperature of $60^{\circ}C$ overall. We could not acquire the good result about solubility, thermal property, and molecular weight to make cast film, we made base for the synthesis of functionalization polymer material.