• Title/Summary/Keyword: Banach algebra.

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A REPRESENTATION FOR NONCOMMUTATIVE BANACH ALGEBRAS

  • PAK HEE CHUL
    • Journal of applied mathematics & informatics
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    • v.17 no.1_2_3
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    • pp.591-603
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    • 2005
  • A representation for non-commutative Banach algebras is discussed, which generalizes the Gelfand representation for commutative Banach algebras and the Gelfand-Naimark representation for $C^{\ast}$-algebras. Its basic properties are also investigated. In appendix, an example of Banach algebra that is neither semi-simple nor radical is presented.

CAUCHY-RASSIAS STABILITY OF A GENERALIZED ADDITIVE MAPPING IN BANACH MODULES AND ISOMORPHISMS IN C*-ALGEBRAS

  • Shin, Dong Yun;Park, Choonkil
    • Journal of the Chungcheong Mathematical Society
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    • v.24 no.4
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    • pp.617-630
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    • 2011
  • Let X, Y be vector spaces, and let r be 2 or 4. It is shown that if an odd mapping $f:X{\rightarrow}Y$ satisfies the functional equation $${\hspace{50}}rf(\frac{\sum_{j=1}^{d}\;x_j} {r})+\;{\sum\limits_{\iota(j)=0,1 \atop {\sum_{j=1}^{d}}\;{\iota}(j)=l}}\;rf(\frac{\sum_{j=1}^{d}{(-1)^{\iota(j)}x_j}}{r}) \\({\ddag}){\hspace{160}}=(_{d-1}C_l-_{d-1}C_{l-1}+1)\;{\sum\limits_{j=1}^{d}\;f(x_j)}$$ then the odd mapping $f:X{\rightarrow}Y$ is additive, and we prove the Cauchy-Rassias stability of the functional equation in Banach modules over a unital $C^*$-algebra. As an application, we show that every almost linear bijection $h:{\mathcal{A}}{\rightarrow}{\mathcal{B}}$ of a unital $C^*$-algebra ${\mathcal{A}}$ onto a unital $C^*$-algebra ${\mathcal{B}}$ is a $C^*$-algebra isomorphism when $h(2^nuy)=h(2^nu)h(y)$ for all unitaries $u{\in}{\mathcal{A}}$, all $y{\in}{\mathcal{A}}$, and $n=0,1,2,{\cdots}$.

A NOTE ON THE NUMERICAL RANGE OF AN OPERATOR

  • Yang, Youngoh
    • Bulletin of the Korean Mathematical Society
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    • v.21 no.1
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    • pp.27-30
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    • 1984
  • The concepts of the numerical range of an operator on a Hillbert space and on a Banach space were introduced by Toeplitz in 1918 and Bauer in 1962 respectively. Bauer's paper was concerned only with finite dimensional Banach spaces, but the concept of numerical range that he introduced is available without restriction of the dimension [1, 2]. In this paper, we define a C-algebra spatial numerical range of an operator on C-algebra valued inner product modules introduced by Paschke [4], and give analogous results on these modules as those on Banach spaces.

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ON DERIVATIONS IN NONCOMMUTATIVE SEMISIMPLE BANACH ALGEBRAS

  • Jung, Yong-Soo
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.3
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    • pp.583-590
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    • 1998
  • The purpose of this paper is to prove the following results: Let A be a noncommutative semisimple Banach algebra. (1) Suppose that a linear derivation D : A $\to A$ is such that [D(x),x]x=0 holds for all $x \in A$. Then we have D=0. (2) Suppose that a linear derivation $D:A\to A$ is such that $D(x)x^2 + x^2D(x)=0$ holds for all $x \in A$. Then we have C=0.

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A RESULT CONCERNING DERIVATIONS IN NONCOMMUTATIVE BANACH ALGERAS

  • Chang, Ick-Soon
    • Journal of the Chungcheong Mathematical Society
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    • v.10 no.1
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    • pp.97-104
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    • 1997
  • The purpose of this paper is to prove the following result: Let A be a noncommutative semisimple Banach algebra. Suppose that $D:A{\rightarrow}A$, $G:A{\rightarrow}A$ are linear derivations such that [G(x), x]D(x) = D(x)[G(x), x] = 0, [D(x), G(x)] = 0 hold for all $x{\in}A$. In this case either D = 0 or G = 0.

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JORDAN DERIVATIONS ON PRIME RINGS AND THEIR APPLICATIONS IN BANACH ALGEBRAS, II

  • Kim, Byung-Do
    • Journal of the Chungcheong Mathematical Society
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    • v.27 no.1
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    • pp.65-87
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    • 2014
  • The purpose of this paper is to prove that the noncommutative version of the Singer-Wermer Conjecture is affirmative under certain conditions. Let A be a noncommutative Banach algebra. We show that if there exists a continuous linear Jordan derivation D : A ${\rightarrow}$ A such that [D(x), x]$D(x)^3{\in}$ rad(A) for all $x{\in}A$, then D(A) ${\subseteq}$ rad(A).

Approximate Jordan mappings on noncommutative Banach algebras

  • Lee, Young-Whan;Kim, Gwang-Hui
    • Communications of the Korean Mathematical Society
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    • v.12 no.1
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    • pp.69-73
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    • 1997
  • We show that if T is an $\varepsilon$-approximate Jordan functional such that T(a) = 0 implies $T(a^2) = 0 (a \in A)$ then T is continuous and $\Vert T \Vert \leq 1 + \varepsilon$. Also we prove that every $\varepsilon$-near Jordan mapping is an $g(\varepsilon)$-approximate Jordan mapping where $g(\varepsilon) \to 0$ as $\varepsilon \to 0$ and for every $\varepsilon > 0$ there is an integer m such that if T is an $\frac {\varepsilon}{m}$-approximate Jordan mapping on a finite dimensional Banach algebra then T is an $\varepsilon$-near Jordan mapping.

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ON DERIVATIONS IN NONCOMMUTATIVE SEMIPRIME RINGS AND BANACH ALGEBRAS

  • PARK, KYOO-HONG
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.4
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    • pp.671-678
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    • 2005
  • Let R be a noncommutative semi prime ring. Suppose that there exists a derivation d : R $\to$ R such that for all x $\in$ R, either [[d(x),x], d(x)] = 0 or $\langle$$\langle(x),\;x\rangle,\;d(x)\rangle$ = 0. In this case [d(x), x] is nilpotent for all x $\in$ R. We also apply the above results to a Banach algebra theory.

SYSTEMS OF DERIVATIONS ON BANACH ALGEBRAS

  • Lee, Eun-Hwi
    • Communications of the Korean Mathematical Society
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    • v.12 no.2
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    • pp.251-256
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    • 1997
  • We show that a strong system of derivations ${D_0, D_1,\cdots,D_m}$ on a commutative Banach algebra A is contained in the radical of A if it satisfies one of the following conditions for separating spaces; (1) $\partial(D_i) \subseteq rad(A) and \partial(D_i) \subseteq K D_i(rad(A))$ for all i, where $K D_i(rad(A)) = {x \in rad(A))$ : for each $m \geq 1, D^m_i(x) \in rad(A)}$. (2) $(D^m_i) \subseteq rad(A)$ for all i and m. (3) $\bar{x\partial(D_i)} = \partial(D_i)$ for all i and all nonzero x in rad(A).

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